src/HOL/Complex.thy
 author hoelzl Fri Feb 19 13:40:50 2016 +0100 (2016-02-19) changeset 62378 85ed00c1fe7c parent 62102 877463945ce9 child 62379 340738057c8c permissions -rw-r--r--
generalize more theorems to support enat and ennreal
 wenzelm@41959  1 (* Title: HOL/Complex.thy  paulson@13957  2  Author: Jacques D. Fleuriot  paulson@13957  3  Copyright: 2001 University of Edinburgh  paulson@14387  4  Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4  paulson@13957  5 *)  paulson@13957  6 wenzelm@60758  7 section \Complex Numbers: Rectangular and Polar Representations\  paulson@14373  8 nipkow@15131  9 theory Complex  haftmann@28952  10 imports Transcendental  nipkow@15131  11 begin  paulson@13957  12 wenzelm@60758  13 text \  wenzelm@61799  14 We use the \codatatype\ command to define the type of complex numbers. This allows us to use  wenzelm@61799  15 \primcorec\ to define complex functions by defining their real and imaginary result  blanchet@58146  16 separately.  wenzelm@60758  17 \  paulson@14373  18 hoelzl@56889  19 codatatype complex = Complex (Re: real) (Im: real)  hoelzl@56889  20 hoelzl@56889  21 lemma complex_surj: "Complex (Re z) (Im z) = z"  hoelzl@56889  22  by (rule complex.collapse)  paulson@13957  23 huffman@44065  24 lemma complex_eqI [intro?]: "\Re x = Re y; Im x = Im y\ \ x = y"  hoelzl@56889  25  by (rule complex.expand) simp  huffman@23125  26 huffman@44065  27 lemma complex_eq_iff: "x = y \ Re x = Re y \ Im x = Im y"  hoelzl@56889  28  by (auto intro: complex.expand)  huffman@23125  29 wenzelm@60758  30 subsection \Addition and Subtraction\  huffman@23125  31 haftmann@25599  32 instantiation complex :: ab_group_add  haftmann@25571  33 begin  haftmann@25571  34 hoelzl@56889  35 primcorec zero_complex where  hoelzl@56889  36  "Re 0 = 0"  hoelzl@56889  37 | "Im 0 = 0"  haftmann@25571  38 hoelzl@56889  39 primcorec plus_complex where  hoelzl@56889  40  "Re (x + y) = Re x + Re y"  hoelzl@56889  41 | "Im (x + y) = Im x + Im y"  haftmann@25712  42 hoelzl@56889  43 primcorec uminus_complex where  hoelzl@56889  44  "Re (- x) = - Re x"  hoelzl@56889  45 | "Im (- x) = - Im x"  huffman@23125  46 hoelzl@56889  47 primcorec minus_complex where  hoelzl@56889  48  "Re (x - y) = Re x - Re y"  hoelzl@56889  49 | "Im (x - y) = Im x - Im y"  huffman@23125  50 haftmann@25712  51 instance  hoelzl@56889  52  by intro_classes (simp_all add: complex_eq_iff)  haftmann@25712  53 haftmann@25712  54 end  haftmann@25712  55 wenzelm@60758  56 subsection \Multiplication and Division\  huffman@23125  57 haftmann@59867  58 instantiation complex :: field  haftmann@25571  59 begin  haftmann@25571  60 hoelzl@56889  61 primcorec one_complex where  hoelzl@56889  62  "Re 1 = 1"  hoelzl@56889  63 | "Im 1 = 0"  paulson@14323  64 hoelzl@56889  65 primcorec times_complex where  hoelzl@56889  66  "Re (x * y) = Re x * Re y - Im x * Im y"  hoelzl@56889  67 | "Im (x * y) = Re x * Im y + Im x * Re y"  paulson@14323  68 hoelzl@56889  69 primcorec inverse_complex where  hoelzl@56889  70  "Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"  hoelzl@56889  71 | "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"  paulson@14335  72 wenzelm@61076  73 definition "x div (y::complex) = x * inverse y"  paulson@14335  74 haftmann@25712  75 instance  lp15@59613  76  by intro_classes  hoelzl@56889  77  (simp_all add: complex_eq_iff divide_complex_def  hoelzl@56889  78  distrib_left distrib_right right_diff_distrib left_diff_distrib  hoelzl@56889  79  power2_eq_square add_divide_distrib [symmetric])  paulson@14335  80 haftmann@25712  81 end  huffman@23125  82 hoelzl@56889  83 lemma Re_divide: "Re (x / y) = (Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"  hoelzl@56889  84  unfolding divide_complex_def by (simp add: add_divide_distrib)  huffman@23125  85 hoelzl@56889  86 lemma Im_divide: "Im (x / y) = (Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"  hoelzl@56889  87  unfolding divide_complex_def times_complex.sel inverse_complex.sel  hoelzl@56889  88  by (simp_all add: divide_simps)  huffman@23125  89 hoelzl@56889  90 lemma Re_power2: "Re (x ^ 2) = (Re x)^2 - (Im x)^2"  hoelzl@56889  91  by (simp add: power2_eq_square)  huffman@20556  92 hoelzl@56889  93 lemma Im_power2: "Im (x ^ 2) = 2 * Re x * Im x"  hoelzl@56889  94  by (simp add: power2_eq_square)  hoelzl@56889  95 lp15@59862  96 lemma Re_power_real [simp]: "Im x = 0 \ Re (x ^ n) = Re x ^ n "  huffman@44724  97  by (induct n) simp_all  huffman@23125  98 lp15@59862  99 lemma Im_power_real [simp]: "Im x = 0 \ Im (x ^ n) = 0"  hoelzl@56889  100  by (induct n) simp_all  huffman@23125  101 wenzelm@60758  102 subsection \Scalar Multiplication\  huffman@20556  103 haftmann@25712  104 instantiation complex :: real_field  haftmann@25571  105 begin  haftmann@25571  106 hoelzl@56889  107 primcorec scaleR_complex where  hoelzl@56889  108  "Re (scaleR r x) = r * Re x"  hoelzl@56889  109 | "Im (scaleR r x) = r * Im x"  huffman@22972  110 haftmann@25712  111 instance  huffman@20556  112 proof  huffman@23125  113  fix a b :: real and x y :: complex  huffman@23125  114  show "scaleR a (x + y) = scaleR a x + scaleR a y"  webertj@49962  115  by (simp add: complex_eq_iff distrib_left)  huffman@23125  116  show "scaleR (a + b) x = scaleR a x + scaleR b x"  webertj@49962  117  by (simp add: complex_eq_iff distrib_right)  huffman@23125  118  show "scaleR a (scaleR b x) = scaleR (a * b) x"  haftmann@57512  119  by (simp add: complex_eq_iff mult.assoc)  huffman@23125  120  show "scaleR 1 x = x"  huffman@44065  121  by (simp add: complex_eq_iff)  huffman@23125  122  show "scaleR a x * y = scaleR a (x * y)"  huffman@44065  123  by (simp add: complex_eq_iff algebra_simps)  huffman@23125  124  show "x * scaleR a y = scaleR a (x * y)"  huffman@44065  125  by (simp add: complex_eq_iff algebra_simps)  huffman@20556  126 qed  huffman@20556  127 haftmann@25712  128 end  haftmann@25712  129 wenzelm@60758  130 subsection \Numerals, Arithmetic, and Embedding from Reals\  paulson@14323  131 huffman@44724  132 abbreviation complex_of_real :: "real \ complex"  huffman@44724  133  where "complex_of_real \ of_real"  huffman@20557  134 hoelzl@59000  135 declare [[coercion "of_real :: real \ complex"]]  hoelzl@59000  136 declare [[coercion "of_rat :: rat \ complex"]]  hoelzl@56889  137 declare [[coercion "of_int :: int \ complex"]]  hoelzl@56889  138 declare [[coercion "of_nat :: nat \ complex"]]  hoelzl@56331  139 hoelzl@56889  140 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"  hoelzl@56889  141  by (induct n) simp_all  hoelzl@56889  142 hoelzl@56889  143 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"  hoelzl@56889  144  by (induct n) simp_all  hoelzl@56889  145 hoelzl@56889  146 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"  hoelzl@56889  147  by (cases z rule: int_diff_cases) simp  hoelzl@56889  148 hoelzl@56889  149 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"  hoelzl@56889  150  by (cases z rule: int_diff_cases) simp  hoelzl@56889  151 hoelzl@56889  152 lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"  hoelzl@56889  153  using complex_Re_of_int [of "numeral v"] by simp  hoelzl@56889  154 hoelzl@56889  155 lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"  hoelzl@56889  156  using complex_Im_of_int [of "numeral v"] by simp  huffman@20557  157 huffman@20557  158 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"  hoelzl@56889  159  by (simp add: of_real_def)  huffman@20557  160 huffman@20557  161 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"  hoelzl@56889  162  by (simp add: of_real_def)  hoelzl@56889  163 lp15@59613  164 lemma Re_divide_numeral [simp]: "Re (z / numeral w) = Re z / numeral w"  lp15@59613  165  by (simp add: Re_divide sqr_conv_mult)  lp15@59613  166 lp15@59613  167 lemma Im_divide_numeral [simp]: "Im (z / numeral w) = Im z / numeral w"  lp15@59613  168  by (simp add: Im_divide sqr_conv_mult)  lp15@61609  169 eberlm@61552  170 lemma Re_divide_of_nat: "Re (z / of_nat n) = Re z / of_nat n"  eberlm@61552  171  by (cases n) (simp_all add: Re_divide divide_simps power2_eq_square del: of_nat_Suc)  eberlm@61552  172 eberlm@61552  173 lemma Im_divide_of_nat: "Im (z / of_nat n) = Im z / of_nat n"  eberlm@61552  174  by (cases n) (simp_all add: Im_divide divide_simps power2_eq_square del: of_nat_Suc)  lp15@59613  175 lp15@60017  176 lemma of_real_Re [simp]:  lp15@60017  177  "z \ \ \ of_real (Re z) = z"  lp15@60017  178  by (auto simp: Reals_def)  lp15@60017  179 eberlm@61531  180 lemma complex_Re_fact [simp]: "Re (fact n) = fact n"  eberlm@61531  181 proof -  eberlm@61531  182  have "(fact n :: complex) = of_real (fact n)" by simp  eberlm@61531  183  also have "Re \ = fact n" by (subst Re_complex_of_real) simp_all  eberlm@61531  184  finally show ?thesis .  eberlm@61531  185 qed  eberlm@61531  186 eberlm@61531  187 lemma complex_Im_fact [simp]: "Im (fact n) = 0"  eberlm@61531  188  by (subst of_nat_fact [symmetric]) (simp only: complex_Im_of_nat)  eberlm@61531  189 eberlm@61531  190 wenzelm@60758  191 subsection \The Complex Number $i$\  hoelzl@56889  192 hoelzl@56889  193 primcorec "ii" :: complex ("\") where  hoelzl@56889  194  "Re ii = 0"  hoelzl@56889  195 | "Im ii = 1"  huffman@20557  196 hoelzl@57259  197 lemma Complex_eq[simp]: "Complex a b = a + \ * b"  hoelzl@57259  198  by (simp add: complex_eq_iff)  hoelzl@57259  199 hoelzl@57259  200 lemma complex_eq: "a = Re a + \ * Im a"  hoelzl@57259  201  by (simp add: complex_eq_iff)  hoelzl@57259  202 hoelzl@57259  203 lemma fun_complex_eq: "f = (\x. Re (f x) + \ * Im (f x))"  hoelzl@57259  204  by (simp add: fun_eq_iff complex_eq)  hoelzl@57259  205 hoelzl@56889  206 lemma i_squared [simp]: "ii * ii = -1"  hoelzl@56889  207  by (simp add: complex_eq_iff)  hoelzl@56889  208 hoelzl@56889  209 lemma power2_i [simp]: "ii\<^sup>2 = -1"  hoelzl@56889  210  by (simp add: power2_eq_square)  paulson@14377  211 hoelzl@56889  212 lemma inverse_i [simp]: "inverse ii = - ii"  hoelzl@56889  213  by (rule inverse_unique) simp  hoelzl@56889  214 hoelzl@56889  215 lemma divide_i [simp]: "x / ii = - ii * x"  hoelzl@56889  216  by (simp add: divide_complex_def)  paulson@14377  217 hoelzl@56889  218 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"  haftmann@57512  219  by (simp add: mult.assoc [symmetric])  paulson@14377  220 hoelzl@56889  221 lemma complex_i_not_zero [simp]: "ii \ 0"  hoelzl@56889  222  by (simp add: complex_eq_iff)  huffman@20557  223 hoelzl@56889  224 lemma complex_i_not_one [simp]: "ii \ 1"  hoelzl@56889  225  by (simp add: complex_eq_iff)  hoelzl@56889  226 hoelzl@56889  227 lemma complex_i_not_numeral [simp]: "ii \ numeral w"  hoelzl@56889  228  by (simp add: complex_eq_iff)  huffman@44841  229 hoelzl@56889  230 lemma complex_i_not_neg_numeral [simp]: "ii \ - numeral w"  hoelzl@56889  231  by (simp add: complex_eq_iff)  hoelzl@56889  232 hoelzl@56889  233 lemma complex_split_polar: "\r a. z = complex_of_real r * (cos a + \ * sin a)"  huffman@44827  234  by (simp add: complex_eq_iff polar_Ex)  huffman@44827  235 lp15@59613  236 lemma i_even_power [simp]: "\ ^ (n * 2) = (-1) ^ n"  lp15@59613  237  by (metis mult.commute power2_i power_mult)  lp15@59613  238 lp15@59741  239 lemma Re_ii_times [simp]: "Re (ii*z) = - Im z"  lp15@59741  240  by simp  lp15@59741  241 lp15@59741  242 lemma Im_ii_times [simp]: "Im (ii*z) = Re z"  lp15@59741  243  by simp  lp15@59741  244 lp15@59741  245 lemma ii_times_eq_iff: "ii*w = z \ w = -(ii*z)"  lp15@59741  246  by auto  lp15@59741  247 lp15@59741  248 lemma divide_numeral_i [simp]: "z / (numeral n * ii) = -(ii*z) / numeral n"  lp15@59741  249  by (metis divide_divide_eq_left divide_i mult.commute mult_minus_right)  lp15@59741  250 wenzelm@60758  251 subsection \Vector Norm\  paulson@14323  252 haftmann@25712  253 instantiation complex :: real_normed_field  haftmann@25571  254 begin  haftmann@25571  255 hoelzl@56889  256 definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"  haftmann@25571  257 huffman@44724  258 abbreviation cmod :: "complex \ real"  huffman@44724  259  where "cmod \ norm"  haftmann@25571  260 huffman@31413  261 definition complex_sgn_def:  huffman@31413  262  "sgn x = x /\<^sub>R cmod x"  haftmann@25571  263 huffman@31413  264 definition dist_complex_def:  huffman@31413  265  "dist x y = cmod (x - y)"  huffman@31413  266 hoelzl@62101  267 definition uniformity_complex_def [code del]:  hoelzl@62101  268  "(uniformity :: (complex \ complex) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})"  hoelzl@62101  269 hoelzl@62101  270 definition open_complex_def [code del]:  hoelzl@62101  271  "open (U :: complex set) \ (\x\U. eventually ($$x', y). x' = x \ y \ U) uniformity)"  huffman@31292  272 huffman@31413  273 instance proof  huffman@31492  274  fix r :: real and x y :: complex and S :: "complex set"  huffman@23125  275  show "(norm x = 0) = (x = 0)"  hoelzl@56889  276  by (simp add: norm_complex_def complex_eq_iff)  huffman@23125  277  show "norm (x + y) \ norm x + norm y"  hoelzl@56889  278  by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq)  huffman@23125  279  show "norm (scaleR r x) = \r\ * norm x"  hoelzl@56889  280  by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] real_sqrt_mult)  huffman@23125  281  show "norm (x * y) = norm x * norm y"  hoelzl@56889  282  by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] power2_eq_square algebra_simps)  hoelzl@62101  283 qed (rule complex_sgn_def dist_complex_def open_complex_def uniformity_complex_def)+  huffman@20557  284 haftmann@25712  285 end  haftmann@25712  286 hoelzl@62102  287 declare uniformity_Abort[where 'a=complex, code]  hoelzl@62102  288 hoelzl@56889  289 lemma norm_ii [simp]: "norm ii = 1"  hoelzl@56889  290  by (simp add: norm_complex_def)  paulson@14323  291 hoelzl@56889  292 lemma cmod_unit_one: "cmod (cos a + \ * sin a) = 1"  hoelzl@56889  293  by (simp add: norm_complex_def)  hoelzl@56889  294 hoelzl@56889  295 lemma cmod_complex_polar: "cmod (r * (cos a + \ * sin a)) = \r\"  hoelzl@56889  296  by (simp add: norm_mult cmod_unit_one)  huffman@22861  297 huffman@22861  298 lemma complex_Re_le_cmod: "Re x \ cmod x"  hoelzl@56889  299  unfolding norm_complex_def  huffman@44724  300  by (rule real_sqrt_sum_squares_ge1)  huffman@22861  301 huffman@44761  302 lemma complex_mod_minus_le_complex_mod: "- cmod x \ cmod x"  hoelzl@56889  303  by (rule order_trans [OF _ norm_ge_zero]) simp  huffman@22861  304 hoelzl@56889  305 lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \ cmod a"  hoelzl@56889  306  by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp  paulson@14323  307 chaieb@26117  308 lemma abs_Re_le_cmod: "\Re x\ \ cmod x"  hoelzl@56889  309  by (simp add: norm_complex_def)  chaieb@26117  310 chaieb@26117  311 lemma abs_Im_le_cmod: "\Im x\ \ cmod x"  hoelzl@56889  312  by (simp add: norm_complex_def)  hoelzl@56889  313 hoelzl@57259  314 lemma cmod_le: "cmod z \ \Re z\ + \Im z\"  hoelzl@57259  315  apply (subst complex_eq)  hoelzl@57259  316  apply (rule order_trans)  hoelzl@57259  317  apply (rule norm_triangle_ineq)  hoelzl@57259  318  apply (simp add: norm_mult)  hoelzl@57259  319  done  hoelzl@57259  320 hoelzl@56889  321 lemma cmod_eq_Re: "Im z = 0 \ cmod z = \Re z\"  hoelzl@56889  322  by (simp add: norm_complex_def)  hoelzl@56889  323 hoelzl@56889  324 lemma cmod_eq_Im: "Re z = 0 \ cmod z = \Im z\"  hoelzl@56889  325  by (simp add: norm_complex_def)  huffman@44724  326 hoelzl@56889  327 lemma cmod_power2: "cmod z ^ 2 = (Re z)^2 + (Im z)^2"  hoelzl@56889  328  by (simp add: norm_complex_def)  hoelzl@56889  329 hoelzl@56889  330 lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \ 0 \ Re z = - cmod z"  hoelzl@56889  331  using abs_Re_le_cmod[of z] by auto  hoelzl@56889  332 hoelzl@56889  333 lemma Im_eq_0: "\Re z\ = cmod z \ Im z = 0"  hoelzl@56889  334  by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2])  hoelzl@56889  335  (auto simp add: norm_complex_def)  hoelzl@56369  336 hoelzl@56369  337 lemma abs_sqrt_wlog:  hoelzl@56369  338  fixes x::"'a::linordered_idom"  hoelzl@56369  339  assumes "\x::'a. x \ 0 \ P x (x\<^sup>2)" shows "P \x\ (x\<^sup>2)"  hoelzl@56369  340 by (metis abs_ge_zero assms power2_abs)  hoelzl@56369  341 hoelzl@56369  342 lemma complex_abs_le_norm: "\Re z\ + \Im z\ \ sqrt 2 * norm z"  hoelzl@56889  343  unfolding norm_complex_def  hoelzl@56369  344  apply (rule abs_sqrt_wlog [where x="Re z"])  hoelzl@56369  345  apply (rule abs_sqrt_wlog [where x="Im z"])  hoelzl@56369  346  apply (rule power2_le_imp_le)  haftmann@57512  347  apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])  hoelzl@56369  348  done  hoelzl@56369  349 lp15@59741  350 lemma complex_unit_circle: "z \ 0 \ (Re z / cmod z)\<^sup>2 + (Im z / cmod z)\<^sup>2 = 1"  lp15@59741  351  by (simp add: norm_complex_def divide_simps complex_eq_iff)  lp15@59741  352 hoelzl@56369  353 wenzelm@60758  354 text \Properties of complex signum.\  huffman@44843  355 huffman@44843  356 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"  haftmann@57512  357  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute)  huffman@44843  358 huffman@44843  359 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"  huffman@44843  360  by (simp add: complex_sgn_def divide_inverse)  huffman@44843  361 huffman@44843  362 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"  huffman@44843  363  by (simp add: complex_sgn_def divide_inverse)  huffman@44843  364 paulson@14354  365 wenzelm@60758  366 subsection \Completeness of the Complexes\  huffman@23123  367 huffman@44290  368 lemma bounded_linear_Re: "bounded_linear Re"  hoelzl@56889  369  by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)  huffman@44290  370 huffman@44290  371 lemma bounded_linear_Im: "bounded_linear Im"  hoelzl@56889  372  by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)  huffman@23123  373 huffman@44290  374 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]  huffman@44290  375 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]  hoelzl@56381  376 lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]  hoelzl@56381  377 lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]  hoelzl@56381  378 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]  hoelzl@56381  379 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]  hoelzl@56381  380 lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]  hoelzl@56381  381 lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]  hoelzl@56381  382 lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]  hoelzl@56381  383 lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]  hoelzl@56381  384 lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]  hoelzl@56381  385 lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]  hoelzl@56381  386 lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]  hoelzl@56381  387 lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]  hoelzl@56369  388 huffman@36825  389 lemma tendsto_Complex [tendsto_intros]:  wenzelm@61973  390  "(f \ a) F \ (g \ b) F \ ((\x. Complex (f x) (g x)) \ Complex a b) F"  hoelzl@56889  391  by (auto intro!: tendsto_intros)  hoelzl@56369  392 hoelzl@56369  393 lemma tendsto_complex_iff:  wenzelm@61973  394  "(f \ x) F \ (((\x. Re (f x)) \ Re x) F \ ((\x. Im (f x)) \ Im x) F)"  hoelzl@56889  395 proof safe  wenzelm@61973  396  assume "((\x. Re (f x)) \ Re x) F" "((\x. Im (f x)) \ Im x) F"  wenzelm@61973  397  from tendsto_Complex[OF this] show "(f \ x) F"  hoelzl@56889  398  unfolding complex.collapse .  hoelzl@56889  399 qed (auto intro: tendsto_intros)  hoelzl@56369  400 hoelzl@57259  401 lemma continuous_complex_iff: "continuous F f \  hoelzl@57259  402  continuous F (\x. Re (f x)) \ continuous F (\x. Im (f x))"  hoelzl@57259  403  unfolding continuous_def tendsto_complex_iff ..  hoelzl@57259  404 hoelzl@57259  405 lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \  hoelzl@57259  406  ((\x. Re (f x)) has_field_derivative (Re x)) F \  hoelzl@57259  407  ((\x. Im (f x)) has_field_derivative (Im x)) F"  hoelzl@57259  408  unfolding has_vector_derivative_def has_field_derivative_def has_derivative_def tendsto_complex_iff  hoelzl@57259  409  by (simp add: field_simps bounded_linear_scaleR_left bounded_linear_mult_right)  hoelzl@57259  410 hoelzl@57259  411 lemma has_field_derivative_Re[derivative_intros]:  hoelzl@57259  412  "(f has_vector_derivative D) F \ ((\x. Re (f x)) has_field_derivative (Re D)) F"  hoelzl@57259  413  unfolding has_vector_derivative_complex_iff by safe  hoelzl@57259  414 hoelzl@57259  415 lemma has_field_derivative_Im[derivative_intros]:  hoelzl@57259  416  "(f has_vector_derivative D) F \ ((\x. Im (f x)) has_field_derivative (Im D)) F"  hoelzl@57259  417  unfolding has_vector_derivative_complex_iff by safe  hoelzl@57259  418 huffman@23123  419 instance complex :: banach  huffman@23123  420 proof  huffman@23123  421  fix X :: "nat \ complex"  huffman@23123  422  assume X: "Cauchy X"  wenzelm@61969  423  then have "(\n. Complex (Re (X n)) (Im (X n))) \ Complex (lim (\n. Re (X n))) (lim (\n. Im (X n)))"  hoelzl@56889  424  by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)  hoelzl@56889  425  then show "convergent X"  hoelzl@56889  426  unfolding complex.collapse by (rule convergentI)  huffman@23123  427 qed  huffman@23123  428 lp15@56238  429 declare  hoelzl@56381  430  DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]  lp15@56238  431 wenzelm@60758  432 subsection \Complex Conjugation\  huffman@23125  433 hoelzl@56889  434 primcorec cnj :: "complex \ complex" where  hoelzl@56889  435  "Re (cnj z) = Re z"  hoelzl@56889  436 | "Im (cnj z) = - Im z"  huffman@23125  437 huffman@23125  438 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"  huffman@44724  439  by (simp add: complex_eq_iff)  huffman@23125  440 huffman@23125  441 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"  hoelzl@56889  442  by (simp add: complex_eq_iff)  huffman@23125  443 huffman@23125  444 lemma complex_cnj_zero [simp]: "cnj 0 = 0"  huffman@44724  445  by (simp add: complex_eq_iff)  huffman@23125  446 huffman@23125  447 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"  huffman@44724  448  by (simp add: complex_eq_iff)  huffman@23125  449 hoelzl@56889  450 lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y"  huffman@44724  451  by (simp add: complex_eq_iff)  huffman@23125  452 hoelzl@56889  453 lemma cnj_setsum [simp]: "cnj (setsum f s) = (\x\s. cnj (f x))"  hoelzl@56889  454  by (induct s rule: infinite_finite_induct) auto  hoelzl@56369  455 hoelzl@56889  456 lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y"  huffman@44724  457  by (simp add: complex_eq_iff)  huffman@23125  458 hoelzl@56889  459 lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x"  huffman@44724  460  by (simp add: complex_eq_iff)  huffman@23125  461 huffman@23125  462 lemma complex_cnj_one [simp]: "cnj 1 = 1"  huffman@44724  463  by (simp add: complex_eq_iff)  huffman@23125  464 hoelzl@56889  465 lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y"  huffman@44724  466  by (simp add: complex_eq_iff)  huffman@23125  467 hoelzl@56889  468 lemma cnj_setprod [simp]: "cnj (setprod f s) = (\x\s. cnj (f x))"  hoelzl@56889  469  by (induct s rule: infinite_finite_induct) auto  hoelzl@56369  470 hoelzl@56889  471 lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)"  hoelzl@56889  472  by (simp add: complex_eq_iff)  paulson@14323  473 hoelzl@56889  474 lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y"  hoelzl@56889  475  by (simp add: divide_complex_def)  huffman@23125  476 hoelzl@56889  477 lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n"  hoelzl@56889  478  by (induct n) simp_all  huffman@23125  479 huffman@23125  480 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"  huffman@44724  481  by (simp add: complex_eq_iff)  huffman@23125  482 huffman@23125  483 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"  huffman@44724  484  by (simp add: complex_eq_iff)  huffman@23125  485 huffman@47108  486 lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"  huffman@47108  487  by (simp add: complex_eq_iff)  huffman@47108  488 haftmann@54489  489 lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"  huffman@44724  490  by (simp add: complex_eq_iff)  huffman@23125  491 hoelzl@56889  492 lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)"  huffman@44724  493  by (simp add: complex_eq_iff)  huffman@23125  494 huffman@23125  495 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"  hoelzl@56889  496  by (simp add: norm_complex_def)  paulson@14323  497 huffman@23125  498 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"  huffman@44724  499  by (simp add: complex_eq_iff)  huffman@23125  500 huffman@23125  501 lemma complex_cnj_i [simp]: "cnj ii = - ii"  huffman@44724  502  by (simp add: complex_eq_iff)  huffman@23125  503 huffman@23125  504 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"  huffman@44724  505  by (simp add: complex_eq_iff)  huffman@23125  506 huffman@23125  507 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"  huffman@44724  508  by (simp add: complex_eq_iff)  paulson@14354  509 wenzelm@53015  510 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"  huffman@44724  511  by (simp add: complex_eq_iff power2_eq_square)  huffman@23125  512 wenzelm@53015  513 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"  huffman@44724  514  by (simp add: norm_mult power2_eq_square)  huffman@23125  515 huffman@44827  516 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"  hoelzl@56889  517  by (simp add: norm_complex_def power2_eq_square)  huffman@44827  518 huffman@44827  519 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"  huffman@44827  520  by simp  huffman@44827  521 eberlm@61531  522 lemma complex_cnj_fact [simp]: "cnj (fact n) = fact n"  eberlm@61531  523  by (subst of_nat_fact [symmetric], subst complex_cnj_of_nat) simp  eberlm@61531  524 eberlm@61531  525 lemma complex_cnj_pochhammer [simp]: "cnj (pochhammer z n) = pochhammer (cnj z) n"  eberlm@61531  526  by (induction n arbitrary: z) (simp_all add: pochhammer_rec)  eberlm@61531  527 huffman@44290  528 lemma bounded_linear_cnj: "bounded_linear cnj"  huffman@44127  529  using complex_cnj_add complex_cnj_scaleR  huffman@44127  530  by (rule bounded_linear_intro [where K=1], simp)  paulson@14354  531 hoelzl@56381  532 lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]  hoelzl@56381  533 lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]  hoelzl@56381  534 lemmas continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]  hoelzl@56381  535 lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]  hoelzl@56381  536 lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]  huffman@44290  537 wenzelm@61973  538 lemma lim_cnj: "((\x. cnj(f x)) \ cnj l) F \ (f \ l) F"  hoelzl@56889  539  by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)  hoelzl@56369  540 hoelzl@56369  541 lemma sums_cnj: "((\x. cnj(f x)) sums cnj l) \ (f sums l)"  hoelzl@56889  542  by (simp add: sums_def lim_cnj cnj_setsum [symmetric] del: cnj_setsum)  hoelzl@56369  543 paulson@14354  544 wenzelm@60758  545 subsection\Basic Lemmas\  lp15@55734  546 lp15@55734  547 lemma complex_eq_0: "z=0 \ (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"  hoelzl@56889  548  by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)  lp15@55734  549 lp15@55734  550 lemma complex_neq_0: "z\0 \ (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"  hoelzl@56889  551  by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)  lp15@55734  552 lp15@55734  553 lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"  hoelzl@56889  554 by (cases z)  hoelzl@56889  555  (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric]  hoelzl@56889  556  simp del: of_real_power)  lp15@55734  557 paulson@61104  558 lemma complex_div_cnj: "a / b = (a * cnj b) / (norm b)^2"  paulson@61104  559  using complex_norm_square by auto  paulson@61104  560 lp15@59741  561 lemma Re_complex_div_eq_0: "Re (a / b) = 0 \ Re (a * cnj b) = 0"  hoelzl@56889  562  by (auto simp add: Re_divide)  lp15@59613  563 lp15@59741  564 lemma Im_complex_div_eq_0: "Im (a / b) = 0 \ Im (a * cnj b) = 0"  hoelzl@56889  565  by (auto simp add: Im_divide)  hoelzl@56889  566 lp15@59613  567 lemma complex_div_gt_0:  hoelzl@56889  568  "(Re (a / b) > 0 \ Re (a * cnj b) > 0) \ (Im (a / b) > 0 \ Im (a * cnj b) > 0)"  hoelzl@56889  569 proof cases  hoelzl@56889  570  assume "b = 0" then show ?thesis by auto  lp15@55734  571 next  hoelzl@56889  572  assume "b \ 0"  hoelzl@56889  573  then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2"  hoelzl@56889  574  by (simp add: complex_eq_iff sum_power2_gt_zero_iff)  hoelzl@56889  575  then show ?thesis  hoelzl@56889  576  by (simp add: Re_divide Im_divide zero_less_divide_iff)  lp15@55734  577 qed  lp15@55734  578 lp15@59741  579 lemma Re_complex_div_gt_0: "Re (a / b) > 0 \ Re (a * cnj b) > 0"  lp15@59741  580  and Im_complex_div_gt_0: "Im (a / b) > 0 \ Im (a * cnj b) > 0"  hoelzl@56889  581  using complex_div_gt_0 by auto  lp15@55734  582 lp15@59741  583 lemma Re_complex_div_ge_0: "Re(a / b) \ 0 \ Re(a * cnj b) \ 0"  lp15@59741  584  by (metis le_less Re_complex_div_eq_0 Re_complex_div_gt_0)  lp15@55734  585 lp15@59741  586 lemma Im_complex_div_ge_0: "Im(a / b) \ 0 \ Im(a * cnj b) \ 0"  lp15@59741  587  by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 le_less)  lp15@55734  588 lp15@59741  589 lemma Re_complex_div_lt_0: "Re(a / b) < 0 \ Re(a * cnj b) < 0"  lp15@59741  590  by (metis less_asym neq_iff Re_complex_div_eq_0 Re_complex_div_gt_0)  lp15@55734  591 lp15@59741  592 lemma Im_complex_div_lt_0: "Im(a / b) < 0 \ Im(a * cnj b) < 0"  lp15@59741  593  by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 less_asym neq_iff)  lp15@55734  594 lp15@59741  595 lemma Re_complex_div_le_0: "Re(a / b) \ 0 \ Re(a * cnj b) \ 0"  lp15@59741  596  by (metis not_le Re_complex_div_gt_0)  lp15@55734  597 lp15@59741  598 lemma Im_complex_div_le_0: "Im(a / b) \ 0 \ Im(a * cnj b) \ 0"  lp15@59741  599  by (metis Im_complex_div_gt_0 not_le)  lp15@55734  600 paulson@61104  601 lemma Re_divide_of_real [simp]: "Re (z / of_real r) = Re z / r"  paulson@61104  602  by (simp add: Re_divide power2_eq_square)  paulson@61104  603 paulson@61104  604 lemma Im_divide_of_real [simp]: "Im (z / of_real r) = Im z / r"  paulson@61104  605  by (simp add: Im_divide power2_eq_square)  paulson@61104  606 paulson@61104  607 lemma Re_divide_Reals: "r \ Reals \ Re (z / r) = Re z / Re r"  paulson@61104  608  by (metis Re_divide_of_real of_real_Re)  paulson@61104  609 paulson@61104  610 lemma Im_divide_Reals: "r \ Reals \ Im (z / r) = Im z / Re r"  paulson@61104  611  by (metis Im_divide_of_real of_real_Re)  paulson@61104  612 hoelzl@56889  613 lemma Re_setsum[simp]: "Re (setsum f s) = (\x\s. Re (f x))"  hoelzl@56369  614  by (induct s rule: infinite_finite_induct) auto  lp15@55734  615 hoelzl@56889  616 lemma Im_setsum[simp]: "Im (setsum f s) = (\x\s. Im(f x))"  hoelzl@56369  617  by (induct s rule: infinite_finite_induct) auto  hoelzl@56369  618 hoelzl@56369  619 lemma sums_complex_iff: "f sums x \ ((\x. Re (f x)) sums Re x) \ ((\x. Im (f x)) sums Im x)"  hoelzl@56369  620  unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..  lp15@59613  621 hoelzl@56369  622 lemma summable_complex_iff: "summable f \ summable (\x. Re (f x)) \ summable (\x. Im (f x))"  hoelzl@56889  623  unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel)  hoelzl@56369  624 hoelzl@56369  625 lemma summable_complex_of_real [simp]: "summable (\n. complex_of_real (f n)) \ summable f"  hoelzl@56369  626  unfolding summable_complex_iff by simp  hoelzl@56369  627 hoelzl@56369  628 lemma summable_Re: "summable f \ summable (\x. Re (f x))"  hoelzl@56369  629  unfolding summable_complex_iff by blast  hoelzl@56369  630 hoelzl@56369  631 lemma summable_Im: "summable f \ summable (\x. Im (f x))"  hoelzl@56369  632  unfolding summable_complex_iff by blast  lp15@56217  633 paulson@61104  634 lemma complex_is_Nat_iff: "z \ \ \ Im z = 0 \ (\i. Re z = of_nat i)"  paulson@61104  635  by (auto simp: Nats_def complex_eq_iff)  paulson@61104  636 paulson@61104  637 lemma complex_is_Int_iff: "z \ \ \ Im z = 0 \ (\i. Re z = of_int i)"  paulson@61104  638  by (auto simp: Ints_def complex_eq_iff)  paulson@61104  639 hoelzl@56889  640 lemma complex_is_Real_iff: "z \ \ \ Im z = 0"  hoelzl@56889  641  by (auto simp: Reals_def complex_eq_iff)  lp15@55734  642 lp15@55734  643 lemma Reals_cnj_iff: "z \ \ \ cnj z = z"  hoelzl@56889  644  by (auto simp: complex_is_Real_iff complex_eq_iff)  lp15@55734  645 wenzelm@61944  646 lemma in_Reals_norm: "z \ \ \ norm z = \Re z\"  hoelzl@56889  647  by (simp add: complex_is_Real_iff norm_complex_def)  hoelzl@56369  648 hoelzl@56369  649 lemma series_comparison_complex:  hoelzl@56369  650  fixes f:: "nat \ 'a::banach"  hoelzl@56369  651  assumes sg: "summable g"  hoelzl@56369  652  and "\n. g n \ \" "\n. Re (g n) \ 0"  hoelzl@56369  653  and fg: "\n. n \ N \ norm(f n) \ norm(g n)"  hoelzl@56369  654  shows "summable f"  hoelzl@56369  655 proof -  hoelzl@56369  656  have g: "\n. cmod (g n) = Re (g n)" using assms  hoelzl@56369  657  by (metis abs_of_nonneg in_Reals_norm)  hoelzl@56369  658  show ?thesis  hoelzl@56369  659  apply (rule summable_comparison_test' [where g = "\n. norm (g n)" and N=N])  hoelzl@56369  660  using sg  hoelzl@56369  661  apply (auto simp: summable_def)  hoelzl@56369  662  apply (rule_tac x="Re s" in exI)  hoelzl@56369  663  apply (auto simp: g sums_Re)  hoelzl@56369  664  apply (metis fg g)  hoelzl@56369  665  done  hoelzl@56369  666 qed  lp15@55734  667 wenzelm@60758  668 subsection\Polar Form for Complex Numbers\  lp15@59746  669 lp15@59746  670 lemma complex_unimodular_polar: "(norm z = 1) \ \x. z = Complex (cos x) (sin x)"  lp15@59746  671  using sincos_total_2pi [of "Re z" "Im z"]  lp15@59746  672  by auto (metis cmod_power2 complex_eq power_one)  paulson@14323  673 wenzelm@60758  674 subsubsection \\cos \theta + i \sin \theta\  huffman@20557  675 hoelzl@56889  676 primcorec cis :: "real \ complex" where  hoelzl@56889  677  "Re (cis a) = cos a"  hoelzl@56889  678 | "Im (cis a) = sin a"  huffman@44827  679 huffman@44827  680 lemma cis_zero [simp]: "cis 0 = 1"  hoelzl@56889  681  by (simp add: complex_eq_iff)  huffman@44827  682 huffman@44828  683 lemma norm_cis [simp]: "norm (cis a) = 1"  hoelzl@56889  684  by (simp add: norm_complex_def)  huffman@44828  685 huffman@44828  686 lemma sgn_cis [simp]: "sgn (cis a) = cis a"  huffman@44828  687  by (simp add: sgn_div_norm)  huffman@44828  688 huffman@44828  689 lemma cis_neq_zero [simp]: "cis a \ 0"  huffman@44828  690  by (metis norm_cis norm_zero zero_neq_one)  huffman@44828  691 huffman@44827  692 lemma cis_mult: "cis a * cis b = cis (a + b)"  hoelzl@56889  693  by (simp add: complex_eq_iff cos_add sin_add)  huffman@44827  694 huffman@44827  695 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"  lp15@61609  696  by (induct n, simp_all add: of_nat_Suc algebra_simps cis_mult)  huffman@44827  697 huffman@44827  698 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"  hoelzl@56889  699  by (simp add: complex_eq_iff)  huffman@44827  700 huffman@44827  701 lemma cis_divide: "cis a / cis b = cis (a - b)"  hoelzl@56889  702  by (simp add: divide_complex_def cis_mult)  huffman@44827  703 huffman@44827  704 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"  huffman@44827  705  by (auto simp add: DeMoivre)  huffman@44827  706 huffman@44827  707 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"  huffman@44827  708  by (auto simp add: DeMoivre)  huffman@44827  709 hoelzl@56889  710 lemma cis_pi: "cis pi = -1"  hoelzl@56889  711  by (simp add: complex_eq_iff)  hoelzl@56889  712 wenzelm@60758  713 subsubsection \r(\cos \theta + i \sin \theta)\  huffman@44715  714 hoelzl@56889  715 definition rcis :: "real \ real \ complex" where  huffman@20557  716  "rcis r a = complex_of_real r * cis a"  huffman@20557  717 huffman@44827  718 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"  huffman@44828  719  by (simp add: rcis_def)  huffman@44827  720 huffman@44827  721 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"  huffman@44828  722  by (simp add: rcis_def)  huffman@44827  723 huffman@44827  724 lemma rcis_Ex: "\r a. z = rcis r a"  huffman@44828  725  by (simp add: complex_eq_iff polar_Ex)  huffman@44827  726 wenzelm@61944  727 lemma complex_mod_rcis [simp]: "cmod (rcis r a) = \r\"  huffman@44828  728  by (simp add: rcis_def norm_mult)  huffman@44827  729 huffman@44827  730 lemma cis_rcis_eq: "cis a = rcis 1 a"  huffman@44827  731  by (simp add: rcis_def)  huffman@44827  732 huffman@44827  733 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"  huffman@44828  734  by (simp add: rcis_def cis_mult)  huffman@44827  735 huffman@44827  736 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"  huffman@44827  737  by (simp add: rcis_def)  huffman@44827  738 huffman@44827  739 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"  huffman@44827  740  by (simp add: rcis_def)  huffman@44827  741 huffman@44828  742 lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \ r = 0"  huffman@44828  743  by (simp add: rcis_def)  huffman@44828  744 huffman@44827  745 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"  huffman@44827  746  by (simp add: rcis_def power_mult_distrib DeMoivre)  huffman@44827  747 huffman@44827  748 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"  huffman@44827  749  by (simp add: divide_inverse rcis_def)  huffman@44827  750 huffman@44827  751 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"  huffman@44828  752  by (simp add: rcis_def cis_divide [symmetric])  huffman@44827  753 wenzelm@60758  754 subsubsection \Complex exponential\  huffman@44827  755 hoelzl@56889  756 lemma cis_conv_exp: "cis b = exp (\ * b)"  hoelzl@56889  757 proof -  hoelzl@56889  758  { fix n :: nat  hoelzl@56889  759  have "\ ^ n = fact n *\<^sub>R (cos_coeff n + \ * sin_coeff n)"  hoelzl@56889  760  by (induct n)  hoelzl@56889  761  (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps  lp15@61609  762  power2_eq_square of_nat_Suc add_nonneg_eq_0_iff)  hoelzl@56889  763  then have "(\ * complex_of_real b) ^ n /\<^sub>R fact n =  hoelzl@56889  764  of_real (cos_coeff n * b^n) + \ * of_real (sin_coeff n * b^n)"  hoelzl@56889  765  by (simp add: field_simps) }  lp15@59658  766  then show ?thesis using sin_converges [of b] cos_converges [of b]  hoelzl@56889  767  by (auto simp add: cis.ctr exp_def simp del: of_real_mult  lp15@59658  768  intro!: sums_unique sums_add sums_mult sums_of_real)  huffman@44291  769 qed  huffman@44291  770 lp15@61762  771 lemma exp_eq_polar: "exp z = exp (Re z) * cis (Im z)"  hoelzl@56889  772  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by (cases z) simp  huffman@20557  773 huffman@44828  774 lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"  lp15@61762  775  unfolding exp_eq_polar by simp  huffman@44828  776 huffman@44828  777 lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"  lp15@61762  778  unfolding exp_eq_polar by simp  huffman@44828  779 lp15@59746  780 lemma norm_cos_sin [simp]: "norm (Complex (cos t) (sin t)) = 1"  lp15@59746  781  by (simp add: norm_complex_def)  lp15@59746  782 lp15@59746  783 lemma norm_exp_eq_Re [simp]: "norm (exp z) = exp (Re z)"  lp15@61762  784  by (simp add: cis.code cmod_complex_polar exp_eq_polar)  lp15@59746  785 lp15@61762  786 lemma complex_exp_exists: "\a r. z = complex_of_real r * exp a"  lp15@59746  787  apply (insert rcis_Ex [of z])  lp15@61762  788  apply (auto simp add: exp_eq_polar rcis_def mult.assoc [symmetric])  lp15@59746  789  apply (rule_tac x = "ii * complex_of_real a" in exI, auto)  lp15@59746  790  done  paulson@14323  791 lp15@61848  792 lemma exp_pi_i [simp]: "exp(of_real pi * ii) = -1"  lp15@61848  793  by (metis cis_conv_exp cis_pi mult.commute)  lp15@61848  794 lp15@61848  795 lemma exp_two_pi_i [simp]: "exp(2 * of_real pi * ii) = 1"  lp15@61762  796  by (simp add: exp_eq_polar complex_eq_iff)  paulson@14387  797 wenzelm@60758  798 subsubsection \Complex argument\  huffman@44844  799 huffman@44844  800 definition arg :: "complex \ real" where  huffman@44844  801  "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \ -pi < a \ a \ pi))"  huffman@44844  802 huffman@44844  803 lemma arg_zero: "arg 0 = 0"  huffman@44844  804  by (simp add: arg_def)  huffman@44844  805 huffman@44844  806 lemma arg_unique:  huffman@44844  807  assumes "sgn z = cis x" and "-pi < x" and "x \ pi"  huffman@44844  808  shows "arg z = x"  huffman@44844  809 proof -  huffman@44844  810  from assms have "z \ 0" by auto  huffman@44844  811  have "(SOME a. sgn z = cis a \ -pi < a \ a \ pi) = x"  huffman@44844  812  proof  huffman@44844  813  fix a def d \ "a - x"  huffman@44844  814  assume a: "sgn z = cis a \ - pi < a \ a \ pi"  huffman@44844  815  from a assms have "- (2*pi) < d \ d < 2*pi"  huffman@44844  816  unfolding d_def by simp  huffman@44844  817  moreover from a assms have "cos a = cos x" and "sin a = sin x"  huffman@44844  818  by (simp_all add: complex_eq_iff)  wenzelm@53374  819  hence cos: "cos d = 1" unfolding d_def cos_diff by simp  wenzelm@53374  820  moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)  huffman@44844  821  ultimately have "d = 0"  haftmann@58709  822  unfolding sin_zero_iff  haftmann@58740  823  by (auto elim!: evenE dest!: less_2_cases)  huffman@44844  824  thus "a = x" unfolding d_def by simp  huffman@44844  825  qed (simp add: assms del: Re_sgn Im_sgn)  wenzelm@60758  826  with \z \ 0\ show "arg z = x"  huffman@44844  827  unfolding arg_def by simp  huffman@44844  828 qed  huffman@44844  829 huffman@44844  830 lemma arg_correct:  huffman@44844  831  assumes "z \ 0" shows "sgn z = cis (arg z) \ -pi < arg z \ arg z \ pi"  huffman@44844  832 proof (simp add: arg_def assms, rule someI_ex)  huffman@44844  833  obtain r a where z: "z = rcis r a" using rcis_Ex by fast  huffman@44844  834  with assms have "r \ 0" by auto  huffman@44844  835  def b \ "if 0 < r then a else a + pi"  huffman@44844  836  have b: "sgn z = cis b"  wenzelm@60758  837  unfolding z b_def rcis_def using \r \ 0\  hoelzl@56889  838  by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff)  huffman@44844  839  have cis_2pi_nat: "\n. cis (2 * pi * real_of_nat n) = 1"  hoelzl@56889  840  by (induct_tac n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)  huffman@44844  841  have cis_2pi_int: "\x. cis (2 * pi * real_of_int x) = 1"  hoelzl@56889  842  by (case_tac x rule: int_diff_cases)  hoelzl@56889  843  (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)  huffman@44844  844  def c \ "b - 2*pi * of_int \(b - pi) / (2*pi)\"  huffman@44844  845  have "sgn z = cis c"  huffman@44844  846  unfolding b c_def  huffman@44844  847  by (simp add: cis_divide [symmetric] cis_2pi_int)  huffman@44844  848  moreover have "- pi < c \ c \ pi"  huffman@44844  849  using ceiling_correct [of "(b - pi) / (2*pi)"]  lp15@61649  850  by (simp add: c_def less_divide_eq divide_le_eq algebra_simps del: le_of_int_ceiling)  huffman@44844  851  ultimately show "\a. sgn z = cis a \ -pi < a \ a \ pi" by fast  huffman@44844  852 qed  huffman@44844  853 huffman@44844  854 lemma arg_bounded: "- pi < arg z \ arg z \ pi"  hoelzl@56889  855  by (cases "z = 0") (simp_all add: arg_zero arg_correct)  huffman@44844  856 huffman@44844  857 lemma cis_arg: "z \ 0 \ cis (arg z) = sgn z"  huffman@44844  858  by (simp add: arg_correct)  huffman@44844  859 huffman@44844  860 lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"  hoelzl@56889  861  by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)  hoelzl@56889  862 hoelzl@56889  863 lemma cos_arg_i_mult_zero [simp]: "y \ 0 \ Re y = 0 \ cos (arg y) = 0"  hoelzl@56889  864  using cis_arg [of y] by (simp add: complex_eq_iff)  hoelzl@56889  865 wenzelm@60758  866 subsection \Square root of complex numbers\  hoelzl@56889  867 hoelzl@56889  868 primcorec csqrt :: "complex \ complex" where  hoelzl@56889  869  "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"  hoelzl@56889  870 | "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)"  hoelzl@56889  871 hoelzl@56889  872 lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \ Re x \ 0 \ csqrt x = sqrt (Re x)"  hoelzl@56889  873  by (simp add: complex_eq_iff norm_complex_def)  hoelzl@56889  874 hoelzl@56889  875 lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \ Re x \ 0 \ csqrt x = \ * sqrt \Re x\"  hoelzl@56889  876  by (simp add: complex_eq_iff norm_complex_def)  hoelzl@56889  877 lp15@59862  878 lemma of_real_sqrt: "x \ 0 \ of_real (sqrt x) = csqrt (of_real x)"  lp15@59862  879  by (simp add: complex_eq_iff norm_complex_def)  lp15@59862  880 hoelzl@56889  881 lemma csqrt_0 [simp]: "csqrt 0 = 0"  hoelzl@56889  882  by simp  hoelzl@56889  883 hoelzl@56889  884 lemma csqrt_1 [simp]: "csqrt 1 = 1"  hoelzl@56889  885  by simp  hoelzl@56889  886 hoelzl@56889  887 lemma csqrt_ii [simp]: "csqrt \ = (1 +$$ / sqrt 2"  hoelzl@56889  888  by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt)  huffman@44844  889 lp15@59741  890 lemma power2_csqrt[simp,algebra]: "(csqrt z)\<^sup>2 = z"  hoelzl@56889  891 proof cases  hoelzl@56889  892  assume "Im z = 0" then show ?thesis  hoelzl@56889  893  using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"]  hoelzl@56889  894  by (cases "0::real" "Re z" rule: linorder_cases)  hoelzl@56889  895  (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)  hoelzl@56889  896 next  hoelzl@56889  897  assume "Im z \ 0"  hoelzl@56889  898  moreover  hoelzl@56889  899  have "cmod z * cmod z - Re z * Re z = Im z * Im z"  hoelzl@56889  900  by (simp add: norm_complex_def power2_eq_square)  hoelzl@56889  901  moreover  hoelzl@56889  902  have "\Re z\ \ cmod z"  hoelzl@56889  903  by (simp add: norm_complex_def)  hoelzl@56889  904  ultimately show ?thesis  hoelzl@56889  905  by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq  hoelzl@56889  906  field_simps real_sqrt_mult[symmetric] real_sqrt_divide)  hoelzl@56889  907 qed  hoelzl@56889  908 hoelzl@56889  909 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \ z = 0"  hoelzl@56889  910  by auto (metis power2_csqrt power_eq_0_iff)  hoelzl@56889  911 hoelzl@56889  912 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \ z = 1"  hoelzl@56889  913  by auto (metis power2_csqrt power2_eq_1_iff)  hoelzl@56889  914 hoelzl@56889  915 lemma csqrt_principal: "0 < Re (csqrt z) \ Re (csqrt z) = 0 \ 0 \ Im (csqrt z)"  hoelzl@56889  916  by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)  hoelzl@56889  917 hoelzl@56889  918 lemma Re_csqrt: "0 \ Re (csqrt z)"  hoelzl@56889  919  by (metis csqrt_principal le_less)  hoelzl@56889  920 hoelzl@56889  921 lemma csqrt_square:  hoelzl@56889  922  assumes "0 < Re b \ (Re b = 0 \ 0 \ Im b)"  hoelzl@56889  923  shows "csqrt (b^2) = b"  hoelzl@56889  924 proof -  hoelzl@56889  925  have "csqrt (b^2) = b \ csqrt (b^2) = - b"  hoelzl@56889  926  unfolding power2_eq_iff[symmetric] by (simp add: power2_csqrt)  hoelzl@56889  927  moreover have "csqrt (b^2) \ -b \ b = 0"  hoelzl@56889  928  using csqrt_principal[of "b ^ 2"] assms by (intro disjCI notI) (auto simp: complex_eq_iff)  hoelzl@56889  929  ultimately show ?thesis  hoelzl@56889  930  by auto  hoelzl@56889  931 qed  hoelzl@56889  932 lp15@59746  933 lemma csqrt_unique:  lp15@59746  934  "w^2 = z \ (0 < Re w \ Re w = 0 \ 0 \ Im w) \ csqrt z = w"  lp15@59746  935  by (auto simp: csqrt_square)  lp15@59746  936 lp15@59613  937 lemma csqrt_minus [simp]:  hoelzl@56889  938  assumes "Im x < 0 \ (Im x = 0 \ 0 \ Re x)"  hoelzl@56889  939  shows "csqrt (- x) = \ * csqrt x"  hoelzl@56889  940 proof -  hoelzl@56889  941  have "csqrt ((\ * csqrt x)^2) = \ * csqrt x"  hoelzl@56889  942  proof (rule csqrt_square)  hoelzl@56889  943  have "Im (csqrt x) \ 0"  hoelzl@56889  944  using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod)  hoelzl@56889  945  then show "0 < Re (\ * csqrt x) \ Re (\ * csqrt x) = 0 \ 0 \ Im (\ * csqrt x)"  hoelzl@56889  946  by (auto simp add: Re_csqrt simp del: csqrt.simps)  hoelzl@56889  947  qed  hoelzl@56889  948  also have "(\ * csqrt x)^2 = - x"  lp15@59746  949  by (simp add: power_mult_distrib)  hoelzl@56889  950  finally show ?thesis .  hoelzl@56889  951 qed  huffman@44844  952 wenzelm@60758  953 text \Legacy theorem names\  huffman@44065  954 huffman@44065  955 lemmas expand_complex_eq = complex_eq_iff  huffman@44065  956 lemmas complex_Re_Im_cancel_iff = complex_eq_iff  huffman@44065  957 lemmas complex_equality = complex_eqI  hoelzl@56889  958 lemmas cmod_def = norm_complex_def  hoelzl@56889  959 lemmas complex_norm_def = norm_complex_def  hoelzl@56889  960 lemmas complex_divide_def = divide_complex_def  hoelzl@56889  961 hoelzl@56889  962 lemma legacy_Complex_simps:  hoelzl@56889  963  shows Complex_eq_0: "Complex a b = 0 \ a = 0 \ b = 0"  hoelzl@56889  964  and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"  hoelzl@56889  965  and complex_minus: "- (Complex a b) = Complex (- a) (- b)"  hoelzl@56889  966  and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"  hoelzl@56889  967  and Complex_eq_1: "Complex a b = 1 \ a = 1 \ b = 0"  hoelzl@56889  968  and Complex_eq_neg_1: "Complex a b = - 1 \ a = - 1 \ b = 0"  hoelzl@56889  969  and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"  hoelzl@56889  970  and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"  hoelzl@56889  971  and Complex_eq_numeral: "Complex a b = numeral w \ a = numeral w \ b = 0"  hoelzl@56889  972  and Complex_eq_neg_numeral: "Complex a b = - numeral w \ a = - numeral w \ b = 0"  hoelzl@56889  973  and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"  hoelzl@56889  974  and Complex_eq_i: "(Complex x y = ii) = (x = 0 \ y = 1)"  hoelzl@56889  975  and i_mult_Complex: "ii * Complex a b = Complex (- b) a"  hoelzl@56889  976  and Complex_mult_i: "Complex a b * ii = Complex (- b) a"  hoelzl@56889  977  and i_complex_of_real: "ii * complex_of_real r = Complex 0 r"  hoelzl@56889  978  and complex_of_real_i: "complex_of_real r * ii = Complex 0 r"  hoelzl@56889  979  and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"  hoelzl@56889  980  and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"  hoelzl@56889  981  and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"  hoelzl@56889  982  and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"  hoelzl@56889  983  and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"  hoelzl@56889  984  and complex_cn: "cnj (Complex a b) = Complex a (- b)"  hoelzl@56889  985  and Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"  hoelzl@56889  986  and Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"  hoelzl@56889  987  and complex_of_real_def: "complex_of_real r = Complex r 0"  hoelzl@56889  988  and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"  hoelzl@56889  989  by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide del: Complex_eq)  hoelzl@56889  990 hoelzl@56889  991 lemma Complex_in_Reals: "Complex x 0 \ \"  hoelzl@56889  992  by (metis Reals_of_real complex_of_real_def)  huffman@44065  993 paulson@13957  994 end