(* Title: 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_equality [intro?]: "\Re x = Re y; Im x = Im y\ \ x = y" by (induct x, induct y) simp lemma expand_complex_eq: "x = y \ Re x = Re y \ Im x = Im y" by (induct x, induct y) simp lemmas complex_Re_Im_cancel_iff = expand_complex_eq 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, division_by_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)\ + (Im x)\)) (- Im x / ((Re x)\ + (Im x)\))" 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_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\ + b\)) (- b / (a\ + b\))" by (simp add: complex_inverse_def) lemma complex_Re_inverse: "Re (inverse x) = Re x / ((Re x)\ + (Im x)\)" by (simp add: complex_inverse_def) lemma complex_Im_inverse: "Im (inverse x) = - Im x / ((Re x)\ + (Im x)\)" by (simp add: complex_inverse_def) instance by intro_classes (simp_all add: complex_mult_def right_distrib left_distrib right_diff_distrib left_diff_distrib complex_inverse_def complex_divide_def power2_eq_square add_divide_distrib [symmetric] expand_complex_eq) end subsection {* Numerals and Arithmetic *} instantiation complex :: number_ring begin definition number_of_complex where complex_number_of_def: "number_of w = (of_int w \ complex)" instance by intro_classes (simp only: complex_number_of_def) end 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_number_of [simp]: "Re (number_of v) = number_of v" unfolding number_of_eq by (rule complex_Re_of_int) lemma complex_Im_number_of [simp]: "Im (number_of v) = 0" unfolding number_of_eq by (rule complex_Im_of_int) lemma Complex_eq_number_of [simp]: "(Complex a b = number_of w) = (a = number_of w \ b = 0)" by (simp add: expand_complex_eq) 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: expand_complex_eq right_distrib) show "scaleR (a + b) x = scaleR a x + scaleR b x" by (simp add: expand_complex_eq left_distrib) show "scaleR a (scaleR b x) = scaleR (a * b) x" by (simp add: expand_complex_eq mult_assoc) show "scaleR 1 x = x" by (simp add: expand_complex_eq) show "scaleR a x * y = scaleR a (x * y)" by (simp add: expand_complex_eq algebra_simps) show "x * scaleR a y = scaleR a (x * y)" by (simp add: expand_complex_eq 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]: "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]: "complex_of_real r + Complex x y = Complex (r+x) y" by (simp add: complex_of_real_def) lemma Complex_mult_complex_of_real: "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: "complex_of_real r * Complex x y = Complex (r*x) (r*y)" by (simp add: complex_of_real_def) subsection {* Vector Norm *} instantiation complex :: real_normed_field begin definition complex_norm_def: "norm z = sqrt ((Re z)\ + (Im z)\)" 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 topo_complex_def [code del]: "topo = {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\ + y\)" by (simp add: complex_norm_def) instance proof fix r :: real and x y :: complex show "0 \ norm x" by (induct x) simp 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 right_distrib [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 "topo = {S::complex set. \x\S. \e>0. \y. dist y x < e \ y \ S}" by (rule topo_complex_def) qed end lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1" by simp lemma cmod_complex_polar [simp]: "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 [simp]: "- cmod x \ cmod x" by (rule order_trans [OF _ norm_ge_zero], simp) lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \ cmod a" by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp) lemmas real_sum_squared_expand = power2_sum [where 'a=real] 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 subsection {* Completeness of the Complexes *} interpretation Re: bounded_linear "Re" apply (unfold_locales, simp, simp) apply (rule_tac x=1 in exI) apply (simp add: complex_norm_def) done interpretation Im: bounded_linear "Im" apply (unfold_locales, simp, simp) apply (rule_tac x=1 in exI) apply (simp add: complex_norm_def) done lemma LIMSEQ_Complex: "\X ----> a; Y ----> b\ \ (\n. Complex (X n) (Y n)) ----> Complex a b" apply (rule LIMSEQ_I) apply (subgoal_tac "0 < r / sqrt 2") apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe) apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe) apply (rename_tac M N, rule_tac x="max M N" in exI, safe) apply (simp add: real_sqrt_sum_squares_less) apply (simp add: divide_pos_pos) done instance complex :: banach proof fix X :: "nat \ complex" assume X: "Cauchy X" from Re.Cauchy [OF X] have 1: "(\n. Re (X n)) ----> lim (\n. Re (X n))" by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) from Im.Cauchy [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 LIMSEQ_Complex [OF 1 2] by simp thus "convergent X" by (rule convergentI) qed subsection {* The Complex Number @{term "\"} *} 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 complex_i_not_zero [simp]: "ii \ 0" by (simp add: expand_complex_eq) lemma complex_i_not_one [simp]: "ii \ 1" by (simp add: expand_complex_eq) lemma complex_i_not_number_of [simp]: "ii \ number_of w" by (simp add: expand_complex_eq) lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a" by (simp add: expand_complex_eq) lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a" by (simp add: expand_complex_eq) 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\ = -1" by (simp add: power2_eq_square) lemma inverse_i [simp]: "inverse ii = - ii" by (rule inverse_unique, simp) 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: expand_complex_eq) 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: expand_complex_eq) lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" by (simp add: expand_complex_eq) lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y" by (simp add: expand_complex_eq) lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y" by (simp add: expand_complex_eq) lemma complex_cnj_minus: "cnj (- x) = - cnj x" by (simp add: expand_complex_eq) lemma complex_cnj_one [simp]: "cnj 1 = 1" by (simp add: expand_complex_eq) lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y" by (simp add: expand_complex_eq) 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: expand_complex_eq) lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" by (simp add: expand_complex_eq) lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w" by (simp add: expand_complex_eq) lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)" by (simp add: expand_complex_eq) 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: expand_complex_eq) lemma complex_cnj_i [simp]: "cnj ii = - ii" by (simp add: expand_complex_eq) lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" by (simp add: expand_complex_eq) lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii" by (simp add: expand_complex_eq) lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\ + (Im z)\)" by (simp add: expand_complex_eq power2_eq_square) lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\" by (simp add: norm_mult power2_eq_square) interpretation cnj: bounded_linear "cnj" apply (unfold_locales) apply (rule complex_cnj_add) apply (rule complex_cnj_scaleR) apply (rule_tac x=1 in exI, simp) done subsection{*The Functions @{term sgn} and @{term arg}*} text {*------------ Argand -------------*} definition arg :: "complex => real" where "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \ pi)" lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute) lemma i_mult_eq: "ii * ii = complex_of_real (-1)" by (simp add: i_def complex_of_real_def) lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)" by (simp add: i_def complex_one_def) lemma complex_eq_cancel_iff2 [simp]: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)" by (simp add: complex_of_real_def) 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) lemma complex_inverse_complex_split: "inverse(complex_of_real x + ii * complex_of_real y) = complex_of_real(x/(x ^ 2 + y ^ 2)) - ii * complex_of_real(y/(x ^ 2 + y ^ 2))" by (simp add: complex_of_real_def i_def diff_minus divide_inverse) (*----------------------------------------------------------------------------*) (* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) (* many of the theorems are not used - so should they be kept? *) (*----------------------------------------------------------------------------*) lemma cos_arg_i_mult_zero_pos: "0 < y ==> cos (arg(Complex 0 y)) = 0" apply (simp add: arg_def abs_if) apply (rule_tac a = "pi/2" in someI2, auto) apply (rule order_less_trans [of _ 0], auto) done lemma cos_arg_i_mult_zero_neg: "y < 0 ==> cos (arg(Complex 0 y)) = 0" apply (simp add: arg_def abs_if) apply (rule_tac a = "- pi/2" in someI2, auto) apply (rule order_trans [of _ 0], auto) done lemma cos_arg_i_mult_zero [simp]: "y \ 0 ==> cos (arg(Complex 0 y)) = 0" by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) subsection{*Finally! Polar Form for Complex Numbers*} definition (* abbreviation for (cos a + i sin a) *) cis :: "real => complex" where "cis a = Complex (cos a) (sin a)" definition (* abbreviation for r*(cos a + i sin a) *) rcis :: "[real, real] => complex" where "rcis r a = complex_of_real r * cis a" definition (* e ^ (x + iy) *) expi :: "complex => complex" where "expi z = complex_of_real(exp (Re z)) * cis (Im z)" lemma complex_split_polar: "\r a. z = complex_of_real r * (Complex (cos a) (sin a))" apply (induct z) apply (auto simp add: polar_Ex complex_of_real_mult_Complex) done lemma rcis_Ex: "\r a. z = rcis r a" apply (induct z) apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex) done lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" by (simp add: rcis_def cis_def) lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" by (simp add: rcis_def cis_def) lemma sin_cos_squared_add2_mult: "(r * cos a)\ + (r * sin a)\ = r\" proof - have "(r * cos a)\ + (r * sin a)\ = r\ * ((cos a)\ + (sin a)\)" by (simp only: power_mult_distrib right_distrib) thus ?thesis by simp qed lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" by (simp add: rcis_def cis_def sin_cos_squared_add2_mult) 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 (*---------------------------------------------------------------------------*) (* (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b) *) (*---------------------------------------------------------------------------*) 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_def cos_add sin_add right_distrib right_diff_distrib complex_of_real_def) lemma cis_mult: "cis a * cis b = cis (a + b)" by (simp add: cis_rcis_eq rcis_mult) lemma cis_zero [simp]: "cis 0 = 1" by (simp add: cis_def complex_one_def) 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 complex_of_real_minus_one: "complex_of_real (-(1::real)) = -(1::complex)" by (simp add: complex_of_real_def complex_one_def) lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x" by (simp add: mult_assoc [symmetric]) lemma cis_real_of_nat_Suc_mult: "cis (real (Suc n) * a) = cis a * cis (real n * a)" by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib) lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" apply (induct_tac "n") apply (auto simp add: cis_real_of_nat_Suc_mult) done lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" by (simp add: rcis_def power_mult_distrib DeMoivre) lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)" by (simp add: cis_def complex_inverse_complex_split diff_minus) lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)" by (simp add: divide_inverse rcis_def) lemma cis_divide: "cis a / cis b = cis (a - b)" by (simp add: complex_divide_def cis_mult real_diff_def) lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)" apply (simp add: complex_divide_def) apply (case_tac "r2=0", simp) apply (simp add: rcis_inverse rcis_mult real_diff_def) done 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 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) lemma expi_add: "expi(a + b) = expi(a) * expi(b)" by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac) lemma expi_zero [simp]: "expi (0::complex) = 1" by (simp add: expi_def) 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) end