author paulson Mon Feb 23 16:35:46 2004 +0100 (2004-02-23) changeset 14408 0cc42bb96330 parent 14407 043bf0d9e9b5 child 14409 91181ee5860c
converted HOL/Complex/NSCA to Isar script
 src/HOL/Complex/NSCA.thy file | annotate | diff | revisions src/HOL/IsaMakefile file | annotate | diff | revisions
```     1.1 --- a/src/HOL/Complex/NSCA.thy	Sat Feb 21 20:05:16 2004 +0100
1.2 +++ b/src/HOL/Complex/NSCA.thy	Mon Feb 23 16:35:46 2004 +0100
1.3 @@ -1,44 +1,1461 @@
1.4  (*  Title       : NSCA.thy
1.5      Author      : Jacques D. Fleuriot
1.6      Copyright   : 2001,2002 University of Edinburgh
1.7 -    Description : Infinite, infinitesimal complex number etc!
1.8  *)
1.9
1.10 -NSCA = NSComplex +
1.11 +header{*Non-Standard Complex Analysis*}
1.12
1.13 -consts
1.14 +theory NSCA = NSComplex:
1.15
1.16 -    (* infinitely close *)
1.17 -    "@c="     :: [hcomplex,hcomplex] => bool  (infixl 50)
1.18 +constdefs
1.19
1.20 +    capprox    :: "[hcomplex,hcomplex] => bool"  (infixl "@c=" 50)
1.21 +      --{*the ``infinitely close'' relation*}
1.22 +      "x @c= y == (x - y) \<in> CInfinitesimal"
1.23
1.24 -constdefs
1.25     (* standard complex numbers reagarded as an embedded subset of NS complex *)
1.26     SComplex  :: "hcomplex set"
1.27 -   "SComplex == {x. EX r. x = hcomplex_of_complex r}"
1.28 +   "SComplex == {x. \<exists>r. x = hcomplex_of_complex r}"
1.29
1.30     CInfinitesimal  :: "hcomplex set"
1.31 -   "CInfinitesimal == {x. ALL r: Reals. 0 < r --> hcmod x < r}"
1.32 +   "CInfinitesimal == {x. \<forall>r \<in> Reals. 0 < r --> hcmod x < r}"
1.33
1.34     CFinite :: "hcomplex set"
1.35 -   "CFinite == {x. EX r: Reals. hcmod x < r}"
1.36 +   "CFinite == {x. \<exists>r \<in> Reals. hcmod x < r}"
1.37
1.38     CInfinite :: "hcomplex set"
1.39 -   "CInfinite == {x. ALL r: Reals. r < hcmod x}"
1.40 +   "CInfinite == {x. \<forall>r \<in> Reals. r < hcmod x}"
1.41
1.42 -   (* standard part map *)
1.43 -   stc :: hcomplex => hcomplex
1.44 -   "stc x == (@r. x : CFinite & r:SComplex & r @c= x)"
1.45 +   stc :: "hcomplex => hcomplex"
1.46 +    --{* standard part map*}
1.47 +   "stc x == (@r. x \<in> CFinite & r:SComplex & r @c= x)"
1.48
1.49 -   cmonad    :: hcomplex => hcomplex set
1.50 +   cmonad    :: "hcomplex => hcomplex set"
1.51     "cmonad x  == {y. x @c= y}"
1.52
1.53 -   cgalaxy   :: hcomplex => hcomplex set
1.54 -   "cgalaxy x == {y. (x - y) : CFinite}"
1.55 +   cgalaxy   :: "hcomplex => hcomplex set"
1.56 +   "cgalaxy x == {y. (x - y) \<in> CFinite}"
1.57 +
1.58 +
1.59 +
1.60 +subsection{*Closure Laws for SComplex, the Standard Complex Numbers*}
1.61 +
1.62 +lemma SComplex_add: "[| x \<in> SComplex; y \<in> SComplex |] ==> x + y \<in> SComplex"
1.63 +apply (simp add: SComplex_def, safe)
1.64 +apply (rule_tac x = "r + ra" in exI, simp)
1.65 +done
1.66 +
1.67 +lemma SComplex_mult: "[| x \<in> SComplex; y \<in> SComplex |] ==> x * y \<in> SComplex"
1.68 +apply (simp add: SComplex_def, safe)
1.69 +apply (rule_tac x = "r * ra" in exI, simp)
1.70 +done
1.71 +
1.72 +lemma SComplex_inverse: "x \<in> SComplex ==> inverse x \<in> SComplex"
1.73 +apply (simp add: SComplex_def)
1.74 +apply (blast intro: hcomplex_of_complex_inverse [symmetric])
1.75 +done
1.76 +
1.77 +lemma SComplex_divide: "[| x \<in> SComplex;  y \<in> SComplex |] ==> x/y \<in> SComplex"
1.78 +by (simp add: SComplex_mult SComplex_inverse divide_inverse_zero)
1.79 +
1.80 +lemma SComplex_minus: "x \<in> SComplex ==> -x \<in> SComplex"
1.81 +apply (simp add: SComplex_def)
1.82 +apply (blast intro: hcomplex_of_complex_minus [symmetric])
1.83 +done
1.84 +
1.85 +lemma SComplex_minus_iff [simp]: "(-x \<in> SComplex) = (x \<in> SComplex)"
1.86 +apply auto
1.87 +apply (erule_tac [2] SComplex_minus)
1.88 +apply (drule SComplex_minus, auto)
1.89 +done
1.90 +
1.91 +lemma SComplex_diff: "[| x \<in> SComplex; y \<in> SComplex |] ==> x - y \<in> SComplex"
1.93 +
1.95 +     "[| x + y \<in> SComplex; y \<in> SComplex |] ==> x \<in> SComplex"
1.96 +by (drule SComplex_diff, assumption, simp)
1.97 +
1.98 +lemma SReal_hcmod_hcomplex_of_complex [simp]:
1.99 +     "hcmod (hcomplex_of_complex r) \<in> Reals"
1.100 +by (simp add: hcomplex_of_complex_def hcmod SReal_def hypreal_of_real_def)
1.101 +
1.102 +lemma SReal_hcmod_number_of [simp]: "hcmod (number_of w ::hcomplex) \<in> Reals"
1.103 +apply (subst hcomplex_number_of [symmetric])
1.104 +apply (rule SReal_hcmod_hcomplex_of_complex)
1.105 +done
1.106 +
1.107 +lemma SReal_hcmod_SComplex: "x \<in> SComplex ==> hcmod x \<in> Reals"
1.108 +by (auto simp add: SComplex_def)
1.109 +
1.110 +lemma SComplex_hcomplex_of_complex [simp]: "hcomplex_of_complex x \<in> SComplex"
1.111 +by (simp add: SComplex_def)
1.112 +
1.113 +lemma SComplex_number_of [simp]: "(number_of w ::hcomplex) \<in> SComplex"
1.114 +apply (subst hcomplex_number_of [symmetric])
1.115 +apply (rule SComplex_hcomplex_of_complex)
1.116 +done
1.117 +
1.118 +lemma SComplex_divide_number_of:
1.119 +     "r \<in> SComplex ==> r/(number_of w::hcomplex) \<in> SComplex"
1.120 +apply (simp only: divide_inverse_zero)
1.121 +apply (blast intro!: SComplex_number_of SComplex_mult SComplex_inverse)
1.122 +done
1.123 +
1.124 +lemma SComplex_UNIV_complex:
1.125 +     "{x. hcomplex_of_complex x \<in> SComplex} = (UNIV::complex set)"
1.126 +by (simp add: SComplex_def)
1.127 +
1.128 +lemma SComplex_iff: "(x \<in> SComplex) = (\<exists>y. x = hcomplex_of_complex y)"
1.129 +by (simp add: SComplex_def)
1.130 +
1.131 +lemma hcomplex_of_complex_image:
1.132 +     "hcomplex_of_complex `(UNIV::complex set) = SComplex"
1.133 +by (auto simp add: SComplex_def)
1.134 +
1.135 +lemma inv_hcomplex_of_complex_image: "inv hcomplex_of_complex `SComplex = UNIV"
1.136 +apply (auto simp add: SComplex_def)
1.137 +apply (rule inj_hcomplex_of_complex [THEN inv_f_f, THEN subst], blast)
1.138 +done
1.139 +
1.140 +lemma SComplex_hcomplex_of_complex_image:
1.141 +      "[| \<exists>x. x: P; P \<le> SComplex |] ==> \<exists>Q. P = hcomplex_of_complex ` Q"
1.142 +apply (simp add: SComplex_def, blast)
1.143 +done
1.144 +
1.145 +lemma SComplex_SReal_dense:
1.146 +     "[| x \<in> SComplex; y \<in> SComplex; hcmod x < hcmod y
1.147 +      |] ==> \<exists>r \<in> Reals. hcmod x< r & r < hcmod y"
1.148 +apply (auto intro: SReal_dense simp add: SReal_hcmod_SComplex)
1.149 +done
1.150 +
1.151 +lemma SComplex_hcmod_SReal:
1.152 +      "z \<in> SComplex ==> hcmod z \<in> Reals"
1.153 +apply (simp add: SComplex_def SReal_def)
1.154 +apply (rule_tac z = z in eq_Abs_hcomplex)
1.155 +apply (auto simp add: hcmod hypreal_of_real_def hcomplex_of_complex_def cmod_def)
1.156 +apply (rule_tac x = "cmod r" in exI)
1.157 +apply (simp add: cmod_def, ultra)
1.158 +done
1.159 +
1.160 +lemma SComplex_zero [simp]: "0 \<in> SComplex"
1.161 +by (simp add: SComplex_def hcomplex_of_complex_zero_iff)
1.162 +
1.163 +lemma SComplex_one [simp]: "1 \<in> SComplex"
1.164 +by (simp add: SComplex_def hcomplex_of_complex_def hcomplex_one_def)
1.165 +
1.166 +(*
1.167 +Goalw [SComplex_def,SReal_def] "hcmod z \<in> Reals ==> z \<in> SComplex"
1.168 +by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
1.169 +by (auto_tac (claset(),simpset() addsimps [hcmod,hypreal_of_real_def,
1.170 +    hcomplex_of_complex_def,cmod_def]));
1.171 +*)
1.172 +
1.173 +
1.174 +subsection{*The Finite Elements form a Subring*}
1.175 +
1.176 +lemma CFinite_add: "[|x \<in> CFinite; y \<in> CFinite|] ==> (x+y) \<in> CFinite"
1.177 +apply (simp add: CFinite_def)
1.179 +done
1.180 +
1.181 +lemma CFinite_mult: "[|x \<in> CFinite; y \<in> CFinite|] ==> x*y \<in> CFinite"
1.182 +apply (simp add: CFinite_def)
1.183 +apply (blast intro!: SReal_mult hcmod_mult_less)
1.184 +done
1.185 +
1.186 +lemma CFinite_minus_iff [simp]: "(-x \<in> CFinite) = (x \<in> CFinite)"
1.187 +by (simp add: CFinite_def)
1.188 +
1.189 +lemma SComplex_subset_CFinite [simp]: "SComplex \<le> CFinite"
1.190 +apply (auto simp add: SComplex_def CFinite_def)
1.191 +apply (rule_tac x = "1 + hcmod (hcomplex_of_complex r) " in bexI)
1.192 +apply (auto intro: SReal_add)
1.193 +done
1.194 +
1.195 +lemma HFinite_hcmod_hcomplex_of_complex [simp]:
1.196 +     "hcmod (hcomplex_of_complex r) \<in> HFinite"
1.197 +by (auto intro!: SReal_subset_HFinite [THEN subsetD])
1.198 +
1.199 +lemma CFinite_hcomplex_of_complex [simp]: "hcomplex_of_complex x \<in> CFinite"
1.200 +by (auto intro!: SComplex_subset_CFinite [THEN subsetD])
1.201 +
1.202 +lemma CFiniteD: "x \<in> CFinite ==> \<exists>t \<in> Reals. hcmod x < t"
1.203 +by (simp add: CFinite_def)
1.204 +
1.205 +lemma CFinite_hcmod_iff: "(x \<in> CFinite) = (hcmod x \<in> HFinite)"
1.206 +by (simp add: CFinite_def HFinite_def)
1.207 +
1.208 +lemma CFinite_number_of [simp]: "number_of w \<in> CFinite"
1.209 +by (rule SComplex_number_of [THEN SComplex_subset_CFinite [THEN subsetD]])
1.210 +
1.211 +lemma CFinite_bounded: "[|x \<in> CFinite; y \<le> hcmod x; 0 \<le> y |] ==> y: HFinite"
1.212 +by (auto intro: HFinite_bounded simp add: CFinite_hcmod_iff)
1.213 +
1.214 +
1.215 +subsection{*The Complex Infinitesimals form a Subring*}
1.216 +
1.217 +lemma CInfinitesimal_zero [iff]: "0 \<in> CInfinitesimal"
1.218 +by (simp add: CInfinitesimal_def)
1.219 +
1.220 +lemma hcomplex_sum_of_halves: "x/(2::hcomplex) + x/(2::hcomplex) = x"
1.221 +by auto
1.222 +
1.223 +lemma CInfinitesimal_hcmod_iff:
1.224 +   "(z \<in> CInfinitesimal) = (hcmod z \<in> Infinitesimal)"
1.225 +by (simp add: CInfinitesimal_def Infinitesimal_def)
1.226 +
1.227 +lemma one_not_CInfinitesimal [simp]: "1 \<notin> CInfinitesimal"
1.228 +by (simp add: CInfinitesimal_hcmod_iff)
1.229 +
1.231 +     "[| x \<in> CInfinitesimal; y \<in> CInfinitesimal |] ==> (x+y) \<in> CInfinitesimal"
1.232 +apply (auto simp add: CInfinitesimal_hcmod_iff)
1.233 +apply (rule hrabs_le_Infinitesimal)
1.234 +apply (rule_tac y = "hcmod y" in Infinitesimal_add, auto)
1.235 +done
1.236 +
1.237 +lemma CInfinitesimal_minus_iff [simp]:
1.238 +     "(-x:CInfinitesimal) = (x:CInfinitesimal)"
1.239 +by (simp add: CInfinitesimal_def)
1.240 +
1.241 +lemma CInfinitesimal_diff:
1.242 +     "[| x \<in> CInfinitesimal;  y \<in> CInfinitesimal |] ==> x-y \<in> CInfinitesimal"
1.244 +
1.245 +lemma CInfinitesimal_mult:
1.246 +     "[| x \<in> CInfinitesimal; y \<in> CInfinitesimal |] ==> x * y \<in> CInfinitesimal"
1.247 +by (auto intro: Infinitesimal_mult simp add: CInfinitesimal_hcmod_iff hcmod_mult)
1.248 +
1.249 +lemma CInfinitesimal_CFinite_mult:
1.250 +     "[| x \<in> CInfinitesimal; y \<in> CFinite |] ==> (x * y) \<in> CInfinitesimal"
1.251 +by (auto intro: Infinitesimal_HFinite_mult simp add: CInfinitesimal_hcmod_iff CFinite_hcmod_iff hcmod_mult)
1.252 +
1.253 +lemma CInfinitesimal_CFinite_mult2:
1.254 +     "[| x \<in> CInfinitesimal; y \<in> CFinite |] ==> (y * x) \<in> CInfinitesimal"
1.255 +by (auto dest: CInfinitesimal_CFinite_mult simp add: hcomplex_mult_commute)
1.256 +
1.257 +lemma CInfinite_hcmod_iff: "(z \<in> CInfinite) = (hcmod z \<in> HInfinite)"
1.258 +by (simp add: CInfinite_def HInfinite_def)
1.259 +
1.260 +lemma CInfinite_inverse_CInfinitesimal:
1.261 +     "x \<in> CInfinite ==> inverse x \<in> CInfinitesimal"
1.262 +by (auto intro: HInfinite_inverse_Infinitesimal simp add: CInfinitesimal_hcmod_iff CInfinite_hcmod_iff hcmod_hcomplex_inverse)
1.263 +
1.264 +lemma CInfinite_mult: "[|x \<in> CInfinite; y \<in> CInfinite|] ==> (x*y): CInfinite"
1.265 +by (auto intro: HInfinite_mult simp add: CInfinite_hcmod_iff hcmod_mult)
1.266 +
1.267 +lemma CInfinite_minus_iff [simp]: "(-x \<in> CInfinite) = (x \<in> CInfinite)"
1.268 +by (simp add: CInfinite_def)
1.269 +
1.270 +lemma CFinite_sum_squares:
1.271 +     "[|a \<in> CFinite; b \<in> CFinite; c \<in> CFinite|]
1.272 +      ==> a*a + b*b + c*c \<in> CFinite"
1.273 +by (auto intro: CFinite_mult CFinite_add)
1.274 +
1.275 +lemma not_CInfinitesimal_not_zero: "x \<notin> CInfinitesimal ==> x \<noteq> 0"
1.276 +by auto
1.277 +
1.278 +lemma not_CInfinitesimal_not_zero2: "x \<in> CFinite - CInfinitesimal ==> x \<noteq> 0"
1.279 +by auto
1.280 +
1.281 +lemma CFinite_diff_CInfinitesimal_hcmod:
1.282 +     "x \<in> CFinite - CInfinitesimal ==> hcmod x \<in> HFinite - Infinitesimal"
1.283 +by (simp add: CFinite_hcmod_iff CInfinitesimal_hcmod_iff)
1.284 +
1.285 +lemma hcmod_less_CInfinitesimal:
1.286 +     "[| e \<in> CInfinitesimal; hcmod x < hcmod e |] ==> x \<in> CInfinitesimal"
1.287 +by (auto intro: hrabs_less_Infinitesimal simp add: CInfinitesimal_hcmod_iff)
1.288 +
1.289 +lemma hcmod_le_CInfinitesimal:
1.290 +     "[| e \<in> CInfinitesimal; hcmod x \<le> hcmod e |] ==> x \<in> CInfinitesimal"
1.291 +by (auto intro: hrabs_le_Infinitesimal simp add: CInfinitesimal_hcmod_iff)
1.292 +
1.293 +lemma CInfinitesimal_interval:
1.294 +     "[| e \<in> CInfinitesimal;
1.295 +          e' \<in> CInfinitesimal;
1.296 +          hcmod e' < hcmod x ; hcmod x < hcmod e
1.297 +       |] ==> x \<in> CInfinitesimal"
1.298 +by (auto intro: Infinitesimal_interval simp add: CInfinitesimal_hcmod_iff)
1.299 +
1.300 +lemma CInfinitesimal_interval2:
1.301 +     "[| e \<in> CInfinitesimal;
1.302 +         e' \<in> CInfinitesimal;
1.303 +         hcmod e' \<le> hcmod x ; hcmod x \<le> hcmod e
1.304 +      |] ==> x \<in> CInfinitesimal"
1.305 +by (auto intro: Infinitesimal_interval2 simp add: CInfinitesimal_hcmod_iff)
1.306 +
1.307 +lemma not_CInfinitesimal_mult:
1.308 +     "[| x \<notin> CInfinitesimal;  y \<notin> CInfinitesimal|] ==> (x*y) \<notin> CInfinitesimal"
1.309 +apply (auto simp add: CInfinitesimal_hcmod_iff hcmod_mult)
1.310 +apply (drule not_Infinitesimal_mult, auto)
1.311 +done
1.312 +
1.313 +lemma CInfinitesimal_mult_disj:
1.314 +     "x*y \<in> CInfinitesimal ==> x \<in> CInfinitesimal | y \<in> CInfinitesimal"
1.315 +by (auto dest: Infinitesimal_mult_disj simp add: CInfinitesimal_hcmod_iff hcmod_mult)
1.316 +
1.317 +lemma CFinite_CInfinitesimal_diff_mult:
1.318 +     "[| x \<in> CFinite - CInfinitesimal; y \<in> CFinite - CInfinitesimal |]
1.319 +      ==> x*y \<in> CFinite - CInfinitesimal"
1.320 +by (blast dest: CFinite_mult not_CInfinitesimal_mult)
1.321 +
1.322 +lemma CInfinitesimal_subset_CFinite: "CInfinitesimal \<le> CFinite"
1.323 +by (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
1.324 +         simp add: CInfinitesimal_hcmod_iff CFinite_hcmod_iff)
1.325 +
1.326 +lemma CInfinitesimal_hcomplex_of_complex_mult:
1.327 +     "x \<in> CInfinitesimal ==> x * hcomplex_of_complex r \<in> CInfinitesimal"
1.328 +by (auto intro!: Infinitesimal_HFinite_mult simp add: CInfinitesimal_hcmod_iff hcmod_mult)
1.329 +
1.330 +lemma CInfinitesimal_hcomplex_of_complex_mult2:
1.331 +     "x \<in> CInfinitesimal ==> hcomplex_of_complex r * x \<in> CInfinitesimal"
1.332 +by (auto intro!: Infinitesimal_HFinite_mult2 simp add: CInfinitesimal_hcmod_iff hcmod_mult)
1.333 +
1.334 +
1.335 +subsection{*The ``Infinitely Close'' Relation*}
1.336 +
1.337 +(*
1.338 +Goalw [capprox_def,approx_def] "(z @c= w) = (hcmod z @= hcmod w)"
1.339 +by (auto_tac (claset(),simpset() addsimps [CInfinitesimal_hcmod_iff]));
1.340 +*)
1.341 +
1.342 +lemma mem_cinfmal_iff: "x:CInfinitesimal = (x @c= 0)"
1.343 +by (simp add: CInfinitesimal_hcmod_iff capprox_def)
1.344 +
1.345 +lemma capprox_minus_iff: "(x @c= y) = (x + -y @c= 0)"
1.346 +by (simp add: capprox_def diff_minus)
1.347 +
1.348 +lemma capprox_minus_iff2: "(x @c= y) = (-y + x @c= 0)"
1.349 +by (simp add: capprox_def diff_minus add_commute)
1.350 +
1.351 +lemma capprox_refl [simp]: "x @c= x"
1.352 +by (simp add: capprox_def)
1.353 +
1.354 +lemma capprox_sym: "x @c= y ==> y @c= x"
1.355 +by (simp add: capprox_def CInfinitesimal_def hcmod_diff_commute)
1.356 +
1.357 +lemma capprox_trans: "[| x @c= y; y @c= z |] ==> x @c= z"
1.358 +apply (simp add: capprox_def)
1.359 +apply (drule CInfinitesimal_add, assumption)
1.360 +apply (simp add: diff_minus)
1.361 +done
1.362 +
1.363 +lemma capprox_trans2: "[| r @c= x; s @c= x |] ==> r @c= s"
1.364 +by (blast intro: capprox_sym capprox_trans)
1.365 +
1.366 +lemma capprox_trans3: "[| x @c= r; x @c= s|] ==> r @c= s"
1.367 +by (blast intro: capprox_sym capprox_trans)
1.368 +
1.369 +lemma number_of_capprox_reorient [simp]:
1.370 +     "(number_of w @c= x) = (x @c= number_of w)"
1.371 +by (blast intro: capprox_sym)
1.372 +
1.373 +lemma CInfinitesimal_capprox_minus: "(x-y \<in> CInfinitesimal) = (x @c= y)"
1.374 +by (simp add: diff_minus capprox_minus_iff [symmetric] mem_cinfmal_iff)
1.375 +
1.377 +by (auto simp add: cmonad_def dest: capprox_sym elim!: capprox_trans equalityCE)
1.378 +
1.379 +lemma Infinitesimal_capprox:
1.380 +     "[| x \<in> CInfinitesimal; y \<in> CInfinitesimal |] ==> x @c= y"
1.381 +apply (simp add: mem_cinfmal_iff)
1.382 +apply (blast intro: capprox_trans capprox_sym)
1.383 +done
1.384 +
1.385 +lemma capprox_add: "[| a @c= b; c @c= d |] ==> a+c @c= b+d"
1.386 +apply (simp add: capprox_def diff_minus)
1.387 +apply (rule minus_add_distrib [THEN ssubst])
1.388 +apply (rule add_assoc [THEN ssubst])
1.389 +apply (rule_tac b1 = c in add_left_commute [THEN subst])
1.390 +apply (rule add_assoc [THEN subst])
1.391 +apply (blast intro: CInfinitesimal_add)
1.392 +done
1.393 +
1.394 +lemma capprox_minus: "a @c= b ==> -a @c= -b"
1.395 +apply (rule capprox_minus_iff [THEN iffD2, THEN capprox_sym])
1.396 +apply (drule capprox_minus_iff [THEN iffD1])
1.398 +done
1.399 +
1.400 +lemma capprox_minus2: "-a @c= -b ==> a @c= b"
1.401 +by (auto dest: capprox_minus)
1.402 +
1.403 +lemma capprox_minus_cancel [simp]: "(-a @c= -b) = (a @c= b)"
1.404 +by (blast intro: capprox_minus capprox_minus2)
1.405 +
1.406 +lemma capprox_add_minus: "[| a @c= b; c @c= d |] ==> a + -c @c= b + -d"
1.407 +by (blast intro!: capprox_add capprox_minus)
1.408 +
1.409 +lemma capprox_mult1:
1.410 +      "[| a @c= b; c \<in> CFinite|] ==> a*c @c= b*c"
1.411 +apply (simp add: capprox_def diff_minus)
1.412 +apply (simp only: CInfinitesimal_CFinite_mult minus_mult_left hcomplex_add_mult_distrib [symmetric])
1.413 +done
1.414 +
1.415 +lemma capprox_mult2: "[|a @c= b; c \<in> CFinite|] ==> c*a @c= c*b"
1.416 +by (simp add: capprox_mult1 hcomplex_mult_commute)
1.417 +
1.418 +lemma capprox_mult_subst:
1.419 +     "[|u @c= v*x; x @c= y; v \<in> CFinite|] ==> u @c= v*y"
1.420 +by (blast intro: capprox_mult2 capprox_trans)
1.421 +
1.422 +lemma capprox_mult_subst2:
1.423 +     "[| u @c= x*v; x @c= y; v \<in> CFinite |] ==> u @c= y*v"
1.424 +by (blast intro: capprox_mult1 capprox_trans)
1.425 +
1.426 +lemma capprox_mult_subst_SComplex:
1.427 +     "[| u @c= x*hcomplex_of_complex v; x @c= y |]
1.428 +      ==> u @c= y*hcomplex_of_complex v"
1.429 +by (auto intro: capprox_mult_subst2)
1.430 +
1.431 +lemma capprox_eq_imp: "a = b ==> a @c= b"
1.432 +by (simp add: capprox_def)
1.433 +
1.434 +lemma CInfinitesimal_minus_capprox: "x \<in> CInfinitesimal ==> -x @c= x"
1.435 +by (blast intro: CInfinitesimal_minus_iff [THEN iffD2] mem_cinfmal_iff [THEN iffD1] capprox_trans2)
1.436 +
1.437 +lemma bex_CInfinitesimal_iff: "(\<exists>y \<in> CInfinitesimal. x - z = y) = (x @c= z)"
1.438 +by (unfold capprox_def, blast)
1.439 +
1.440 +lemma bex_CInfinitesimal_iff2: "(\<exists>y \<in> CInfinitesimal. x = z + y) = (x @c= z)"
1.441 +by (simp add: bex_CInfinitesimal_iff [symmetric], force)
1.442 +
1.444 +     "[| y \<in> CInfinitesimal; x + y = z |] ==> x @c= z"
1.445 +apply (rule bex_CInfinitesimal_iff [THEN iffD1])
1.446 +apply (drule CInfinitesimal_minus_iff [THEN iffD2])
1.447 +apply (simp add: eq_commute compare_rls)
1.448 +done
1.449 +
1.450 +lemma CInfinitesimal_add_capprox_self: "y \<in> CInfinitesimal ==> x @c= x + y"
1.451 +apply (rule bex_CInfinitesimal_iff [THEN iffD1])
1.452 +apply (drule CInfinitesimal_minus_iff [THEN iffD2])
1.453 +apply (simp add: eq_commute compare_rls)
1.454 +done
1.455 +
1.456 +lemma CInfinitesimal_add_capprox_self2: "y \<in> CInfinitesimal ==> x @c= y + x"
1.458 +
1.460 +     "y \<in> CInfinitesimal ==> x @c= x + -y"
1.461 +by (blast intro!: CInfinitesimal_add_capprox_self CInfinitesimal_minus_iff [THEN iffD2])
1.462 +
1.464 +     "[| y \<in> CInfinitesimal; x+y @c= z|] ==> x @c= z"
1.465 +apply (drule_tac x = x in CInfinitesimal_add_capprox_self [THEN capprox_sym])
1.466 +apply (erule capprox_trans3 [THEN capprox_sym], assumption)
1.467 +done
1.468 +
1.470 +     "[| y \<in> CInfinitesimal; x @c= z + y|] ==> x @c= z"
1.471 +apply (drule_tac x = z in CInfinitesimal_add_capprox_self2 [THEN capprox_sym])
1.472 +apply (erule capprox_trans3 [THEN capprox_sym])
1.474 +apply (erule capprox_sym)
1.475 +done
1.476 +
1.477 +lemma capprox_add_left_cancel: "d + b  @c= d + c ==> b @c= c"
1.478 +apply (drule capprox_minus_iff [THEN iffD1])
1.480 +done
1.481 +
1.482 +lemma capprox_add_right_cancel: "b + d @c= c + d ==> b @c= c"
1.483 +apply (rule capprox_add_left_cancel)
1.485 +done
1.486 +
1.487 +lemma capprox_add_mono1: "b @c= c ==> d + b @c= d + c"
1.488 +apply (rule capprox_minus_iff [THEN iffD2])
1.489 +apply (simp add: capprox_minus_iff [symmetric] add_ac)
1.490 +done
1.491 +
1.492 +lemma capprox_add_mono2: "b @c= c ==> b + a @c= c + a"
1.494 +done
1.495 +
1.496 +lemma capprox_add_left_iff [iff]: "(a + b @c= a + c) = (b @c= c)"
1.498 +
1.499 +lemma capprox_add_right_iff [iff]: "(b + a @c= c + a) = (b @c= c)"
1.501 +
1.502 +lemma capprox_CFinite: "[| x \<in> CFinite; x @c= y |] ==> y \<in> CFinite"
1.503 +apply (drule bex_CInfinitesimal_iff2 [THEN iffD2], safe)
1.504 +apply (drule CInfinitesimal_subset_CFinite [THEN subsetD, THEN CFinite_minus_iff [THEN iffD2]])
1.505 +apply (drule CFinite_add)
1.506 +apply (assumption, auto)
1.507 +done
1.508 +
1.509 +lemma capprox_hcomplex_of_complex_CFinite:
1.510 +     "x @c= hcomplex_of_complex D ==> x \<in> CFinite"
1.511 +by (rule capprox_sym [THEN [2] capprox_CFinite], auto)
1.512 +
1.513 +lemma capprox_mult_CFinite:
1.514 +     "[|a @c= b; c @c= d; b \<in> CFinite; d \<in> CFinite|] ==> a*c @c= b*d"
1.515 +apply (rule capprox_trans)
1.516 +apply (rule_tac [2] capprox_mult2)
1.517 +apply (rule capprox_mult1)
1.518 +prefer 2 apply (blast intro: capprox_CFinite capprox_sym, auto)
1.519 +done
1.520 +
1.521 +lemma capprox_mult_hcomplex_of_complex:
1.522 +     "[|a @c= hcomplex_of_complex b; c @c= hcomplex_of_complex d |]
1.523 +      ==> a*c @c= hcomplex_of_complex b * hcomplex_of_complex d"
1.524 +apply (blast intro!: capprox_mult_CFinite capprox_hcomplex_of_complex_CFinite CFinite_hcomplex_of_complex)
1.525 +done
1.526 +
1.527 +lemma capprox_SComplex_mult_cancel_zero:
1.528 +     "[| a \<in> SComplex; a \<noteq> 0; a*x @c= 0 |] ==> x @c= 0"
1.529 +apply (drule SComplex_inverse [THEN SComplex_subset_CFinite [THEN subsetD]])
1.530 +apply (auto dest: capprox_mult2 simp add: hcomplex_mult_assoc [symmetric])
1.531 +done
1.532 +
1.533 +lemma capprox_mult_SComplex1: "[| a \<in> SComplex; x @c= 0 |] ==> x*a @c= 0"
1.534 +by (auto dest: SComplex_subset_CFinite [THEN subsetD] capprox_mult1)
1.535 +
1.536 +lemma capprox_mult_SComplex2: "[| a \<in> SComplex; x @c= 0 |] ==> a*x @c= 0"
1.537 +by (auto dest: SComplex_subset_CFinite [THEN subsetD] capprox_mult2)
1.538 +
1.539 +lemma capprox_mult_SComplex_zero_cancel_iff [simp]:
1.540 +     "[|a \<in> SComplex; a \<noteq> 0 |] ==> (a*x @c= 0) = (x @c= 0)"
1.541 +by (blast intro: capprox_SComplex_mult_cancel_zero capprox_mult_SComplex2)
1.542 +
1.543 +lemma capprox_SComplex_mult_cancel:
1.544 +     "[| a \<in> SComplex; a \<noteq> 0; a* w @c= a*z |] ==> w @c= z"
1.545 +apply (drule SComplex_inverse [THEN SComplex_subset_CFinite [THEN subsetD]])
1.546 +apply (auto dest: capprox_mult2 simp add: hcomplex_mult_assoc [symmetric])
1.547 +done
1.548 +
1.549 +lemma capprox_SComplex_mult_cancel_iff1 [simp]:
1.550 +     "[| a \<in> SComplex; a \<noteq> 0|] ==> (a* w @c= a*z) = (w @c= z)"
1.551 +by (auto intro!: capprox_mult2 SComplex_subset_CFinite [THEN subsetD]
1.552 +            intro: capprox_SComplex_mult_cancel)
1.553 +
1.554 +lemma capprox_hcmod_approx_zero: "(x @c= y) = (hcmod (y - x) @= 0)"
1.555 +apply (rule capprox_minus_iff [THEN ssubst])
1.556 +apply (simp add: capprox_def CInfinitesimal_hcmod_iff mem_infmal_iff diff_minus [symmetric] hcmod_diff_commute)
1.557 +done
1.558 +
1.559 +lemma capprox_approx_zero_iff: "(x @c= 0) = (hcmod x @= 0)"
1.560 +by (simp add: capprox_hcmod_approx_zero)
1.561 +
1.562 +lemma capprox_minus_zero_cancel_iff [simp]: "(-x @c= 0) = (x @c= 0)"
1.563 +by (simp add: capprox_hcmod_approx_zero)
1.564 +
1.566 +     "u @c= 0 ==> hcmod(x + u) - hcmod x \<in> Infinitesimal"
1.567 +apply (rule_tac e = "hcmod u" and e' = "- hcmod u" in Infinitesimal_interval2)
1.568 +apply (auto dest: capprox_approx_zero_iff [THEN iffD1]
1.569 +             simp add: mem_infmal_iff [symmetric] hypreal_diff_def)
1.570 +apply (rule_tac c1 = "hcmod x" in add_le_cancel_left [THEN iffD1])
1.571 +apply (auto simp add: diff_minus [symmetric])
1.572 +done
1.573 +
1.574 +lemma approx_hcmod_add_hcmod: "u @c= 0 ==> hcmod(x + u) @= hcmod x"
1.575 +apply (rule approx_minus_iff [THEN iffD2])
1.576 +apply (auto intro: Infinitesimal_hcmod_add_diff simp add: mem_infmal_iff [symmetric] diff_minus [symmetric])
1.577 +done
1.578 +
1.579 +lemma capprox_hcmod_approx: "x @c= y ==> hcmod x @= hcmod y"
1.580 +by (auto intro: approx_hcmod_add_hcmod
1.581 +         dest!: bex_CInfinitesimal_iff2 [THEN iffD2]
1.582 +         simp add: mem_cinfmal_iff)
1.583
1.584
1.585 -defs
1.586 +subsection{*Zero is the Only Infinitesimal Complex Number*}
1.587 +
1.588 +lemma CInfinitesimal_less_SComplex:
1.589 +   "[| x \<in> SComplex; y \<in> CInfinitesimal; 0 < hcmod x |] ==> hcmod y < hcmod x"
1.590 +by (auto intro!: Infinitesimal_less_SReal SComplex_hcmod_SReal simp add: CInfinitesimal_hcmod_iff)
1.591 +
1.592 +lemma SComplex_Int_CInfinitesimal_zero: "SComplex Int CInfinitesimal = {0}"
1.593 +apply (auto simp add: SComplex_def CInfinitesimal_hcmod_iff)
1.594 +apply (cut_tac r = r in SReal_hcmod_hcomplex_of_complex)
1.595 +apply (drule_tac A = Reals in IntI, assumption)
1.596 +apply (subgoal_tac "hcmod (hcomplex_of_complex r) = 0")
1.597 +apply simp
1.598 +apply (simp add: SReal_Int_Infinitesimal_zero)
1.599 +done
1.600 +
1.601 +lemma SComplex_CInfinitesimal_zero:
1.602 +     "[| x \<in> SComplex; x \<in> CInfinitesimal|] ==> x = 0"
1.603 +by (cut_tac SComplex_Int_CInfinitesimal_zero, blast)
1.604 +
1.605 +lemma SComplex_CFinite_diff_CInfinitesimal:
1.606 +     "[| x \<in> SComplex; x \<noteq> 0 |] ==> x \<in> CFinite - CInfinitesimal"
1.607 +by (auto dest: SComplex_CInfinitesimal_zero SComplex_subset_CFinite [THEN subsetD])
1.608 +
1.609 +lemma hcomplex_of_complex_CFinite_diff_CInfinitesimal:
1.610 +     "hcomplex_of_complex x \<noteq> 0
1.611 +      ==> hcomplex_of_complex x \<in> CFinite - CInfinitesimal"
1.612 +by (rule SComplex_CFinite_diff_CInfinitesimal, auto)
1.613 +
1.614 +lemma hcomplex_of_complex_CInfinitesimal_iff_0 [iff]:
1.615 +     "(hcomplex_of_complex x \<in> CInfinitesimal) = (x=0)"
1.616 +apply (auto simp add: hcomplex_of_complex_zero)
1.617 +apply (rule ccontr)
1.618 +apply (rule hcomplex_of_complex_CFinite_diff_CInfinitesimal [THEN DiffD2], auto)
1.619 +done
1.620 +
1.621 +lemma number_of_not_CInfinitesimal [simp]:
1.622 +     "number_of w \<noteq> (0::hcomplex) ==> number_of w \<notin> CInfinitesimal"
1.623 +by (fast dest: SComplex_number_of [THEN SComplex_CInfinitesimal_zero])
1.624 +
1.625 +lemma capprox_SComplex_not_zero:
1.626 +     "[| y \<in> SComplex; x @c= y; y\<noteq> 0 |] ==> x \<noteq> 0"
1.627 +by (auto dest: SComplex_CInfinitesimal_zero capprox_sym [THEN mem_cinfmal_iff [THEN iffD2]])
1.628 +
1.629 +lemma CFinite_diff_CInfinitesimal_capprox:
1.630 +     "[| x @c= y; y \<in> CFinite - CInfinitesimal |]
1.631 +      ==> x \<in> CFinite - CInfinitesimal"
1.632 +apply (auto intro: capprox_sym [THEN [2] capprox_CFinite]
1.633 +            simp add: mem_cinfmal_iff)
1.634 +apply (drule capprox_trans3, assumption)
1.635 +apply (blast dest: capprox_sym)
1.636 +done
1.637 +
1.638 +lemma CInfinitesimal_ratio:
1.639 +     "[| y \<noteq> 0;  y \<in> CInfinitesimal;  x/y \<in> CFinite |] ==> x \<in> CInfinitesimal"
1.640 +apply (drule CInfinitesimal_CFinite_mult2, assumption)
1.641 +apply (simp add: divide_inverse_zero hcomplex_mult_assoc)
1.642 +done
1.643 +
1.644 +lemma SComplex_capprox_iff:
1.645 +     "[|x \<in> SComplex; y \<in> SComplex|] ==> (x @c= y) = (x = y)"
1.646 +apply auto
1.647 +apply (simp add: capprox_def)
1.648 +apply (subgoal_tac "x-y = 0", simp)
1.649 +apply (rule SComplex_CInfinitesimal_zero)
1.650 +apply (simp add: SComplex_diff, assumption)
1.651 +done
1.652 +
1.653 +lemma number_of_capprox_iff [simp]:
1.654 +    "(number_of v @c= number_of w) = (number_of v = (number_of w :: hcomplex))"
1.655 +by (rule SComplex_capprox_iff, auto)
1.656 +
1.657 +lemma number_of_CInfinitesimal_iff [simp]:
1.658 +     "(number_of w \<in> CInfinitesimal) = (number_of w = (0::hcomplex))"
1.659 +apply (rule iffI)
1.660 +apply (fast dest: SComplex_number_of [THEN SComplex_CInfinitesimal_zero])
1.661 +apply (simp (no_asm_simp))
1.662 +done
1.663 +
1.664 +lemma hcomplex_of_complex_approx_iff [simp]:
1.665 +     "(hcomplex_of_complex k @c= hcomplex_of_complex m) = (k = m)"
1.666 +apply auto
1.667 +apply (rule inj_hcomplex_of_complex [THEN injD])
1.668 +apply (rule SComplex_capprox_iff [THEN iffD1], auto)
1.669 +done
1.670 +
1.671 +lemma hcomplex_of_complex_capprox_number_of_iff [simp]:
1.672 +     "(hcomplex_of_complex k @c= number_of w) = (k = number_of w)"
1.673 +by (subst hcomplex_of_complex_approx_iff [symmetric], auto)
1.674 +
1.675 +lemma capprox_unique_complex:
1.676 +     "[| r \<in> SComplex; s \<in> SComplex; r @c= x; s @c= x|] ==> r = s"
1.677 +by (blast intro: SComplex_capprox_iff [THEN iffD1] capprox_trans2)
1.678 +
1.679 +lemma hcomplex_capproxD1:
1.680 +     "Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n})
1.681 +      ==> Abs_hypreal(hyprel `` {%n. Re(X n)}) @=
1.682 +          Abs_hypreal(hyprel `` {%n. Re(Y n)})"
1.683 +apply (auto simp add: approx_FreeUltrafilterNat_iff)
1.684 +apply (drule capprox_minus_iff [THEN iffD1])
1.685 +apply (auto simp add: hcomplex_minus hcomplex_add mem_cinfmal_iff [symmetric] CInfinitesimal_hcmod_iff hcmod Infinitesimal_FreeUltrafilterNat_iff2)
1.686 +apply (drule_tac x = m in spec, ultra)
1.687 +apply (rename_tac Z x)
1.688 +apply (case_tac "X x")
1.689 +apply (case_tac "Y x")
1.690 +apply (auto simp add: complex_minus complex_add complex_mod
1.691 +           simp del: realpow_Suc)
1.692 +apply (rule_tac y="abs(Z x)" in order_le_less_trans)
1.693 +apply (drule_tac t = "Z x" in sym)
1.694 +apply (auto simp add: abs_eqI1 simp del: realpow_Suc)
1.695 +done
1.696 +
1.697 +(* same proof *)
1.698 +lemma hcomplex_capproxD2:
1.699 +     "Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n})
1.700 +      ==> Abs_hypreal(hyprel `` {%n. Im(X n)}) @=
1.701 +          Abs_hypreal(hyprel `` {%n. Im(Y n)})"
1.702 +apply (auto simp add: approx_FreeUltrafilterNat_iff)
1.703 +apply (drule capprox_minus_iff [THEN iffD1])
1.704 +apply (auto simp add: hcomplex_minus hcomplex_add mem_cinfmal_iff [symmetric] CInfinitesimal_hcmod_iff hcmod Infinitesimal_FreeUltrafilterNat_iff2)
1.705 +apply (drule_tac x = m in spec, ultra)
1.706 +apply (rename_tac Z x)
1.707 +apply (case_tac "X x")
1.708 +apply (case_tac "Y x")
1.709 +apply (auto simp add: complex_minus complex_add complex_mod simp del: realpow_Suc)
1.710 +apply (rule_tac y="abs(Z x)" in order_le_less_trans)
1.711 +apply (drule_tac t = "Z x" in sym)
1.712 +apply (auto simp add: abs_eqI1 simp del: realpow_Suc)
1.713 +done
1.714 +
1.715 +lemma hcomplex_capproxI:
1.716 +     "[| Abs_hypreal(hyprel `` {%n. Re(X n)}) @=
1.717 +         Abs_hypreal(hyprel `` {%n. Re(Y n)});
1.718 +         Abs_hypreal(hyprel `` {%n. Im(X n)}) @=
1.719 +         Abs_hypreal(hyprel `` {%n. Im(Y n)})
1.720 +      |] ==> Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n})"
1.721 +apply (drule approx_minus_iff [THEN iffD1])
1.722 +apply (drule approx_minus_iff [THEN iffD1])
1.723 +apply (rule capprox_minus_iff [THEN iffD2])
1.724 +apply (auto simp add: mem_cinfmal_iff [symmetric] mem_infmal_iff [symmetric] hypreal_minus hypreal_add hcomplex_minus hcomplex_add CInfinitesimal_hcmod_iff hcmod Infinitesimal_FreeUltrafilterNat_iff)
1.725 +apply (rule bexI, rule_tac [2] lemma_hyprel_refl, auto)
1.726 +apply (drule_tac x = "u/2" in spec)
1.727 +apply (drule_tac x = "u/2" in spec, auto, ultra)
1.728 +apply (drule sym, drule sym)
1.729 +apply (case_tac "X x")
1.730 +apply (case_tac "Y x")
1.731 +apply (auto simp add: complex_minus complex_add complex_mod snd_conv fst_conv numeral_2_eq_2)
1.732 +apply (rename_tac a b c d)
1.733 +apply (subgoal_tac "sqrt (abs (a + - c) ^ 2 + abs (b + - d) ^ 2) < u")
1.734 +apply (rule_tac [2] lemma_sqrt_hcomplex_capprox, auto)
1.735 +apply (simp add: power2_eq_square)
1.736 +done
1.737 +
1.738 +lemma capprox_approx_iff:
1.739 +     "(Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n})) =
1.740 +       (Abs_hypreal(hyprel `` {%n. Re(X n)}) @= Abs_hypreal(hyprel `` {%n. Re(Y n)}) &
1.741 +        Abs_hypreal(hyprel `` {%n. Im(X n)}) @= Abs_hypreal(hyprel `` {%n. Im(Y n)}))"
1.742 +apply (blast intro: hcomplex_capproxI hcomplex_capproxD1 hcomplex_capproxD2)
1.743 +done
1.744 +
1.745 +lemma hcomplex_of_hypreal_capprox_iff [simp]:
1.746 +     "(hcomplex_of_hypreal x @c= hcomplex_of_hypreal z) = (x @= z)"
1.747 +apply (rule eq_Abs_hypreal [of x])
1.748 +apply (rule eq_Abs_hypreal [of z])
1.749 +apply (simp add: hcomplex_of_hypreal capprox_approx_iff)
1.750 +done
1.751 +
1.752 +lemma CFinite_HFinite_Re:
1.753 +     "Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CFinite
1.754 +      ==> Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> HFinite"
1.755 +apply (auto simp add: CFinite_hcmod_iff hcmod HFinite_FreeUltrafilterNat_iff)
1.756 +apply (rule bexI, rule_tac [2] lemma_hyprel_refl)
1.757 +apply (rule_tac x = u in exI, ultra)
1.758 +apply (drule sym, case_tac "X x")
1.759 +apply (auto simp add: complex_mod numeral_2_eq_2 simp del: realpow_Suc)
1.760 +apply (rule ccontr, drule linorder_not_less [THEN iffD1])
1.761 +apply (drule order_less_le_trans, assumption)
1.762 +apply (drule real_sqrt_ge_abs1 [THEN [2] order_less_le_trans])
1.763 +apply (auto simp add: numeral_2_eq_2 [symmetric])
1.764 +done
1.765 +
1.766 +lemma CFinite_HFinite_Im:
1.767 +     "Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CFinite
1.768 +      ==> Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> HFinite"
1.769 +apply (auto simp add: CFinite_hcmod_iff hcmod HFinite_FreeUltrafilterNat_iff)
1.770 +apply (rule bexI, rule_tac [2] lemma_hyprel_refl)
1.771 +apply (rule_tac x = u in exI, ultra)
1.772 +apply (drule sym, case_tac "X x")
1.773 +apply (auto simp add: complex_mod simp del: realpow_Suc)
1.774 +apply (rule ccontr, drule linorder_not_less [THEN iffD1])
1.775 +apply (drule order_less_le_trans, assumption)
1.776 +apply (drule real_sqrt_ge_abs2 [THEN [2] order_less_le_trans], auto)
1.777 +done
1.778 +
1.779 +lemma HFinite_Re_Im_CFinite:
1.780 +     "[| Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> HFinite;
1.781 +         Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> HFinite
1.782 +      |] ==> Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CFinite"
1.783 +apply (auto simp add: CFinite_hcmod_iff hcmod HFinite_FreeUltrafilterNat_iff)
1.784 +apply (rename_tac Y Z u v)
1.785 +apply (rule bexI, rule_tac [2] lemma_hyprel_refl)
1.786 +apply (rule_tac x = "2* (u + v) " in exI)
1.787 +apply ultra
1.788 +apply (drule sym, case_tac "X x")
1.789 +apply (auto simp add: complex_mod numeral_2_eq_2 simp del: realpow_Suc)
1.790 +apply (subgoal_tac "0 < u")
1.791 + prefer 2 apply arith
1.792 +apply (subgoal_tac "0 < v")
1.793 + prefer 2 apply arith
1.794 +apply (subgoal_tac "sqrt (abs (Y x) ^ 2 + abs (Z x) ^ 2) < 2*u + 2*v")
1.795 +apply (rule_tac [2] lemma_sqrt_hcomplex_capprox, auto)
1.796 +apply (simp add: power2_eq_square)
1.797 +done
1.798 +
1.799 +lemma CFinite_HFinite_iff:
1.800 +     "(Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CFinite) =
1.801 +      (Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> HFinite &
1.802 +       Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> HFinite)"
1.803 +by (blast intro: HFinite_Re_Im_CFinite CFinite_HFinite_Im CFinite_HFinite_Re)
1.804 +
1.805 +lemma SComplex_Re_SReal:
1.806 +     "Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> SComplex
1.807 +      ==> Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> Reals"
1.808 +apply (auto simp add: SComplex_def hcomplex_of_complex_def SReal_def hypreal_of_real_def)
1.809 +apply (rule_tac x = "Re r" in exI, ultra)
1.810 +done
1.811 +
1.812 +lemma SComplex_Im_SReal:
1.813 +     "Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> SComplex
1.814 +      ==> Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> Reals"
1.815 +apply (auto simp add: SComplex_def hcomplex_of_complex_def SReal_def hypreal_of_real_def)
1.816 +apply (rule_tac x = "Im r" in exI, ultra)
1.817 +done
1.818 +
1.819 +lemma Reals_Re_Im_SComplex:
1.820 +     "[| Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> Reals;
1.821 +         Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> Reals
1.822 +      |] ==> Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> SComplex"
1.823 +apply (auto simp add: SComplex_def hcomplex_of_complex_def SReal_def hypreal_of_real_def)
1.824 +apply (rule_tac x = "Complex r ra" in exI, ultra)
1.825 +done
1.826 +
1.827 +lemma SComplex_SReal_iff:
1.828 +     "(Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> SComplex) =
1.829 +      (Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> Reals &
1.830 +       Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> Reals)"
1.831 +by (blast intro: SComplex_Re_SReal SComplex_Im_SReal Reals_Re_Im_SComplex)
1.832 +
1.833 +lemma CInfinitesimal_Infinitesimal_iff:
1.834 +     "(Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CInfinitesimal) =
1.835 +      (Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> Infinitesimal &
1.836 +       Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> Infinitesimal)"
1.837 +by (simp add: mem_cinfmal_iff mem_infmal_iff hcomplex_zero_num hypreal_zero_num capprox_approx_iff)
1.838 +
1.839 +lemma eq_Abs_hcomplex_EX:
1.840 +     "(\<exists>t. P t) = (\<exists>X. P (Abs_hcomplex(hcomplexrel `` {X})))"
1.841 +apply auto
1.842 +apply (rule_tac z = t in eq_Abs_hcomplex, auto)
1.843 +done
1.844 +
1.845 +lemma eq_Abs_hcomplex_Bex:
1.846 +     "(\<exists>t \<in> A. P t) = (\<exists>X. (Abs_hcomplex(hcomplexrel `` {X})) \<in> A &
1.847 +                         P (Abs_hcomplex(hcomplexrel `` {X})))"
1.848 +apply auto
1.849 +apply (rule_tac z = t in eq_Abs_hcomplex, auto)
1.850 +done
1.851 +
1.852 +(* Here we go - easy proof now!! *)
1.853 +lemma stc_part_Ex: "x:CFinite ==> \<exists>t \<in> SComplex. x @c= t"
1.854 +apply (rule_tac z = x in eq_Abs_hcomplex)
1.855 +apply (auto simp add: CFinite_HFinite_iff eq_Abs_hcomplex_Bex SComplex_SReal_iff capprox_approx_iff)
1.856 +apply (drule st_part_Ex, safe)+
1.857 +apply (rule_tac z = t in eq_Abs_hypreal)
1.858 +apply (rule_tac z = ta in eq_Abs_hypreal, auto)
1.859 +apply (rule_tac x = "%n. Complex (xa n) (xb n) " in exI)
1.860 +apply auto
1.861 +done
1.862 +
1.863 +lemma stc_part_Ex1: "x:CFinite ==> EX! t. t \<in> SComplex &  x @c= t"
1.864 +apply (drule stc_part_Ex, safe)
1.865 +apply (drule_tac [2] capprox_sym, drule_tac [2] capprox_sym, drule_tac [2] capprox_sym)
1.866 +apply (auto intro!: capprox_unique_complex)
1.867 +done
1.868 +
1.869 +lemma CFinite_Int_CInfinite_empty: "CFinite Int CInfinite = {}"
1.870 +by (simp add: CFinite_def CInfinite_def, auto)
1.871 +
1.872 +lemma CFinite_not_CInfinite: "x \<in> CFinite ==> x \<notin> CInfinite"
1.873 +by (insert CFinite_Int_CInfinite_empty, blast)
1.874 +
1.875 +text{*Not sure this is a good idea!*}
1.876 +declare CFinite_Int_CInfinite_empty [simp]
1.877 +
1.878 +lemma not_CFinite_CInfinite: "x\<notin> CFinite ==> x \<in> CInfinite"
1.879 +by (auto intro: not_HFinite_HInfinite simp add: CFinite_hcmod_iff CInfinite_hcmod_iff)
1.880 +
1.881 +lemma CInfinite_CFinite_disj: "x \<in> CInfinite | x \<in> CFinite"
1.882 +by (blast intro: not_CFinite_CInfinite)
1.883 +
1.884 +lemma CInfinite_CFinite_iff: "(x \<in> CInfinite) = (x \<notin> CFinite)"
1.885 +by (blast dest: CFinite_not_CInfinite not_CFinite_CInfinite)
1.886 +
1.887 +lemma CFinite_CInfinite_iff: "(x \<in> CFinite) = (x \<notin> CInfinite)"
1.888 +by (simp add: CInfinite_CFinite_iff)
1.889 +
1.890 +lemma CInfinite_diff_CFinite_CInfinitesimal_disj:
1.891 +     "x \<notin> CInfinitesimal ==> x \<in> CInfinite | x \<in> CFinite - CInfinitesimal"
1.892 +by (fast intro: not_CFinite_CInfinite)
1.893 +
1.894 +lemma CFinite_inverse:
1.895 +     "[| x \<in> CFinite; x \<notin> CInfinitesimal |] ==> inverse x \<in> CFinite"
1.896 +apply (cut_tac x = "inverse x" in CInfinite_CFinite_disj)
1.897 +apply (auto dest!: CInfinite_inverse_CInfinitesimal)
1.898 +done
1.899 +
1.900 +lemma CFinite_inverse2: "x \<in> CFinite - CInfinitesimal ==> inverse x \<in> CFinite"
1.901 +by (blast intro: CFinite_inverse)
1.902 +
1.903 +lemma CInfinitesimal_inverse_CFinite:
1.904 +     "x \<notin> CInfinitesimal ==> inverse(x) \<in> CFinite"
1.905 +apply (drule CInfinite_diff_CFinite_CInfinitesimal_disj)
1.906 +apply (blast intro: CFinite_inverse CInfinite_inverse_CInfinitesimal CInfinitesimal_subset_CFinite [THEN subsetD])
1.907 +done
1.908 +
1.909 +
1.910 +lemma CFinite_not_CInfinitesimal_inverse:
1.911 +     "x \<in> CFinite - CInfinitesimal ==> inverse x \<in> CFinite - CInfinitesimal"
1.912 +apply (auto intro: CInfinitesimal_inverse_CFinite)
1.913 +apply (drule CInfinitesimal_CFinite_mult2, assumption)
1.914 +apply (simp add: not_CInfinitesimal_not_zero)
1.915 +done
1.916 +
1.917 +lemma capprox_inverse:
1.918 +     "[| x @c= y; y \<in>  CFinite - CInfinitesimal |] ==> inverse x @c= inverse y"
1.919 +apply (frule CFinite_diff_CInfinitesimal_capprox, assumption)
1.920 +apply (frule not_CInfinitesimal_not_zero2)
1.921 +apply (frule_tac x = x in not_CInfinitesimal_not_zero2)
1.922 +apply (drule CFinite_inverse2)+
1.923 +apply (drule capprox_mult2, assumption, auto)
1.924 +apply (drule_tac c = "inverse x" in capprox_mult1, assumption)
1.925 +apply (auto intro: capprox_sym simp add: hcomplex_mult_assoc)
1.926 +done
1.927 +
1.928 +lemmas hcomplex_of_complex_capprox_inverse =  hcomplex_of_complex_CFinite_diff_CInfinitesimal [THEN [2] capprox_inverse]
1.929 +
1.931 +     "[| x \<in> CFinite - CInfinitesimal;
1.932 +         h \<in> CInfinitesimal |] ==> inverse(x + h) @c= inverse x"
1.933 +by (auto intro: capprox_inverse capprox_sym CInfinitesimal_add_capprox_self)
1.934 +
1.936 +     "[| x \<in> CFinite - CInfinitesimal;
1.937 +         h \<in> CInfinitesimal |] ==> inverse(h + x) @c= inverse x"
1.938 +apply (rule add_commute [THEN subst])
1.939 +apply (blast intro: inverse_add_CInfinitesimal_capprox)
1.940 +done
1.941 +
1.943 +     "[| x \<in> CFinite - CInfinitesimal;
1.944 +         h \<in> CInfinitesimal |] ==> inverse(x + h) - inverse x @c= h"
1.945 +apply (rule capprox_trans2)
1.946 +apply (auto intro: inverse_add_CInfinitesimal_capprox
1.947 +       simp add: mem_cinfmal_iff diff_minus capprox_minus_iff [symmetric])
1.948 +done
1.949 +
1.950 +lemma CInfinitesimal_square_iff [iff]:
1.951 +     "(x*x \<in> CInfinitesimal) = (x \<in> CInfinitesimal)"
1.952 +by (simp add: CInfinitesimal_hcmod_iff hcmod_mult)
1.953 +
1.954 +lemma capprox_CFinite_mult_cancel:
1.955 +     "[| a \<in> CFinite-CInfinitesimal; a*w @c= a*z |] ==> w @c= z"
1.956 +apply safe
1.957 +apply (frule CFinite_inverse, assumption)
1.958 +apply (drule not_CInfinitesimal_not_zero)
1.959 +apply (auto dest: capprox_mult2 simp add: hcomplex_mult_assoc [symmetric])
1.960 +done
1.961 +
1.962 +lemma capprox_CFinite_mult_cancel_iff1:
1.963 +     "a \<in> CFinite-CInfinitesimal ==> (a * w @c= a * z) = (w @c= z)"
1.964 +by (auto intro: capprox_mult2 capprox_CFinite_mult_cancel)
1.965 +
1.966 +
1.968 +
1.971 +apply (auto dest: capprox_sym elim!: capprox_trans equalityCE)
1.972 +done
1.973 +
1.975 +     "e \<in> CInfinitesimal ==> cmonad (x+e) = cmonad x"
1.976 +by (fast intro!: CInfinitesimal_add_capprox_self [THEN capprox_sym] capprox_cmonad_iff [THEN iffD1])
1.977 +
1.978 +lemma mem_cmonad_iff: "(u \<in> cmonad x) = (-u \<in> cmonad (-x))"
1.980 +
1.981 +lemma CInfinitesimal_cmonad_zero_iff: "(x:CInfinitesimal) = (x \<in> cmonad 0)"
1.982 +by (auto intro: capprox_sym simp add: mem_cinfmal_iff cmonad_def)
1.983 +
1.984 +lemma cmonad_zero_minus_iff: "(x \<in> cmonad 0) = (-x \<in> cmonad 0)"
1.986 +
1.987 +lemma cmonad_zero_hcmod_iff: "(x \<in> cmonad 0) = (hcmod x:monad 0)"
1.989 +
1.990 +lemma mem_cmonad_self [simp]: "x \<in> cmonad x"
1.992 +
1.993 +
1.994 +subsection{*Theorems About Standard Part*}
1.995 +
1.996 +lemma stc_capprox_self: "x \<in> CFinite ==> stc x @c= x"
1.997 +apply (simp add: stc_def)
1.998 +apply (frule stc_part_Ex, safe)
1.999 +apply (rule someI2)
1.1000 +apply (auto intro: capprox_sym)
1.1001 +done
1.1002 +
1.1003 +lemma stc_SComplex: "x \<in> CFinite ==> stc x \<in> SComplex"
1.1004 +apply (simp add: stc_def)
1.1005 +apply (frule stc_part_Ex, safe)
1.1006 +apply (rule someI2)
1.1007 +apply (auto intro: capprox_sym)
1.1008 +done
1.1009 +
1.1010 +lemma stc_CFinite: "x \<in> CFinite ==> stc x \<in> CFinite"
1.1011 +by (erule stc_SComplex [THEN SComplex_subset_CFinite [THEN subsetD]])
1.1012 +
1.1013 +lemma stc_SComplex_eq [simp]: "x \<in> SComplex ==> stc x = x"
1.1014 +apply (simp add: stc_def)
1.1015 +apply (rule some_equality)
1.1016 +apply (auto intro: SComplex_subset_CFinite [THEN subsetD])
1.1017 +apply (blast dest: SComplex_capprox_iff [THEN iffD1])
1.1018 +done
1.1019 +
1.1020 +lemma stc_hcomplex_of_complex:
1.1021 +     "stc (hcomplex_of_complex x) = hcomplex_of_complex x"
1.1022 +by auto
1.1023 +
1.1024 +lemma stc_eq_capprox:
1.1025 +     "[| x \<in> CFinite; y \<in> CFinite; stc x = stc y |] ==> x @c= y"
1.1026 +by (auto dest!: stc_capprox_self elim!: capprox_trans3)
1.1027 +
1.1028 +lemma capprox_stc_eq:
1.1029 +     "[| x \<in> CFinite; y \<in> CFinite; x @c= y |] ==> stc x = stc y"
1.1030 +by (blast intro: capprox_trans capprox_trans2 SComplex_capprox_iff [THEN iffD1]
1.1031 +          dest: stc_capprox_self stc_SComplex)
1.1032
1.1033 -   capprox_def  "x @c= y == (x - y) : CInfinitesimal"
1.1034 +lemma stc_eq_capprox_iff:
1.1035 +     "[| x \<in> CFinite; y \<in> CFinite|] ==> (x @c= y) = (stc x = stc y)"
1.1036 +by (blast intro: capprox_stc_eq stc_eq_capprox)
1.1037 +
1.1039 +     "[| x \<in> SComplex; e \<in> CInfinitesimal |] ==> stc(x + e) = x"
1.1040 +apply (frule stc_SComplex_eq [THEN subst])
1.1041 +prefer 2 apply assumption
1.1042 +apply (frule SComplex_subset_CFinite [THEN subsetD])
1.1043 +apply (frule CInfinitesimal_subset_CFinite [THEN subsetD])
1.1044 +apply (drule stc_SComplex_eq)
1.1045 +apply (rule capprox_stc_eq)
1.1046 +apply (auto intro: CFinite_add simp add: CInfinitesimal_add_capprox_self [THEN capprox_sym])
1.1047 +done
1.1048 +
1.1050 +     "[| x \<in> SComplex; e \<in> CInfinitesimal |] ==> stc(e + x) = x"
1.1051 +apply (rule add_commute [THEN subst])
1.1052 +apply (blast intro!: stc_CInfinitesimal_add_SComplex)
1.1053 +done
1.1054 +
1.1056 +     "x \<in> CFinite ==> \<exists>e \<in> CInfinitesimal. x = stc(x) + e"
1.1057 +by (blast dest!: stc_capprox_self [THEN capprox_sym] bex_CInfinitesimal_iff2 [THEN iffD2])
1.1058 +
1.1060 +     "[| x \<in> CFinite; y \<in> CFinite |] ==> stc (x + y) = stc(x) + stc(y)"
1.1061 +apply (frule CFinite_stc_CInfinitesimal_add)
1.1062 +apply (frule_tac x = y in CFinite_stc_CInfinitesimal_add, safe)
1.1063 +apply (subgoal_tac "stc (x + y) = stc ((stc x + e) + (stc y + ea))")
1.1064 +apply (drule_tac [2] sym, drule_tac [2] sym)
1.1065 + prefer 2 apply simp
1.1067 +apply (drule stc_SComplex)+
1.1068 +apply (drule SComplex_add, assumption)
1.1069 +apply (drule CInfinitesimal_add, assumption)
1.1070 +apply (rule add_assoc [THEN subst])
1.1071 +apply (blast intro!: stc_CInfinitesimal_add_SComplex2)
1.1072 +done
1.1073 +
1.1074 +lemma stc_number_of [simp]: "stc (number_of w) = number_of w"
1.1075 +by (rule SComplex_number_of [THEN stc_SComplex_eq])
1.1076 +
1.1077 +lemma stc_zero [simp]: "stc 0 = 0"
1.1078 +by simp
1.1079 +
1.1080 +lemma stc_one [simp]: "stc 1 = 1"
1.1081 +by simp
1.1082 +
1.1083 +lemma stc_minus: "y \<in> CFinite ==> stc(-y) = -stc(y)"
1.1084 +apply (frule CFinite_minus_iff [THEN iffD2])
1.1085 +apply (rule hcomplex_add_minus_eq_minus)
1.1086 +apply (drule stc_add [symmetric], assumption)
1.1088 +done
1.1089 +
1.1090 +lemma stc_diff:
1.1091 +     "[| x \<in> CFinite; y \<in> CFinite |] ==> stc (x-y) = stc(x) - stc(y)"
1.1092 +apply (simp add: diff_minus)
1.1093 +apply (frule_tac y1 = y in stc_minus [symmetric])
1.1094 +apply (drule_tac x1 = y in CFinite_minus_iff [THEN iffD2])
1.1095 +apply (auto intro: stc_add)
1.1096 +done
1.1097 +
1.1098 +lemma lemma_stc_mult:
1.1099 +     "[| x \<in> CFinite; y \<in> CFinite;
1.1100 +         e \<in> CInfinitesimal;
1.1101 +         ea: CInfinitesimal |]
1.1102 +       ==> e*y + x*ea + e*ea: CInfinitesimal"
1.1103 +apply (frule_tac x = e and y = y in CInfinitesimal_CFinite_mult)
1.1104 +apply (frule_tac [2] x = ea and y = x in CInfinitesimal_CFinite_mult)
1.1105 +apply (drule_tac [3] CInfinitesimal_mult)
1.1107 +done
1.1108 +
1.1109 +lemma stc_mult:
1.1110 +     "[| x \<in> CFinite; y \<in> CFinite |]
1.1111 +               ==> stc (x * y) = stc(x) * stc(y)"
1.1112 +apply (frule CFinite_stc_CInfinitesimal_add)
1.1113 +apply (frule_tac x = y in CFinite_stc_CInfinitesimal_add, safe)
1.1114 +apply (subgoal_tac "stc (x * y) = stc ((stc x + e) * (stc y + ea))")
1.1115 +apply (drule_tac [2] sym, drule_tac [2] sym)
1.1116 + prefer 2 apply simp
1.1117 +apply (erule_tac V = "x = stc x + e" in thin_rl)
1.1118 +apply (erule_tac V = "y = stc y + ea" in thin_rl)
1.1120 +apply (drule stc_SComplex)+
1.1122 +apply (rule stc_CInfinitesimal_add_SComplex)
1.1123 +apply (blast intro!: SComplex_mult)
1.1124 +apply (drule SComplex_subset_CFinite [THEN subsetD])+
1.1125 +apply (rule add_assoc [THEN subst])
1.1126 +apply (blast intro!: lemma_stc_mult)
1.1127 +done
1.1128 +
1.1129 +lemma stc_CInfinitesimal: "x \<in> CInfinitesimal ==> stc x = 0"
1.1130 +apply (rule stc_zero [THEN subst])
1.1131 +apply (rule capprox_stc_eq)
1.1132 +apply (auto intro: CInfinitesimal_subset_CFinite [THEN subsetD]
1.1133 +                 simp add: mem_cinfmal_iff [symmetric])
1.1134 +done
1.1135 +
1.1136 +lemma stc_not_CInfinitesimal: "stc(x) \<noteq> 0 ==> x \<notin> CInfinitesimal"
1.1137 +by (fast intro: stc_CInfinitesimal)
1.1138 +
1.1139 +lemma stc_inverse:
1.1140 +     "[| x \<in> CFinite; stc x \<noteq> 0 |]
1.1141 +      ==> stc(inverse x) = inverse (stc x)"
1.1142 +apply (rule_tac c1 = "stc x" in hcomplex_mult_left_cancel [THEN iffD1])
1.1143 +apply (auto simp add: stc_mult [symmetric] stc_not_CInfinitesimal CFinite_inverse)
1.1144 +apply (subst right_inverse, auto)
1.1145 +done
1.1146 +
1.1147 +lemma stc_divide [simp]:
1.1148 +     "[| x \<in> CFinite; y \<in> CFinite; stc y \<noteq> 0 |]
1.1149 +      ==> stc(x/y) = (stc x) / (stc y)"
1.1150 +by (simp add: divide_inverse_zero stc_mult stc_not_CInfinitesimal CFinite_inverse stc_inverse)
1.1151 +
1.1152 +lemma stc_idempotent [simp]: "x \<in> CFinite ==> stc(stc(x)) = stc(x)"
1.1153 +by (blast intro: stc_CFinite stc_capprox_self capprox_stc_eq)
1.1154 +
1.1155 +lemma CFinite_HFinite_hcomplex_of_hypreal:
1.1156 +     "z \<in> HFinite ==> hcomplex_of_hypreal z \<in> CFinite"
1.1157 +apply (rule eq_Abs_hypreal [of z])
1.1158 +apply (simp add: hcomplex_of_hypreal CFinite_HFinite_iff hypreal_zero_def [symmetric])
1.1159 +done
1.1160 +
1.1161 +lemma SComplex_SReal_hcomplex_of_hypreal:
1.1162 +     "x \<in> Reals ==>  hcomplex_of_hypreal x \<in> SComplex"
1.1163 +apply (rule eq_Abs_hypreal [of x])
1.1164 +apply (simp add: hcomplex_of_hypreal SComplex_SReal_iff hypreal_zero_def [symmetric])
1.1165 +done
1.1166 +
1.1167 +lemma stc_hcomplex_of_hypreal:
1.1168 + "z \<in> HFinite ==> stc(hcomplex_of_hypreal z) = hcomplex_of_hypreal (st z)"
1.1169 +apply (simp add: st_def stc_def)
1.1170 +apply (frule st_part_Ex, safe)
1.1171 +apply (rule someI2)
1.1172 +apply (auto intro: approx_sym)
1.1173 +apply (drule CFinite_HFinite_hcomplex_of_hypreal)
1.1174 +apply (frule stc_part_Ex, safe)
1.1175 +apply (rule someI2)
1.1176 +apply (auto intro: capprox_sym intro!: capprox_unique_complex dest: SComplex_SReal_hcomplex_of_hypreal)
1.1177 +done
1.1178 +
1.1179 +(*
1.1180 +Goal "x \<in> CFinite ==> hcmod(stc x) = st(hcmod x)"
1.1181 +by (dtac stc_capprox_self 1)
1.1182 +by (auto_tac (claset(),simpset() addsimps [bex_CInfinitesimal_iff2 RS sym]));
1.1183 +
1.1184 +
1.1186 +*)
1.1187 +
1.1188 +lemma CInfinitesimal_hcnj_iff [simp]:
1.1189 +     "(hcnj z \<in> CInfinitesimal) = (z \<in> CInfinitesimal)"
1.1190 +by (simp add: CInfinitesimal_hcmod_iff)
1.1191 +
1.1192 +lemma CInfinite_HInfinite_iff:
1.1193 +     "(Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CInfinite) =
1.1194 +      (Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> HInfinite |
1.1195 +       Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> HInfinite)"
1.1196 +by (simp add: CInfinite_CFinite_iff HInfinite_HFinite_iff CFinite_HFinite_iff)
1.1197 +
1.1198 +text{*These theorems should probably be deleted*}
1.1199 +lemma hcomplex_split_CInfinitesimal_iff:
1.1200 +     "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y \<in> CInfinitesimal) =
1.1201 +      (x \<in> Infinitesimal & y \<in> Infinitesimal)"
1.1202 +apply (rule eq_Abs_hypreal [of x])
1.1203 +apply (rule eq_Abs_hypreal [of y])
1.1204 +apply (simp add: iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal CInfinitesimal_Infinitesimal_iff)
1.1205 +done
1.1206 +
1.1207 +lemma hcomplex_split_CFinite_iff:
1.1208 +     "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y \<in> CFinite) =
1.1209 +      (x \<in> HFinite & y \<in> HFinite)"
1.1210 +apply (rule eq_Abs_hypreal [of x])
1.1211 +apply (rule eq_Abs_hypreal [of y])
1.1212 +apply (simp add: iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal CFinite_HFinite_iff)
1.1213 +done
1.1214 +
1.1215 +lemma hcomplex_split_SComplex_iff:
1.1216 +     "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y \<in> SComplex) =
1.1217 +      (x \<in> Reals & y \<in> Reals)"
1.1218 +apply (rule eq_Abs_hypreal [of x])
1.1219 +apply (rule eq_Abs_hypreal [of y])
1.1220 +apply (simp add: iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal SComplex_SReal_iff)
1.1221 +done
1.1222 +
1.1223 +lemma hcomplex_split_CInfinite_iff:
1.1224 +     "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y \<in> CInfinite) =
1.1225 +      (x \<in> HInfinite | y \<in> HInfinite)"
1.1226 +apply (rule eq_Abs_hypreal [of x])
1.1227 +apply (rule eq_Abs_hypreal [of y])
1.1228 +apply (simp add: iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal CInfinite_HInfinite_iff)
1.1229 +done
1.1230 +
1.1231 +lemma hcomplex_split_capprox_iff:
1.1232 +     "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y @c=
1.1233 +       hcomplex_of_hypreal x' + iii * hcomplex_of_hypreal y') =
1.1234 +      (x @= x' & y @= y')"
1.1235 +apply (rule eq_Abs_hypreal [of x])
1.1236 +apply (rule eq_Abs_hypreal [of y])
1.1237 +apply (rule eq_Abs_hypreal [of x'])
1.1238 +apply (rule eq_Abs_hypreal [of y'])
1.1239 +apply (simp add: iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal capprox_approx_iff)
1.1240 +done
1.1241 +
1.1242 +lemma complex_seq_to_hcomplex_CInfinitesimal:
1.1243 +     "\<forall>n. cmod (X n - x) < inverse (real (Suc n)) ==>
1.1244 +      Abs_hcomplex(hcomplexrel``{X}) - hcomplex_of_complex x \<in> CInfinitesimal"
1.1245 +apply (simp add: hcomplex_diff CInfinitesimal_hcmod_iff hcomplex_of_complex_def Infinitesimal_FreeUltrafilterNat_iff hcmod)
1.1246 +apply (rule bexI, auto)
1.1247 +apply (auto dest: FreeUltrafilterNat_inverse_real_of_posnat FreeUltrafilterNat_all FreeUltrafilterNat_Int intro: order_less_trans FreeUltrafilterNat_subset)
1.1248 +done
1.1249 +
1.1250 +lemma CInfinitesimal_hcomplex_of_hypreal_epsilon [simp]:
1.1251 +     "hcomplex_of_hypreal epsilon \<in> CInfinitesimal"
1.1252 +by (simp add: CInfinitesimal_hcmod_iff)
1.1253 +
1.1254 +lemma hcomplex_of_complex_approx_zero_iff [simp]:
1.1255 +     "(hcomplex_of_complex z @c= 0) = (z = 0)"
1.1256 +by (simp add: hcomplex_of_complex_zero [symmetric]
1.1257 +         del: hcomplex_of_complex_zero)
1.1258 +
1.1259 +lemma hcomplex_of_complex_approx_zero_iff2 [simp]:
1.1260 +     "(0 @c= hcomplex_of_complex z) = (z = 0)"
1.1261 +by (simp add: hcomplex_of_complex_zero [symmetric]
1.1262 +         del: hcomplex_of_complex_zero)
1.1263 +
1.1264 +
1.1265 +ML
1.1266 +{*
1.1268 +val SComplex_mult = thm "SComplex_mult";
1.1269 +val SComplex_inverse = thm "SComplex_inverse";
1.1270 +val SComplex_divide = thm "SComplex_divide";
1.1271 +val SComplex_minus = thm "SComplex_minus";
1.1272 +val SComplex_minus_iff = thm "SComplex_minus_iff";
1.1273 +val SComplex_diff = thm "SComplex_diff";
1.1275 +val SReal_hcmod_hcomplex_of_complex = thm "SReal_hcmod_hcomplex_of_complex";
1.1276 +val SReal_hcmod_number_of = thm "SReal_hcmod_number_of";
1.1277 +val SReal_hcmod_SComplex = thm "SReal_hcmod_SComplex";
1.1278 +val SComplex_hcomplex_of_complex = thm "SComplex_hcomplex_of_complex";
1.1279 +val SComplex_number_of = thm "SComplex_number_of";
1.1280 +val SComplex_divide_number_of = thm "SComplex_divide_number_of";
1.1281 +val SComplex_UNIV_complex = thm "SComplex_UNIV_complex";
1.1282 +val SComplex_iff = thm "SComplex_iff";
1.1283 +val hcomplex_of_complex_image = thm "hcomplex_of_complex_image";
1.1284 +val inv_hcomplex_of_complex_image = thm "inv_hcomplex_of_complex_image";
1.1285 +val SComplex_hcomplex_of_complex_image = thm "SComplex_hcomplex_of_complex_image";
1.1286 +val SComplex_SReal_dense = thm "SComplex_SReal_dense";
1.1287 +val SComplex_hcmod_SReal = thm "SComplex_hcmod_SReal";
1.1288 +val SComplex_zero = thm "SComplex_zero";
1.1289 +val SComplex_one = thm "SComplex_one";
1.1291 +val CFinite_mult = thm "CFinite_mult";
1.1292 +val CFinite_minus_iff = thm "CFinite_minus_iff";
1.1293 +val SComplex_subset_CFinite = thm "SComplex_subset_CFinite";
1.1294 +val HFinite_hcmod_hcomplex_of_complex = thm "HFinite_hcmod_hcomplex_of_complex";
1.1295 +val CFinite_hcomplex_of_complex = thm "CFinite_hcomplex_of_complex";
1.1296 +val CFiniteD = thm "CFiniteD";
1.1297 +val CFinite_hcmod_iff = thm "CFinite_hcmod_iff";
1.1298 +val CFinite_number_of = thm "CFinite_number_of";
1.1299 +val CFinite_bounded = thm "CFinite_bounded";
1.1300 +val CInfinitesimal_zero = thm "CInfinitesimal_zero";
1.1301 +val hcomplex_sum_of_halves = thm "hcomplex_sum_of_halves";
1.1302 +val CInfinitesimal_hcmod_iff = thm "CInfinitesimal_hcmod_iff";
1.1303 +val one_not_CInfinitesimal = thm "one_not_CInfinitesimal";
1.1305 +val CInfinitesimal_minus_iff = thm "CInfinitesimal_minus_iff";
1.1306 +val CInfinitesimal_diff = thm "CInfinitesimal_diff";
1.1307 +val CInfinitesimal_mult = thm "CInfinitesimal_mult";
1.1308 +val CInfinitesimal_CFinite_mult = thm "CInfinitesimal_CFinite_mult";
1.1309 +val CInfinitesimal_CFinite_mult2 = thm "CInfinitesimal_CFinite_mult2";
1.1310 +val CInfinite_hcmod_iff = thm "CInfinite_hcmod_iff";
1.1311 +val CInfinite_inverse_CInfinitesimal = thm "CInfinite_inverse_CInfinitesimal";
1.1312 +val CInfinite_mult = thm "CInfinite_mult";
1.1313 +val CInfinite_minus_iff = thm "CInfinite_minus_iff";
1.1314 +val CFinite_sum_squares = thm "CFinite_sum_squares";
1.1315 +val not_CInfinitesimal_not_zero = thm "not_CInfinitesimal_not_zero";
1.1316 +val not_CInfinitesimal_not_zero2 = thm "not_CInfinitesimal_not_zero2";
1.1317 +val CFinite_diff_CInfinitesimal_hcmod = thm "CFinite_diff_CInfinitesimal_hcmod";
1.1318 +val hcmod_less_CInfinitesimal = thm "hcmod_less_CInfinitesimal";
1.1319 +val hcmod_le_CInfinitesimal = thm "hcmod_le_CInfinitesimal";
1.1320 +val CInfinitesimal_interval = thm "CInfinitesimal_interval";
1.1321 +val CInfinitesimal_interval2 = thm "CInfinitesimal_interval2";
1.1322 +val not_CInfinitesimal_mult = thm "not_CInfinitesimal_mult";
1.1323 +val CInfinitesimal_mult_disj = thm "CInfinitesimal_mult_disj";
1.1324 +val CFinite_CInfinitesimal_diff_mult = thm "CFinite_CInfinitesimal_diff_mult";
1.1325 +val CInfinitesimal_subset_CFinite = thm "CInfinitesimal_subset_CFinite";
1.1326 +val CInfinitesimal_hcomplex_of_complex_mult = thm "CInfinitesimal_hcomplex_of_complex_mult";
1.1327 +val CInfinitesimal_hcomplex_of_complex_mult2 = thm "CInfinitesimal_hcomplex_of_complex_mult2";
1.1328 +val mem_cinfmal_iff = thm "mem_cinfmal_iff";
1.1329 +val capprox_minus_iff = thm "capprox_minus_iff";
1.1330 +val capprox_minus_iff2 = thm "capprox_minus_iff2";
1.1331 +val capprox_refl = thm "capprox_refl";
1.1332 +val capprox_sym = thm "capprox_sym";
1.1333 +val capprox_trans = thm "capprox_trans";
1.1334 +val capprox_trans2 = thm "capprox_trans2";
1.1335 +val capprox_trans3 = thm "capprox_trans3";
1.1336 +val number_of_capprox_reorient = thm "number_of_capprox_reorient";
1.1337 +val CInfinitesimal_capprox_minus = thm "CInfinitesimal_capprox_minus";
1.1339 +val Infinitesimal_capprox = thm "Infinitesimal_capprox";
1.1341 +val capprox_minus = thm "capprox_minus";
1.1342 +val capprox_minus2 = thm "capprox_minus2";
1.1343 +val capprox_minus_cancel = thm "capprox_minus_cancel";
1.1345 +val capprox_mult1 = thm "capprox_mult1";
1.1346 +val capprox_mult2 = thm "capprox_mult2";
1.1347 +val capprox_mult_subst = thm "capprox_mult_subst";
1.1348 +val capprox_mult_subst2 = thm "capprox_mult_subst2";
1.1349 +val capprox_mult_subst_SComplex = thm "capprox_mult_subst_SComplex";
1.1350 +val capprox_eq_imp = thm "capprox_eq_imp";
1.1351 +val CInfinitesimal_minus_capprox = thm "CInfinitesimal_minus_capprox";
1.1352 +val bex_CInfinitesimal_iff = thm "bex_CInfinitesimal_iff";
1.1353 +val bex_CInfinitesimal_iff2 = thm "bex_CInfinitesimal_iff2";
1.1366 +val capprox_CFinite = thm "capprox_CFinite";
1.1367 +val capprox_hcomplex_of_complex_CFinite = thm "capprox_hcomplex_of_complex_CFinite";
1.1368 +val capprox_mult_CFinite = thm "capprox_mult_CFinite";
1.1369 +val capprox_mult_hcomplex_of_complex = thm "capprox_mult_hcomplex_of_complex";
1.1370 +val capprox_SComplex_mult_cancel_zero = thm "capprox_SComplex_mult_cancel_zero";
1.1371 +val capprox_mult_SComplex1 = thm "capprox_mult_SComplex1";
1.1372 +val capprox_mult_SComplex2 = thm "capprox_mult_SComplex2";
1.1373 +val capprox_mult_SComplex_zero_cancel_iff = thm "capprox_mult_SComplex_zero_cancel_iff";
1.1374 +val capprox_SComplex_mult_cancel = thm "capprox_SComplex_mult_cancel";
1.1375 +val capprox_SComplex_mult_cancel_iff1 = thm "capprox_SComplex_mult_cancel_iff1";
1.1376 +val capprox_hcmod_approx_zero = thm "capprox_hcmod_approx_zero";
1.1377 +val capprox_approx_zero_iff = thm "capprox_approx_zero_iff";
1.1378 +val capprox_minus_zero_cancel_iff = thm "capprox_minus_zero_cancel_iff";
1.1381 +val capprox_hcmod_approx = thm "capprox_hcmod_approx";
1.1382 +val CInfinitesimal_less_SComplex = thm "CInfinitesimal_less_SComplex";
1.1383 +val SComplex_Int_CInfinitesimal_zero = thm "SComplex_Int_CInfinitesimal_zero";
1.1384 +val SComplex_CInfinitesimal_zero = thm "SComplex_CInfinitesimal_zero";
1.1385 +val SComplex_CFinite_diff_CInfinitesimal = thm "SComplex_CFinite_diff_CInfinitesimal";
1.1386 +val hcomplex_of_complex_CFinite_diff_CInfinitesimal = thm "hcomplex_of_complex_CFinite_diff_CInfinitesimal";
1.1387 +val hcomplex_of_complex_CInfinitesimal_iff_0 = thm "hcomplex_of_complex_CInfinitesimal_iff_0";
1.1388 +val number_of_not_CInfinitesimal = thm "number_of_not_CInfinitesimal";
1.1389 +val capprox_SComplex_not_zero = thm "capprox_SComplex_not_zero";
1.1390 +val CFinite_diff_CInfinitesimal_capprox = thm "CFinite_diff_CInfinitesimal_capprox";
1.1391 +val CInfinitesimal_ratio = thm "CInfinitesimal_ratio";
1.1392 +val SComplex_capprox_iff = thm "SComplex_capprox_iff";
1.1393 +val number_of_capprox_iff = thm "number_of_capprox_iff";
1.1394 +val number_of_CInfinitesimal_iff = thm "number_of_CInfinitesimal_iff";
1.1395 +val hcomplex_of_complex_approx_iff = thm "hcomplex_of_complex_approx_iff";
1.1396 +val hcomplex_of_complex_capprox_number_of_iff = thm "hcomplex_of_complex_capprox_number_of_iff";
1.1397 +val capprox_unique_complex = thm "capprox_unique_complex";
1.1398 +val hcomplex_capproxD1 = thm "hcomplex_capproxD1";
1.1399 +val hcomplex_capproxD2 = thm "hcomplex_capproxD2";
1.1400 +val hcomplex_capproxI = thm "hcomplex_capproxI";
1.1401 +val capprox_approx_iff = thm "capprox_approx_iff";
1.1402 +val hcomplex_of_hypreal_capprox_iff = thm "hcomplex_of_hypreal_capprox_iff";
1.1403 +val CFinite_HFinite_Re = thm "CFinite_HFinite_Re";
1.1404 +val CFinite_HFinite_Im = thm "CFinite_HFinite_Im";
1.1405 +val HFinite_Re_Im_CFinite = thm "HFinite_Re_Im_CFinite";
1.1406 +val CFinite_HFinite_iff = thm "CFinite_HFinite_iff";
1.1407 +val SComplex_Re_SReal = thm "SComplex_Re_SReal";
1.1408 +val SComplex_Im_SReal = thm "SComplex_Im_SReal";
1.1409 +val Reals_Re_Im_SComplex = thm "Reals_Re_Im_SComplex";
1.1410 +val SComplex_SReal_iff = thm "SComplex_SReal_iff";
1.1411 +val CInfinitesimal_Infinitesimal_iff = thm "CInfinitesimal_Infinitesimal_iff";
1.1412 +val eq_Abs_hcomplex_Bex = thm "eq_Abs_hcomplex_Bex";
1.1413 +val stc_part_Ex = thm "stc_part_Ex";
1.1414 +val stc_part_Ex1 = thm "stc_part_Ex1";
1.1415 +val CFinite_Int_CInfinite_empty = thm "CFinite_Int_CInfinite_empty";
1.1416 +val CFinite_not_CInfinite = thm "CFinite_not_CInfinite";
1.1417 +val not_CFinite_CInfinite = thm "not_CFinite_CInfinite";
1.1418 +val CInfinite_CFinite_disj = thm "CInfinite_CFinite_disj";
1.1419 +val CInfinite_CFinite_iff = thm "CInfinite_CFinite_iff";
1.1420 +val CFinite_CInfinite_iff = thm "CFinite_CInfinite_iff";
1.1421 +val CInfinite_diff_CFinite_CInfinitesimal_disj = thm "CInfinite_diff_CFinite_CInfinitesimal_disj";
1.1422 +val CFinite_inverse = thm "CFinite_inverse";
1.1423 +val CFinite_inverse2 = thm "CFinite_inverse2";
1.1424 +val CInfinitesimal_inverse_CFinite = thm "CInfinitesimal_inverse_CFinite";
1.1425 +val CFinite_not_CInfinitesimal_inverse = thm "CFinite_not_CInfinitesimal_inverse";
1.1426 +val capprox_inverse = thm "capprox_inverse";
1.1427 +val hcomplex_of_complex_capprox_inverse = thms "hcomplex_of_complex_capprox_inverse";
1.1431 +val CInfinitesimal_square_iff = thm "CInfinitesimal_square_iff";
1.1432 +val capprox_CFinite_mult_cancel = thm "capprox_CFinite_mult_cancel";
1.1433 +val capprox_CFinite_mult_cancel_iff1 = thm "capprox_CFinite_mult_cancel_iff1";
1.1441 +val stc_capprox_self = thm "stc_capprox_self";
1.1442 +val stc_SComplex = thm "stc_SComplex";
1.1443 +val stc_CFinite = thm "stc_CFinite";
1.1444 +val stc_SComplex_eq = thm "stc_SComplex_eq";
1.1445 +val stc_hcomplex_of_complex = thm "stc_hcomplex_of_complex";
1.1446 +val stc_eq_capprox = thm "stc_eq_capprox";
1.1447 +val capprox_stc_eq = thm "capprox_stc_eq";
1.1448 +val stc_eq_capprox_iff = thm "stc_eq_capprox_iff";
1.1453 +val stc_number_of = thm "stc_number_of";
1.1454 +val stc_zero = thm "stc_zero";
1.1455 +val stc_one = thm "stc_one";
1.1456 +val stc_minus = thm "stc_minus";
1.1457 +val stc_diff = thm "stc_diff";
1.1458 +val lemma_stc_mult = thm "lemma_stc_mult";
1.1459 +val stc_mult = thm "stc_mult";
1.1460 +val stc_CInfinitesimal = thm "stc_CInfinitesimal";
1.1461 +val stc_not_CInfinitesimal = thm "stc_not_CInfinitesimal";
1.1462 +val stc_inverse = thm "stc_inverse";
1.1463 +val stc_divide = thm "stc_divide";
1.1464 +val stc_idempotent = thm "stc_idempotent";
1.1465 +val CFinite_HFinite_hcomplex_of_hypreal = thm "CFinite_HFinite_hcomplex_of_hypreal";
1.1466 +val SComplex_SReal_hcomplex_of_hypreal = thm "SComplex_SReal_hcomplex_of_hypreal";
1.1467 +val stc_hcomplex_of_hypreal = thm "stc_hcomplex_of_hypreal";
1.1468 +val CInfinitesimal_hcnj_iff = thm "CInfinitesimal_hcnj_iff";
1.1469 +val CInfinite_HInfinite_iff = thm "CInfinite_HInfinite_iff";
1.1470 +val hcomplex_split_CInfinitesimal_iff = thm "hcomplex_split_CInfinitesimal_iff";
1.1471 +val hcomplex_split_CFinite_iff = thm "hcomplex_split_CFinite_iff";
1.1472 +val hcomplex_split_SComplex_iff = thm "hcomplex_split_SComplex_iff";
1.1473 +val hcomplex_split_CInfinite_iff = thm "hcomplex_split_CInfinite_iff";
1.1474 +val hcomplex_split_capprox_iff = thm "hcomplex_split_capprox_iff";
1.1475 +val complex_seq_to_hcomplex_CInfinitesimal = thm "complex_seq_to_hcomplex_CInfinitesimal";
1.1476 +val CInfinitesimal_hcomplex_of_hypreal_epsilon = thm "CInfinitesimal_hcomplex_of_hypreal_epsilon";
1.1477 +val hcomplex_of_complex_approx_zero_iff = thm "hcomplex_of_complex_approx_zero_iff";
1.1478 +val hcomplex_of_complex_approx_zero_iff2 = thm "hcomplex_of_complex_approx_zero_iff2";
1.1479 +*}
1.1480 +
1.1481
1.1482  end
```
```     2.1 --- a/src/HOL/IsaMakefile	Sat Feb 21 20:05:16 2004 +0100
2.2 +++ b/src/HOL/IsaMakefile	Mon Feb 23 16:35:46 2004 +0100
2.3 @@ -160,8 +160,7 @@
2.4    Hyperreal/Transcendental.thy Hyperreal/fuf.ML Hyperreal/hypreal_arith.ML \
2.5    Complex/Complex_Main.thy Complex/CLim.thy Complex/CSeries.thy\
2.6    Complex/CStar.thy Complex/Complex.thy Complex/ComplexBin.thy\
2.7 -  Complex/NSCA.ML Complex/NSCA.thy\
2.8 -  Complex/NSComplex.thy
2.9 +  Complex/NSCA.thy Complex/NSComplex.thy
2.10  	@cd Complex; \$(ISATOOL) usedir -b \$(OUT)/HOL HOL-Complex
2.11
2.12
```