src/HOL/Complex.thy
 author huffman Fri Mar 30 12:32:35 2012 +0200 (2012-03-30) changeset 47220 52426c62b5d0 parent 47108 2a1953f0d20d child 49962 a8cc904a6820 permissions -rw-r--r--
replace lemmas eval_nat_numeral with a simpler reformulation
 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 huffman@23125  154 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"  huffman@44724  155  by (induct n) simp_all  huffman@20556  156 huffman@23125  157 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"  huffman@44724  158  by (induct n) simp_all  huffman@23125  159 huffman@23125  160 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"  huffman@44724  161  by (cases z rule: int_diff_cases) simp  huffman@23125  162 huffman@23125  163 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"  huffman@44724  164  by (cases z rule: int_diff_cases) simp  huffman@23125  165 huffman@47108  166 lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"  huffman@47108  167  using complex_Re_of_int [of "numeral v"] by simp  huffman@47108  168 huffman@47108  169 lemma complex_Re_neg_numeral [simp]: "Re (neg_numeral v) = neg_numeral v"  huffman@47108  170  using complex_Re_of_int [of "neg_numeral v"] by simp  huffman@47108  171 huffman@47108  172 lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"  huffman@47108  173  using complex_Im_of_int [of "numeral v"] by simp  huffman@20556  174 huffman@47108  175 lemma complex_Im_neg_numeral [simp]: "Im (neg_numeral v) = 0"  huffman@47108  176  using complex_Im_of_int [of "neg_numeral v"] by simp  huffman@23125  177 huffman@47108  178 lemma Complex_eq_numeral [simp]:  huffman@47108  179  "(Complex a b = numeral w) = (a = numeral w \ b = 0)"  huffman@47108  180  by (simp add: complex_eq_iff)  huffman@47108  181 huffman@47108  182 lemma Complex_eq_neg_numeral [simp]:  huffman@47108  183  "(Complex a b = neg_numeral w) = (a = neg_numeral w \ b = 0)"  huffman@44724  184  by (simp add: complex_eq_iff)  huffman@23125  185 huffman@23125  186 huffman@23125  187 subsection {* Scalar Multiplication *}  huffman@20556  188 haftmann@25712  189 instantiation complex :: real_field  haftmann@25571  190 begin  haftmann@25571  191 huffman@44724  192 definition complex_scaleR_def:  huffman@44724  193  "scaleR r x = Complex (r * Re x) (r * Im x)"  haftmann@25571  194 huffman@23125  195 lemma complex_scaleR [simp]:  huffman@23125  196  "scaleR r (Complex a b) = Complex (r * a) (r * b)"  haftmann@25712  197  unfolding complex_scaleR_def by simp  huffman@23125  198 huffman@23125  199 lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"  haftmann@25712  200  unfolding complex_scaleR_def by simp  huffman@23125  201 huffman@23125  202 lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"  haftmann@25712  203  unfolding complex_scaleR_def by simp  huffman@22972  204 haftmann@25712  205 instance  huffman@20556  206 proof  huffman@23125  207  fix a b :: real and x y :: complex  huffman@23125  208  show "scaleR a (x + y) = scaleR a x + scaleR a y"  huffman@44065  209  by (simp add: complex_eq_iff right_distrib)  huffman@23125  210  show "scaleR (a + b) x = scaleR a x + scaleR b x"  huffman@44065  211  by (simp add: complex_eq_iff left_distrib)  huffman@23125  212  show "scaleR a (scaleR b x) = scaleR (a * b) x"  huffman@44065  213  by (simp add: complex_eq_iff mult_assoc)  huffman@23125  214  show "scaleR 1 x = x"  huffman@44065  215  by (simp add: complex_eq_iff)  huffman@23125  216  show "scaleR a x * y = scaleR a (x * y)"  huffman@44065  217  by (simp add: complex_eq_iff algebra_simps)  huffman@23125  218  show "x * scaleR a y = scaleR a (x * y)"  huffman@44065  219  by (simp add: complex_eq_iff algebra_simps)  huffman@20556  220 qed  huffman@20556  221 haftmann@25712  222 end  haftmann@25712  223 huffman@20556  224 huffman@23125  225 subsection{* Properties of Embedding from Reals *}  paulson@14323  226 huffman@44724  227 abbreviation complex_of_real :: "real \ complex"  huffman@44724  228  where "complex_of_real \ of_real"  huffman@20557  229 huffman@20557  230 lemma complex_of_real_def: "complex_of_real r = Complex r 0"  huffman@44724  231  by (simp add: of_real_def complex_scaleR_def)  huffman@20557  232 huffman@20557  233 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"  huffman@44724  234  by (simp add: complex_of_real_def)  huffman@20557  235 huffman@20557  236 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"  huffman@44724  237  by (simp add: complex_of_real_def)  huffman@20557  238 paulson@14377  239 lemma Complex_add_complex_of_real [simp]:  huffman@44724  240  shows "Complex x y + complex_of_real r = Complex (x+r) y"  huffman@44724  241  by (simp add: complex_of_real_def)  paulson@14377  242 paulson@14377  243 lemma complex_of_real_add_Complex [simp]:  huffman@44724  244  shows "complex_of_real r + Complex x y = Complex (r+x) y"  huffman@44724  245  by (simp add: complex_of_real_def)  paulson@14377  246 paulson@14377  247 lemma Complex_mult_complex_of_real:  huffman@44724  248  shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"  huffman@44724  249  by (simp add: complex_of_real_def)  paulson@14377  250 paulson@14377  251 lemma complex_of_real_mult_Complex:  huffman@44724  252  shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"  huffman@44724  253  by (simp add: complex_of_real_def)  huffman@20557  254 huffman@44841  255 lemma complex_eq_cancel_iff2 [simp]:  huffman@44841  256  shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"  huffman@44841  257  by (simp add: complex_of_real_def)  huffman@44841  258 huffman@44827  259 lemma complex_split_polar:  huffman@44827  260  "\r a. z = complex_of_real r * (Complex (cos a) (sin a))"  huffman@44827  261  by (simp add: complex_eq_iff polar_Ex)  huffman@44827  262 paulson@14377  263 huffman@23125  264 subsection {* Vector Norm *}  paulson@14323  265 haftmann@25712  266 instantiation complex :: real_normed_field  haftmann@25571  267 begin  haftmann@25571  268 huffman@31413  269 definition complex_norm_def:  huffman@31413  270  "norm z = sqrt ((Re z)\ + (Im z)\)"  haftmann@25571  271 huffman@44724  272 abbreviation cmod :: "complex \ real"  huffman@44724  273  where "cmod \ norm"  haftmann@25571  274 huffman@31413  275 definition complex_sgn_def:  huffman@31413  276  "sgn x = x /\<^sub>R cmod x"  haftmann@25571  277 huffman@31413  278 definition dist_complex_def:  huffman@31413  279  "dist x y = cmod (x - y)"  huffman@31413  280 haftmann@37767  281 definition open_complex_def:  huffman@31492  282  "open (S :: complex set) \ (\x\S. \e>0. \y. dist y x < e \ y \ S)"  huffman@31292  283 huffman@20557  284 lemmas cmod_def = complex_norm_def  huffman@20557  285 huffman@23125  286 lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\ + y\)"  haftmann@25712  287  by (simp add: complex_norm_def)  huffman@22852  288 huffman@31413  289 instance proof  huffman@31492  290  fix r :: real and x y :: complex and S :: "complex set"  huffman@23125  291  show "0 \ norm x"  huffman@22861  292  by (induct x) simp  huffman@23125  293  show "(norm x = 0) = (x = 0)"  huffman@22861  294  by (induct x) simp  huffman@23125  295  show "norm (x + y) \ norm x + norm y"  huffman@23125  296  by (induct x, induct y)  huffman@23125  297  (simp add: real_sqrt_sum_squares_triangle_ineq)  huffman@23125  298  show "norm (scaleR r x) = \r\ * norm x"  huffman@23125  299  by (induct x)  huffman@23125  300  (simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult)  huffman@23125  301  show "norm (x * y) = norm x * norm y"  huffman@23125  302  by (induct x, induct y)  nipkow@29667  303  (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)  huffman@31292  304  show "sgn x = x /\<^sub>R cmod x"  huffman@31292  305  by (rule complex_sgn_def)  huffman@31292  306  show "dist x y = cmod (x - y)"  huffman@31292  307  by (rule dist_complex_def)  huffman@31492  308  show "open S \ (\x\S. \e>0. \y. dist y x < e \ y \ S)"  huffman@31492  309  by (rule open_complex_def)  huffman@24520  310 qed  huffman@20557  311 haftmann@25712  312 end  haftmann@25712  313 huffman@44761  314 lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1"  huffman@44724  315  by simp  paulson@14323  316 huffman@44761  317 lemma cmod_complex_polar:  huffman@44724  318  "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"  huffman@44724  319  by (simp add: norm_mult)  huffman@22861  320 huffman@22861  321 lemma complex_Re_le_cmod: "Re x \ cmod x"  huffman@44724  322  unfolding complex_norm_def  huffman@44724  323  by (rule real_sqrt_sum_squares_ge1)  huffman@22861  324 huffman@44761  325 lemma complex_mod_minus_le_complex_mod: "- cmod x \ cmod x"  huffman@44724  326  by (rule order_trans [OF _ norm_ge_zero], simp)  huffman@22861  327 huffman@44761  328 lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \ cmod a"  huffman@44724  329  by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)  paulson@14323  330 chaieb@26117  331 lemma abs_Re_le_cmod: "\Re x\ \ cmod x"  huffman@44724  332  by (cases x) simp  chaieb@26117  333 chaieb@26117  334 lemma abs_Im_le_cmod: "\Im x\ \ cmod x"  huffman@44724  335  by (cases x) simp  huffman@44724  336 huffman@44843  337 text {* Properties of complex signum. *}  huffman@44843  338 huffman@44843  339 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"  huffman@44843  340  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)  huffman@44843  341 huffman@44843  342 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"  huffman@44843  343  by (simp add: complex_sgn_def divide_inverse)  huffman@44843  344 huffman@44843  345 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"  huffman@44843  346  by (simp add: complex_sgn_def divide_inverse)  huffman@44843  347 paulson@14354  348 huffman@23123  349 subsection {* Completeness of the Complexes *}  huffman@23123  350 huffman@44290  351 lemma bounded_linear_Re: "bounded_linear Re"  huffman@44290  352  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)  huffman@44290  353 huffman@44290  354 lemma bounded_linear_Im: "bounded_linear Im"  huffman@44127  355  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)  huffman@23123  356 huffman@44290  357 lemmas tendsto_Re [tendsto_intros] =  huffman@44290  358  bounded_linear.tendsto [OF bounded_linear_Re]  huffman@44290  359 huffman@44290  360 lemmas tendsto_Im [tendsto_intros] =  huffman@44290  361  bounded_linear.tendsto [OF bounded_linear_Im]  huffman@44290  362 huffman@44290  363 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]  huffman@44290  364 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]  huffman@44290  365 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]  huffman@44290  366 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]  huffman@23123  367 huffman@36825  368 lemma tendsto_Complex [tendsto_intros]:  huffman@44724  369  assumes "(f ---> a) F" and "(g ---> b) F"  huffman@44724  370  shows "((\x. Complex (f x) (g x)) ---> Complex a b) F"  huffman@36825  371 proof (rule tendstoI)  huffman@36825  372  fix r :: real assume "0 < r"  huffman@36825  373  hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)  huffman@44724  374  have "eventually (\x. dist (f x) a < r / sqrt 2) F"  huffman@44724  375  using (f ---> a) F and 0 < r / sqrt 2 by (rule tendstoD)  huffman@36825  376  moreover  huffman@44724  377  have "eventually (\x. dist (g x) b < r / sqrt 2) F"  huffman@44724  378  using (g ---> b) F and 0 < r / sqrt 2 by (rule tendstoD)  huffman@36825  379  ultimately  huffman@44724  380  show "eventually (\x. dist (Complex (f x) (g x)) (Complex a b) < r) F"  huffman@36825  381  by (rule eventually_elim2)  huffman@36825  382  (simp add: dist_norm real_sqrt_sum_squares_less)  huffman@36825  383 qed  huffman@36825  384 huffman@23123  385 instance complex :: banach  huffman@23123  386 proof  huffman@23123  387  fix X :: "nat \ complex"  huffman@23123  388  assume X: "Cauchy X"  huffman@44290  389  from Cauchy_Re [OF X] have 1: "(\n. Re (X n)) ----> lim (\n. Re (X n))"  huffman@23123  390  by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)  huffman@44290  391  from Cauchy_Im [OF X] have 2: "(\n. Im (X n)) ----> lim (\n. Im (X n))"  huffman@23123  392  by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)  huffman@23123  393  have "X ----> Complex (lim (\n. Re (X n))) (lim (\n. Im (X n)))"  huffman@44748  394  using tendsto_Complex [OF 1 2] by simp  huffman@23123  395  thus "convergent X"  huffman@23123  396  by (rule convergentI)  huffman@23123  397 qed  huffman@23123  398 huffman@23123  399 huffman@44827  400 subsection {* The Complex Number $i$ *}  huffman@23125  401 huffman@44724  402 definition "ii" :: complex ("\")  huffman@44724  403  where i_def: "ii \ Complex 0 1"  huffman@23125  404 huffman@23125  405 lemma complex_Re_i [simp]: "Re ii = 0"  huffman@44724  406  by (simp add: i_def)  paulson@14354  407 huffman@23125  408 lemma complex_Im_i [simp]: "Im ii = 1"  huffman@44724  409  by (simp add: i_def)  huffman@23125  410 huffman@23125  411 lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \ y = 1)"  huffman@44724  412  by (simp add: i_def)  huffman@23125  413 huffman@44902  414 lemma norm_ii [simp]: "norm ii = 1"  huffman@44902  415  by (simp add: i_def)  huffman@44902  416 huffman@23125  417 lemma complex_i_not_zero [simp]: "ii \ 0"  huffman@44724  418  by (simp add: complex_eq_iff)  huffman@23125  419 huffman@23125  420 lemma complex_i_not_one [simp]: "ii \ 1"  huffman@44724  421  by (simp add: complex_eq_iff)  huffman@23124  422 huffman@47108  423 lemma complex_i_not_numeral [simp]: "ii \ numeral w"  huffman@47108  424  by (simp add: complex_eq_iff)  huffman@47108  425 huffman@47108  426 lemma complex_i_not_neg_numeral [simp]: "ii \ neg_numeral w"  huffman@44724  427  by (simp add: complex_eq_iff)  huffman@23125  428 huffman@23125  429 lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"  huffman@44724  430  by (simp add: complex_eq_iff)  huffman@23125  431 huffman@23125  432 lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"  huffman@44724  433  by (simp add: complex_eq_iff)  huffman@23125  434 huffman@23125  435 lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"  huffman@44724  436  by (simp add: i_def complex_of_real_def)  huffman@23125  437 huffman@23125  438 lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"  huffman@44724  439  by (simp add: i_def complex_of_real_def)  huffman@23125  440 huffman@23125  441 lemma i_squared [simp]: "ii * ii = -1"  huffman@44724  442  by (simp add: i_def)  huffman@23125  443 huffman@23125  444 lemma power2_i [simp]: "ii\ = -1"  huffman@44724  445  by (simp add: power2_eq_square)  huffman@23125  446 huffman@23125  447 lemma inverse_i [simp]: "inverse ii = - ii"  huffman@44724  448  by (rule inverse_unique, simp)  paulson@14354  449 huffman@44827  450 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"  huffman@44827  451  by (simp add: mult_assoc [symmetric])  huffman@44827  452 paulson@14354  453 huffman@23125  454 subsection {* Complex Conjugation *}  huffman@23125  455 huffman@44724  456 definition cnj :: "complex \ complex" where  huffman@23125  457  "cnj z = Complex (Re z) (- Im z)"  huffman@23125  458 huffman@23125  459 lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"  huffman@44724  460  by (simp add: cnj_def)  huffman@23125  461 huffman@23125  462 lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"  huffman@44724  463  by (simp add: cnj_def)  huffman@23125  464 huffman@23125  465 lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"  huffman@44724  466  by (simp add: cnj_def)  huffman@23125  467 huffman@23125  468 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"  huffman@44724  469  by (simp add: complex_eq_iff)  huffman@23125  470 huffman@23125  471 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"  huffman@44724  472  by (simp add: cnj_def)  huffman@23125  473 huffman@23125  474 lemma complex_cnj_zero [simp]: "cnj 0 = 0"  huffman@44724  475  by (simp add: complex_eq_iff)  huffman@23125  476 huffman@23125  477 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"  huffman@44724  478  by (simp add: complex_eq_iff)  huffman@23125  479 huffman@23125  480 lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"  huffman@44724  481  by (simp add: complex_eq_iff)  huffman@23125  482 huffman@23125  483 lemma complex_cnj_diff: "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_minus: "cnj (- x) = - cnj x"  huffman@44724  487  by (simp add: complex_eq_iff)  huffman@23125  488 huffman@23125  489 lemma complex_cnj_one [simp]: "cnj 1 = 1"  huffman@44724  490  by (simp add: complex_eq_iff)  huffman@23125  491 huffman@23125  492 lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"  huffman@44724  493  by (simp add: complex_eq_iff)  huffman@23125  494 huffman@23125  495 lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"  huffman@44724  496  by (simp add: complex_inverse_def)  paulson@14323  497 huffman@23125  498 lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"  huffman@44724  499  by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)  huffman@23125  500 huffman@23125  501 lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"  huffman@44724  502  by (induct n, simp_all add: complex_cnj_mult)  huffman@23125  503 huffman@23125  504 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"  huffman@44724  505  by (simp add: complex_eq_iff)  huffman@23125  506 huffman@23125  507 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"  huffman@44724  508  by (simp add: complex_eq_iff)  huffman@23125  509 huffman@47108  510 lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"  huffman@47108  511  by (simp add: complex_eq_iff)  huffman@47108  512 huffman@47108  513 lemma complex_cnj_neg_numeral [simp]: "cnj (neg_numeral w) = neg_numeral w"  huffman@44724  514  by (simp add: complex_eq_iff)  huffman@23125  515 huffman@23125  516 lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"  huffman@44724  517  by (simp add: complex_eq_iff)  huffman@23125  518 huffman@23125  519 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"  huffman@44724  520  by (simp add: complex_norm_def)  paulson@14323  521 huffman@23125  522 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"  huffman@44724  523  by (simp add: complex_eq_iff)  huffman@23125  524 huffman@23125  525 lemma complex_cnj_i [simp]: "cnj ii = - ii"  huffman@44724  526  by (simp add: complex_eq_iff)  huffman@23125  527 huffman@23125  528 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"  huffman@44724  529  by (simp add: complex_eq_iff)  huffman@23125  530 huffman@23125  531 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"  huffman@44724  532  by (simp add: complex_eq_iff)  paulson@14354  533 huffman@23125  534 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\ + (Im z)\)"  huffman@44724  535  by (simp add: complex_eq_iff power2_eq_square)  huffman@23125  536 huffman@23125  537 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\"  huffman@44724  538  by (simp add: norm_mult power2_eq_square)  huffman@23125  539 huffman@44827  540 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"  huffman@44827  541  by (simp add: cmod_def power2_eq_square)  huffman@44827  542 huffman@44827  543 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"  huffman@44827  544  by simp  huffman@44827  545 huffman@44290  546 lemma bounded_linear_cnj: "bounded_linear cnj"  huffman@44127  547  using complex_cnj_add complex_cnj_scaleR  huffman@44127  548  by (rule bounded_linear_intro [where K=1], simp)  paulson@14354  549 huffman@44290  550 lemmas tendsto_cnj [tendsto_intros] =  huffman@44290  551  bounded_linear.tendsto [OF bounded_linear_cnj]  huffman@44290  552 huffman@44290  553 lemmas isCont_cnj [simp] =  huffman@44290  554  bounded_linear.isCont [OF bounded_linear_cnj]  huffman@44290  555 paulson@14354  556 paulson@14323  557 subsection{*Finally! Polar Form for Complex Numbers*}  paulson@14323  558 huffman@44827  559 subsubsection {* $\cos \theta + i \sin \theta$ *}  huffman@20557  560 huffman@44715  561 definition cis :: "real \ complex" where  huffman@20557  562  "cis a = Complex (cos a) (sin a)"  huffman@20557  563 huffman@44827  564 lemma Re_cis [simp]: "Re (cis a) = cos a"  huffman@44827  565  by (simp add: cis_def)  huffman@44827  566 huffman@44827  567 lemma Im_cis [simp]: "Im (cis a) = sin a"  huffman@44827  568  by (simp add: cis_def)  huffman@44827  569 huffman@44827  570 lemma cis_zero [simp]: "cis 0 = 1"  huffman@44827  571  by (simp add: cis_def)  huffman@44827  572 huffman@44828  573 lemma norm_cis [simp]: "norm (cis a) = 1"  huffman@44828  574  by (simp add: cis_def)  huffman@44828  575 huffman@44828  576 lemma sgn_cis [simp]: "sgn (cis a) = cis a"  huffman@44828  577  by (simp add: sgn_div_norm)  huffman@44828  578 huffman@44828  579 lemma cis_neq_zero [simp]: "cis a \ 0"  huffman@44828  580  by (metis norm_cis norm_zero zero_neq_one)  huffman@44828  581 huffman@44827  582 lemma cis_mult: "cis a * cis b = cis (a + b)"  huffman@44827  583  by (simp add: cis_def cos_add sin_add)  huffman@44827  584 huffman@44827  585 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"  huffman@44827  586  by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)  huffman@44827  587 huffman@44827  588 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"  huffman@44827  589  by (simp add: cis_def)  huffman@44827  590 huffman@44827  591 lemma cis_divide: "cis a / cis b = cis (a - b)"  huffman@44827  592  by (simp add: complex_divide_def cis_mult diff_minus)  huffman@44827  593 huffman@44827  594 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"  huffman@44827  595  by (auto simp add: DeMoivre)  huffman@44827  596 huffman@44827  597 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"  huffman@44827  598  by (auto simp add: DeMoivre)  huffman@44827  599 huffman@44827  600 subsubsection {* $r(\cos \theta + i \sin \theta)$ *}  huffman@44715  601 huffman@44715  602 definition rcis :: "[real, real] \ complex" where  huffman@20557  603  "rcis r a = complex_of_real r * cis a"  huffman@20557  604 huffman@44827  605 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"  huffman@44828  606  by (simp add: rcis_def)  huffman@44827  607 huffman@44827  608 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"  huffman@44828  609  by (simp add: rcis_def)  huffman@44827  610 huffman@44827  611 lemma rcis_Ex: "\r a. z = rcis r a"  huffman@44828  612  by (simp add: complex_eq_iff polar_Ex)  huffman@44827  613 huffman@44827  614 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"  huffman@44828  615  by (simp add: rcis_def norm_mult)  huffman@44827  616 huffman@44827  617 lemma cis_rcis_eq: "cis a = rcis 1 a"  huffman@44827  618  by (simp add: rcis_def)  huffman@44827  619 huffman@44827  620 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"  huffman@44828  621  by (simp add: rcis_def cis_mult)  huffman@44827  622 huffman@44827  623 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"  huffman@44827  624  by (simp add: rcis_def)  huffman@44827  625 huffman@44827  626 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"  huffman@44827  627  by (simp add: rcis_def)  huffman@44827  628 huffman@44828  629 lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \ r = 0"  huffman@44828  630  by (simp add: rcis_def)  huffman@44828  631 huffman@44827  632 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"  huffman@44827  633  by (simp add: rcis_def power_mult_distrib DeMoivre)  huffman@44827  634 huffman@44827  635 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"  huffman@44827  636  by (simp add: divide_inverse rcis_def)  huffman@44827  637 huffman@44827  638 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"  huffman@44828  639  by (simp add: rcis_def cis_divide [symmetric])  huffman@44827  640 huffman@44827  641 subsubsection {* Complex exponential *}  huffman@44827  642 huffman@44291  643 abbreviation expi :: "complex \ complex"  huffman@44291  644  where "expi \ exp"  huffman@44291  645 huffman@44712  646 lemma cis_conv_exp: "cis b = exp (Complex 0 b)"  huffman@44291  647 proof (rule complex_eqI)  huffman@44291  648  { fix n have "Complex 0 b ^ n =  huffman@44291  649  real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"  huffman@44291  650  apply (induct n)  huffman@44291  651  apply (simp add: cos_coeff_def sin_coeff_def)  huffman@44291  652  apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)  huffman@44291  653  done } note * = this  huffman@44712  654  show "Re (cis b) = Re (exp (Complex 0 b))"  huffman@44291  655  unfolding exp_def cis_def cos_def  huffman@44291  656  by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],  huffman@44291  657  simp add: * mult_assoc [symmetric])  huffman@44712  658  show "Im (cis b) = Im (exp (Complex 0 b))"  huffman@44291  659  unfolding exp_def cis_def sin_def  huffman@44291  660  by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],  huffman@44291  661  simp add: * mult_assoc [symmetric])  huffman@44291  662 qed  huffman@44291  663 huffman@44291  664 lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"  huffman@44712  665  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp  huffman@20557  666 huffman@44828  667 lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"  huffman@44828  668  unfolding expi_def by simp  huffman@44828  669 huffman@44828  670 lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"  huffman@44828  671  unfolding expi_def by simp  huffman@44828  672 paulson@14374  673 lemma complex_expi_Ex: "\a r. z = complex_of_real r * expi a"  paulson@14373  674 apply (insert rcis_Ex [of z])  huffman@23125  675 apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])  paulson@14334  676 apply (rule_tac x = "ii * complex_of_real a" in exI, auto)  paulson@14323  677 done  paulson@14323  678 paulson@14387  679 lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"  huffman@44724  680  by (simp add: expi_def cis_def)  paulson@14387  681 huffman@44844  682 subsubsection {* Complex argument *}  huffman@44844  683 huffman@44844  684 definition arg :: "complex \ real" where  huffman@44844  685  "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \ -pi < a \ a \ pi))"  huffman@44844  686 huffman@44844  687 lemma arg_zero: "arg 0 = 0"  huffman@44844  688  by (simp add: arg_def)  huffman@44844  689 huffman@44844  690 lemma of_nat_less_of_int_iff: (* TODO: move *)  huffman@44844  691  "(of_nat n :: 'a::linordered_idom) < of_int x \ int n < x"  huffman@44844  692  by (metis of_int_of_nat_eq of_int_less_iff)  huffman@44844  693 huffman@47108  694 lemma real_of_nat_less_numeral_iff [simp]: (* TODO: move *)  huffman@47108  695  "real (n::nat) < numeral w \ n < numeral w"  huffman@47108  696  using of_nat_less_of_int_iff [of n "numeral w", where 'a=real]  huffman@47108  697  by (simp add: real_of_nat_def zless_nat_eq_int_zless [symmetric])  huffman@44844  698 huffman@44844  699 lemma arg_unique:  huffman@44844  700  assumes "sgn z = cis x" and "-pi < x" and "x \ pi"  huffman@44844  701  shows "arg z = x"  huffman@44844  702 proof -  huffman@44844  703  from assms have "z \ 0" by auto  huffman@44844  704  have "(SOME a. sgn z = cis a \ -pi < a \ a \ pi) = x"  huffman@44844  705  proof  huffman@44844  706  fix a def d \ "a - x"  huffman@44844  707  assume a: "sgn z = cis a \ - pi < a \ a \ pi"  huffman@44844  708  from a assms have "- (2*pi) < d \ d < 2*pi"  huffman@44844  709  unfolding d_def by simp  huffman@44844  710  moreover from a assms have "cos a = cos x" and "sin a = sin x"  huffman@44844  711  by (simp_all add: complex_eq_iff)  huffman@44844  712  hence "cos d = 1" unfolding d_def cos_diff by simp  huffman@44844  713  moreover hence "sin d = 0" by (rule cos_one_sin_zero)  huffman@44844  714  ultimately have "d = 0"  huffman@44844  715  unfolding sin_zero_iff even_mult_two_ex  huffman@44844  716  by (safe, auto simp add: numeral_2_eq_2 less_Suc_eq)  huffman@44844  717  thus "a = x" unfolding d_def by simp  huffman@44844  718  qed (simp add: assms del: Re_sgn Im_sgn)  huffman@44844  719  with z \ 0 show "arg z = x"  huffman@44844  720  unfolding arg_def by simp  huffman@44844  721 qed  huffman@44844  722 huffman@44844  723 lemma arg_correct:  huffman@44844  724  assumes "z \ 0" shows "sgn z = cis (arg z) \ -pi < arg z \ arg z \ pi"  huffman@44844  725 proof (simp add: arg_def assms, rule someI_ex)  huffman@44844  726  obtain r a where z: "z = rcis r a" using rcis_Ex by fast  huffman@44844  727  with assms have "r \ 0" by auto  huffman@44844  728  def b \ "if 0 < r then a else a + pi"  huffman@44844  729  have b: "sgn z = cis b"  huffman@44844  730  unfolding z b_def rcis_def using r \ 0  huffman@44844  731  by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def)  huffman@44844  732  have cis_2pi_nat: "\n. cis (2 * pi * real_of_nat n) = 1"  huffman@44844  733  by (induct_tac n, simp_all add: right_distrib cis_mult [symmetric],  huffman@44844  734  simp add: cis_def)  huffman@44844  735  have cis_2pi_int: "\x. cis (2 * pi * real_of_int x) = 1"  huffman@44844  736  by (case_tac x rule: int_diff_cases,  huffman@44844  737  simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)  huffman@44844  738  def c \ "b - 2*pi * of_int \(b - pi) / (2*pi)\"  huffman@44844  739  have "sgn z = cis c"  huffman@44844  740  unfolding b c_def  huffman@44844  741  by (simp add: cis_divide [symmetric] cis_2pi_int)  huffman@44844  742  moreover have "- pi < c \ c \ pi"  huffman@44844  743  using ceiling_correct [of "(b - pi) / (2*pi)"]  huffman@44844  744  by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)  huffman@44844  745  ultimately show "\a. sgn z = cis a \ -pi < a \ a \ pi" by fast  huffman@44844  746 qed  huffman@44844  747 huffman@44844  748 lemma arg_bounded: "- pi < arg z \ arg z \ pi"  huffman@44844  749  by (cases "z = 0", simp_all add: arg_zero arg_correct)  huffman@44844  750 huffman@44844  751 lemma cis_arg: "z \ 0 \ cis (arg z) = sgn z"  huffman@44844  752  by (simp add: arg_correct)  huffman@44844  753 huffman@44844  754 lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"  huffman@44844  755  by (cases "z = 0", simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)  huffman@44844  756 huffman@44844  757 lemma cos_arg_i_mult_zero [simp]:  huffman@44844  758  "y \ 0 ==> cos (arg(Complex 0 y)) = 0"  huffman@44844  759  using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff)  huffman@44844  760 huffman@44065  761 text {* Legacy theorem names *}  huffman@44065  762 huffman@44065  763 lemmas expand_complex_eq = complex_eq_iff  huffman@44065  764 lemmas complex_Re_Im_cancel_iff = complex_eq_iff  huffman@44065  765 lemmas complex_equality = complex_eqI  huffman@44065  766 paulson@13957  767 end