src/HOL/Complex.thy
 author huffman Wed Sep 07 20:44:39 2011 -0700 (2011-09-07) changeset 44827 4d1384a1fc82 parent 44825 353ddca2e4c0 child 44828 3d6a79e0e1d0 permissions -rw-r--r--
Complex.thy: move theorems into appropriate subsections
 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 =  haftmann@25571  106  Complex (Re x / ((Re x)\ + (Im x)\)) (- Im x / ((Re x)\ + (Im x)\))"  huffman@23125  107 huffman@44724  108 definition complex_divide_def:  huffman@44724  109  "x / (y\complex) = x * inverse y"  haftmann@25571  110 huffman@23125  111 lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \ b = 0)"  haftmann@25712  112  by (simp add: complex_one_def)  huffman@22861  113 paulson@14374  114 lemma complex_Re_one [simp]: "Re 1 = 1"  haftmann@25712  115  by (simp add: complex_one_def)  paulson@14323  116 paulson@14374  117 lemma complex_Im_one [simp]: "Im 1 = 0"  haftmann@25712  118  by (simp add: complex_one_def)  paulson@14323  119 huffman@23125  120 lemma complex_mult [simp]:  huffman@23125  121  "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"  haftmann@25712  122  by (simp add: complex_mult_def)  paulson@14323  123 huffman@23125  124 lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"  haftmann@25712  125  by (simp add: complex_mult_def)  paulson@14323  126 huffman@23125  127 lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"  haftmann@25712  128  by (simp add: complex_mult_def)  paulson@14323  129 paulson@14377  130 lemma complex_inverse [simp]:  huffman@23125  131  "inverse (Complex a b) = Complex (a / (a\ + b\)) (- b / (a\ + b\))"  haftmann@25712  132  by (simp add: complex_inverse_def)  paulson@14335  133 huffman@23125  134 lemma complex_Re_inverse:  huffman@23125  135  "Re (inverse x) = Re x / ((Re x)\ + (Im x)\)"  haftmann@25712  136  by (simp add: complex_inverse_def)  paulson@14323  137 huffman@23125  138 lemma complex_Im_inverse:  huffman@23125  139  "Im (inverse x) = - Im x / ((Re x)\ + (Im x)\)"  haftmann@25712  140  by (simp add: complex_inverse_def)  paulson@14335  141 haftmann@25712  142 instance  haftmann@25712  143  by intro_classes (simp_all add: complex_mult_def  huffman@44724  144  right_distrib left_distrib right_diff_distrib left_diff_distrib  huffman@44724  145  complex_inverse_def complex_divide_def  huffman@44724  146  power2_eq_square add_divide_distrib [symmetric]  huffman@44724  147  complex_eq_iff)  paulson@14335  148 haftmann@25712  149 end  huffman@23125  150 huffman@23125  151 huffman@23125  152 subsection {* Numerals and Arithmetic *}  huffman@23125  153 haftmann@25571  154 instantiation complex :: number_ring  haftmann@25571  155 begin  huffman@23125  156 huffman@44724  157 definition complex_number_of_def:  huffman@44724  158  "number_of w = (of_int w \ complex)"  haftmann@25571  159 haftmann@25571  160 instance  haftmann@25712  161  by intro_classes (simp only: complex_number_of_def)  haftmann@25571  162 haftmann@25571  163 end  huffman@23125  164 huffman@23125  165 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"  huffman@44724  166  by (induct n) simp_all  huffman@20556  167 huffman@23125  168 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"  huffman@44724  169  by (induct n) simp_all  huffman@23125  170 huffman@23125  171 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"  huffman@44724  172  by (cases z rule: int_diff_cases) simp  huffman@23125  173 huffman@23125  174 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"  huffman@44724  175  by (cases z rule: int_diff_cases) simp  huffman@23125  176 huffman@23125  177 lemma complex_Re_number_of [simp]: "Re (number_of v) = number_of v"  huffman@44724  178  unfolding number_of_eq by (rule complex_Re_of_int)  huffman@20556  179 huffman@23125  180 lemma complex_Im_number_of [simp]: "Im (number_of v) = 0"  huffman@44724  181  unfolding number_of_eq by (rule complex_Im_of_int)  huffman@23125  182 huffman@23125  183 lemma Complex_eq_number_of [simp]:  huffman@23125  184  "(Complex a b = number_of w) = (a = number_of w \ b = 0)"  huffman@44724  185  by (simp add: complex_eq_iff)  huffman@23125  186 huffman@23125  187 huffman@23125  188 subsection {* Scalar Multiplication *}  huffman@20556  189 haftmann@25712  190 instantiation complex :: real_field  haftmann@25571  191 begin  haftmann@25571  192 huffman@44724  193 definition complex_scaleR_def:  huffman@44724  194  "scaleR r x = Complex (r * Re x) (r * Im x)"  haftmann@25571  195 huffman@23125  196 lemma complex_scaleR [simp]:  huffman@23125  197  "scaleR r (Complex a b) = Complex (r * a) (r * b)"  haftmann@25712  198  unfolding complex_scaleR_def by simp  huffman@23125  199 huffman@23125  200 lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"  haftmann@25712  201  unfolding complex_scaleR_def by simp  huffman@23125  202 huffman@23125  203 lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"  haftmann@25712  204  unfolding complex_scaleR_def by simp  huffman@22972  205 haftmann@25712  206 instance  huffman@20556  207 proof  huffman@23125  208  fix a b :: real and x y :: complex  huffman@23125  209  show "scaleR a (x + y) = scaleR a x + scaleR a y"  huffman@44065  210  by (simp add: complex_eq_iff right_distrib)  huffman@23125  211  show "scaleR (a + b) x = scaleR a x + scaleR b x"  huffman@44065  212  by (simp add: complex_eq_iff left_distrib)  huffman@23125  213  show "scaleR a (scaleR b x) = scaleR (a * b) x"  huffman@44065  214  by (simp add: complex_eq_iff mult_assoc)  huffman@23125  215  show "scaleR 1 x = x"  huffman@44065  216  by (simp add: complex_eq_iff)  huffman@23125  217  show "scaleR a x * y = scaleR a (x * y)"  huffman@44065  218  by (simp add: complex_eq_iff algebra_simps)  huffman@23125  219  show "x * scaleR a y = scaleR a (x * y)"  huffman@44065  220  by (simp add: complex_eq_iff algebra_simps)  huffman@20556  221 qed  huffman@20556  222 haftmann@25712  223 end  haftmann@25712  224 huffman@20556  225 huffman@23125  226 subsection{* Properties of Embedding from Reals *}  paulson@14323  227 huffman@44724  228 abbreviation complex_of_real :: "real \ complex"  huffman@44724  229  where "complex_of_real \ of_real"  huffman@20557  230 huffman@20557  231 lemma complex_of_real_def: "complex_of_real r = Complex r 0"  huffman@44724  232  by (simp add: of_real_def complex_scaleR_def)  huffman@20557  233 huffman@20557  234 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"  huffman@44724  235  by (simp add: complex_of_real_def)  huffman@20557  236 huffman@20557  237 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"  huffman@44724  238  by (simp add: complex_of_real_def)  huffman@20557  239 paulson@14377  240 lemma Complex_add_complex_of_real [simp]:  huffman@44724  241  shows "Complex x y + complex_of_real r = Complex (x+r) y"  huffman@44724  242  by (simp add: complex_of_real_def)  paulson@14377  243 paulson@14377  244 lemma complex_of_real_add_Complex [simp]:  huffman@44724  245  shows "complex_of_real r + Complex x y = Complex (r+x) y"  huffman@44724  246  by (simp add: complex_of_real_def)  paulson@14377  247 paulson@14377  248 lemma Complex_mult_complex_of_real:  huffman@44724  249  shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"  huffman@44724  250  by (simp add: complex_of_real_def)  paulson@14377  251 paulson@14377  252 lemma complex_of_real_mult_Complex:  huffman@44724  253  shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"  huffman@44724  254  by (simp add: complex_of_real_def)  huffman@20557  255 huffman@44827  256 lemma complex_split_polar:  huffman@44827  257  "\r a. z = complex_of_real r * (Complex (cos a) (sin a))"  huffman@44827  258  by (simp add: complex_eq_iff polar_Ex)  huffman@44827  259 paulson@14377  260 huffman@23125  261 subsection {* Vector Norm *}  paulson@14323  262 haftmann@25712  263 instantiation complex :: real_normed_field  haftmann@25571  264 begin  haftmann@25571  265 huffman@31413  266 definition complex_norm_def:  huffman@31413  267  "norm z = sqrt ((Re z)\ + (Im z)\)"  haftmann@25571  268 huffman@44724  269 abbreviation cmod :: "complex \ real"  huffman@44724  270  where "cmod \ norm"  haftmann@25571  271 huffman@31413  272 definition complex_sgn_def:  huffman@31413  273  "sgn x = x /\<^sub>R cmod x"  haftmann@25571  274 huffman@31413  275 definition dist_complex_def:  huffman@31413  276  "dist x y = cmod (x - y)"  huffman@31413  277 haftmann@37767  278 definition open_complex_def:  huffman@31492  279  "open (S :: complex set) \ (\x\S. \e>0. \y. dist y x < e \ y \ S)"  huffman@31292  280 huffman@20557  281 lemmas cmod_def = complex_norm_def  huffman@20557  282 huffman@23125  283 lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\ + y\)"  haftmann@25712  284  by (simp add: complex_norm_def)  huffman@22852  285 huffman@31413  286 instance proof  huffman@31492  287  fix r :: real and x y :: complex and S :: "complex set"  huffman@23125  288  show "0 \ norm x"  huffman@22861  289  by (induct x) simp  huffman@23125  290  show "(norm x = 0) = (x = 0)"  huffman@22861  291  by (induct x) simp  huffman@23125  292  show "norm (x + y) \ norm x + norm y"  huffman@23125  293  by (induct x, induct y)  huffman@23125  294  (simp add: real_sqrt_sum_squares_triangle_ineq)  huffman@23125  295  show "norm (scaleR r x) = \r\ * norm x"  huffman@23125  296  by (induct x)  huffman@23125  297  (simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult)  huffman@23125  298  show "norm (x * y) = norm x * norm y"  huffman@23125  299  by (induct x, induct y)  nipkow@29667  300  (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)  huffman@31292  301  show "sgn x = x /\<^sub>R cmod x"  huffman@31292  302  by (rule complex_sgn_def)  huffman@31292  303  show "dist x y = cmod (x - y)"  huffman@31292  304  by (rule dist_complex_def)  huffman@31492  305  show "open S \ (\x\S. \e>0. \y. dist y x < e \ y \ S)"  huffman@31492  306  by (rule open_complex_def)  huffman@24520  307 qed  huffman@20557  308 haftmann@25712  309 end  haftmann@25712  310 huffman@44761  311 lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1"  huffman@44724  312  by simp  paulson@14323  313 huffman@44761  314 lemma cmod_complex_polar:  huffman@44724  315  "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"  huffman@44724  316  by (simp add: norm_mult)  huffman@22861  317 huffman@22861  318 lemma complex_Re_le_cmod: "Re x \ cmod x"  huffman@44724  319  unfolding complex_norm_def  huffman@44724  320  by (rule real_sqrt_sum_squares_ge1)  huffman@22861  321 huffman@44761  322 lemma complex_mod_minus_le_complex_mod: "- cmod x \ cmod x"  huffman@44724  323  by (rule order_trans [OF _ norm_ge_zero], simp)  huffman@22861  324 huffman@44761  325 lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \ cmod a"  huffman@44724  326  by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)  paulson@14323  327 chaieb@26117  328 lemma abs_Re_le_cmod: "\Re x\ \ cmod x"  huffman@44724  329  by (cases x) simp  chaieb@26117  330 chaieb@26117  331 lemma abs_Im_le_cmod: "\Im x\ \ cmod x"  huffman@44724  332  by (cases x) simp  huffman@44724  333 paulson@14354  334 huffman@23123  335 subsection {* Completeness of the Complexes *}  huffman@23123  336 huffman@44290  337 lemma bounded_linear_Re: "bounded_linear Re"  huffman@44290  338  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)  huffman@44290  339 huffman@44290  340 lemma bounded_linear_Im: "bounded_linear Im"  huffman@44127  341  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)  huffman@23123  342 huffman@44290  343 lemmas tendsto_Re [tendsto_intros] =  huffman@44290  344  bounded_linear.tendsto [OF bounded_linear_Re]  huffman@44290  345 huffman@44290  346 lemmas tendsto_Im [tendsto_intros] =  huffman@44290  347  bounded_linear.tendsto [OF bounded_linear_Im]  huffman@44290  348 huffman@44290  349 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]  huffman@44290  350 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]  huffman@44290  351 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]  huffman@44290  352 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]  huffman@23123  353 huffman@36825  354 lemma tendsto_Complex [tendsto_intros]:  huffman@44724  355  assumes "(f ---> a) F" and "(g ---> b) F"  huffman@44724  356  shows "((\x. Complex (f x) (g x)) ---> Complex a b) F"  huffman@36825  357 proof (rule tendstoI)  huffman@36825  358  fix r :: real assume "0 < r"  huffman@36825  359  hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)  huffman@44724  360  have "eventually (\x. dist (f x) a < r / sqrt 2) F"  huffman@44724  361  using (f ---> a) F and 0 < r / sqrt 2 by (rule tendstoD)  huffman@36825  362  moreover  huffman@44724  363  have "eventually (\x. dist (g x) b < r / sqrt 2) F"  huffman@44724  364  using (g ---> b) F and 0 < r / sqrt 2 by (rule tendstoD)  huffman@36825  365  ultimately  huffman@44724  366  show "eventually (\x. dist (Complex (f x) (g x)) (Complex a b) < r) F"  huffman@36825  367  by (rule eventually_elim2)  huffman@36825  368  (simp add: dist_norm real_sqrt_sum_squares_less)  huffman@36825  369 qed  huffman@36825  370 huffman@23123  371 instance complex :: banach  huffman@23123  372 proof  huffman@23123  373  fix X :: "nat \ complex"  huffman@23123  374  assume X: "Cauchy X"  huffman@44290  375  from Cauchy_Re [OF X] have 1: "(\n. Re (X n)) ----> lim (\n. Re (X n))"  huffman@23123  376  by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)  huffman@44290  377  from Cauchy_Im [OF X] have 2: "(\n. Im (X n)) ----> lim (\n. Im (X n))"  huffman@23123  378  by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)  huffman@23123  379  have "X ----> Complex (lim (\n. Re (X n))) (lim (\n. Im (X n)))"  huffman@44748  380  using tendsto_Complex [OF 1 2] by simp  huffman@23123  381  thus "convergent X"  huffman@23123  382  by (rule convergentI)  huffman@23123  383 qed  huffman@23123  384 huffman@23123  385 huffman@44827  386 subsection {* The Complex Number $i$ *}  huffman@23125  387 huffman@44724  388 definition "ii" :: complex ("\")  huffman@44724  389  where i_def: "ii \ Complex 0 1"  huffman@23125  390 huffman@23125  391 lemma complex_Re_i [simp]: "Re ii = 0"  huffman@44724  392  by (simp add: i_def)  paulson@14354  393 huffman@23125  394 lemma complex_Im_i [simp]: "Im ii = 1"  huffman@44724  395  by (simp add: i_def)  huffman@23125  396 huffman@23125  397 lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \ y = 1)"  huffman@44724  398  by (simp add: i_def)  huffman@23125  399 huffman@23125  400 lemma complex_i_not_zero [simp]: "ii \ 0"  huffman@44724  401  by (simp add: complex_eq_iff)  huffman@23125  402 huffman@23125  403 lemma complex_i_not_one [simp]: "ii \ 1"  huffman@44724  404  by (simp add: complex_eq_iff)  huffman@23124  405 huffman@23125  406 lemma complex_i_not_number_of [simp]: "ii \ number_of w"  huffman@44724  407  by (simp add: complex_eq_iff)  huffman@23125  408 huffman@23125  409 lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"  huffman@44724  410  by (simp add: complex_eq_iff)  huffman@23125  411 huffman@23125  412 lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"  huffman@44724  413  by (simp add: complex_eq_iff)  huffman@23125  414 huffman@23125  415 lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"  huffman@44724  416  by (simp add: i_def complex_of_real_def)  huffman@23125  417 huffman@23125  418 lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"  huffman@44724  419  by (simp add: i_def complex_of_real_def)  huffman@23125  420 huffman@23125  421 lemma i_squared [simp]: "ii * ii = -1"  huffman@44724  422  by (simp add: i_def)  huffman@23125  423 huffman@23125  424 lemma power2_i [simp]: "ii\ = -1"  huffman@44724  425  by (simp add: power2_eq_square)  huffman@23125  426 huffman@23125  427 lemma inverse_i [simp]: "inverse ii = - ii"  huffman@44724  428  by (rule inverse_unique, simp)  paulson@14354  429 huffman@44827  430 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"  huffman@44827  431  by (simp add: mult_assoc [symmetric])  huffman@44827  432 paulson@14354  433 huffman@23125  434 subsection {* Complex Conjugation *}  huffman@23125  435 huffman@44724  436 definition cnj :: "complex \ complex" where  huffman@23125  437  "cnj z = Complex (Re z) (- Im z)"  huffman@23125  438 huffman@23125  439 lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"  huffman@44724  440  by (simp add: cnj_def)  huffman@23125  441 huffman@23125  442 lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"  huffman@44724  443  by (simp add: cnj_def)  huffman@23125  444 huffman@23125  445 lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"  huffman@44724  446  by (simp add: cnj_def)  huffman@23125  447 huffman@23125  448 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"  huffman@44724  449  by (simp add: complex_eq_iff)  huffman@23125  450 huffman@23125  451 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"  huffman@44724  452  by (simp add: cnj_def)  huffman@23125  453 huffman@23125  454 lemma complex_cnj_zero [simp]: "cnj 0 = 0"  huffman@44724  455  by (simp add: complex_eq_iff)  huffman@23125  456 huffman@23125  457 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"  huffman@44724  458  by (simp add: complex_eq_iff)  huffman@23125  459 huffman@23125  460 lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"  huffman@44724  461  by (simp add: complex_eq_iff)  huffman@23125  462 huffman@23125  463 lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"  huffman@44724  464  by (simp add: complex_eq_iff)  huffman@23125  465 huffman@23125  466 lemma complex_cnj_minus: "cnj (- x) = - cnj x"  huffman@44724  467  by (simp add: complex_eq_iff)  huffman@23125  468 huffman@23125  469 lemma complex_cnj_one [simp]: "cnj 1 = 1"  huffman@44724  470  by (simp add: complex_eq_iff)  huffman@23125  471 huffman@23125  472 lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"  huffman@44724  473  by (simp add: complex_eq_iff)  huffman@23125  474 huffman@23125  475 lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"  huffman@44724  476  by (simp add: complex_inverse_def)  paulson@14323  477 huffman@23125  478 lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"  huffman@44724  479  by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)  huffman@23125  480 huffman@23125  481 lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"  huffman@44724  482  by (induct n, simp_all add: complex_cnj_mult)  huffman@23125  483 huffman@23125  484 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"  huffman@44724  485  by (simp add: complex_eq_iff)  huffman@23125  486 huffman@23125  487 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"  huffman@44724  488  by (simp add: complex_eq_iff)  huffman@23125  489 huffman@23125  490 lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w"  huffman@44724  491  by (simp add: complex_eq_iff)  huffman@23125  492 huffman@23125  493 lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"  huffman@44724  494  by (simp add: complex_eq_iff)  huffman@23125  495 huffman@23125  496 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"  huffman@44724  497  by (simp add: complex_norm_def)  paulson@14323  498 huffman@23125  499 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"  huffman@44724  500  by (simp add: complex_eq_iff)  huffman@23125  501 huffman@23125  502 lemma complex_cnj_i [simp]: "cnj ii = - ii"  huffman@44724  503  by (simp add: complex_eq_iff)  huffman@23125  504 huffman@23125  505 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"  huffman@44724  506  by (simp add: complex_eq_iff)  huffman@23125  507 huffman@23125  508 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"  huffman@44724  509  by (simp add: complex_eq_iff)  paulson@14354  510 huffman@23125  511 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\ + (Im z)\)"  huffman@44724  512  by (simp add: complex_eq_iff power2_eq_square)  huffman@23125  513 huffman@23125  514 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\"  huffman@44724  515  by (simp add: norm_mult power2_eq_square)  huffman@23125  516 huffman@44827  517 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"  huffman@44827  518  by (simp add: cmod_def power2_eq_square)  huffman@44827  519 huffman@44827  520 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"  huffman@44827  521  by simp  huffman@44827  522 huffman@44290  523 lemma bounded_linear_cnj: "bounded_linear cnj"  huffman@44127  524  using complex_cnj_add complex_cnj_scaleR  huffman@44127  525  by (rule bounded_linear_intro [where K=1], simp)  paulson@14354  526 huffman@44290  527 lemmas tendsto_cnj [tendsto_intros] =  huffman@44290  528  bounded_linear.tendsto [OF bounded_linear_cnj]  huffman@44290  529 huffman@44290  530 lemmas isCont_cnj [simp] =  huffman@44290  531  bounded_linear.isCont [OF bounded_linear_cnj]  huffman@44290  532 paulson@14354  533 huffman@44827  534 subsection {* Complex Signum and Argument *}  huffman@20557  535 huffman@44724  536 definition arg :: "complex => real" where  huffman@20557  537  "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \ pi)"  huffman@20557  538 paulson@14374  539 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"  huffman@44724  540  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)  paulson@14323  541 paulson@14374  542 lemma complex_eq_cancel_iff2 [simp]:  huffman@44724  543  shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"  huffman@44724  544  by (simp add: complex_of_real_def)  paulson@14323  545 paulson@14374  546 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"  huffman@44724  547  by (simp add: complex_sgn_def divide_inverse)  paulson@14323  548 paulson@14374  549 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"  huffman@44724  550  by (simp add: complex_sgn_def divide_inverse)  paulson@14323  551 paulson@14323  552 lemma complex_inverse_complex_split:  paulson@14323  553  "inverse(complex_of_real x + ii * complex_of_real y) =  paulson@14323  554  complex_of_real(x/(x ^ 2 + y ^ 2)) -  paulson@14323  555  ii * complex_of_real(y/(x ^ 2 + y ^ 2))"  huffman@44724  556  by (simp add: complex_of_real_def i_def diff_minus divide_inverse)  paulson@14323  557 paulson@14323  558 (*----------------------------------------------------------------------------*)  paulson@14323  559 (* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)  paulson@14323  560 (* many of the theorems are not used - so should they be kept? *)  paulson@14323  561 (*----------------------------------------------------------------------------*)  paulson@14323  562 paulson@14354  563 lemma cos_arg_i_mult_zero_pos:  paulson@14377  564  "0 < y ==> cos (arg(Complex 0 y)) = 0"  paulson@14373  565 apply (simp add: arg_def abs_if)  paulson@14334  566 apply (rule_tac a = "pi/2" in someI2, auto)  paulson@14334  567 apply (rule order_less_trans [of _ 0], auto)  paulson@14323  568 done  paulson@14323  569 paulson@14354  570 lemma cos_arg_i_mult_zero_neg:  paulson@14377  571  "y < 0 ==> cos (arg(Complex 0 y)) = 0"  paulson@14373  572 apply (simp add: arg_def abs_if)  paulson@14334  573 apply (rule_tac a = "- pi/2" in someI2, auto)  paulson@14334  574 apply (rule order_trans [of _ 0], auto)  paulson@14323  575 done  paulson@14323  576 paulson@14374  577 lemma cos_arg_i_mult_zero [simp]:  paulson@14377  578  "y \ 0 ==> cos (arg(Complex 0 y)) = 0"  paulson@14377  579 by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)  paulson@14323  580 paulson@14323  581 paulson@14323  582 subsection{*Finally! Polar Form for Complex Numbers*}  paulson@14323  583 huffman@44827  584 subsubsection {* $\cos \theta + i \sin \theta$ *}  huffman@20557  585 huffman@44715  586 definition cis :: "real \ complex" where  huffman@20557  587  "cis a = Complex (cos a) (sin a)"  huffman@20557  588 huffman@44827  589 lemma Re_cis [simp]: "Re (cis a) = cos a"  huffman@44827  590  by (simp add: cis_def)  huffman@44827  591 huffman@44827  592 lemma Im_cis [simp]: "Im (cis a) = sin a"  huffman@44827  593  by (simp add: cis_def)  huffman@44827  594 huffman@44827  595 lemma cis_zero [simp]: "cis 0 = 1"  huffman@44827  596  by (simp add: cis_def)  huffman@44827  597 huffman@44827  598 lemma cis_mult: "cis a * cis b = cis (a + b)"  huffman@44827  599  by (simp add: cis_def cos_add sin_add)  huffman@44827  600 huffman@44827  601 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"  huffman@44827  602  by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)  huffman@44827  603 huffman@44827  604 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"  huffman@44827  605  by (simp add: cis_def)  huffman@44827  606 huffman@44827  607 lemma cis_divide: "cis a / cis b = cis (a - b)"  huffman@44827  608  by (simp add: complex_divide_def cis_mult diff_minus)  huffman@44827  609 huffman@44827  610 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"  huffman@44827  611  by (auto simp add: DeMoivre)  huffman@44827  612 huffman@44827  613 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"  huffman@44827  614  by (auto simp add: DeMoivre)  huffman@44827  615 huffman@44827  616 subsubsection {* $r(\cos \theta + i \sin \theta)$ *}  huffman@44715  617 huffman@44715  618 definition rcis :: "[real, real] \ complex" where  huffman@20557  619  "rcis r a = complex_of_real r * cis a"  huffman@20557  620 huffman@44827  621 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"  huffman@44827  622  by (simp add: rcis_def cis_def)  huffman@44827  623 huffman@44827  624 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"  huffman@44827  625  by (simp add: rcis_def cis_def)  huffman@44827  626 huffman@44827  627 lemma rcis_Ex: "\r a. z = rcis r a"  huffman@44827  628 apply (induct z)  huffman@44827  629 apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex)  huffman@44827  630 done  huffman@44827  631 huffman@44827  632 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"  huffman@44827  633  by (simp add: rcis_def cis_def norm_mult)  huffman@44827  634 huffman@44827  635 lemma cis_rcis_eq: "cis a = rcis 1 a"  huffman@44827  636  by (simp add: rcis_def)  huffman@44827  637 huffman@44827  638 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"  huffman@44827  639  by (simp add: rcis_def cis_def cos_add sin_add right_distrib  huffman@44827  640  right_diff_distrib complex_of_real_def)  huffman@44827  641 huffman@44827  642 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"  huffman@44827  643  by (simp add: rcis_def)  huffman@44827  644 huffman@44827  645 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"  huffman@44827  646  by (simp add: rcis_def)  huffman@44827  647 huffman@44827  648 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"  huffman@44827  649  by (simp add: rcis_def power_mult_distrib DeMoivre)  huffman@44827  650 huffman@44827  651 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"  huffman@44827  652  by (simp add: divide_inverse rcis_def)  huffman@44827  653 huffman@44827  654 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"  huffman@44827  655 apply (simp add: complex_divide_def)  huffman@44827  656 apply (case_tac "r2=0", simp)  huffman@44827  657 apply (simp add: rcis_inverse rcis_mult diff_minus)  huffman@44827  658 done  huffman@44827  659 huffman@44827  660 subsubsection {* Complex exponential *}  huffman@44827  661 huffman@44291  662 abbreviation expi :: "complex \ complex"  huffman@44291  663  where "expi \ exp"  huffman@44291  664 huffman@44712  665 lemma cis_conv_exp: "cis b = exp (Complex 0 b)"  huffman@44291  666 proof (rule complex_eqI)  huffman@44291  667  { fix n have "Complex 0 b ^ n =  huffman@44291  668  real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"  huffman@44291  669  apply (induct n)  huffman@44291  670  apply (simp add: cos_coeff_def sin_coeff_def)  huffman@44291  671  apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)  huffman@44291  672  done } note * = this  huffman@44712  673  show "Re (cis b) = Re (exp (Complex 0 b))"  huffman@44291  674  unfolding exp_def cis_def cos_def  huffman@44291  675  by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],  huffman@44291  676  simp add: * mult_assoc [symmetric])  huffman@44712  677  show "Im (cis b) = Im (exp (Complex 0 b))"  huffman@44291  678  unfolding exp_def cis_def sin_def  huffman@44291  679  by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],  huffman@44291  680  simp add: * mult_assoc [symmetric])  huffman@44291  681 qed  huffman@44291  682 huffman@44291  683 lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"  huffman@44712  684  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp  huffman@20557  685 paulson@14374  686 lemma complex_expi_Ex: "\a r. z = complex_of_real r * expi a"  paulson@14373  687 apply (insert rcis_Ex [of z])  huffman@23125  688 apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])  paulson@14334  689 apply (rule_tac x = "ii * complex_of_real a" in exI, auto)  paulson@14323  690 done  paulson@14323  691 paulson@14387  692 lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"  huffman@44724  693  by (simp add: expi_def cis_def)  paulson@14387  694 huffman@44065  695 text {* Legacy theorem names *}  huffman@44065  696 huffman@44065  697 lemmas expand_complex_eq = complex_eq_iff  huffman@44065  698 lemmas complex_Re_Im_cancel_iff = complex_eq_iff  huffman@44065  699 lemmas complex_equality = complex_eqI  huffman@44065  700 paulson@13957  701 end