(* Title: HOL/Complex.thy Author: Jacques D. Fleuriot Copyright: 2001 University of Edinburgh Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 *) header {* Complex Numbers: Rectangular and Polar Representations *} theory Complex imports Transcendental begin datatype complex = Complex real real primrec Re :: "complex \ real" where Re: "Re (Complex x y) = x" primrec Im :: "complex \ real" where Im: "Im (Complex x y) = y" lemma complex_surj [simp]: "Complex (Re z) (Im z) = z" by (induct z) simp lemma complex_eqI [intro?]: "\Re x = Re y; Im x = Im y\ \ x = y" by (induct x, induct y) simp lemma complex_eq_iff: "x = y \ Re x = Re y \ Im x = Im y" by (induct x, induct y) simp subsection {* Addition and Subtraction *} instantiation complex :: ab_group_add begin definition complex_zero_def: "0 = Complex 0 0" definition complex_add_def: "x + y = Complex (Re x + Re y) (Im x + Im y)" definition complex_minus_def: "- x = Complex (- Re x) (- Im x)" definition complex_diff_def: "x - (y\complex) = x + - y" lemma Complex_eq_0 [simp]: "Complex a b = 0 \ a = 0 \ b = 0" by (simp add: complex_zero_def) lemma complex_Re_zero [simp]: "Re 0 = 0" by (simp add: complex_zero_def) lemma complex_Im_zero [simp]: "Im 0 = 0" by (simp add: complex_zero_def) lemma complex_add [simp]: "Complex a b + Complex c d = Complex (a + c) (b + d)" by (simp add: complex_add_def) lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y" by (simp add: complex_add_def) lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y" by (simp add: complex_add_def) lemma complex_minus [simp]: "- (Complex a b) = Complex (- a) (- b)" by (simp add: complex_minus_def) lemma complex_Re_minus [simp]: "Re (- x) = - Re x" by (simp add: complex_minus_def) lemma complex_Im_minus [simp]: "Im (- x) = - Im x" by (simp add: complex_minus_def) lemma complex_diff [simp]: "Complex a b - Complex c d = Complex (a - c) (b - d)" by (simp add: complex_diff_def) lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y" by (simp add: complex_diff_def) lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y" by (simp add: complex_diff_def) instance by intro_classes (simp_all add: complex_add_def complex_diff_def) end subsection {* Multiplication and Division *} instantiation complex :: field_inverse_zero begin definition complex_one_def: "1 = Complex 1 0" definition complex_mult_def: "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)" definition complex_inverse_def: "inverse x = Complex (Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)) (- Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2))" definition complex_divide_def: "x / (y\complex) = x * inverse y" lemma Complex_eq_1 [simp]: "Complex a b = 1 \ a = 1 \ b = 0" by (simp add: complex_one_def) lemma Complex_eq_neg_1 [simp]: "Complex a b = - 1 \ a = - 1 \ b = 0" by (simp add: complex_one_def) lemma complex_Re_one [simp]: "Re 1 = 1" by (simp add: complex_one_def) lemma complex_Im_one [simp]: "Im 1 = 0" by (simp add: complex_one_def) lemma complex_mult [simp]: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)" by (simp add: complex_mult_def) lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y" by (simp add: complex_mult_def) lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y" by (simp add: complex_mult_def) lemma complex_inverse [simp]: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))" by (simp add: complex_inverse_def) lemma complex_Re_inverse: "Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)" by (simp add: complex_inverse_def) lemma complex_Im_inverse: "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)" by (simp add: complex_inverse_def) instance by intro_classes (simp_all add: complex_mult_def distrib_left distrib_right right_diff_distrib left_diff_distrib complex_inverse_def complex_divide_def power2_eq_square add_divide_distrib [symmetric] complex_eq_iff) end subsection {* Numerals and Arithmetic *} lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n" by (induct n) simp_all lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0" by (induct n) simp_all lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z" by (cases z rule: int_diff_cases) simp lemma complex_Im_of_int [simp]: "Im (of_int z) = 0" by (cases z rule: int_diff_cases) simp lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v" using complex_Re_of_int [of "numeral v"] by simp lemma complex_Re_neg_numeral [simp]: "Re (- numeral v) = - numeral v" using complex_Re_of_int [of "- numeral v"] by simp lemma complex_Im_numeral [simp]: "Im (numeral v) = 0" using complex_Im_of_int [of "numeral v"] by simp lemma complex_Im_neg_numeral [simp]: "Im (- numeral v) = 0" using complex_Im_of_int [of "- numeral v"] by simp lemma Complex_eq_numeral [simp]: "Complex a b = numeral w \ a = numeral w \ b = 0" by (simp add: complex_eq_iff) lemma Complex_eq_neg_numeral [simp]: "Complex a b = - numeral w \ a = - numeral w \ b = 0" by (simp add: complex_eq_iff) subsection {* Scalar Multiplication *} instantiation complex :: real_field begin definition complex_scaleR_def: "scaleR r x = Complex (r * Re x) (r * Im x)" lemma complex_scaleR [simp]: "scaleR r (Complex a b) = Complex (r * a) (r * b)" unfolding complex_scaleR_def by simp lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x" unfolding complex_scaleR_def by simp lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x" unfolding complex_scaleR_def by simp instance proof fix a b :: real and x y :: complex show "scaleR a (x + y) = scaleR a x + scaleR a y" by (simp add: complex_eq_iff distrib_left) show "scaleR (a + b) x = scaleR a x + scaleR b x" by (simp add: complex_eq_iff distrib_right) show "scaleR a (scaleR b x) = scaleR (a * b) x" by (simp add: complex_eq_iff mult_assoc) show "scaleR 1 x = x" by (simp add: complex_eq_iff) show "scaleR a x * y = scaleR a (x * y)" by (simp add: complex_eq_iff algebra_simps) show "x * scaleR a y = scaleR a (x * y)" by (simp add: complex_eq_iff algebra_simps) qed end subsection{* Properties of Embedding from Reals *} abbreviation complex_of_real :: "real \ complex" where "complex_of_real \ of_real" lemma complex_of_real_def: "complex_of_real r = Complex r 0" by (simp add: of_real_def complex_scaleR_def) lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" by (simp add: complex_of_real_def) lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" by (simp add: complex_of_real_def) lemma Complex_add_complex_of_real [simp]: shows "Complex x y + complex_of_real r = Complex (x+r) y" by (simp add: complex_of_real_def) lemma complex_of_real_add_Complex [simp]: shows "complex_of_real r + Complex x y = Complex (r+x) y" by (simp add: complex_of_real_def) lemma Complex_mult_complex_of_real: shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)" by (simp add: complex_of_real_def) lemma complex_of_real_mult_Complex: shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)" by (simp add: complex_of_real_def) lemma complex_eq_cancel_iff2 [simp]: shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)" by (simp add: complex_of_real_def) lemma complex_split_polar: "\r a. z = complex_of_real r * (Complex (cos a) (sin a))" by (simp add: complex_eq_iff polar_Ex) subsection {* Vector Norm *} instantiation complex :: real_normed_field begin definition complex_norm_def: "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)" abbreviation cmod :: "complex \ real" where "cmod \ norm" definition complex_sgn_def: "sgn x = x /\<^sub>R cmod x" definition dist_complex_def: "dist x y = cmod (x - y)" definition open_complex_def: "open (S :: complex set) \ (\x\S. \e>0. \y. dist y x < e \ y \ S)" lemmas cmod_def = complex_norm_def lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)" by (simp add: complex_norm_def) instance proof fix r :: real and x y :: complex and S :: "complex set" show "(norm x = 0) = (x = 0)" by (induct x) simp show "norm (x + y) \ norm x + norm y" by (induct x, induct y) (simp add: real_sqrt_sum_squares_triangle_ineq) show "norm (scaleR r x) = \r\ * norm x" by (induct x) (simp add: power_mult_distrib distrib_left [symmetric] real_sqrt_mult) show "norm (x * y) = norm x * norm y" by (induct x, induct y) (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps) show "sgn x = x /\<^sub>R cmod x" by (rule complex_sgn_def) show "dist x y = cmod (x - y)" by (rule dist_complex_def) show "open S \ (\x\S. \e>0. \y. dist y x < e \ y \ S)" by (rule open_complex_def) qed end lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1" by simp lemma cmod_complex_polar: "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r" by (simp add: norm_mult) lemma complex_Re_le_cmod: "Re x \ cmod x" unfolding complex_norm_def by (rule real_sqrt_sum_squares_ge1) lemma complex_mod_minus_le_complex_mod: "- cmod x \ cmod x" by (rule order_trans [OF _ norm_ge_zero], simp) lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \ cmod a" by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp) lemma abs_Re_le_cmod: "\Re x\ \ cmod x" by (cases x) simp lemma abs_Im_le_cmod: "\Im x\ \ cmod x" by (cases x) simp text {* Properties of complex signum. *} lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute) lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" by (simp add: complex_sgn_def divide_inverse) lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" by (simp add: complex_sgn_def divide_inverse) subsection {* Completeness of the Complexes *} lemma bounded_linear_Re: "bounded_linear Re" by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def) lemma bounded_linear_Im: "bounded_linear Im" by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def) lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re] lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im] lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re] lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im] lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re] lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im] lemma tendsto_Complex [tendsto_intros]: assumes "(f ---> a) F" and "(g ---> b) F" shows "((\x. Complex (f x) (g x)) ---> Complex a b) F" proof (rule tendstoI) fix r :: real assume "0 < r" hence "0 < r / sqrt 2" by (simp add: divide_pos_pos) have "eventually (\x. dist (f x) a < r / sqrt 2) F" using `(f ---> a) F` and `0 < r / sqrt 2` by (rule tendstoD) moreover have "eventually (\x. dist (g x) b < r / sqrt 2) F" using `(g ---> b) F` and `0 < r / sqrt 2` by (rule tendstoD) ultimately show "eventually (\x. dist (Complex (f x) (g x)) (Complex a b) < r) F" by (rule eventually_elim2) (simp add: dist_norm real_sqrt_sum_squares_less) qed instance complex :: banach proof fix X :: "nat \ complex" assume X: "Cauchy X" from Cauchy_Re [OF X] have 1: "(\n. Re (X n)) ----> lim (\n. Re (X n))" by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) from Cauchy_Im [OF X] have 2: "(\n. Im (X n)) ----> lim (\n. Im (X n))" by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) have "X ----> Complex (lim (\n. Re (X n))) (lim (\n. Im (X n)))" using tendsto_Complex [OF 1 2] by simp thus "convergent X" by (rule convergentI) qed declare DERIV_power[where 'a=complex, THEN DERIV_cong, unfolded of_nat_def[symmetric], DERIV_intros] subsection {* The Complex Number $i$ *} definition "ii" :: complex ("\") where i_def: "ii \ Complex 0 1" lemma complex_Re_i [simp]: "Re ii = 0" by (simp add: i_def) lemma complex_Im_i [simp]: "Im ii = 1" by (simp add: i_def) lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \ y = 1)" by (simp add: i_def) lemma norm_ii [simp]: "norm ii = 1" by (simp add: i_def) lemma complex_i_not_zero [simp]: "ii \ 0" by (simp add: complex_eq_iff) lemma complex_i_not_one [simp]: "ii \ 1" by (simp add: complex_eq_iff) lemma complex_i_not_numeral [simp]: "ii \ numeral w" by (simp add: complex_eq_iff) lemma complex_i_not_neg_numeral [simp]: "ii \ - numeral w" by (simp add: complex_eq_iff) lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a" by (simp add: complex_eq_iff) lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a" by (simp add: complex_eq_iff) lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" by (simp add: i_def complex_of_real_def) lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" by (simp add: i_def complex_of_real_def) lemma i_squared [simp]: "ii * ii = -1" by (simp add: i_def) lemma power2_i [simp]: "ii\<^sup>2 = -1" by (simp add: power2_eq_square) lemma inverse_i [simp]: "inverse ii = - ii" by (rule inverse_unique, simp) lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x" by (simp add: mult_assoc [symmetric]) subsection {* Complex Conjugation *} definition cnj :: "complex \ complex" where "cnj z = Complex (Re z) (- Im z)" lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)" by (simp add: cnj_def) lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x" by (simp add: cnj_def) lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x" by (simp add: cnj_def) lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" by (simp add: complex_eq_iff) lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" by (simp add: cnj_def) lemma complex_cnj_zero [simp]: "cnj 0 = 0" by (simp add: complex_eq_iff) lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" by (simp add: complex_eq_iff) lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y" by (simp add: complex_eq_iff) lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y" by (simp add: complex_eq_iff) lemma complex_cnj_minus: "cnj (- x) = - cnj x" by (simp add: complex_eq_iff) lemma complex_cnj_one [simp]: "cnj 1 = 1" by (simp add: complex_eq_iff) lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y" by (simp add: complex_eq_iff) lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)" by (simp add: complex_inverse_def) lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y" by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n" by (induct n, simp_all add: complex_cnj_mult) lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" by (simp add: complex_eq_iff) lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" by (simp add: complex_eq_iff) lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w" by (simp add: complex_eq_iff) lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w" by (simp add: complex_eq_iff) lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)" by (simp add: complex_eq_iff) lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" by (simp add: complex_norm_def) lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" by (simp add: complex_eq_iff) lemma complex_cnj_i [simp]: "cnj ii = - ii" by (simp add: complex_eq_iff) lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" by (simp add: complex_eq_iff) lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii" by (simp add: complex_eq_iff) lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)" by (simp add: complex_eq_iff power2_eq_square) lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2" by (simp add: norm_mult power2_eq_square) lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" by (simp add: cmod_def power2_eq_square) lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" by simp lemma bounded_linear_cnj: "bounded_linear cnj" using complex_cnj_add complex_cnj_scaleR by (rule bounded_linear_intro [where K=1], simp) lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj] lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj] subsection{*Basic Lemmas*} lemma complex_eq_0: "z=0 \ (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0" by (metis complex_Im_zero complex_Re_zero complex_eqI sum_power2_eq_zero_iff) lemma complex_neq_0: "z\0 \ (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0" by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff) lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z" apply (cases z, auto) by (metis complex_of_real_def of_real_add of_real_power power2_eq_square) lemma complex_div_eq_0: "(Re(a / b) = 0 \ Re(a * cnj b) = 0) & (Im(a / b) = 0 \ Im(a * cnj b) = 0)" proof (cases "b=0") case True then show ?thesis by auto next case False show ?thesis proof (cases b) case (Complex x y) then have "x\<^sup>2 + y\<^sup>2 > 0" by (metis Complex_eq_0 False sum_power2_gt_zero_iff) then have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)" by (metis add_divide_distrib) with Complex False show ?thesis by (auto simp: complex_divide_def) qed qed lemma re_complex_div_eq_0: "Re(a / b) = 0 \ Re(a * cnj b) = 0" and im_complex_div_eq_0: "Im(a / b) = 0 \ Im(a * cnj b) = 0" using complex_div_eq_0 by auto lemma complex_div_gt_0: "(Re(a / b) > 0 \ Re(a * cnj b) > 0) & (Im(a / b) > 0 \ Im(a * cnj b) > 0)" proof (cases "b=0") case True then show ?thesis by auto next case False show ?thesis proof (cases b) case (Complex x y) then have "x\<^sup>2 + y\<^sup>2 > 0" by (metis Complex_eq_0 False sum_power2_gt_zero_iff) moreover have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)" by (metis add_divide_distrib) ultimately show ?thesis using Complex False `0 < x\<^sup>2 + y\<^sup>2` apply (simp add: complex_divide_def zero_less_divide_iff less_divide_eq) apply (metis less_divide_eq mult_zero_left diff_conv_add_uminus minus_divide_left) done qed qed lemma re_complex_div_gt_0: "Re(a / b) > 0 \ Re(a * cnj b) > 0" and im_complex_div_gt_0: "Im(a / b) > 0 \ Im(a * cnj b) > 0" using complex_div_gt_0 by auto lemma re_complex_div_ge_0: "Re(a / b) \ 0 \ Re(a * cnj b) \ 0" by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0) lemma im_complex_div_ge_0: "Im(a / b) \ 0 \ Im(a * cnj b) \ 0" by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less) lemma re_complex_div_lt_0: "Re(a / b) < 0 \ Re(a * cnj b) < 0" by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0) lemma im_complex_div_lt_0: "Im(a / b) < 0 \ Im(a * cnj b) < 0" by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff) lemma re_complex_div_le_0: "Re(a / b) \ 0 \ Re(a * cnj b) \ 0" by (metis not_le re_complex_div_gt_0) lemma im_complex_div_le_0: "Im(a / b) \ 0 \ Im(a * cnj b) \ 0" by (metis im_complex_div_gt_0 not_le) lemma Re_setsum: "Re(setsum f s) = setsum (%x. Re(f x)) s" apply (cases "finite s") by (induct s rule: finite_induct) auto lemma Im_setsum: "Im(setsum f s) = setsum (%x. Im(f x)) s" apply (cases "finite s") by (induct s rule: finite_induct) auto lemma Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0" apply (cases "finite s") by (induct s rule: finite_induct) auto lemma Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s" by (metis Complex_setsum') lemma cnj_setsum: "cnj (setsum f s) = setsum (%x. cnj (f x)) s" apply (cases "finite s") by (induct s rule: finite_induct) (auto simp: complex_cnj_add) lemma of_real_setsum: "of_real (setsum f s) = setsum (%x. of_real(f x)) s" apply (cases "finite s") by (induct s rule: finite_induct) auto lemma of_real_setprod: "of_real (setprod f s) = setprod (%x. of_real(f x)) s" apply (cases "finite s") by (induct s rule: finite_induct) auto lemma Reals_cnj_iff: "z \ \ \ cnj z = z" by (metis Reals_cases Reals_of_real complex.exhaust complex.inject complex_cnj complex_of_real_def equal_neg_zero) lemma Complex_in_Reals: "Complex x 0 \ \" by (metis Reals_of_real complex_of_real_def) lemma in_Reals_norm: "z \ \ \ norm(z) = abs(Re z)" by (metis Re_complex_of_real Reals_cases norm_of_real) subsection{*Finally! Polar Form for Complex Numbers*} subsubsection {* $\cos \theta + i \sin \theta$ *} definition cis :: "real \ complex" where "cis a = Complex (cos a) (sin a)" lemma Re_cis [simp]: "Re (cis a) = cos a" by (simp add: cis_def) lemma Im_cis [simp]: "Im (cis a) = sin a" by (simp add: cis_def) lemma cis_zero [simp]: "cis 0 = 1" by (simp add: cis_def) lemma norm_cis [simp]: "norm (cis a) = 1" by (simp add: cis_def) lemma sgn_cis [simp]: "sgn (cis a) = cis a" by (simp add: sgn_div_norm) lemma cis_neq_zero [simp]: "cis a \ 0" by (metis norm_cis norm_zero zero_neq_one) lemma cis_mult: "cis a * cis b = cis (a + b)" by (simp add: cis_def cos_add sin_add) lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult) lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)" by (simp add: cis_def) lemma cis_divide: "cis a / cis b = cis (a - b)" by (simp add: complex_divide_def cis_mult) lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" by (auto simp add: DeMoivre) lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" by (auto simp add: DeMoivre) subsubsection {* $r(\cos \theta + i \sin \theta)$ *} definition rcis :: "[real, real] \ complex" where "rcis r a = complex_of_real r * cis a" lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" by (simp add: rcis_def) lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" by (simp add: rcis_def) lemma rcis_Ex: "\r a. z = rcis r a" by (simp add: complex_eq_iff polar_Ex) lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" by (simp add: rcis_def norm_mult) lemma cis_rcis_eq: "cis a = rcis 1 a" by (simp add: rcis_def) lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" by (simp add: rcis_def cis_mult) lemma rcis_zero_mod [simp]: "rcis 0 a = 0" by (simp add: rcis_def) lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" by (simp add: rcis_def) lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \ r = 0" by (simp add: rcis_def) lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" by (simp add: rcis_def power_mult_distrib DeMoivre) lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)" by (simp add: divide_inverse rcis_def) lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)" by (simp add: rcis_def cis_divide [symmetric]) subsubsection {* Complex exponential *} abbreviation expi :: "complex \ complex" where "expi \ exp" lemma cis_conv_exp: "cis b = exp (Complex 0 b)" proof (rule complex_eqI) { fix n have "Complex 0 b ^ n = real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)" apply (induct n) apply (simp add: cos_coeff_def sin_coeff_def) apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc) done } note * = this show "Re (cis b) = Re (exp (Complex 0 b))" unfolding exp_def cis_def cos_def by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic], simp add: * mult_assoc [symmetric]) show "Im (cis b) = Im (exp (Complex 0 b))" unfolding exp_def cis_def sin_def by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic], simp add: * mult_assoc [symmetric]) qed lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)" unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)" unfolding expi_def by simp lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)" unfolding expi_def by simp lemma complex_expi_Ex: "\a r. z = complex_of_real r * expi a" apply (insert rcis_Ex [of z]) apply (auto simp add: expi_def rcis_def mult_assoc [symmetric]) apply (rule_tac x = "ii * complex_of_real a" in exI, auto) done lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1" by (simp add: expi_def cis_def) subsubsection {* Complex argument *} definition arg :: "complex \ real" where "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \ -pi < a \ a \ pi))" lemma arg_zero: "arg 0 = 0" by (simp add: arg_def) lemma of_nat_less_of_int_iff: (* TODO: move *) "(of_nat n :: 'a::linordered_idom) < of_int x \ int n < x" by (metis of_int_of_nat_eq of_int_less_iff) lemma real_of_nat_less_numeral_iff [simp]: (* TODO: move *) "real (n::nat) < numeral w \ n < numeral w" using of_nat_less_of_int_iff [of n "numeral w", where 'a=real] by (simp add: real_of_nat_def zless_nat_eq_int_zless [symmetric]) lemma arg_unique: assumes "sgn z = cis x" and "-pi < x" and "x \ pi" shows "arg z = x" proof - from assms have "z \ 0" by auto have "(SOME a. sgn z = cis a \ -pi < a \ a \ pi) = x" proof fix a def d \ "a - x" assume a: "sgn z = cis a \ - pi < a \ a \ pi" from a assms have "- (2*pi) < d \ d < 2*pi" unfolding d_def by simp moreover from a assms have "cos a = cos x" and "sin a = sin x" by (simp_all add: complex_eq_iff) hence cos: "cos d = 1" unfolding d_def cos_diff by simp moreover from cos have "sin d = 0" by (rule cos_one_sin_zero) ultimately have "d = 0" unfolding sin_zero_iff even_mult_two_ex by (auto simp add: numeral_2_eq_2 less_Suc_eq) thus "a = x" unfolding d_def by simp qed (simp add: assms del: Re_sgn Im_sgn) with `z \ 0` show "arg z = x" unfolding arg_def by simp qed lemma arg_correct: assumes "z \ 0" shows "sgn z = cis (arg z) \ -pi < arg z \ arg z \ pi" proof (simp add: arg_def assms, rule someI_ex) obtain r a where z: "z = rcis r a" using rcis_Ex by fast with assms have "r \ 0" by auto def b \ "if 0 < r then a else a + pi" have b: "sgn z = cis b" unfolding z b_def rcis_def using `r \ 0` by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def) have cis_2pi_nat: "\n. cis (2 * pi * real_of_nat n) = 1" by (induct_tac n, simp_all add: distrib_left cis_mult [symmetric], simp add: cis_def) have cis_2pi_int: "\x. cis (2 * pi * real_of_int x) = 1" by (case_tac x rule: int_diff_cases, simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat) def c \ "b - 2*pi * of_int \(b - pi) / (2*pi)\" have "sgn z = cis c" unfolding b c_def by (simp add: cis_divide [symmetric] cis_2pi_int) moreover have "- pi < c \ c \ pi" using ceiling_correct [of "(b - pi) / (2*pi)"] by (simp add: c_def less_divide_eq divide_le_eq algebra_simps) ultimately show "\a. sgn z = cis a \ -pi < a \ a \ pi" by fast qed lemma arg_bounded: "- pi < arg z \ arg z \ pi" by (cases "z = 0", simp_all add: arg_zero arg_correct) lemma cis_arg: "z \ 0 \ cis (arg z) = sgn z" by (simp add: arg_correct) lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z" by (cases "z = 0", simp_all add: rcis_def cis_arg sgn_div_norm of_real_def) lemma cos_arg_i_mult_zero [simp]: "y \ 0 ==> cos (arg(Complex 0 y)) = 0" using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff) text {* Legacy theorem names *} lemmas expand_complex_eq = complex_eq_iff lemmas complex_Re_Im_cancel_iff = complex_eq_iff lemmas complex_equality = complex_eqI end