paulson@13957: (* Title: Complex.thy paulson@14430: ID: $Id$ paulson@13957: Author: Jacques D. Fleuriot paulson@13957: Copyright: 2001 University of Edinburgh paulson@14387: Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 paulson@13957: *) paulson@13957: paulson@14377: header {* Complex Numbers: Rectangular and Polar Representations *} paulson@14373: nipkow@15131: theory Complex huffman@22655: imports "../Hyperreal/Transcendental" nipkow@15131: begin paulson@13957: paulson@14373: datatype complex = Complex real real paulson@13957: wenzelm@14691: instance complex :: "{zero, one, plus, times, minus, inverse, power}" .. paulson@13957: paulson@13957: consts paulson@14373: "ii" :: complex ("\") paulson@14373: paulson@14373: consts Re :: "complex => real" huffman@20557: primrec Re: "Re (Complex x y) = x" paulson@14373: paulson@14373: consts Im :: "complex => real" huffman@20557: primrec Im: "Im (Complex x y) = y" paulson@14373: paulson@14373: lemma complex_surj [simp]: "Complex (Re z) (Im z) = z" paulson@14373: by (induct z) simp paulson@13957: paulson@14323: defs (overloaded) paulson@14323: paulson@14323: complex_zero_def: paulson@14373: "0 == Complex 0 0" paulson@13957: paulson@14323: complex_one_def: paulson@14373: "1 == Complex 1 0" paulson@14323: paulson@14373: i_def: "ii == Complex 0 1" paulson@14323: paulson@14373: complex_minus_def: "- z == Complex (- Re z) (- Im z)" paulson@14323: paulson@14323: complex_inverse_def: paulson@14373: "inverse z == paulson@14373: Complex (Re z / ((Re z)\ + (Im z)\)) (- Im z / ((Re z)\ + (Im z)\))" paulson@13957: paulson@14323: complex_add_def: paulson@14373: "z + w == Complex (Re z + Re w) (Im z + Im w)" paulson@13957: paulson@14323: complex_diff_def: paulson@14373: "z - w == z + - (w::complex)" paulson@13957: paulson@14374: complex_mult_def: paulson@14373: "z * w == Complex (Re z * Re w - Im z * Im w) (Re z * Im w + Im z * Re w)" paulson@13957: paulson@14373: complex_divide_def: "w / (z::complex) == w * inverse z" paulson@14323: paulson@13957: paulson@14373: lemma complex_equality [intro?]: "Re z = Re w ==> Im z = Im w ==> z = w" paulson@14373: by (induct z, induct w) simp paulson@14323: paulson@14323: lemma complex_Re_Im_cancel_iff: "(w=z) = (Re(w) = Re(z) & Im(w) = Im(z))" paulson@14373: by (induct w, induct z, simp) paulson@14323: paulson@14374: lemma complex_Re_zero [simp]: "Re 0 = 0" paulson@14374: by (simp add: complex_zero_def) paulson@14374: paulson@14374: lemma complex_Im_zero [simp]: "Im 0 = 0" paulson@14373: by (simp add: complex_zero_def) paulson@14323: paulson@14374: lemma complex_Re_one [simp]: "Re 1 = 1" paulson@14374: by (simp add: complex_one_def) paulson@14323: paulson@14374: lemma complex_Im_one [simp]: "Im 1 = 0" paulson@14373: by (simp add: complex_one_def) paulson@14323: paulson@14374: lemma complex_Re_i [simp]: "Re(ii) = 0" paulson@14373: by (simp add: i_def) paulson@14323: paulson@14374: lemma complex_Im_i [simp]: "Im(ii) = 1" paulson@14373: by (simp add: i_def) paulson@14323: paulson@14323: paulson@14374: subsection{*Unary Minus*} paulson@14323: paulson@14377: lemma complex_minus [simp]: "- (Complex x y) = Complex (-x) (-y)" paulson@14373: by (simp add: complex_minus_def) paulson@14323: paulson@14374: lemma complex_Re_minus [simp]: "Re (-z) = - Re z" paulson@14373: by (simp add: complex_minus_def) paulson@14323: paulson@14374: lemma complex_Im_minus [simp]: "Im (-z) = - Im z" paulson@14374: by (simp add: complex_minus_def) paulson@14323: paulson@14323: paulson@14323: subsection{*Addition*} paulson@14323: paulson@14377: lemma complex_add [simp]: paulson@14377: "Complex x1 y1 + Complex x2 y2 = Complex (x1+x2) (y1+y2)" paulson@14373: by (simp add: complex_add_def) paulson@14323: paulson@14374: lemma complex_Re_add [simp]: "Re(x + y) = Re(x) + Re(y)" paulson@14373: by (simp add: complex_add_def) paulson@14323: paulson@14374: lemma complex_Im_add [simp]: "Im(x + y) = Im(x) + Im(y)" paulson@14373: by (simp add: complex_add_def) paulson@14323: paulson@14323: lemma complex_add_commute: "(u::complex) + v = v + u" paulson@14373: by (simp add: complex_add_def add_commute) paulson@14323: paulson@14323: lemma complex_add_assoc: "((u::complex) + v) + w = u + (v + w)" paulson@14373: by (simp add: complex_add_def add_assoc) paulson@14323: paulson@14323: lemma complex_add_zero_left: "(0::complex) + z = z" paulson@14373: by (simp add: complex_add_def complex_zero_def) paulson@14323: paulson@14323: lemma complex_add_zero_right: "z + (0::complex) = z" paulson@14373: by (simp add: complex_add_def complex_zero_def) paulson@14323: paulson@14373: lemma complex_add_minus_left: "-z + z = (0::complex)" paulson@14373: by (simp add: complex_add_def complex_minus_def complex_zero_def) paulson@14323: paulson@14323: lemma complex_diff: paulson@14373: "Complex x1 y1 - Complex x2 y2 = Complex (x1-x2) (y1-y2)" paulson@14373: by (simp add: complex_add_def complex_minus_def complex_diff_def) paulson@14323: paulson@14374: lemma complex_Re_diff [simp]: "Re(x - y) = Re(x) - Re(y)" paulson@14374: by (simp add: complex_diff_def) paulson@14374: paulson@14374: lemma complex_Im_diff [simp]: "Im(x - y) = Im(x) - Im(y)" paulson@14374: by (simp add: complex_diff_def) paulson@14374: paulson@14374: paulson@14323: subsection{*Multiplication*} paulson@14323: paulson@14377: lemma complex_mult [simp]: paulson@14373: "Complex x1 y1 * Complex x2 y2 = Complex (x1*x2 - y1*y2) (x1*y2 + y1*x2)" paulson@14373: by (simp add: complex_mult_def) paulson@14323: paulson@14323: lemma complex_mult_commute: "(w::complex) * z = z * w" paulson@14373: by (simp add: complex_mult_def mult_commute add_commute) paulson@14323: paulson@14323: lemma complex_mult_assoc: "((u::complex) * v) * w = u * (v * w)" paulson@14374: by (simp add: complex_mult_def mult_ac add_ac paulson@14373: right_diff_distrib right_distrib left_diff_distrib left_distrib) paulson@14323: paulson@14323: lemma complex_mult_one_left: "(1::complex) * z = z" paulson@14373: by (simp add: complex_mult_def complex_one_def) paulson@14323: paulson@14323: lemma complex_mult_one_right: "z * (1::complex) = z" paulson@14373: by (simp add: complex_mult_def complex_one_def) paulson@14323: paulson@14323: paulson@14323: subsection{*Inverse*} paulson@14323: paulson@14377: lemma complex_inverse [simp]: paulson@14373: "inverse (Complex x y) = Complex (x/(x ^ 2 + y ^ 2)) (-y/(x ^ 2 + y ^ 2))" paulson@14373: by (simp add: complex_inverse_def) paulson@14335: paulson@14354: lemma complex_mult_inv_left: "z \ (0::complex) ==> inverse(z) * z = 1" paulson@14374: apply (induct z) paulson@14374: apply (rename_tac x y) huffman@20725: apply (auto simp add: paulson@15234: complex_one_def complex_zero_def add_divide_distrib [symmetric] paulson@15234: power2_eq_square mult_ac) paulson@15234: apply (simp_all add: real_sum_squares_not_zero real_sum_squares_not_zero2) paulson@14323: done paulson@14323: paulson@14335: paulson@14335: subsection {* The field of complex numbers *} paulson@14335: paulson@14335: instance complex :: field paulson@14335: proof paulson@14335: fix z u v w :: complex paulson@14335: show "(u + v) + w = u + (v + w)" paulson@14374: by (rule complex_add_assoc) paulson@14335: show "z + w = w + z" paulson@14374: by (rule complex_add_commute) paulson@14335: show "0 + z = z" paulson@14374: by (rule complex_add_zero_left) paulson@14335: show "-z + z = 0" paulson@14374: by (rule complex_add_minus_left) paulson@14335: show "z - w = z + -w" paulson@14335: by (simp add: complex_diff_def) paulson@14335: show "(u * v) * w = u * (v * w)" paulson@14374: by (rule complex_mult_assoc) paulson@14335: show "z * w = w * z" paulson@14374: by (rule complex_mult_commute) paulson@14335: show "1 * z = z" paulson@14374: by (rule complex_mult_one_left) paulson@14341: show "0 \ (1::complex)" paulson@14373: by (simp add: complex_zero_def complex_one_def) paulson@14335: show "(u + v) * w = u * w + v * w" paulson@14421: by (simp add: complex_mult_def complex_add_def left_distrib paulson@14421: diff_minus add_ac) paulson@14430: show "z / w = z * inverse w" paulson@14335: by (simp add: complex_divide_def) paulson@14430: assume "w \ 0" paulson@14430: thus "inverse w * w = 1" paulson@14430: by (simp add: complex_mult_inv_left) paulson@14335: qed paulson@14335: paulson@14373: instance complex :: division_by_zero paulson@14373: proof paulson@14430: show "inverse 0 = (0::complex)" paulson@14373: by (simp add: complex_inverse_def complex_zero_def) paulson@14373: qed paulson@14335: paulson@14323: huffman@20556: subsection{*The real algebra of complex numbers*} huffman@20556: huffman@20556: instance complex :: scaleR .. huffman@20556: huffman@20556: defs (overloaded) huffman@20556: complex_scaleR_def: "r *# x == Complex r 0 * x" huffman@20556: huffman@20725: instance complex :: real_field huffman@20556: proof huffman@20556: fix a b :: real huffman@20556: fix x y :: complex huffman@20556: show "a *# (x + y) = a *# x + a *# y" huffman@20556: by (simp add: complex_scaleR_def right_distrib) huffman@20556: show "(a + b) *# x = a *# x + b *# x" huffman@20556: by (simp add: complex_scaleR_def left_distrib [symmetric]) huffman@20763: show "a *# b *# x = (a * b) *# x" huffman@20556: by (simp add: complex_scaleR_def mult_assoc [symmetric]) huffman@20556: show "1 *# x = x" huffman@20556: by (simp add: complex_scaleR_def complex_one_def [symmetric]) huffman@20556: show "a *# x * y = a *# (x * y)" huffman@20556: by (simp add: complex_scaleR_def mult_assoc) huffman@20556: show "x * a *# y = a *# (x * y)" huffman@20556: by (simp add: complex_scaleR_def mult_left_commute) huffman@20556: qed huffman@20556: huffman@20556: paulson@14323: subsection{*Embedding Properties for @{term complex_of_real} Map*} paulson@14323: huffman@20557: abbreviation wenzelm@21404: complex_of_real :: "real => complex" where huffman@20557: "complex_of_real == of_real" huffman@20557: huffman@20557: lemma complex_of_real_def: "complex_of_real r = Complex r 0" huffman@20557: by (simp add: of_real_def complex_scaleR_def) huffman@20557: huffman@20557: lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" huffman@20557: by (simp add: complex_of_real_def) huffman@20557: huffman@20557: lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" huffman@20557: by (simp add: complex_of_real_def) huffman@20557: paulson@14377: lemma Complex_add_complex_of_real [simp]: paulson@14377: "Complex x y + complex_of_real r = Complex (x+r) y" paulson@14377: by (simp add: complex_of_real_def) paulson@14377: paulson@14377: lemma complex_of_real_add_Complex [simp]: paulson@14377: "complex_of_real r + Complex x y = Complex (r+x) y" paulson@14377: by (simp add: i_def complex_of_real_def) paulson@14377: paulson@14377: lemma Complex_mult_complex_of_real: paulson@14377: "Complex x y * complex_of_real r = Complex (x*r) (y*r)" paulson@14377: by (simp add: complex_of_real_def) paulson@14377: paulson@14377: lemma complex_of_real_mult_Complex: paulson@14377: "complex_of_real r * Complex x y = Complex (r*x) (r*y)" paulson@14377: by (simp add: i_def complex_of_real_def) paulson@14377: paulson@14377: lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" paulson@14377: by (simp add: i_def complex_of_real_def) paulson@14377: paulson@14377: lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" paulson@14377: by (simp add: i_def complex_of_real_def) paulson@14377: huffman@20725: lemma complex_of_real_inverse: paulson@14374: "complex_of_real(inverse x) = inverse(complex_of_real x)" huffman@20725: by (rule of_real_inverse) paulson@14323: huffman@20725: lemma complex_of_real_divide: paulson@15013: "complex_of_real(x/y) = complex_of_real x / complex_of_real y" huffman@20725: by (rule of_real_divide) paulson@14323: paulson@14323: paulson@14377: subsection{*The Functions @{term Re} and @{term Im}*} paulson@14377: paulson@14377: lemma complex_Re_mult_eq: "Re (w * z) = Re w * Re z - Im w * Im z" huffman@20725: by (induct z, induct w, simp) paulson@14377: paulson@14377: lemma complex_Im_mult_eq: "Im (w * z) = Re w * Im z + Im w * Re z" huffman@20725: by (induct z, induct w, simp) paulson@14377: paulson@14377: lemma Re_i_times [simp]: "Re(ii * z) = - Im z" huffman@20725: by (simp add: complex_Re_mult_eq) paulson@14377: paulson@14377: lemma Re_times_i [simp]: "Re(z * ii) = - Im z" huffman@20725: by (simp add: complex_Re_mult_eq) paulson@14377: paulson@14377: lemma Im_i_times [simp]: "Im(ii * z) = Re z" huffman@20725: by (simp add: complex_Im_mult_eq) paulson@14377: paulson@14377: lemma Im_times_i [simp]: "Im(z * ii) = Re z" huffman@20725: by (simp add: complex_Im_mult_eq) paulson@14377: paulson@14377: lemma complex_Re_mult: "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)" paulson@14377: by (simp add: complex_Re_mult_eq) paulson@14377: paulson@14377: lemma complex_Re_mult_complex_of_real [simp]: paulson@14377: "Re (z * complex_of_real c) = Re(z) * c" paulson@14377: by (simp add: complex_Re_mult_eq) paulson@14377: paulson@14377: lemma complex_Im_mult_complex_of_real [simp]: paulson@14377: "Im (z * complex_of_real c) = Im(z) * c" paulson@14377: by (simp add: complex_Im_mult_eq) paulson@14377: paulson@14377: lemma complex_Re_mult_complex_of_real2 [simp]: paulson@14377: "Re (complex_of_real c * z) = c * Re(z)" paulson@14377: by (simp add: complex_Re_mult_eq) paulson@14377: paulson@14377: lemma complex_Im_mult_complex_of_real2 [simp]: paulson@14377: "Im (complex_of_real c * z) = c * Im(z)" paulson@14377: by (simp add: complex_Im_mult_eq) huffman@20557: paulson@14377: paulson@14323: subsection{*Conjugation is an Automorphism*} paulson@14323: huffman@20557: definition wenzelm@21404: cnj :: "complex => complex" where huffman@20557: "cnj z = Complex (Re z) (-Im z)" huffman@20557: paulson@14373: lemma complex_cnj: "cnj (Complex x y) = Complex x (-y)" paulson@14373: by (simp add: cnj_def) paulson@14323: paulson@14374: lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" paulson@14373: by (simp add: cnj_def complex_Re_Im_cancel_iff) paulson@14323: paulson@14374: lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" paulson@14373: by (simp add: cnj_def) paulson@14323: paulson@14374: lemma complex_cnj_complex_of_real [simp]: paulson@14373: "cnj (complex_of_real x) = complex_of_real x" paulson@14373: by (simp add: complex_of_real_def complex_cnj) paulson@14323: paulson@14323: lemma complex_cnj_minus: "cnj (-z) = - cnj z" huffman@20725: by (simp add: cnj_def) paulson@14323: paulson@14323: lemma complex_cnj_inverse: "cnj(inverse z) = inverse(cnj z)" huffman@20725: by (induct z, simp add: complex_cnj power2_eq_square) paulson@14323: paulson@14323: lemma complex_cnj_add: "cnj(w + z) = cnj(w) + cnj(z)" huffman@20725: by (induct w, induct z, simp add: complex_cnj) paulson@14323: paulson@14323: lemma complex_cnj_diff: "cnj(w - z) = cnj(w) - cnj(z)" paulson@15013: by (simp add: diff_minus complex_cnj_add complex_cnj_minus) paulson@14323: paulson@14323: lemma complex_cnj_mult: "cnj(w * z) = cnj(w) * cnj(z)" huffman@20725: by (induct w, induct z, simp add: complex_cnj) paulson@14323: paulson@14323: lemma complex_cnj_divide: "cnj(w / z) = (cnj w)/(cnj z)" paulson@14373: by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) paulson@14323: paulson@14374: lemma complex_cnj_one [simp]: "cnj 1 = 1" paulson@14373: by (simp add: cnj_def complex_one_def) paulson@14323: paulson@14323: lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re(z))" huffman@20725: by (induct z, simp add: complex_cnj complex_of_real_def) paulson@14323: paulson@14323: lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im(z)) * ii" paulson@14373: apply (induct z) paulson@15013: apply (simp add: complex_add complex_cnj complex_of_real_def diff_minus paulson@14354: complex_minus i_def complex_mult) paulson@14323: done paulson@14323: paulson@14354: lemma complex_cnj_zero [simp]: "cnj 0 = 0" paulson@14334: by (simp add: cnj_def complex_zero_def) paulson@14323: paulson@14374: lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" paulson@14373: by (induct z, simp add: complex_zero_def complex_cnj) paulson@14323: paulson@14323: lemma complex_mult_cnj: "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)" huffman@20725: by (induct z, simp add: complex_cnj complex_of_real_def power2_eq_square) paulson@14323: paulson@14323: paulson@14323: subsection{*Modulus*} paulson@14323: huffman@20557: instance complex :: norm .. huffman@20557: huffman@20557: defs (overloaded) huffman@20557: complex_norm_def: "norm z == sqrt(Re(z) ^ 2 + Im(z) ^ 2)" huffman@20557: huffman@20557: abbreviation wenzelm@21404: cmod :: "complex => real" where huffman@20557: "cmod == norm" huffman@20557: huffman@20557: lemmas cmod_def = complex_norm_def huffman@20557: huffman@20557: lemma complex_mod [simp]: "cmod (Complex x y) = sqrt(x ^ 2 + y ^ 2)" huffman@20557: by (simp add: cmod_def) huffman@20557: huffman@20557: lemma complex_mod_zero [simp]: "cmod(0) = 0" huffman@20557: by (simp add: cmod_def) huffman@20557: huffman@20557: lemma complex_mod_one [simp]: "cmod(1) = 1" huffman@20557: by (simp add: cmod_def power2_eq_square) huffman@20557: huffman@20557: lemma complex_mod_complex_of_real [simp]: "cmod(complex_of_real x) = abs x" huffman@20725: by (simp add: complex_of_real_def power2_eq_square) huffman@20557: huffman@20557: lemma complex_of_real_abs: huffman@20557: "complex_of_real (abs x) = complex_of_real(cmod(complex_of_real x))" huffman@20557: by simp huffman@20557: paulson@14374: lemma complex_mod_eq_zero_cancel [simp]: "(cmod x = 0) = (x = 0)" paulson@14373: apply (induct x) paulson@15085: apply (auto iff: real_0_le_add_iff paulson@15085: intro: real_sum_squares_cancel real_sum_squares_cancel2 paulson@14373: simp add: complex_mod complex_zero_def power2_eq_square) paulson@14323: done paulson@14323: paulson@14374: lemma complex_mod_complex_of_real_of_nat [simp]: paulson@14373: "cmod (complex_of_real(real (n::nat))) = real n" paulson@14373: by simp paulson@14323: paulson@14374: lemma complex_mod_minus [simp]: "cmod (-x) = cmod(x)" huffman@20725: by (induct x, simp add: power2_eq_square) paulson@14323: huffman@20557: lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" huffman@20725: by (induct z, simp add: complex_cnj power2_eq_square) huffman@20557: paulson@14323: lemma complex_mod_mult_cnj: "cmod(z * cnj(z)) = cmod(z) ^ 2" paulson@14373: apply (induct z, simp add: complex_mod complex_cnj complex_mult) paulson@15085: apply (simp add: power2_eq_square abs_if linorder_not_less real_0_le_add_iff) paulson@14323: done paulson@14323: paulson@14373: lemma complex_mod_squared: "cmod(Complex x y) ^ 2 = x ^ 2 + y ^ 2" paulson@14373: by (simp add: cmod_def) paulson@14323: paulson@14374: lemma complex_mod_ge_zero [simp]: "0 \ cmod x" paulson@14373: by (simp add: cmod_def) paulson@14323: paulson@14374: lemma abs_cmod_cancel [simp]: "abs(cmod x) = cmod x" paulson@14374: by (simp add: abs_if linorder_not_less) paulson@14323: paulson@14323: lemma complex_mod_mult: "cmod(x*y) = cmod(x) * cmod(y)" paulson@14373: apply (induct x, induct y) paulson@14377: apply (auto simp add: complex_mult complex_mod real_sqrt_mult_distrib2[symmetric]) paulson@14348: apply (rule_tac n = 1 in power_inject_base) paulson@14353: apply (auto simp add: power2_eq_square [symmetric] simp del: realpow_Suc) paulson@14374: apply (auto simp add: real_diff_def power2_eq_square right_distrib left_distrib paulson@14374: add_ac mult_ac) paulson@14323: done paulson@14323: paulson@14377: lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1" paulson@14377: by (simp add: cmod_def) paulson@14377: paulson@14377: lemma cmod_complex_polar [simp]: paulson@14377: "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r" paulson@14377: by (simp only: cmod_unit_one complex_mod_mult, simp) paulson@14377: paulson@14374: lemma complex_mod_add_squared_eq: paulson@14374: "cmod(x + y) ^ 2 = cmod(x) ^ 2 + cmod(y) ^ 2 + 2 * Re(x * cnj y)" paulson@14373: apply (induct x, induct y) huffman@20725: apply (auto simp add: complex_mod_squared complex_cnj real_diff_def simp del: realpow_Suc) paulson@14353: apply (auto simp add: right_distrib left_distrib power2_eq_square mult_ac add_ac) paulson@14323: done paulson@14323: paulson@14374: lemma complex_Re_mult_cnj_le_cmod [simp]: "Re(x * cnj y) \ cmod(x * cnj y)" paulson@14373: apply (induct x, induct y) huffman@20725: apply (auto simp add: complex_mod complex_cnj diff_def simp del: realpow_Suc) paulson@14323: done paulson@14323: paulson@14374: lemma complex_Re_mult_cnj_le_cmod2 [simp]: "Re(x * cnj y) \ cmod(x * y)" paulson@14373: by (insert complex_Re_mult_cnj_le_cmod [of x y], simp add: complex_mod_mult) paulson@14323: paulson@14374: lemma real_sum_squared_expand: paulson@14374: "((x::real) + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y" paulson@14373: by (simp add: left_distrib right_distrib power2_eq_square) paulson@14323: paulson@14374: lemma complex_mod_triangle_squared [simp]: paulson@14374: "cmod (x + y) ^ 2 \ (cmod(x) + cmod(y)) ^ 2" paulson@14373: by (simp add: real_sum_squared_expand complex_mod_add_squared_eq real_mult_assoc complex_mod_mult [symmetric]) paulson@14323: paulson@14374: lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \ cmod x" paulson@14373: by (rule order_trans [OF _ complex_mod_ge_zero], simp) paulson@14323: paulson@14374: lemma complex_mod_triangle_ineq [simp]: "cmod (x + y) \ cmod(x) + cmod(y)" paulson@14334: apply (rule_tac n = 1 in realpow_increasing) paulson@14323: apply (auto intro: order_trans [OF _ complex_mod_ge_zero] paulson@15085: simp add: add_increasing power2_eq_square [symmetric]) paulson@14323: done paulson@14323: huffman@22852: lemma complex_norm_scaleR: huffman@22852: "norm (scaleR a x) = \a\ * norm (x::complex)" huffman@22852: by (simp only: huffman@22852: scaleR_conv_of_real complex_mod_mult complex_mod_complex_of_real) huffman@22852: huffman@20725: instance complex :: real_normed_field huffman@20557: proof huffman@20557: fix r :: real huffman@20557: fix x y :: complex huffman@20557: show "0 \ cmod x" huffman@20557: by (rule complex_mod_ge_zero) huffman@20557: show "(cmod x = 0) = (x = 0)" huffman@20557: by (rule complex_mod_eq_zero_cancel) huffman@20557: show "cmod (x + y) \ cmod x + cmod y" huffman@20557: by (rule complex_mod_triangle_ineq) huffman@22852: show "cmod (scaleR r x) = \r\ * cmod x" huffman@22852: by (rule complex_norm_scaleR) huffman@20557: show "cmod (x * y) = cmod x * cmod y" huffman@20557: by (rule complex_mod_mult) huffman@20557: qed huffman@20557: paulson@14374: lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \ cmod a" paulson@14373: by (insert complex_mod_triangle_ineq [THEN add_right_mono, of b a"-cmod b"], simp) paulson@14323: paulson@14323: lemma complex_mod_diff_commute: "cmod (x - y) = cmod (y - x)" huffman@20557: by (rule norm_minus_commute) paulson@14323: paulson@14374: lemma complex_mod_add_less: paulson@14374: "[| cmod x < r; cmod y < s |] ==> cmod (x + y) < r + s" paulson@14334: by (auto intro: order_le_less_trans complex_mod_triangle_ineq) paulson@14323: paulson@14374: lemma complex_mod_mult_less: paulson@14374: "[| cmod x < r; cmod y < s |] ==> cmod (x * y) < r * s" paulson@14334: by (auto intro: real_mult_less_mono' simp add: complex_mod_mult) paulson@14323: paulson@14374: lemma complex_mod_diff_ineq [simp]: "cmod(a) - cmod(b) \ cmod(a + b)" huffman@20725: proof - huffman@20725: have "cmod a - cmod b = cmod a - cmod (- b)" by simp huffman@20725: also have "\ \ cmod (a - - b)" by (rule norm_triangle_ineq2) huffman@20725: also have "\ = cmod (a + b)" by simp huffman@20725: finally show ?thesis . huffman@20725: qed paulson@14323: paulson@14374: lemma complex_Re_le_cmod [simp]: "Re z \ cmod z" huffman@20725: by (induct z, simp) paulson@14323: paulson@14354: lemma complex_mod_gt_zero: "z \ 0 ==> 0 < cmod z" huffman@20557: by (rule zero_less_norm_iff [THEN iffD2]) paulson@14323: paulson@14323: lemma complex_mod_inverse: "cmod(inverse x) = inverse(cmod x)" huffman@20557: by (rule norm_inverse) paulson@14323: paulson@14373: lemma complex_mod_divide: "cmod(x/y) = cmod(x)/(cmod y)" huffman@20725: by (rule norm_divide) paulson@14323: paulson@14354: paulson@14354: subsection{*Exponentiation*} paulson@14354: paulson@14354: primrec paulson@14354: complexpow_0: "z ^ 0 = 1" paulson@14354: complexpow_Suc: "z ^ (Suc n) = (z::complex) * (z ^ n)" paulson@14354: paulson@14354: paulson@15003: instance complex :: recpower paulson@14354: proof paulson@14354: fix z :: complex paulson@14354: fix n :: nat paulson@14354: show "z^0 = 1" by simp paulson@14354: show "z^(Suc n) = z * (z^n)" by simp paulson@14354: qed paulson@14323: paulson@14323: paulson@14354: lemma complex_of_real_pow: "complex_of_real (x ^ n) = (complex_of_real x) ^ n" huffman@20725: by (rule of_real_power) paulson@14323: paulson@14354: lemma complex_cnj_pow: "cnj(z ^ n) = cnj(z) ^ n" paulson@14323: apply (induct_tac "n") paulson@14354: apply (auto simp add: complex_cnj_mult) paulson@14323: done paulson@14323: paulson@14354: lemma complex_mod_complexpow: "cmod(x ^ n) = cmod(x) ^ n" huffman@20725: by (rule norm_power) paulson@14354: paulson@14354: lemma complexpow_i_squared [simp]: "ii ^ 2 = -(1::complex)" huffman@20725: by (simp add: i_def complex_one_def numeral_2_eq_2) paulson@14354: paulson@14354: lemma complex_i_not_zero [simp]: "ii \ 0" paulson@14373: by (simp add: i_def complex_zero_def) paulson@14354: paulson@14354: paulson@14354: subsection{*The Function @{term sgn}*} paulson@14323: huffman@20557: definition huffman@20557: (*------------ Argand -------------*) huffman@20557: wenzelm@21404: sgn :: "complex => complex" where huffman@20557: "sgn z = z / complex_of_real(cmod z)" huffman@20557: wenzelm@21404: definition wenzelm@21404: arg :: "complex => real" where huffman@20557: "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \ pi)" huffman@20557: paulson@14374: lemma sgn_zero [simp]: "sgn 0 = 0" paulson@14373: by (simp add: sgn_def) paulson@14323: paulson@14374: lemma sgn_one [simp]: "sgn 1 = 1" paulson@14373: by (simp add: sgn_def) paulson@14323: paulson@14323: lemma sgn_minus: "sgn (-z) = - sgn(z)" paulson@14373: by (simp add: sgn_def) paulson@14323: paulson@14374: lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" paulson@14377: by (simp add: sgn_def) paulson@14323: paulson@14323: lemma i_mult_eq: "ii * ii = complex_of_real (-1)" huffman@20725: by (simp add: i_def complex_of_real_def) paulson@14323: paulson@14374: lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)" huffman@20725: by (simp add: i_def complex_one_def) paulson@14323: paulson@14374: lemma complex_eq_cancel_iff2 [simp]: paulson@14377: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)" paulson@14377: by (simp add: complex_of_real_def) paulson@14323: paulson@14377: lemma Complex_eq_0 [simp]: "(Complex x y = 0) = (x = 0 & y = 0)" paulson@14377: by (simp add: complex_zero_def) paulson@14323: paulson@14377: lemma Complex_eq_1 [simp]: "(Complex x y = 1) = (x = 1 & y = 0)" paulson@14377: by (simp add: complex_one_def) paulson@14323: paulson@14377: lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 & y = 1)" paulson@14377: by (simp add: i_def) paulson@14323: paulson@15013: paulson@15013: paulson@14374: lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" paulson@15013: proof (induct z) paulson@15013: case (Complex x y) paulson@15013: have "sqrt (x\ + y\) * inverse (x\ + y\) = inverse (sqrt (x\ + y\))" paulson@15013: by (simp add: divide_inverse [symmetric] sqrt_divide_self_eq) paulson@15013: thus "Re (sgn (Complex x y)) = Re (Complex x y) /cmod (Complex x y)" paulson@15013: by (simp add: sgn_def complex_of_real_def divide_inverse) paulson@15013: qed paulson@15013: paulson@14323: paulson@14374: lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" paulson@15013: proof (induct z) paulson@15013: case (Complex x y) paulson@15013: have "sqrt (x\ + y\) * inverse (x\ + y\) = inverse (sqrt (x\ + y\))" paulson@15013: by (simp add: divide_inverse [symmetric] sqrt_divide_self_eq) paulson@15013: thus "Im (sgn (Complex x y)) = Im (Complex x y) /cmod (Complex x y)" paulson@15013: by (simp add: sgn_def complex_of_real_def divide_inverse) paulson@15013: qed paulson@14323: paulson@14323: lemma complex_inverse_complex_split: paulson@14323: "inverse(complex_of_real x + ii * complex_of_real y) = paulson@14323: complex_of_real(x/(x ^ 2 + y ^ 2)) - paulson@14323: ii * complex_of_real(y/(x ^ 2 + y ^ 2))" huffman@20725: by (simp add: complex_of_real_def i_def diff_minus divide_inverse) paulson@14323: paulson@14323: (*----------------------------------------------------------------------------*) paulson@14323: (* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) paulson@14323: (* many of the theorems are not used - so should they be kept? *) paulson@14323: (*----------------------------------------------------------------------------*) paulson@14323: huffman@20725: lemma complex_of_real_zero_iff: "(complex_of_real y = 0) = (y = 0)" huffman@20725: by (rule of_real_eq_0_iff) paulson@14354: paulson@14354: lemma cos_arg_i_mult_zero_pos: paulson@14377: "0 < y ==> cos (arg(Complex 0 y)) = 0" paulson@14373: apply (simp add: arg_def abs_if) paulson@14334: apply (rule_tac a = "pi/2" in someI2, auto) paulson@14334: apply (rule order_less_trans [of _ 0], auto) paulson@14323: done paulson@14323: paulson@14354: lemma cos_arg_i_mult_zero_neg: paulson@14377: "y < 0 ==> cos (arg(Complex 0 y)) = 0" paulson@14373: apply (simp add: arg_def abs_if) paulson@14334: apply (rule_tac a = "- pi/2" in someI2, auto) paulson@14334: apply (rule order_trans [of _ 0], auto) paulson@14323: done paulson@14323: paulson@14374: lemma cos_arg_i_mult_zero [simp]: paulson@14377: "y \ 0 ==> cos (arg(Complex 0 y)) = 0" paulson@14377: by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) paulson@14323: paulson@14323: paulson@14323: subsection{*Finally! Polar Form for Complex Numbers*} paulson@14323: huffman@20557: definition huffman@20557: huffman@20557: (* abbreviation for (cos a + i sin a) *) wenzelm@21404: cis :: "real => complex" where huffman@20557: "cis a = Complex (cos a) (sin a)" huffman@20557: wenzelm@21404: definition huffman@20557: (* abbreviation for r*(cos a + i sin a) *) wenzelm@21404: rcis :: "[real, real] => complex" where huffman@20557: "rcis r a = complex_of_real r * cis a" huffman@20557: wenzelm@21404: definition huffman@20557: (* e ^ (x + iy) *) wenzelm@21404: expi :: "complex => complex" where huffman@20557: "expi z = complex_of_real(exp (Re z)) * cis (Im z)" huffman@20557: paulson@14374: lemma complex_split_polar: paulson@14377: "\r a. z = complex_of_real r * (Complex (cos a) (sin a))" huffman@20725: apply (induct z) paulson@14377: apply (auto simp add: polar_Ex complex_of_real_mult_Complex) paulson@14323: done paulson@14323: paulson@14354: lemma rcis_Ex: "\r a. z = rcis r a" huffman@20725: apply (induct z) paulson@14377: apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex) paulson@14323: done paulson@14323: paulson@14374: lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" paulson@14373: by (simp add: rcis_def cis_def) paulson@14323: paulson@14348: lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" paulson@14373: by (simp add: rcis_def cis_def) paulson@14323: paulson@14377: lemma sin_cos_squared_add2_mult: "(r * cos a)\ + (r * sin a)\ = r\" paulson@14377: proof - paulson@14377: have "(r * cos a)\ + (r * sin a)\ = r\ * ((cos a)\ + (sin a)\)" huffman@20725: by (simp only: power_mult_distrib right_distrib) paulson@14377: thus ?thesis by simp paulson@14377: qed paulson@14323: paulson@14374: lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" paulson@14377: by (simp add: rcis_def cis_def sin_cos_squared_add2_mult) paulson@14323: paulson@14323: lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" paulson@14373: apply (simp add: cmod_def) paulson@14323: apply (rule real_sqrt_eq_iff [THEN iffD2]) huffman@20725: apply (auto simp add: complex_mult_cnj huffman@20725: simp del: of_real_add) paulson@14323: done paulson@14323: paulson@14374: lemma complex_Re_cnj [simp]: "Re(cnj z) = Re z" paulson@14373: by (induct z, simp add: complex_cnj) paulson@14323: paulson@14374: lemma complex_Im_cnj [simp]: "Im(cnj z) = - Im z" paulson@14374: by (induct z, simp add: complex_cnj) paulson@14374: paulson@14374: lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" paulson@14373: by (induct z, simp add: complex_cnj complex_mult) paulson@14323: paulson@14323: paulson@14323: (*---------------------------------------------------------------------------*) paulson@14323: (* (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b) *) paulson@14323: (*---------------------------------------------------------------------------*) paulson@14323: paulson@14323: lemma cis_rcis_eq: "cis a = rcis 1 a" paulson@14373: by (simp add: rcis_def) paulson@14323: paulson@14374: lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" paulson@15013: by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib paulson@15013: complex_of_real_def) paulson@14323: paulson@14323: lemma cis_mult: "cis a * cis b = cis (a + b)" paulson@14373: by (simp add: cis_rcis_eq rcis_mult) paulson@14323: paulson@14374: lemma cis_zero [simp]: "cis 0 = 1" paulson@14377: by (simp add: cis_def complex_one_def) paulson@14323: paulson@14374: lemma rcis_zero_mod [simp]: "rcis 0 a = 0" paulson@14373: by (simp add: rcis_def) paulson@14323: paulson@14374: lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" paulson@14373: by (simp add: rcis_def) paulson@14323: paulson@14323: lemma complex_of_real_minus_one: paulson@14323: "complex_of_real (-(1::real)) = -(1::complex)" huffman@20725: by (simp add: complex_of_real_def complex_one_def) paulson@14323: paulson@14374: lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x" paulson@14373: by (simp add: complex_mult_assoc [symmetric]) paulson@14323: paulson@14323: paulson@14323: lemma cis_real_of_nat_Suc_mult: paulson@14323: "cis (real (Suc n) * a) = cis a * cis (real n * a)" paulson@14377: by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib) paulson@14323: paulson@14323: lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" paulson@14323: apply (induct_tac "n") paulson@14323: apply (auto simp add: cis_real_of_nat_Suc_mult) paulson@14323: done paulson@14323: paulson@14374: lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" paulson@14374: by (simp add: rcis_def power_mult_distrib DeMoivre complex_of_real_pow) paulson@14323: paulson@14374: lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)" huffman@20725: by (simp add: cis_def complex_inverse_complex_split diff_minus) paulson@14323: paulson@14323: lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)" paulson@14430: by (simp add: divide_inverse rcis_def complex_of_real_inverse) paulson@14323: paulson@14323: lemma cis_divide: "cis a / cis b = cis (a - b)" paulson@14373: by (simp add: complex_divide_def cis_mult real_diff_def) paulson@14323: paulson@14354: lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)" paulson@14373: apply (simp add: complex_divide_def) paulson@14373: apply (case_tac "r2=0", simp) paulson@14373: apply (simp add: rcis_inverse rcis_mult real_diff_def) paulson@14323: done paulson@14323: paulson@14374: lemma Re_cis [simp]: "Re(cis a) = cos a" paulson@14373: by (simp add: cis_def) paulson@14323: paulson@14374: lemma Im_cis [simp]: "Im(cis a) = sin a" paulson@14373: by (simp add: cis_def) paulson@14323: paulson@14323: lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" paulson@14334: by (auto simp add: DeMoivre) paulson@14323: paulson@14323: lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" paulson@14334: by (auto simp add: DeMoivre) paulson@14323: paulson@14323: lemma expi_add: "expi(a + b) = expi(a) * expi(b)" huffman@20725: by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac) paulson@14323: paulson@14374: lemma expi_zero [simp]: "expi (0::complex) = 1" paulson@14373: by (simp add: expi_def) paulson@14323: paulson@14374: lemma complex_expi_Ex: "\a r. z = complex_of_real r * expi a" paulson@14373: apply (insert rcis_Ex [of z]) huffman@20557: apply (auto simp add: expi_def rcis_def complex_mult_assoc [symmetric] of_real_mult) paulson@14334: apply (rule_tac x = "ii * complex_of_real a" in exI, auto) paulson@14323: done paulson@14323: paulson@14323: paulson@14387: subsection{*Numerals and Arithmetic*} paulson@14387: paulson@14387: instance complex :: number .. paulson@14387: paulson@15013: defs (overloaded) haftmann@20485: complex_number_of_def: "(number_of w :: complex) == of_int w" paulson@15013: --{*the type constraint is essential!*} paulson@14387: paulson@14387: instance complex :: number_ring huffman@20725: by (intro_classes, simp add: complex_number_of_def) paulson@15013: paulson@15013: paulson@14387: text{*Collapse applications of @{term complex_of_real} to @{term number_of}*} paulson@14387: lemma complex_number_of [simp]: "complex_of_real (number_of w) = number_of w" huffman@20557: by (rule of_real_number_of_eq) paulson@14387: paulson@14387: text{*This theorem is necessary because theorems such as paulson@14387: @{text iszero_number_of_0} only hold for ordered rings. They cannot paulson@14387: be generalized to fields in general because they fail for finite fields. paulson@14387: They work for type complex because the reals can be embedded in them.*} huffman@20557: (* TODO: generalize and move to Real/RealVector.thy *) paulson@14387: lemma iszero_complex_number_of [simp]: paulson@14387: "iszero (number_of w :: complex) = iszero (number_of w :: real)" paulson@14387: by (simp only: complex_of_real_zero_iff complex_number_of [symmetric] huffman@20725: iszero_def) paulson@14387: paulson@14387: lemma complex_number_of_cnj [simp]: "cnj(number_of v :: complex) = number_of v" paulson@15481: by (simp only: complex_number_of [symmetric] complex_cnj_complex_of_real) paulson@14387: paulson@14387: lemma complex_number_of_cmod: paulson@14387: "cmod(number_of v :: complex) = abs (number_of v :: real)" paulson@14387: by (simp only: complex_number_of [symmetric] complex_mod_complex_of_real) paulson@14387: paulson@14387: lemma complex_number_of_Re [simp]: "Re(number_of v :: complex) = number_of v" paulson@14387: by (simp only: complex_number_of [symmetric] Re_complex_of_real) paulson@14387: paulson@14387: lemma complex_number_of_Im [simp]: "Im(number_of v :: complex) = 0" paulson@14387: by (simp only: complex_number_of [symmetric] Im_complex_of_real) paulson@14387: paulson@14387: lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1" paulson@14387: by (simp add: expi_def complex_Re_mult_eq complex_Im_mult_eq cis_def) paulson@14387: paulson@14387: paulson@14387: (*examples: paulson@14387: print_depth 22 paulson@14387: set timing; paulson@14387: set trace_simp; paulson@14387: fun test s = (Goal s, by (Simp_tac 1)); paulson@14387: paulson@14387: test "23 * ii + 45 * ii= (x::complex)"; paulson@14387: paulson@14387: test "5 * ii + 12 - 45 * ii= (x::complex)"; paulson@14387: test "5 * ii + 40 - 12 * ii + 9 = (x::complex) + 89 * ii"; paulson@14387: test "5 * ii + 40 - 12 * ii + 9 - 78 = (x::complex) + 89 * ii"; paulson@14387: paulson@14387: test "l + 10 * ii + 90 + 3*l + 9 + 45 * ii= (x::complex)"; paulson@14387: test "87 + 10 * ii + 90 + 3*7 + 9 + 45 * ii= (x::complex)"; paulson@14387: paulson@14387: paulson@14387: fun test s = (Goal s; by (Asm_simp_tac 1)); paulson@14387: paulson@14387: test "x*k = k*(y::complex)"; paulson@14387: test "k = k*(y::complex)"; paulson@14387: test "a*(b*c) = (b::complex)"; paulson@14387: test "a*(b*c) = d*(b::complex)*(x*a)"; paulson@14387: paulson@14387: paulson@14387: test "(x*k) / (k*(y::complex)) = (uu::complex)"; paulson@14387: test "(k) / (k*(y::complex)) = (uu::complex)"; paulson@14387: test "(a*(b*c)) / ((b::complex)) = (uu::complex)"; paulson@14387: test "(a*(b*c)) / (d*(b::complex)*(x*a)) = (uu::complex)"; paulson@14387: paulson@15003: FIXME: what do we do about this? paulson@14387: test "a*(b*c)/(y*z) = d*(b::complex)*(x*a)/z"; paulson@14387: *) paulson@14387: paulson@13957: end