src/HOL/Complex.thy
 author paulson Thu Apr 03 23:51:52 2014 +0100 (2014-04-03) changeset 56409 36489d77c484 parent 56381 0556204bc230 child 56479 91958d4b30f7 permissions -rw-r--r--
removing simprule status for divide_minus_left and divide_minus_right
 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 paulson@14377  7 header {* 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 paulson@14373  13 datatype complex = Complex real real  paulson@13957  14 huffman@44724  15 primrec Re :: "complex \ real"  huffman@44724  16  where Re: "Re (Complex x y) = x"  paulson@14373  17 huffman@44724  18 primrec Im :: "complex \ real"  huffman@44724  19  where Im: "Im (Complex x y) = y"  paulson@14373  20 paulson@14373  21 lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"  paulson@14373  22  by (induct z) simp  paulson@13957  23 huffman@44065  24 lemma complex_eqI [intro?]: "\Re x = Re y; Im x = Im y\ \ x = y"  haftmann@25712  25  by (induct x, induct y) simp  huffman@23125  26 huffman@44065  27 lemma complex_eq_iff: "x = y \ Re x = Re y \ Im x = Im y"  haftmann@25712  28  by (induct x, induct y) simp  huffman@23125  29 huffman@23125  30 huffman@23125  31 subsection {* Addition and Subtraction *}  huffman@23125  32 haftmann@25599  33 instantiation complex :: ab_group_add  haftmann@25571  34 begin  haftmann@25571  35 huffman@44724  36 definition complex_zero_def:  huffman@44724  37  "0 = Complex 0 0"  haftmann@25571  38 huffman@44724  39 definition complex_add_def:  huffman@44724  40  "x + y = Complex (Re x + Re y) (Im x + Im y)"  huffman@23124  41 huffman@44724  42 definition complex_minus_def:  huffman@44724  43  "- x = Complex (- Re x) (- Im x)"  paulson@14323  44 huffman@44724  45 definition complex_diff_def:  huffman@44724  46  "x - (y\complex) = x + - y"  haftmann@25571  47 haftmann@25599  48 lemma Complex_eq_0 [simp]: "Complex a b = 0 \ a = 0 \ b = 0"  haftmann@25599  49  by (simp add: complex_zero_def)  paulson@14323  50 paulson@14374  51 lemma complex_Re_zero [simp]: "Re 0 = 0"  haftmann@25599  52  by (simp add: complex_zero_def)  paulson@14374  53 paulson@14374  54 lemma complex_Im_zero [simp]: "Im 0 = 0"  haftmann@25599  55  by (simp add: complex_zero_def)  haftmann@25599  56 haftmann@25712  57 lemma complex_add [simp]:  haftmann@25712  58  "Complex a b + Complex c d = Complex (a + c) (b + d)"  haftmann@25712  59  by (simp add: complex_add_def)  haftmann@25712  60 haftmann@25599  61 lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y"  haftmann@25599  62  by (simp add: complex_add_def)  haftmann@25599  63 haftmann@25599  64 lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y"  haftmann@25599  65  by (simp add: complex_add_def)  paulson@14323  66 haftmann@25712  67 lemma complex_minus [simp]:  haftmann@25712  68  "- (Complex a b) = Complex (- a) (- b)"  haftmann@25599  69  by (simp add: complex_minus_def)  huffman@23125  70 huffman@23125  71 lemma complex_Re_minus [simp]: "Re (- x) = - Re x"  haftmann@25599  72  by (simp add: complex_minus_def)  huffman@23125  73 huffman@23125  74 lemma complex_Im_minus [simp]: "Im (- x) = - Im x"  haftmann@25599  75  by (simp add: complex_minus_def)  huffman@23125  76 huffman@23275  77 lemma complex_diff [simp]:  huffman@23125  78  "Complex a b - Complex c d = Complex (a - c) (b - d)"  haftmann@25599  79  by (simp add: complex_diff_def)  huffman@23125  80 huffman@23125  81 lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y"  haftmann@25599  82  by (simp add: complex_diff_def)  huffman@23125  83 huffman@23125  84 lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y"  haftmann@25599  85  by (simp add: complex_diff_def)  huffman@23125  86 haftmann@25712  87 instance  haftmann@25712  88  by intro_classes (simp_all add: complex_add_def complex_diff_def)  haftmann@25712  89 haftmann@25712  90 end  haftmann@25712  91 haftmann@25712  92 huffman@23125  93 subsection {* Multiplication and Division *}  huffman@23125  94 haftmann@36409  95 instantiation complex :: field_inverse_zero  haftmann@25571  96 begin  haftmann@25571  97 huffman@44724  98 definition complex_one_def:  huffman@44724  99  "1 = Complex 1 0"  haftmann@25571  100 huffman@44724  101 definition complex_mult_def:  huffman@44724  102  "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"  huffman@23125  103 huffman@44724  104 definition complex_inverse_def:  huffman@44724  105  "inverse x =  wenzelm@53015  106  Complex (Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)) (- Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2))"  huffman@23125  107 huffman@44724  108 definition complex_divide_def:  huffman@44724  109  "x / (y\complex) = x * inverse y"  haftmann@25571  110 haftmann@54489  111 lemma Complex_eq_1 [simp]:  haftmann@54489  112  "Complex a b = 1 \ a = 1 \ b = 0"  haftmann@54489  113  by (simp add: complex_one_def)  haftmann@54489  114 haftmann@54489  115 lemma Complex_eq_neg_1 [simp]:  haftmann@54489  116  "Complex a b = - 1 \ a = - 1 \ b = 0"  haftmann@25712  117  by (simp add: complex_one_def)  huffman@22861  118 paulson@14374  119 lemma complex_Re_one [simp]: "Re 1 = 1"  haftmann@25712  120  by (simp add: complex_one_def)  paulson@14323  121 paulson@14374  122 lemma complex_Im_one [simp]: "Im 1 = 0"  haftmann@25712  123  by (simp add: complex_one_def)  paulson@14323  124 huffman@23125  125 lemma complex_mult [simp]:  huffman@23125  126  "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"  haftmann@25712  127  by (simp add: complex_mult_def)  paulson@14323  128 huffman@23125  129 lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"  haftmann@25712  130  by (simp add: complex_mult_def)  paulson@14323  131 huffman@23125  132 lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"  haftmann@25712  133  by (simp add: complex_mult_def)  paulson@14323  134 paulson@14377  135 lemma complex_inverse [simp]:  wenzelm@53015  136  "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"  haftmann@25712  137  by (simp add: complex_inverse_def)  paulson@14335  138 huffman@23125  139 lemma complex_Re_inverse:  wenzelm@53015  140  "Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"  haftmann@25712  141  by (simp add: complex_inverse_def)  paulson@14323  142 huffman@23125  143 lemma complex_Im_inverse:  wenzelm@53015  144  "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"  haftmann@25712  145  by (simp add: complex_inverse_def)  paulson@14335  146 haftmann@25712  147 instance  lp15@56409  148  by intro_classes (simp_all add: complex_mult_def divide_minus_left  webertj@49962  149  distrib_left distrib_right right_diff_distrib left_diff_distrib  huffman@44724  150  complex_inverse_def complex_divide_def  huffman@44724  151  power2_eq_square add_divide_distrib [symmetric]  huffman@44724  152  complex_eq_iff)  paulson@14335  153 haftmann@25712  154 end  huffman@23125  155 huffman@23125  156 huffman@23125  157 subsection {* Numerals and Arithmetic *}  huffman@23125  158 huffman@23125  159 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"  huffman@44724  160  by (induct n) simp_all  huffman@20556  161 huffman@23125  162 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"  huffman@44724  163  by (induct n) simp_all  huffman@23125  164 huffman@23125  165 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"  huffman@44724  166  by (cases z rule: int_diff_cases) simp  huffman@23125  167 huffman@23125  168 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"  huffman@44724  169  by (cases z rule: int_diff_cases) simp  huffman@23125  170 huffman@47108  171 lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"  huffman@47108  172  using complex_Re_of_int [of "numeral v"] by simp  huffman@47108  173 haftmann@54489  174 lemma complex_Re_neg_numeral [simp]: "Re (- numeral v) = - numeral v"  haftmann@54489  175  using complex_Re_of_int [of "- numeral v"] by simp  huffman@47108  176 huffman@47108  177 lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"  huffman@47108  178  using complex_Im_of_int [of "numeral v"] by simp  huffman@20556  179 haftmann@54489  180 lemma complex_Im_neg_numeral [simp]: "Im (- numeral v) = 0"  haftmann@54489  181  using complex_Im_of_int [of "- numeral v"] by simp  huffman@23125  182 huffman@47108  183 lemma Complex_eq_numeral [simp]:  haftmann@54489  184  "Complex a b = numeral w \ a = numeral w \ b = 0"  huffman@47108  185  by (simp add: complex_eq_iff)  huffman@47108  186 huffman@47108  187 lemma Complex_eq_neg_numeral [simp]:  haftmann@54489  188  "Complex a b = - numeral w \ a = - numeral w \ b = 0"  huffman@44724  189  by (simp add: complex_eq_iff)  huffman@23125  190 huffman@23125  191 huffman@23125  192 subsection {* Scalar Multiplication *}  huffman@20556  193 haftmann@25712  194 instantiation complex :: real_field  haftmann@25571  195 begin  haftmann@25571  196 huffman@44724  197 definition complex_scaleR_def:  huffman@44724  198  "scaleR r x = Complex (r * Re x) (r * Im x)"  haftmann@25571  199 huffman@23125  200 lemma complex_scaleR [simp]:  huffman@23125  201  "scaleR r (Complex a b) = Complex (r * a) (r * b)"  haftmann@25712  202  unfolding complex_scaleR_def by simp  huffman@23125  203 huffman@23125  204 lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"  haftmann@25712  205  unfolding complex_scaleR_def by simp  huffman@23125  206 huffman@23125  207 lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"  haftmann@25712  208  unfolding complex_scaleR_def by simp  huffman@22972  209 haftmann@25712  210 instance  huffman@20556  211 proof  huffman@23125  212  fix a b :: real and x y :: complex  huffman@23125  213  show "scaleR a (x + y) = scaleR a x + scaleR a y"  webertj@49962  214  by (simp add: complex_eq_iff distrib_left)  huffman@23125  215  show "scaleR (a + b) x = scaleR a x + scaleR b x"  webertj@49962  216  by (simp add: complex_eq_iff distrib_right)  huffman@23125  217  show "scaleR a (scaleR b x) = scaleR (a * b) x"  huffman@44065  218  by (simp add: complex_eq_iff mult_assoc)  huffman@23125  219  show "scaleR 1 x = x"  huffman@44065  220  by (simp add: complex_eq_iff)  huffman@23125  221  show "scaleR a x * y = scaleR a (x * y)"  huffman@44065  222  by (simp add: complex_eq_iff algebra_simps)  huffman@23125  223  show "x * scaleR a y = scaleR a (x * y)"  huffman@44065  224  by (simp add: complex_eq_iff algebra_simps)  huffman@20556  225 qed  huffman@20556  226 haftmann@25712  227 end  haftmann@25712  228 huffman@20556  229 huffman@23125  230 subsection{* Properties of Embedding from Reals *}  paulson@14323  231 huffman@44724  232 abbreviation complex_of_real :: "real \ complex"  huffman@44724  233  where "complex_of_real \ of_real"  huffman@20557  234 hoelzl@56331  235 declare [[coercion complex_of_real]]  hoelzl@56331  236 huffman@20557  237 lemma complex_of_real_def: "complex_of_real r = Complex r 0"  huffman@44724  238  by (simp add: of_real_def complex_scaleR_def)  huffman@20557  239 huffman@20557  240 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"  huffman@44724  241  by (simp add: complex_of_real_def)  huffman@20557  242 huffman@20557  243 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"  huffman@44724  244  by (simp add: complex_of_real_def)  huffman@20557  245 paulson@14377  246 lemma Complex_add_complex_of_real [simp]:  huffman@44724  247  shows "Complex x y + complex_of_real r = Complex (x+r) y"  huffman@44724  248  by (simp add: complex_of_real_def)  paulson@14377  249 paulson@14377  250 lemma complex_of_real_add_Complex [simp]:  huffman@44724  251  shows "complex_of_real r + Complex x y = Complex (r+x) y"  huffman@44724  252  by (simp add: complex_of_real_def)  paulson@14377  253 paulson@14377  254 lemma Complex_mult_complex_of_real:  huffman@44724  255  shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"  huffman@44724  256  by (simp add: complex_of_real_def)  paulson@14377  257 paulson@14377  258 lemma complex_of_real_mult_Complex:  huffman@44724  259  shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"  huffman@44724  260  by (simp add: complex_of_real_def)  huffman@20557  261 huffman@44841  262 lemma complex_eq_cancel_iff2 [simp]:  huffman@44841  263  shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"  huffman@44841  264  by (simp add: complex_of_real_def)  huffman@44841  265 huffman@44827  266 lemma complex_split_polar:  huffman@44827  267  "\r a. z = complex_of_real r * (Complex (cos a) (sin a))"  huffman@44827  268  by (simp add: complex_eq_iff polar_Ex)  huffman@44827  269 paulson@14377  270 huffman@23125  271 subsection {* Vector Norm *}  paulson@14323  272 haftmann@25712  273 instantiation complex :: real_normed_field  haftmann@25571  274 begin  haftmann@25571  275 huffman@31413  276 definition complex_norm_def:  wenzelm@53015  277  "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"  haftmann@25571  278 huffman@44724  279 abbreviation cmod :: "complex \ real"  huffman@44724  280  where "cmod \ norm"  haftmann@25571  281 huffman@31413  282 definition complex_sgn_def:  huffman@31413  283  "sgn x = x /\<^sub>R cmod x"  haftmann@25571  284 huffman@31413  285 definition dist_complex_def:  huffman@31413  286  "dist x y = cmod (x - y)"  huffman@31413  287 haftmann@37767  288 definition open_complex_def:  huffman@31492  289  "open (S :: complex set) \ (\x\S. \e>0. \y. dist y x < e \ y \ S)"  huffman@31292  290 huffman@20557  291 lemmas cmod_def = complex_norm_def  huffman@20557  292 wenzelm@53015  293 lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"  haftmann@25712  294  by (simp add: complex_norm_def)  huffman@22852  295 huffman@31413  296 instance proof  huffman@31492  297  fix r :: real and x y :: complex and S :: "complex set"  huffman@23125  298  show "(norm x = 0) = (x = 0)"  huffman@22861  299  by (induct x) simp  huffman@23125  300  show "norm (x + y) \ norm x + norm y"  huffman@23125  301  by (induct x, induct y)  huffman@23125  302  (simp add: real_sqrt_sum_squares_triangle_ineq)  huffman@23125  303  show "norm (scaleR r x) = \r\ * norm x"  huffman@23125  304  by (induct x)  webertj@49962  305  (simp add: power_mult_distrib distrib_left [symmetric] real_sqrt_mult)  huffman@23125  306  show "norm (x * y) = norm x * norm y"  huffman@23125  307  by (induct x, induct y)  nipkow@29667  308  (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)  huffman@31292  309  show "sgn x = x /\<^sub>R cmod x"  huffman@31292  310  by (rule complex_sgn_def)  huffman@31292  311  show "dist x y = cmod (x - y)"  huffman@31292  312  by (rule dist_complex_def)  huffman@31492  313  show "open S \ (\x\S. \e>0. \y. dist y x < e \ y \ S)"  huffman@31492  314  by (rule open_complex_def)  huffman@24520  315 qed  huffman@20557  316 haftmann@25712  317 end  haftmann@25712  318 huffman@44761  319 lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1"  huffman@44724  320  by simp  paulson@14323  321 huffman@44761  322 lemma cmod_complex_polar:  huffman@44724  323  "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"  huffman@44724  324  by (simp add: norm_mult)  huffman@22861  325 huffman@22861  326 lemma complex_Re_le_cmod: "Re x \ cmod x"  huffman@44724  327  unfolding complex_norm_def  huffman@44724  328  by (rule real_sqrt_sum_squares_ge1)  huffman@22861  329 huffman@44761  330 lemma complex_mod_minus_le_complex_mod: "- cmod x \ cmod x"  huffman@44724  331  by (rule order_trans [OF _ norm_ge_zero], simp)  huffman@22861  332 huffman@44761  333 lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \ cmod a"  huffman@44724  334  by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)  paulson@14323  335 chaieb@26117  336 lemma abs_Re_le_cmod: "\Re x\ \ cmod x"  huffman@44724  337  by (cases x) simp  chaieb@26117  338 chaieb@26117  339 lemma abs_Im_le_cmod: "\Im x\ \ cmod x"  huffman@44724  340  by (cases x) simp  huffman@44724  341 hoelzl@56369  342 hoelzl@56369  343 lemma abs_sqrt_wlog:  hoelzl@56369  344  fixes x::"'a::linordered_idom"  hoelzl@56369  345  assumes "\x::'a. x \ 0 \ P x (x\<^sup>2)" shows "P \x\ (x\<^sup>2)"  hoelzl@56369  346 by (metis abs_ge_zero assms power2_abs)  hoelzl@56369  347 hoelzl@56369  348 lemma complex_abs_le_norm: "\Re z\ + \Im z\ \ sqrt 2 * norm z"  hoelzl@56369  349  unfolding complex_norm_def  hoelzl@56369  350  apply (rule abs_sqrt_wlog [where x="Re z"])  hoelzl@56369  351  apply (rule abs_sqrt_wlog [where x="Im z"])  hoelzl@56369  352  apply (rule power2_le_imp_le)  hoelzl@56369  353  apply (simp_all add: power2_sum add_commute sum_squares_bound real_sqrt_mult [symmetric])  hoelzl@56369  354  done  hoelzl@56369  355 hoelzl@56369  356 huffman@44843  357 text {* Properties of complex signum. *}  huffman@44843  358 huffman@44843  359 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"  huffman@44843  360  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)  huffman@44843  361 huffman@44843  362 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"  huffman@44843  363  by (simp add: complex_sgn_def divide_inverse)  huffman@44843  364 huffman@44843  365 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"  huffman@44843  366  by (simp add: complex_sgn_def divide_inverse)  huffman@44843  367 paulson@14354  368 huffman@23123  369 subsection {* Completeness of the Complexes *}  huffman@23123  370 huffman@44290  371 lemma bounded_linear_Re: "bounded_linear Re"  huffman@44290  372  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)  huffman@44290  373 huffman@44290  374 lemma bounded_linear_Im: "bounded_linear Im"  huffman@44127  375  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)  huffman@23123  376 huffman@44290  377 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]  huffman@44290  378 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]  hoelzl@56381  379 lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]  hoelzl@56381  380 lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]  hoelzl@56381  381 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]  hoelzl@56381  382 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]  hoelzl@56381  383 lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]  hoelzl@56381  384 lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]  hoelzl@56381  385 lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]  hoelzl@56381  386 lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]  hoelzl@56381  387 lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]  hoelzl@56381  388 lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]  hoelzl@56381  389 lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]  hoelzl@56381  390 lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]  hoelzl@56369  391 huffman@36825  392 lemma tendsto_Complex [tendsto_intros]:  huffman@44724  393  assumes "(f ---> a) F" and "(g ---> b) F"  huffman@44724  394  shows "((\x. Complex (f x) (g x)) ---> Complex a b) F"  huffman@36825  395 proof (rule tendstoI)  huffman@36825  396  fix r :: real assume "0 < r"  huffman@36825  397  hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)  huffman@44724  398  have "eventually (\x. dist (f x) a < r / sqrt 2) F"  huffman@44724  399  using (f ---> a) F and 0 < r / sqrt 2 by (rule tendstoD)  huffman@36825  400  moreover  huffman@44724  401  have "eventually (\x. dist (g x) b < r / sqrt 2) F"  huffman@44724  402  using (g ---> b) F and 0 < r / sqrt 2 by (rule tendstoD)  huffman@36825  403  ultimately  huffman@44724  404  show "eventually (\x. dist (Complex (f x) (g x)) (Complex a b) < r) F"  huffman@36825  405  by (rule eventually_elim2)  huffman@36825  406  (simp add: dist_norm real_sqrt_sum_squares_less)  huffman@36825  407 qed  huffman@36825  408 hoelzl@56369  409 hoelzl@56369  410 lemma tendsto_complex_iff:  hoelzl@56369  411  "(f ---> x) F \ (((\x. Re (f x)) ---> Re x) F \ ((\x. Im (f x)) ---> Im x) F)"  hoelzl@56369  412 proof -  hoelzl@56369  413  have f: "f = (\x. Complex (Re (f x)) (Im (f x)))" and x: "x = Complex (Re x) (Im x)"  hoelzl@56369  414  by simp_all  hoelzl@56369  415  show ?thesis  hoelzl@56369  416  apply (subst f)  hoelzl@56369  417  apply (subst x)  hoelzl@56369  418  apply (intro iffI tendsto_Complex conjI)  hoelzl@56369  419  apply (simp_all add: tendsto_Re tendsto_Im)  hoelzl@56369  420  done  hoelzl@56369  421 qed  hoelzl@56369  422 huffman@23123  423 instance complex :: banach  huffman@23123  424 proof  huffman@23123  425  fix X :: "nat \ complex"  huffman@23123  426  assume X: "Cauchy X"  huffman@44290  427  from Cauchy_Re [OF X] have 1: "(\n. Re (X n)) ----> lim (\n. Re (X n))"  huffman@23123  428  by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)  huffman@44290  429  from Cauchy_Im [OF X] have 2: "(\n. Im (X n)) ----> lim (\n. Im (X n))"  huffman@23123  430  by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)  huffman@23123  431  have "X ----> Complex (lim (\n. Re (X n))) (lim (\n. Im (X n)))"  huffman@44748  432  using tendsto_Complex [OF 1 2] by simp  huffman@23123  433  thus "convergent X"  huffman@23123  434  by (rule convergentI)  huffman@23123  435 qed  huffman@23123  436 lp15@56238  437 declare  hoelzl@56381  438  DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]  lp15@56238  439 huffman@23123  440 huffman@44827  441 subsection {* The Complex Number $i$ *}  huffman@23125  442 huffman@44724  443 definition "ii" :: complex ("\")  huffman@44724  444  where i_def: "ii \ Complex 0 1"  huffman@23125  445 huffman@23125  446 lemma complex_Re_i [simp]: "Re ii = 0"  huffman@44724  447  by (simp add: i_def)  paulson@14354  448 huffman@23125  449 lemma complex_Im_i [simp]: "Im ii = 1"  huffman@44724  450  by (simp add: i_def)  huffman@23125  451 huffman@23125  452 lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \ y = 1)"  huffman@44724  453  by (simp add: i_def)  huffman@23125  454 huffman@44902  455 lemma norm_ii [simp]: "norm ii = 1"  huffman@44902  456  by (simp add: i_def)  huffman@44902  457 huffman@23125  458 lemma complex_i_not_zero [simp]: "ii \ 0"  huffman@44724  459  by (simp add: complex_eq_iff)  huffman@23125  460 huffman@23125  461 lemma complex_i_not_one [simp]: "ii \ 1"  huffman@44724  462  by (simp add: complex_eq_iff)  huffman@23124  463 huffman@47108  464 lemma complex_i_not_numeral [simp]: "ii \ numeral w"  huffman@47108  465  by (simp add: complex_eq_iff)  huffman@47108  466 haftmann@54489  467 lemma complex_i_not_neg_numeral [simp]: "ii \ - numeral w"  huffman@44724  468  by (simp add: complex_eq_iff)  huffman@23125  469 huffman@23125  470 lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"  huffman@44724  471  by (simp add: complex_eq_iff)  huffman@23125  472 huffman@23125  473 lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"  huffman@44724  474  by (simp add: complex_eq_iff)  huffman@23125  475 huffman@23125  476 lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"  huffman@44724  477  by (simp add: i_def complex_of_real_def)  huffman@23125  478 huffman@23125  479 lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"  huffman@44724  480  by (simp add: i_def complex_of_real_def)  huffman@23125  481 huffman@23125  482 lemma i_squared [simp]: "ii * ii = -1"  huffman@44724  483  by (simp add: i_def)  huffman@23125  484 wenzelm@53015  485 lemma power2_i [simp]: "ii\<^sup>2 = -1"  huffman@44724  486  by (simp add: power2_eq_square)  huffman@23125  487 huffman@23125  488 lemma inverse_i [simp]: "inverse ii = - ii"  huffman@44724  489  by (rule inverse_unique, simp)  paulson@14354  490 huffman@44827  491 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"  huffman@44827  492  by (simp add: mult_assoc [symmetric])  huffman@44827  493 paulson@14354  494 huffman@23125  495 subsection {* Complex Conjugation *}  huffman@23125  496 huffman@44724  497 definition cnj :: "complex \ complex" where  huffman@23125  498  "cnj z = Complex (Re z) (- Im z)"  huffman@23125  499 huffman@23125  500 lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"  huffman@44724  501  by (simp add: cnj_def)  huffman@23125  502 huffman@23125  503 lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"  huffman@44724  504  by (simp add: cnj_def)  huffman@23125  505 huffman@23125  506 lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"  huffman@44724  507  by (simp add: cnj_def)  huffman@23125  508 huffman@23125  509 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"  huffman@44724  510  by (simp add: complex_eq_iff)  huffman@23125  511 huffman@23125  512 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"  huffman@44724  513  by (simp add: cnj_def)  huffman@23125  514 huffman@23125  515 lemma complex_cnj_zero [simp]: "cnj 0 = 0"  huffman@44724  516  by (simp add: complex_eq_iff)  huffman@23125  517 huffman@23125  518 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"  huffman@44724  519  by (simp add: complex_eq_iff)  huffman@23125  520 huffman@23125  521 lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"  huffman@44724  522  by (simp add: complex_eq_iff)  huffman@23125  523 hoelzl@56369  524 lemma cnj_setsum: "cnj (setsum f s) = (\x\s. cnj (f x))"  hoelzl@56369  525  by (induct s rule: infinite_finite_induct) (auto simp: complex_cnj_add)  hoelzl@56369  526 huffman@23125  527 lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"  huffman@44724  528  by (simp add: complex_eq_iff)  huffman@23125  529 huffman@23125  530 lemma complex_cnj_minus: "cnj (- x) = - cnj x"  huffman@44724  531  by (simp add: complex_eq_iff)  huffman@23125  532 huffman@23125  533 lemma complex_cnj_one [simp]: "cnj 1 = 1"  huffman@44724  534  by (simp add: complex_eq_iff)  huffman@23125  535 huffman@23125  536 lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"  huffman@44724  537  by (simp add: complex_eq_iff)  huffman@23125  538 hoelzl@56369  539 lemma cnj_setprod: "cnj (setprod f s) = (\x\s. cnj (f x))"  hoelzl@56369  540  by (induct s rule: infinite_finite_induct) (auto simp: complex_cnj_mult)  hoelzl@56369  541 huffman@23125  542 lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"  huffman@44724  543  by (simp add: complex_inverse_def)  paulson@14323  544 huffman@23125  545 lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"  huffman@44724  546  by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)  huffman@23125  547 huffman@23125  548 lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"  huffman@44724  549  by (induct n, simp_all add: complex_cnj_mult)  huffman@23125  550 huffman@23125  551 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"  huffman@44724  552  by (simp add: complex_eq_iff)  huffman@23125  553 huffman@23125  554 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"  huffman@44724  555  by (simp add: complex_eq_iff)  huffman@23125  556 huffman@47108  557 lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"  huffman@47108  558  by (simp add: complex_eq_iff)  huffman@47108  559 haftmann@54489  560 lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"  huffman@44724  561  by (simp add: complex_eq_iff)  huffman@23125  562 huffman@23125  563 lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"  huffman@44724  564  by (simp add: complex_eq_iff)  huffman@23125  565 huffman@23125  566 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"  huffman@44724  567  by (simp add: complex_norm_def)  paulson@14323  568 huffman@23125  569 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"  huffman@44724  570  by (simp add: complex_eq_iff)  huffman@23125  571 huffman@23125  572 lemma complex_cnj_i [simp]: "cnj ii = - ii"  huffman@44724  573  by (simp add: complex_eq_iff)  huffman@23125  574 huffman@23125  575 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"  huffman@44724  576  by (simp add: complex_eq_iff)  huffman@23125  577 huffman@23125  578 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"  huffman@44724  579  by (simp add: complex_eq_iff)  paulson@14354  580 wenzelm@53015  581 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"  huffman@44724  582  by (simp add: complex_eq_iff power2_eq_square)  huffman@23125  583 wenzelm@53015  584 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"  huffman@44724  585  by (simp add: norm_mult power2_eq_square)  huffman@23125  586 huffman@44827  587 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"  huffman@44827  588  by (simp add: cmod_def power2_eq_square)  huffman@44827  589 huffman@44827  590 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"  huffman@44827  591  by simp  huffman@44827  592 huffman@44290  593 lemma bounded_linear_cnj: "bounded_linear cnj"  huffman@44127  594  using complex_cnj_add complex_cnj_scaleR  huffman@44127  595  by (rule bounded_linear_intro [where K=1], simp)  paulson@14354  596 hoelzl@56381  597 lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]  hoelzl@56381  598 lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]  hoelzl@56381  599 lemmas continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]  hoelzl@56381  600 lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]  hoelzl@56381  601 lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]  huffman@44290  602 hoelzl@56369  603 lemma lim_cnj: "((\x. cnj(f x)) ---> cnj l) F \ (f ---> l) F"  hoelzl@56369  604  by (simp add: tendsto_iff dist_complex_def Complex.complex_cnj_diff [symmetric])  hoelzl@56369  605 hoelzl@56369  606 lemma sums_cnj: "((\x. cnj(f x)) sums cnj l) \ (f sums l)"  hoelzl@56369  607  by (simp add: sums_def lim_cnj cnj_setsum [symmetric])  hoelzl@56369  608 paulson@14354  609 lp15@55734  610 subsection{*Basic Lemmas*}  lp15@55734  611 lp15@55734  612 lemma complex_eq_0: "z=0 \ (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"  lp15@55734  613  by (metis complex_Im_zero complex_Re_zero complex_eqI sum_power2_eq_zero_iff)  lp15@55734  614 lp15@55734  615 lemma complex_neq_0: "z\0 \ (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"  lp15@55734  616 by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)  lp15@55734  617 lp15@55734  618 lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"  lp15@55734  619 apply (cases z, auto)  lp15@55734  620 by (metis complex_of_real_def of_real_add of_real_power power2_eq_square)  lp15@55734  621 lp15@55734  622 lemma complex_div_eq_0:  lp15@55734  623  "(Re(a / b) = 0 \ Re(a * cnj b) = 0) & (Im(a / b) = 0 \ Im(a * cnj b) = 0)"  lp15@55734  624 proof (cases "b=0")  lp15@55734  625  case True then show ?thesis by auto  lp15@55734  626 next  lp15@55734  627  case False  lp15@55734  628  show ?thesis  lp15@55734  629  proof (cases b)  lp15@55734  630  case (Complex x y)  lp15@55734  631  then have "x\<^sup>2 + y\<^sup>2 > 0"  lp15@55734  632  by (metis Complex_eq_0 False sum_power2_gt_zero_iff)  lp15@55734  633  then have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)"  lp15@55734  634  by (metis add_divide_distrib)  lp15@55734  635  with Complex False show ?thesis  lp15@55734  636  by (auto simp: complex_divide_def)  lp15@55734  637  qed  lp15@55734  638 qed  lp15@55734  639 lp15@55734  640 lemma re_complex_div_eq_0: "Re(a / b) = 0 \ Re(a * cnj b) = 0"  lp15@55734  641  and im_complex_div_eq_0: "Im(a / b) = 0 \ Im(a * cnj b) = 0"  lp15@55734  642 using complex_div_eq_0 by auto  lp15@55734  643 lp15@55734  644 lp15@55734  645 lemma complex_div_gt_0:  lp15@55734  646  "(Re(a / b) > 0 \ Re(a * cnj b) > 0) & (Im(a / b) > 0 \ Im(a * cnj b) > 0)"  lp15@55734  647 proof (cases "b=0")  lp15@55734  648  case True then show ?thesis by auto  lp15@55734  649 next  lp15@55734  650  case False  lp15@55734  651  show ?thesis  lp15@55734  652  proof (cases b)  lp15@55734  653  case (Complex x y)  lp15@55734  654  then have "x\<^sup>2 + y\<^sup>2 > 0"  lp15@55734  655  by (metis Complex_eq_0 False sum_power2_gt_zero_iff)  lp15@55734  656  moreover have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)"  lp15@55734  657  by (metis add_divide_distrib)  lp15@55734  658  ultimately show ?thesis using Complex False 0 < x\<^sup>2 + y\<^sup>2  lp15@56409  659  apply (simp add: complex_divide_def divide_minus_left zero_less_divide_iff less_divide_eq)  lp15@55734  660  apply (metis less_divide_eq mult_zero_left diff_conv_add_uminus minus_divide_left)  lp15@55734  661  done  lp15@55734  662  qed  lp15@55734  663 qed  lp15@55734  664 lp15@55734  665 lemma re_complex_div_gt_0: "Re(a / b) > 0 \ Re(a * cnj b) > 0"  lp15@55734  666  and im_complex_div_gt_0: "Im(a / b) > 0 \ Im(a * cnj b) > 0"  lp15@55734  667 using complex_div_gt_0 by auto  lp15@55734  668 lp15@55734  669 lemma re_complex_div_ge_0: "Re(a / b) \ 0 \ Re(a * cnj b) \ 0"  lp15@55734  670  by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0)  lp15@55734  671 lp15@55734  672 lemma im_complex_div_ge_0: "Im(a / b) \ 0 \ Im(a * cnj b) \ 0"  lp15@55734  673  by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less)  lp15@55734  674 lp15@55734  675 lemma re_complex_div_lt_0: "Re(a / b) < 0 \ Re(a * cnj b) < 0"  boehmes@55759  676  by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0)  lp15@55734  677 lp15@55734  678 lemma im_complex_div_lt_0: "Im(a / b) < 0 \ Im(a * cnj b) < 0"  lp15@55734  679  by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff)  lp15@55734  680 lp15@55734  681 lemma re_complex_div_le_0: "Re(a / b) \ 0 \ Re(a * cnj b) \ 0"  lp15@55734  682  by (metis not_le re_complex_div_gt_0)  lp15@55734  683 lp15@55734  684 lemma im_complex_div_le_0: "Im(a / b) \ 0 \ Im(a * cnj b) \ 0"  lp15@55734  685  by (metis im_complex_div_gt_0 not_le)  lp15@55734  686 lp15@56217  687 lemma Re_setsum: "Re(setsum f s) = setsum (%x. Re(f x)) s"  hoelzl@56369  688  by (induct s rule: infinite_finite_induct) auto  lp15@55734  689 lp15@56217  690 lemma Im_setsum: "Im(setsum f s) = setsum (%x. Im(f x)) s"  hoelzl@56369  691  by (induct s rule: infinite_finite_induct) auto  hoelzl@56369  692 hoelzl@56369  693 lemma sums_complex_iff: "f sums x \ ((\x. Re (f x)) sums Re x) \ ((\x. Im (f x)) sums Im x)"  hoelzl@56369  694  unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..  hoelzl@56369  695   hoelzl@56369  696 lemma summable_complex_iff: "summable f \ summable (\x. Re (f x)) \ summable (\x. Im (f x))"  hoelzl@56369  697  unfolding summable_def sums_complex_iff[abs_def] by (metis Im.simps Re.simps)  hoelzl@56369  698 hoelzl@56369  699 lemma summable_complex_of_real [simp]: "summable (\n. complex_of_real (f n)) \ summable f"  hoelzl@56369  700  unfolding summable_complex_iff by simp  hoelzl@56369  701 hoelzl@56369  702 lemma summable_Re: "summable f \ summable (\x. Re (f x))"  hoelzl@56369  703  unfolding summable_complex_iff by blast  hoelzl@56369  704 hoelzl@56369  705 lemma summable_Im: "summable f \ summable (\x. Im (f x))"  hoelzl@56369  706  unfolding summable_complex_iff by blast  lp15@56217  707 lp15@56217  708 lemma Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"  hoelzl@56369  709  by (induct s rule: infinite_finite_induct) auto  lp15@55734  710 lp15@56217  711 lemma Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"  lp15@56217  712  by (metis Complex_setsum')  lp15@56217  713 lp15@56217  714 lemma of_real_setsum: "of_real (setsum f s) = setsum (%x. of_real(f x)) s"  hoelzl@56369  715  by (induct s rule: infinite_finite_induct) auto  lp15@55734  716 lp15@56217  717 lemma of_real_setprod: "of_real (setprod f s) = setprod (%x. of_real(f x)) s"  hoelzl@56369  718  by (induct s rule: infinite_finite_induct) auto  lp15@55734  719 lp15@55734  720 lemma Reals_cnj_iff: "z \ \ \ cnj z = z"  lp15@55734  721 by (metis Reals_cases Reals_of_real complex.exhaust complex.inject complex_cnj  lp15@55734  722  complex_of_real_def equal_neg_zero)  lp15@55734  723 lp15@55734  724 lemma Complex_in_Reals: "Complex x 0 \ \"  lp15@55734  725  by (metis Reals_of_real complex_of_real_def)  lp15@55734  726 lp15@55734  727 lemma in_Reals_norm: "z \ \ \ norm(z) = abs(Re z)"  lp15@55734  728  by (metis Re_complex_of_real Reals_cases norm_of_real)  lp15@55734  729 hoelzl@56369  730 lemma complex_is_Real_iff: "z \ \ \ Im z = 0"  hoelzl@56369  731  by (metis Complex_in_Reals Im_complex_of_real Reals_cases complex_surj)  hoelzl@56369  732 hoelzl@56369  733 lemma series_comparison_complex:  hoelzl@56369  734  fixes f:: "nat \ 'a::banach"  hoelzl@56369  735  assumes sg: "summable g"  hoelzl@56369  736  and "\n. g n \ \" "\n. Re (g n) \ 0"  hoelzl@56369  737  and fg: "\n. n \ N \ norm(f n) \ norm(g n)"  hoelzl@56369  738  shows "summable f"  hoelzl@56369  739 proof -  hoelzl@56369  740  have g: "\n. cmod (g n) = Re (g n)" using assms  hoelzl@56369  741  by (metis abs_of_nonneg in_Reals_norm)  hoelzl@56369  742  show ?thesis  hoelzl@56369  743  apply (rule summable_comparison_test' [where g = "\n. norm (g n)" and N=N])  hoelzl@56369  744  using sg  hoelzl@56369  745  apply (auto simp: summable_def)  hoelzl@56369  746  apply (rule_tac x="Re s" in exI)  hoelzl@56369  747  apply (auto simp: g sums_Re)  hoelzl@56369  748  apply (metis fg g)  hoelzl@56369  749  done  hoelzl@56369  750 qed  lp15@55734  751 paulson@14323  752 subsection{*Finally! Polar Form for Complex Numbers*}  paulson@14323  753 huffman@44827  754 subsubsection {* $\cos \theta + i \sin \theta$ *}  huffman@20557  755 huffman@44715  756 definition cis :: "real \ complex" where  huffman@20557  757  "cis a = Complex (cos a) (sin a)"  huffman@20557  758 huffman@44827  759 lemma Re_cis [simp]: "Re (cis a) = cos a"  huffman@44827  760  by (simp add: cis_def)  huffman@44827  761 huffman@44827  762 lemma Im_cis [simp]: "Im (cis a) = sin a"  huffman@44827  763  by (simp add: cis_def)  huffman@44827  764 huffman@44827  765 lemma cis_zero [simp]: "cis 0 = 1"  huffman@44827  766  by (simp add: cis_def)  huffman@44827  767 huffman@44828  768 lemma norm_cis [simp]: "norm (cis a) = 1"  huffman@44828  769  by (simp add: cis_def)  huffman@44828  770 huffman@44828  771 lemma sgn_cis [simp]: "sgn (cis a) = cis a"  huffman@44828  772  by (simp add: sgn_div_norm)  huffman@44828  773 huffman@44828  774 lemma cis_neq_zero [simp]: "cis a \ 0"  huffman@44828  775  by (metis norm_cis norm_zero zero_neq_one)  huffman@44828  776 huffman@44827  777 lemma cis_mult: "cis a * cis b = cis (a + b)"  huffman@44827  778  by (simp add: cis_def cos_add sin_add)  huffman@44827  779 huffman@44827  780 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"  huffman@44827  781  by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)  huffman@44827  782 huffman@44827  783 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"  huffman@44827  784  by (simp add: cis_def)  huffman@44827  785 huffman@44827  786 lemma cis_divide: "cis a / cis b = cis (a - b)"  haftmann@54230  787  by (simp add: complex_divide_def cis_mult)  huffman@44827  788 huffman@44827  789 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"  huffman@44827  790  by (auto simp add: DeMoivre)  huffman@44827  791 huffman@44827  792 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"  huffman@44827  793  by (auto simp add: DeMoivre)  huffman@44827  794 huffman@44827  795 subsubsection {* $r(\cos \theta + i \sin \theta)$ *}  huffman@44715  796 huffman@44715  797 definition rcis :: "[real, real] \ complex" where  huffman@20557  798  "rcis r a = complex_of_real r * cis a"  huffman@20557  799 huffman@44827  800 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"  huffman@44828  801  by (simp add: rcis_def)  huffman@44827  802 huffman@44827  803 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"  huffman@44828  804  by (simp add: rcis_def)  huffman@44827  805 huffman@44827  806 lemma rcis_Ex: "\r a. z = rcis r a"  huffman@44828  807  by (simp add: complex_eq_iff polar_Ex)  huffman@44827  808 huffman@44827  809 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"  huffman@44828  810  by (simp add: rcis_def norm_mult)  huffman@44827  811 huffman@44827  812 lemma cis_rcis_eq: "cis a = rcis 1 a"  huffman@44827  813  by (simp add: rcis_def)  huffman@44827  814 huffman@44827  815 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"  huffman@44828  816  by (simp add: rcis_def cis_mult)  huffman@44827  817 huffman@44827  818 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"  huffman@44827  819  by (simp add: rcis_def)  huffman@44827  820 huffman@44827  821 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"  huffman@44827  822  by (simp add: rcis_def)  huffman@44827  823 huffman@44828  824 lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \ r = 0"  huffman@44828  825  by (simp add: rcis_def)  huffman@44828  826 huffman@44827  827 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"  huffman@44827  828  by (simp add: rcis_def power_mult_distrib DeMoivre)  huffman@44827  829 huffman@44827  830 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"  huffman@44827  831  by (simp add: divide_inverse rcis_def)  huffman@44827  832 huffman@44827  833 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"  huffman@44828  834  by (simp add: rcis_def cis_divide [symmetric])  huffman@44827  835 huffman@44827  836 subsubsection {* Complex exponential *}  huffman@44827  837 huffman@44291  838 abbreviation expi :: "complex \ complex"  huffman@44291  839  where "expi \ exp"  huffman@44291  840 huffman@44712  841 lemma cis_conv_exp: "cis b = exp (Complex 0 b)"  huffman@44291  842 proof (rule complex_eqI)  huffman@44291  843  { fix n have "Complex 0 b ^ n =  huffman@44291  844  real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"  huffman@44291  845  apply (induct n)  huffman@44291  846  apply (simp add: cos_coeff_def sin_coeff_def)  lp15@56409  847  apply (simp add: sin_coeff_Suc cos_coeff_Suc divide_minus_left del: mult_Suc)  huffman@44291  848  done } note * = this  huffman@44712  849  show "Re (cis b) = Re (exp (Complex 0 b))"  huffman@44291  850  unfolding exp_def cis_def cos_def  huffman@44291  851  by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],  huffman@44291  852  simp add: * mult_assoc [symmetric])  huffman@44712  853  show "Im (cis b) = Im (exp (Complex 0 b))"  huffman@44291  854  unfolding exp_def cis_def sin_def  huffman@44291  855  by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],  huffman@44291  856  simp add: * mult_assoc [symmetric])  huffman@44291  857 qed  huffman@44291  858 huffman@44291  859 lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"  huffman@44712  860  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp  huffman@20557  861 huffman@44828  862 lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"  huffman@44828  863  unfolding expi_def by simp  huffman@44828  864 huffman@44828  865 lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"  huffman@44828  866  unfolding expi_def by simp  huffman@44828  867 paulson@14374  868 lemma complex_expi_Ex: "\a r. z = complex_of_real r * expi a"  paulson@14373  869 apply (insert rcis_Ex [of z])  huffman@23125  870 apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])  paulson@14334  871 apply (rule_tac x = "ii * complex_of_real a" in exI, auto)  paulson@14323  872 done  paulson@14323  873 paulson@14387  874 lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"  huffman@44724  875  by (simp add: expi_def cis_def)  paulson@14387  876 huffman@44844  877 subsubsection {* Complex argument *}  huffman@44844  878 huffman@44844  879 definition arg :: "complex \ real" where  huffman@44844  880  "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \ -pi < a \ a \ pi))"  huffman@44844  881 huffman@44844  882 lemma arg_zero: "arg 0 = 0"  huffman@44844  883  by (simp add: arg_def)  huffman@44844  884 huffman@44844  885 lemma of_nat_less_of_int_iff: (* TODO: move *)  huffman@44844  886  "(of_nat n :: 'a::linordered_idom) < of_int x \ int n < x"  huffman@44844  887  by (metis of_int_of_nat_eq of_int_less_iff)  huffman@44844  888 huffman@47108  889 lemma real_of_nat_less_numeral_iff [simp]: (* TODO: move *)  huffman@47108  890  "real (n::nat) < numeral w \ n < numeral w"  huffman@47108  891  using of_nat_less_of_int_iff [of n "numeral w", where 'a=real]  huffman@47108  892  by (simp add: real_of_nat_def zless_nat_eq_int_zless [symmetric])  huffman@44844  893 huffman@44844  894 lemma arg_unique:  huffman@44844  895  assumes "sgn z = cis x" and "-pi < x" and "x \ pi"  huffman@44844  896  shows "arg z = x"  huffman@44844  897 proof -  huffman@44844  898  from assms have "z \ 0" by auto  huffman@44844  899  have "(SOME a. sgn z = cis a \ -pi < a \ a \ pi) = x"  huffman@44844  900  proof  huffman@44844  901  fix a def d \ "a - x"  huffman@44844  902  assume a: "sgn z = cis a \ - pi < a \ a \ pi"  huffman@44844  903  from a assms have "- (2*pi) < d \ d < 2*pi"  huffman@44844  904  unfolding d_def by simp  huffman@44844  905  moreover from a assms have "cos a = cos x" and "sin a = sin x"  huffman@44844  906  by (simp_all add: complex_eq_iff)  wenzelm@53374  907  hence cos: "cos d = 1" unfolding d_def cos_diff by simp  wenzelm@53374  908  moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)  huffman@44844  909  ultimately have "d = 0"  huffman@44844  910  unfolding sin_zero_iff even_mult_two_ex  wenzelm@53374  911  by (auto simp add: numeral_2_eq_2 less_Suc_eq)  huffman@44844  912  thus "a = x" unfolding d_def by simp  huffman@44844  913  qed (simp add: assms del: Re_sgn Im_sgn)  huffman@44844  914  with z \ 0 show "arg z = x"  huffman@44844  915  unfolding arg_def by simp  huffman@44844  916 qed  huffman@44844  917 huffman@44844  918 lemma arg_correct:  huffman@44844  919  assumes "z \ 0" shows "sgn z = cis (arg z) \ -pi < arg z \ arg z \ pi"  huffman@44844  920 proof (simp add: arg_def assms, rule someI_ex)  huffman@44844  921  obtain r a where z: "z = rcis r a" using rcis_Ex by fast  huffman@44844  922  with assms have "r \ 0" by auto  huffman@44844  923  def b \ "if 0 < r then a else a + pi"  huffman@44844  924  have b: "sgn z = cis b"  huffman@44844  925  unfolding z b_def rcis_def using r \ 0  huffman@44844  926  by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def)  huffman@44844  927  have cis_2pi_nat: "\n. cis (2 * pi * real_of_nat n) = 1"  webertj@49962  928  by (induct_tac n, simp_all add: distrib_left cis_mult [symmetric],  huffman@44844  929  simp add: cis_def)  huffman@44844  930  have cis_2pi_int: "\x. cis (2 * pi * real_of_int x) = 1"  huffman@44844  931  by (case_tac x rule: int_diff_cases,  huffman@44844  932  simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)  huffman@44844  933  def c \ "b - 2*pi * of_int \(b - pi) / (2*pi)\"  huffman@44844  934  have "sgn z = cis c"  huffman@44844  935  unfolding b c_def  huffman@44844  936  by (simp add: cis_divide [symmetric] cis_2pi_int)  huffman@44844  937  moreover have "- pi < c \ c \ pi"  huffman@44844  938  using ceiling_correct [of "(b - pi) / (2*pi)"]  huffman@44844  939  by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)  huffman@44844  940  ultimately show "\a. sgn z = cis a \ -pi < a \ a \ pi" by fast  huffman@44844  941 qed  huffman@44844  942 huffman@44844  943 lemma arg_bounded: "- pi < arg z \ arg z \ pi"  huffman@44844  944  by (cases "z = 0", simp_all add: arg_zero arg_correct)  huffman@44844  945 huffman@44844  946 lemma cis_arg: "z \ 0 \ cis (arg z) = sgn z"  huffman@44844  947  by (simp add: arg_correct)  huffman@44844  948 huffman@44844  949 lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"  huffman@44844  950  by (cases "z = 0", simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)  huffman@44844  951 huffman@44844  952 lemma cos_arg_i_mult_zero [simp]:  huffman@44844  953  "y \ 0 ==> cos (arg(Complex 0 y)) = 0"  huffman@44844  954  using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff)  huffman@44844  955 huffman@44065  956 text {* Legacy theorem names *}  huffman@44065  957 huffman@44065  958 lemmas expand_complex_eq = complex_eq_iff  huffman@44065  959 lemmas complex_Re_Im_cancel_iff = complex_eq_iff  huffman@44065  960 lemmas complex_equality = complex_eqI  huffman@44065  961 paulson@13957  962 end