src/HOL/Complex.thy
 author haftmann Tue Nov 19 10:05:53 2013 +0100 (2013-11-19) changeset 54489 03ff4d1e6784 parent 54230 b1d955791529 child 55734 3f5b2745d659 permissions -rw-r--r--
eliminiated neg_numeral in favour of - (numeral _)
 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  haftmann@25712  148  by intro_classes (simp_all add: complex_mult_def  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 huffman@20557  235 lemma complex_of_real_def: "complex_of_real r = Complex r 0"  huffman@44724  236  by (simp add: of_real_def complex_scaleR_def)  huffman@20557  237 huffman@20557  238 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"  huffman@44724  239  by (simp add: complex_of_real_def)  huffman@20557  240 huffman@20557  241 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"  huffman@44724  242  by (simp add: complex_of_real_def)  huffman@20557  243 paulson@14377  244 lemma Complex_add_complex_of_real [simp]:  huffman@44724  245  shows "Complex x y + complex_of_real r = Complex (x+r) y"  huffman@44724  246  by (simp add: complex_of_real_def)  paulson@14377  247 paulson@14377  248 lemma complex_of_real_add_Complex [simp]:  huffman@44724  249  shows "complex_of_real r + Complex x y = Complex (r+x) y"  huffman@44724  250  by (simp add: complex_of_real_def)  paulson@14377  251 paulson@14377  252 lemma Complex_mult_complex_of_real:  huffman@44724  253  shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"  huffman@44724  254  by (simp add: complex_of_real_def)  paulson@14377  255 paulson@14377  256 lemma complex_of_real_mult_Complex:  huffman@44724  257  shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"  huffman@44724  258  by (simp add: complex_of_real_def)  huffman@20557  259 huffman@44841  260 lemma complex_eq_cancel_iff2 [simp]:  huffman@44841  261  shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"  huffman@44841  262  by (simp add: complex_of_real_def)  huffman@44841  263 huffman@44827  264 lemma complex_split_polar:  huffman@44827  265  "\r a. z = complex_of_real r * (Complex (cos a) (sin a))"  huffman@44827  266  by (simp add: complex_eq_iff polar_Ex)  huffman@44827  267 paulson@14377  268 huffman@23125  269 subsection {* Vector Norm *}  paulson@14323  270 haftmann@25712  271 instantiation complex :: real_normed_field  haftmann@25571  272 begin  haftmann@25571  273 huffman@31413  274 definition complex_norm_def:  wenzelm@53015  275  "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"  haftmann@25571  276 huffman@44724  277 abbreviation cmod :: "complex \ real"  huffman@44724  278  where "cmod \ norm"  haftmann@25571  279 huffman@31413  280 definition complex_sgn_def:  huffman@31413  281  "sgn x = x /\<^sub>R cmod x"  haftmann@25571  282 huffman@31413  283 definition dist_complex_def:  huffman@31413  284  "dist x y = cmod (x - y)"  huffman@31413  285 haftmann@37767  286 definition open_complex_def:  huffman@31492  287  "open (S :: complex set) \ (\x\S. \e>0. \y. dist y x < e \ y \ S)"  huffman@31292  288 huffman@20557  289 lemmas cmod_def = complex_norm_def  huffman@20557  290 wenzelm@53015  291 lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"  haftmann@25712  292  by (simp add: complex_norm_def)  huffman@22852  293 huffman@31413  294 instance proof  huffman@31492  295  fix r :: real and x y :: complex and S :: "complex set"  huffman@23125  296  show "(norm x = 0) = (x = 0)"  huffman@22861  297  by (induct x) simp  huffman@23125  298  show "norm (x + y) \ norm x + norm y"  huffman@23125  299  by (induct x, induct y)  huffman@23125  300  (simp add: real_sqrt_sum_squares_triangle_ineq)  huffman@23125  301  show "norm (scaleR r x) = \r\ * norm x"  huffman@23125  302  by (induct x)  webertj@49962  303  (simp add: power_mult_distrib distrib_left [symmetric] real_sqrt_mult)  huffman@23125  304  show "norm (x * y) = norm x * norm y"  huffman@23125  305  by (induct x, induct y)  nipkow@29667  306  (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)  huffman@31292  307  show "sgn x = x /\<^sub>R cmod x"  huffman@31292  308  by (rule complex_sgn_def)  huffman@31292  309  show "dist x y = cmod (x - y)"  huffman@31292  310  by (rule dist_complex_def)  huffman@31492  311  show "open S \ (\x\S. \e>0. \y. dist y x < e \ y \ S)"  huffman@31492  312  by (rule open_complex_def)  huffman@24520  313 qed  huffman@20557  314 haftmann@25712  315 end  haftmann@25712  316 huffman@44761  317 lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1"  huffman@44724  318  by simp  paulson@14323  319 huffman@44761  320 lemma cmod_complex_polar:  huffman@44724  321  "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"  huffman@44724  322  by (simp add: norm_mult)  huffman@22861  323 huffman@22861  324 lemma complex_Re_le_cmod: "Re x \ cmod x"  huffman@44724  325  unfolding complex_norm_def  huffman@44724  326  by (rule real_sqrt_sum_squares_ge1)  huffman@22861  327 huffman@44761  328 lemma complex_mod_minus_le_complex_mod: "- cmod x \ cmod x"  huffman@44724  329  by (rule order_trans [OF _ norm_ge_zero], simp)  huffman@22861  330 huffman@44761  331 lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \ cmod a"  huffman@44724  332  by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)  paulson@14323  333 chaieb@26117  334 lemma abs_Re_le_cmod: "\Re x\ \ cmod x"  huffman@44724  335  by (cases x) simp  chaieb@26117  336 chaieb@26117  337 lemma abs_Im_le_cmod: "\Im x\ \ cmod x"  huffman@44724  338  by (cases x) simp  huffman@44724  339 huffman@44843  340 text {* Properties of complex signum. *}  huffman@44843  341 huffman@44843  342 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"  huffman@44843  343  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)  huffman@44843  344 huffman@44843  345 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"  huffman@44843  346  by (simp add: complex_sgn_def divide_inverse)  huffman@44843  347 huffman@44843  348 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"  huffman@44843  349  by (simp add: complex_sgn_def divide_inverse)  huffman@44843  350 paulson@14354  351 huffman@23123  352 subsection {* Completeness of the Complexes *}  huffman@23123  353 huffman@44290  354 lemma bounded_linear_Re: "bounded_linear Re"  huffman@44290  355  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)  huffman@44290  356 huffman@44290  357 lemma bounded_linear_Im: "bounded_linear Im"  huffman@44127  358  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)  huffman@23123  359 huffman@44290  360 lemmas tendsto_Re [tendsto_intros] =  huffman@44290  361  bounded_linear.tendsto [OF bounded_linear_Re]  huffman@44290  362 huffman@44290  363 lemmas tendsto_Im [tendsto_intros] =  huffman@44290  364  bounded_linear.tendsto [OF bounded_linear_Im]  huffman@44290  365 huffman@44290  366 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]  huffman@44290  367 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]  huffman@44290  368 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]  huffman@44290  369 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]  huffman@23123  370 huffman@36825  371 lemma tendsto_Complex [tendsto_intros]:  huffman@44724  372  assumes "(f ---> a) F" and "(g ---> b) F"  huffman@44724  373  shows "((\x. Complex (f x) (g x)) ---> Complex a b) F"  huffman@36825  374 proof (rule tendstoI)  huffman@36825  375  fix r :: real assume "0 < r"  huffman@36825  376  hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)  huffman@44724  377  have "eventually (\x. dist (f x) a < r / sqrt 2) F"  huffman@44724  378  using (f ---> a) F and 0 < r / sqrt 2 by (rule tendstoD)  huffman@36825  379  moreover  huffman@44724  380  have "eventually (\x. dist (g x) b < r / sqrt 2) F"  huffman@44724  381  using (g ---> b) F and 0 < r / sqrt 2 by (rule tendstoD)  huffman@36825  382  ultimately  huffman@44724  383  show "eventually (\x. dist (Complex (f x) (g x)) (Complex a b) < r) F"  huffman@36825  384  by (rule eventually_elim2)  huffman@36825  385  (simp add: dist_norm real_sqrt_sum_squares_less)  huffman@36825  386 qed  huffman@36825  387 huffman@23123  388 instance complex :: banach  huffman@23123  389 proof  huffman@23123  390  fix X :: "nat \ complex"  huffman@23123  391  assume X: "Cauchy X"  huffman@44290  392  from Cauchy_Re [OF X] have 1: "(\n. Re (X n)) ----> lim (\n. Re (X n))"  huffman@23123  393  by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)  huffman@44290  394  from Cauchy_Im [OF X] have 2: "(\n. Im (X n)) ----> lim (\n. Im (X n))"  huffman@23123  395  by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)  huffman@23123  396  have "X ----> Complex (lim (\n. Re (X n))) (lim (\n. Im (X n)))"  huffman@44748  397  using tendsto_Complex [OF 1 2] by simp  huffman@23123  398  thus "convergent X"  huffman@23123  399  by (rule convergentI)  huffman@23123  400 qed  huffman@23123  401 huffman@23123  402 huffman@44827  403 subsection {* The Complex Number $i$ *}  huffman@23125  404 huffman@44724  405 definition "ii" :: complex ("\")  huffman@44724  406  where i_def: "ii \ Complex 0 1"  huffman@23125  407 huffman@23125  408 lemma complex_Re_i [simp]: "Re ii = 0"  huffman@44724  409  by (simp add: i_def)  paulson@14354  410 huffman@23125  411 lemma complex_Im_i [simp]: "Im ii = 1"  huffman@44724  412  by (simp add: i_def)  huffman@23125  413 huffman@23125  414 lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \ y = 1)"  huffman@44724  415  by (simp add: i_def)  huffman@23125  416 huffman@44902  417 lemma norm_ii [simp]: "norm ii = 1"  huffman@44902  418  by (simp add: i_def)  huffman@44902  419 huffman@23125  420 lemma complex_i_not_zero [simp]: "ii \ 0"  huffman@44724  421  by (simp add: complex_eq_iff)  huffman@23125  422 huffman@23125  423 lemma complex_i_not_one [simp]: "ii \ 1"  huffman@44724  424  by (simp add: complex_eq_iff)  huffman@23124  425 huffman@47108  426 lemma complex_i_not_numeral [simp]: "ii \ numeral w"  huffman@47108  427  by (simp add: complex_eq_iff)  huffman@47108  428 haftmann@54489  429 lemma complex_i_not_neg_numeral [simp]: "ii \ - numeral w"  huffman@44724  430  by (simp add: complex_eq_iff)  huffman@23125  431 huffman@23125  432 lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"  huffman@44724  433  by (simp add: complex_eq_iff)  huffman@23125  434 huffman@23125  435 lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"  huffman@44724  436  by (simp add: complex_eq_iff)  huffman@23125  437 huffman@23125  438 lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"  huffman@44724  439  by (simp add: i_def complex_of_real_def)  huffman@23125  440 huffman@23125  441 lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"  huffman@44724  442  by (simp add: i_def complex_of_real_def)  huffman@23125  443 huffman@23125  444 lemma i_squared [simp]: "ii * ii = -1"  huffman@44724  445  by (simp add: i_def)  huffman@23125  446 wenzelm@53015  447 lemma power2_i [simp]: "ii\<^sup>2 = -1"  huffman@44724  448  by (simp add: power2_eq_square)  huffman@23125  449 huffman@23125  450 lemma inverse_i [simp]: "inverse ii = - ii"  huffman@44724  451  by (rule inverse_unique, simp)  paulson@14354  452 huffman@44827  453 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"  huffman@44827  454  by (simp add: mult_assoc [symmetric])  huffman@44827  455 paulson@14354  456 huffman@23125  457 subsection {* Complex Conjugation *}  huffman@23125  458 huffman@44724  459 definition cnj :: "complex \ complex" where  huffman@23125  460  "cnj z = Complex (Re z) (- Im z)"  huffman@23125  461 huffman@23125  462 lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"  huffman@44724  463  by (simp add: cnj_def)  huffman@23125  464 huffman@23125  465 lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"  huffman@44724  466  by (simp add: cnj_def)  huffman@23125  467 huffman@23125  468 lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"  huffman@44724  469  by (simp add: cnj_def)  huffman@23125  470 huffman@23125  471 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"  huffman@44724  472  by (simp add: complex_eq_iff)  huffman@23125  473 huffman@23125  474 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"  huffman@44724  475  by (simp add: cnj_def)  huffman@23125  476 huffman@23125  477 lemma complex_cnj_zero [simp]: "cnj 0 = 0"  huffman@44724  478  by (simp add: complex_eq_iff)  huffman@23125  479 huffman@23125  480 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"  huffman@44724  481  by (simp add: complex_eq_iff)  huffman@23125  482 huffman@23125  483 lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"  huffman@44724  484  by (simp add: complex_eq_iff)  huffman@23125  485 huffman@23125  486 lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"  huffman@44724  487  by (simp add: complex_eq_iff)  huffman@23125  488 huffman@23125  489 lemma complex_cnj_minus: "cnj (- x) = - cnj x"  huffman@44724  490  by (simp add: complex_eq_iff)  huffman@23125  491 huffman@23125  492 lemma complex_cnj_one [simp]: "cnj 1 = 1"  huffman@44724  493  by (simp add: complex_eq_iff)  huffman@23125  494 huffman@23125  495 lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"  huffman@44724  496  by (simp add: complex_eq_iff)  huffman@23125  497 huffman@23125  498 lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"  huffman@44724  499  by (simp add: complex_inverse_def)  paulson@14323  500 huffman@23125  501 lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"  huffman@44724  502  by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)  huffman@23125  503 huffman@23125  504 lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"  huffman@44724  505  by (induct n, simp_all add: complex_cnj_mult)  huffman@23125  506 huffman@23125  507 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"  huffman@44724  508  by (simp add: complex_eq_iff)  huffman@23125  509 huffman@23125  510 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"  huffman@44724  511  by (simp add: complex_eq_iff)  huffman@23125  512 huffman@47108  513 lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"  huffman@47108  514  by (simp add: complex_eq_iff)  huffman@47108  515 haftmann@54489  516 lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"  huffman@44724  517  by (simp add: complex_eq_iff)  huffman@23125  518 huffman@23125  519 lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"  huffman@44724  520  by (simp add: complex_eq_iff)  huffman@23125  521 huffman@23125  522 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"  huffman@44724  523  by (simp add: complex_norm_def)  paulson@14323  524 huffman@23125  525 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"  huffman@44724  526  by (simp add: complex_eq_iff)  huffman@23125  527 huffman@23125  528 lemma complex_cnj_i [simp]: "cnj ii = - ii"  huffman@44724  529  by (simp add: complex_eq_iff)  huffman@23125  530 huffman@23125  531 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"  huffman@44724  532  by (simp add: complex_eq_iff)  huffman@23125  533 huffman@23125  534 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"  huffman@44724  535  by (simp add: complex_eq_iff)  paulson@14354  536 wenzelm@53015  537 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"  huffman@44724  538  by (simp add: complex_eq_iff power2_eq_square)  huffman@23125  539 wenzelm@53015  540 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"  huffman@44724  541  by (simp add: norm_mult power2_eq_square)  huffman@23125  542 huffman@44827  543 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"  huffman@44827  544  by (simp add: cmod_def power2_eq_square)  huffman@44827  545 huffman@44827  546 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"  huffman@44827  547  by simp  huffman@44827  548 huffman@44290  549 lemma bounded_linear_cnj: "bounded_linear cnj"  huffman@44127  550  using complex_cnj_add complex_cnj_scaleR  huffman@44127  551  by (rule bounded_linear_intro [where K=1], simp)  paulson@14354  552 huffman@44290  553 lemmas tendsto_cnj [tendsto_intros] =  huffman@44290  554  bounded_linear.tendsto [OF bounded_linear_cnj]  huffman@44290  555 huffman@44290  556 lemmas isCont_cnj [simp] =  huffman@44290  557  bounded_linear.isCont [OF bounded_linear_cnj]  huffman@44290  558 paulson@14354  559 paulson@14323  560 subsection{*Finally! Polar Form for Complex Numbers*}  paulson@14323  561 huffman@44827  562 subsubsection {* $\cos \theta + i \sin \theta$ *}  huffman@20557  563 huffman@44715  564 definition cis :: "real \ complex" where  huffman@20557  565  "cis a = Complex (cos a) (sin a)"  huffman@20557  566 huffman@44827  567 lemma Re_cis [simp]: "Re (cis a) = cos a"  huffman@44827  568  by (simp add: cis_def)  huffman@44827  569 huffman@44827  570 lemma Im_cis [simp]: "Im (cis a) = sin a"  huffman@44827  571  by (simp add: cis_def)  huffman@44827  572 huffman@44827  573 lemma cis_zero [simp]: "cis 0 = 1"  huffman@44827  574  by (simp add: cis_def)  huffman@44827  575 huffman@44828  576 lemma norm_cis [simp]: "norm (cis a) = 1"  huffman@44828  577  by (simp add: cis_def)  huffman@44828  578 huffman@44828  579 lemma sgn_cis [simp]: "sgn (cis a) = cis a"  huffman@44828  580  by (simp add: sgn_div_norm)  huffman@44828  581 huffman@44828  582 lemma cis_neq_zero [simp]: "cis a \ 0"  huffman@44828  583  by (metis norm_cis norm_zero zero_neq_one)  huffman@44828  584 huffman@44827  585 lemma cis_mult: "cis a * cis b = cis (a + b)"  huffman@44827  586  by (simp add: cis_def cos_add sin_add)  huffman@44827  587 huffman@44827  588 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"  huffman@44827  589  by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)  huffman@44827  590 huffman@44827  591 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"  huffman@44827  592  by (simp add: cis_def)  huffman@44827  593 huffman@44827  594 lemma cis_divide: "cis a / cis b = cis (a - b)"  haftmann@54230  595  by (simp add: complex_divide_def cis_mult)  huffman@44827  596 huffman@44827  597 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"  huffman@44827  598  by (auto simp add: DeMoivre)  huffman@44827  599 huffman@44827  600 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"  huffman@44827  601  by (auto simp add: DeMoivre)  huffman@44827  602 huffman@44827  603 subsubsection {* $r(\cos \theta + i \sin \theta)$ *}  huffman@44715  604 huffman@44715  605 definition rcis :: "[real, real] \ complex" where  huffman@20557  606  "rcis r a = complex_of_real r * cis a"  huffman@20557  607 huffman@44827  608 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"  huffman@44828  609  by (simp add: rcis_def)  huffman@44827  610 huffman@44827  611 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"  huffman@44828  612  by (simp add: rcis_def)  huffman@44827  613 huffman@44827  614 lemma rcis_Ex: "\r a. z = rcis r a"  huffman@44828  615  by (simp add: complex_eq_iff polar_Ex)  huffman@44827  616 huffman@44827  617 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"  huffman@44828  618  by (simp add: rcis_def norm_mult)  huffman@44827  619 huffman@44827  620 lemma cis_rcis_eq: "cis a = rcis 1 a"  huffman@44827  621  by (simp add: rcis_def)  huffman@44827  622 huffman@44827  623 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"  huffman@44828  624  by (simp add: rcis_def cis_mult)  huffman@44827  625 huffman@44827  626 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"  huffman@44827  627  by (simp add: rcis_def)  huffman@44827  628 huffman@44827  629 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"  huffman@44827  630  by (simp add: rcis_def)  huffman@44827  631 huffman@44828  632 lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \ r = 0"  huffman@44828  633  by (simp add: rcis_def)  huffman@44828  634 huffman@44827  635 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"  huffman@44827  636  by (simp add: rcis_def power_mult_distrib DeMoivre)  huffman@44827  637 huffman@44827  638 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"  huffman@44827  639  by (simp add: divide_inverse rcis_def)  huffman@44827  640 huffman@44827  641 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"  huffman@44828  642  by (simp add: rcis_def cis_divide [symmetric])  huffman@44827  643 huffman@44827  644 subsubsection {* Complex exponential *}  huffman@44827  645 huffman@44291  646 abbreviation expi :: "complex \ complex"  huffman@44291  647  where "expi \ exp"  huffman@44291  648 huffman@44712  649 lemma cis_conv_exp: "cis b = exp (Complex 0 b)"  huffman@44291  650 proof (rule complex_eqI)  huffman@44291  651  { fix n have "Complex 0 b ^ n =  huffman@44291  652  real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"  huffman@44291  653  apply (induct n)  huffman@44291  654  apply (simp add: cos_coeff_def sin_coeff_def)  huffman@44291  655  apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)  huffman@44291  656  done } note * = this  huffman@44712  657  show "Re (cis b) = Re (exp (Complex 0 b))"  huffman@44291  658  unfolding exp_def cis_def cos_def  huffman@44291  659  by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],  huffman@44291  660  simp add: * mult_assoc [symmetric])  huffman@44712  661  show "Im (cis b) = Im (exp (Complex 0 b))"  huffman@44291  662  unfolding exp_def cis_def sin_def  huffman@44291  663  by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],  huffman@44291  664  simp add: * mult_assoc [symmetric])  huffman@44291  665 qed  huffman@44291  666 huffman@44291  667 lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"  huffman@44712  668  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp  huffman@20557  669 huffman@44828  670 lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"  huffman@44828  671  unfolding expi_def by simp  huffman@44828  672 huffman@44828  673 lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"  huffman@44828  674  unfolding expi_def by simp  huffman@44828  675 paulson@14374  676 lemma complex_expi_Ex: "\a r. z = complex_of_real r * expi a"  paulson@14373  677 apply (insert rcis_Ex [of z])  huffman@23125  678 apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])  paulson@14334  679 apply (rule_tac x = "ii * complex_of_real a" in exI, auto)  paulson@14323  680 done  paulson@14323  681 paulson@14387  682 lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"  huffman@44724  683  by (simp add: expi_def cis_def)  paulson@14387  684 huffman@44844  685 subsubsection {* Complex argument *}  huffman@44844  686 huffman@44844  687 definition arg :: "complex \ real" where  huffman@44844  688  "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \ -pi < a \ a \ pi))"  huffman@44844  689 huffman@44844  690 lemma arg_zero: "arg 0 = 0"  huffman@44844  691  by (simp add: arg_def)  huffman@44844  692 huffman@44844  693 lemma of_nat_less_of_int_iff: (* TODO: move *)  huffman@44844  694  "(of_nat n :: 'a::linordered_idom) < of_int x \ int n < x"  huffman@44844  695  by (metis of_int_of_nat_eq of_int_less_iff)  huffman@44844  696 huffman@47108  697 lemma real_of_nat_less_numeral_iff [simp]: (* TODO: move *)  huffman@47108  698  "real (n::nat) < numeral w \ n < numeral w"  huffman@47108  699  using of_nat_less_of_int_iff [of n "numeral w", where 'a=real]  huffman@47108  700  by (simp add: real_of_nat_def zless_nat_eq_int_zless [symmetric])  huffman@44844  701 huffman@44844  702 lemma arg_unique:  huffman@44844  703  assumes "sgn z = cis x" and "-pi < x" and "x \ pi"  huffman@44844  704  shows "arg z = x"  huffman@44844  705 proof -  huffman@44844  706  from assms have "z \ 0" by auto  huffman@44844  707  have "(SOME a. sgn z = cis a \ -pi < a \ a \ pi) = x"  huffman@44844  708  proof  huffman@44844  709  fix a def d \ "a - x"  huffman@44844  710  assume a: "sgn z = cis a \ - pi < a \ a \ pi"  huffman@44844  711  from a assms have "- (2*pi) < d \ d < 2*pi"  huffman@44844  712  unfolding d_def by simp  huffman@44844  713  moreover from a assms have "cos a = cos x" and "sin a = sin x"  huffman@44844  714  by (simp_all add: complex_eq_iff)  wenzelm@53374  715  hence cos: "cos d = 1" unfolding d_def cos_diff by simp  wenzelm@53374  716  moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)  huffman@44844  717  ultimately have "d = 0"  huffman@44844  718  unfolding sin_zero_iff even_mult_two_ex  wenzelm@53374  719  by (auto simp add: numeral_2_eq_2 less_Suc_eq)  huffman@44844  720  thus "a = x" unfolding d_def by simp  huffman@44844  721  qed (simp add: assms del: Re_sgn Im_sgn)  huffman@44844  722  with z \ 0 show "arg z = x"  huffman@44844  723  unfolding arg_def by simp  huffman@44844  724 qed  huffman@44844  725 huffman@44844  726 lemma arg_correct:  huffman@44844  727  assumes "z \ 0" shows "sgn z = cis (arg z) \ -pi < arg z \ arg z \ pi"  huffman@44844  728 proof (simp add: arg_def assms, rule someI_ex)  huffman@44844  729  obtain r a where z: "z = rcis r a" using rcis_Ex by fast  huffman@44844  730  with assms have "r \ 0" by auto  huffman@44844  731  def b \ "if 0 < r then a else a + pi"  huffman@44844  732  have b: "sgn z = cis b"  huffman@44844  733  unfolding z b_def rcis_def using r \ 0  huffman@44844  734  by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def)  huffman@44844  735  have cis_2pi_nat: "\n. cis (2 * pi * real_of_nat n) = 1"  webertj@49962  736  by (induct_tac n, simp_all add: distrib_left cis_mult [symmetric],  huffman@44844  737  simp add: cis_def)  huffman@44844  738  have cis_2pi_int: "\x. cis (2 * pi * real_of_int x) = 1"  huffman@44844  739  by (case_tac x rule: int_diff_cases,  huffman@44844  740  simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)  huffman@44844  741  def c \ "b - 2*pi * of_int \(b - pi) / (2*pi)\"  huffman@44844  742  have "sgn z = cis c"  huffman@44844  743  unfolding b c_def  huffman@44844  744  by (simp add: cis_divide [symmetric] cis_2pi_int)  huffman@44844  745  moreover have "- pi < c \ c \ pi"  huffman@44844  746  using ceiling_correct [of "(b - pi) / (2*pi)"]  huffman@44844  747  by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)  huffman@44844  748  ultimately show "\a. sgn z = cis a \ -pi < a \ a \ pi" by fast  huffman@44844  749 qed  huffman@44844  750 huffman@44844  751 lemma arg_bounded: "- pi < arg z \ arg z \ pi"  huffman@44844  752  by (cases "z = 0", simp_all add: arg_zero arg_correct)  huffman@44844  753 huffman@44844  754 lemma cis_arg: "z \ 0 \ cis (arg z) = sgn z"  huffman@44844  755  by (simp add: arg_correct)  huffman@44844  756 huffman@44844  757 lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"  huffman@44844  758  by (cases "z = 0", simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)  huffman@44844  759 huffman@44844  760 lemma cos_arg_i_mult_zero [simp]:  huffman@44844  761  "y \ 0 ==> cos (arg(Complex 0 y)) = 0"  huffman@44844  762  using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff)  huffman@44844  763 huffman@44065  764 text {* Legacy theorem names *}  huffman@44065  765 huffman@44065  766 lemmas expand_complex_eq = complex_eq_iff  huffman@44065  767 lemmas complex_Re_Im_cancel_iff = complex_eq_iff  huffman@44065  768 lemmas complex_equality = complex_eqI  huffman@44065  769 paulson@13957  770 end