(* Title: Complex.thy
Author: Jacques D. Fleuriot
Copyright: 2001 University of Edinburgh
*)
header {* Complex numbers *}
theory Complex = HLog:
subsection {* Representation of complex numbers *}
datatype complex = Complex real real
instance complex :: zero ..
instance complex :: one ..
instance complex :: plus ..
instance complex :: times ..
instance complex :: minus ..
instance complex :: inverse ..
instance complex :: power ..
consts
"ii" :: complex ("\<i>")
consts Re :: "complex => real"
primrec "Re (Complex x y) = x"
consts Im :: "complex => real"
primrec "Im (Complex x y) = y"
lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"
by (induct z) simp
constdefs
(*----------- modulus ------------*)
cmod :: "complex => real"
"cmod z == sqrt(Re(z) ^ 2 + Im(z) ^ 2)"
(*----- injection from reals -----*)
complex_of_real :: "real => complex"
"complex_of_real r == Complex r 0"
(*------- complex conjugate ------*)
cnj :: "complex => complex"
"cnj z == Complex (Re z) (-Im z)"
(*------------ Argand -------------*)
sgn :: "complex => complex"
"sgn z == z / complex_of_real(cmod z)"
arg :: "complex => real"
"arg z == @a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi"
defs (overloaded)
complex_zero_def:
"0 == Complex 0 0"
complex_one_def:
"1 == Complex 1 0"
i_def: "ii == Complex 0 1"
complex_minus_def: "- z == Complex (- Re z) (- Im z)"
complex_inverse_def:
"inverse z ==
Complex (Re z / ((Re z)\<twosuperior> + (Im z)\<twosuperior>)) (- Im z / ((Re z)\<twosuperior> + (Im z)\<twosuperior>))"
complex_add_def:
"z + w == Complex (Re z + Re w) (Im z + Im w)"
complex_diff_def:
"z - w == z + - (w::complex)"
complex_mult_def:
"z * w == Complex (Re z * Re w - Im z * Im w) (Re z * Im w + Im z * Re w)"
complex_divide_def: "w / (z::complex) == w * inverse z"
constdefs
(* abbreviation for (cos a + i sin a) *)
cis :: "real => complex"
"cis a == complex_of_real(cos a) + ii * complex_of_real(sin a)"
(* abbreviation for r*(cos a + i sin a) *)
rcis :: "[real, real] => complex"
"rcis r a == complex_of_real r * cis a"
(* e ^ (x + iy) *)
expi :: "complex => complex"
"expi z == complex_of_real(exp (Re z)) * cis (Im z)"
lemma complex_equality [intro?]: "Re z = Re w ==> Im z = Im w ==> z = w"
by (induct z, induct w) simp
lemma Re: "Re(Complex x y) = x"
by simp
declare Re [simp]
lemma Im: "Im(Complex x y) = y"
by simp
declare Im [simp]
lemma complex_Re_Im_cancel_iff: "(w=z) = (Re(w) = Re(z) & Im(w) = Im(z))"
by (induct w, induct z, simp)
lemma complex_Re_zero: "Re 0 = 0"
by (simp add: complex_zero_def)
lemma complex_Im_zero: "Im 0 = 0"
by (simp add: complex_zero_def)
declare complex_Re_zero [simp] complex_Im_zero [simp]
lemma complex_Re_one: "Re 1 = 1"
by (simp add: complex_one_def)
declare complex_Re_one [simp]
lemma complex_Im_one: "Im 1 = 0"
by (simp add: complex_one_def)
declare complex_Im_one [simp]
lemma complex_Re_i: "Re(ii) = 0"
by (simp add: i_def)
declare complex_Re_i [simp]
lemma complex_Im_i: "Im(ii) = 1"
by (simp add: i_def)
declare complex_Im_i [simp]
lemma Re_complex_of_real_zero: "Re(complex_of_real 0) = 0"
by (simp add: complex_of_real_def)
declare Re_complex_of_real_zero [simp]
lemma Im_complex_of_real_zero: "Im(complex_of_real 0) = 0"
by (simp add: complex_of_real_def)
declare Im_complex_of_real_zero [simp]
lemma Re_complex_of_real_one: "Re(complex_of_real 1) = 1"
by (simp add: complex_of_real_def)
declare Re_complex_of_real_one [simp]
lemma Im_complex_of_real_one: "Im(complex_of_real 1) = 0"
by (simp add: complex_of_real_def)
declare Im_complex_of_real_one [simp]
lemma Re_complex_of_real: "Re(complex_of_real z) = z"
by (simp add: complex_of_real_def)
declare Re_complex_of_real [simp]
lemma Im_complex_of_real: "Im(complex_of_real z) = 0"
by (simp add: complex_of_real_def)
declare Im_complex_of_real [simp]
subsection{*Negation*}
lemma complex_minus: "- (Complex x y) = Complex (-x) (-y)"
by (simp add: complex_minus_def)
lemma complex_Re_minus: "Re (-z) = - Re z"
by (simp add: complex_minus_def)
lemma complex_Im_minus: "Im (-z) = - Im z"
by (simp add: complex_minus_def)
lemma complex_minus_zero: "-(0::complex) = 0"
by (simp add: complex_minus_def complex_zero_def)
declare complex_minus_zero [simp]
lemma complex_minus_zero_iff: "(-x = 0) = (x = (0::complex))"
by (induct x, simp add: complex_minus_def complex_zero_def)
declare complex_minus_zero_iff [simp]
subsection{*Addition*}
lemma complex_add: "Complex x1 y1 + Complex x2 y2 = Complex (x1+x2) (y1+y2)"
by (simp add: complex_add_def)
lemma complex_Re_add: "Re(x + y) = Re(x) + Re(y)"
by (simp add: complex_add_def)
lemma complex_Im_add: "Im(x + y) = Im(x) + Im(y)"
by (simp add: complex_add_def)
lemma complex_add_commute: "(u::complex) + v = v + u"
by (simp add: complex_add_def add_commute)
lemma complex_add_assoc: "((u::complex) + v) + w = u + (v + w)"
by (simp add: complex_add_def add_assoc)
lemma complex_add_zero_left: "(0::complex) + z = z"
by (simp add: complex_add_def complex_zero_def)
lemma complex_add_zero_right: "z + (0::complex) = z"
by (simp add: complex_add_def complex_zero_def)
lemma complex_add_minus_left: "-z + z = (0::complex)"
by (simp add: complex_add_def complex_minus_def complex_zero_def)
lemma complex_diff:
"Complex x1 y1 - Complex x2 y2 = Complex (x1-x2) (y1-y2)"
by (simp add: complex_add_def complex_minus_def complex_diff_def)
subsection{*Multiplication*}
lemma complex_mult:
"Complex x1 y1 * Complex x2 y2 = Complex (x1*x2 - y1*y2) (x1*y2 + y1*x2)"
by (simp add: complex_mult_def)
lemma complex_mult_commute: "(w::complex) * z = z * w"
by (simp add: complex_mult_def mult_commute add_commute)
lemma complex_mult_assoc: "((u::complex) * v) * w = u * (v * w)"
by (simp add: complex_mult_def mult_ac add_ac
right_diff_distrib right_distrib left_diff_distrib left_distrib)
lemma complex_mult_one_left: "(1::complex) * z = z"
by (simp add: complex_mult_def complex_one_def)
lemma complex_mult_one_right: "z * (1::complex) = z"
by (simp add: complex_mult_def complex_one_def)
subsection{*Inverse*}
lemma complex_inverse:
"inverse (Complex x y) = Complex (x/(x ^ 2 + y ^ 2)) (-y/(x ^ 2 + y ^ 2))"
by (simp add: complex_inverse_def)
lemma complex_mult_inv_left: "z \<noteq> (0::complex) ==> inverse(z) * z = 1"
apply (induct z)
apply (rename_tac x y)
apply (auto simp add: complex_mult complex_inverse complex_one_def
complex_zero_def add_divide_distrib [symmetric] power2_eq_square mult_ac)
apply (drule_tac y = y in real_sum_squares_not_zero)
apply (drule_tac [2] x = x in real_sum_squares_not_zero2, auto)
done
subsection {* The field of complex numbers *}
instance complex :: field
proof
fix z u v w :: complex
show "(u + v) + w = u + (v + w)"
by (rule complex_add_assoc)
show "z + w = w + z"
by (rule complex_add_commute)
show "0 + z = z"
by (rule complex_add_zero_left)
show "-z + z = 0"
by (rule complex_add_minus_left)
show "z - w = z + -w"
by (simp add: complex_diff_def)
show "(u * v) * w = u * (v * w)"
by (rule complex_mult_assoc)
show "z * w = w * z"
by (rule complex_mult_commute)
show "1 * z = z"
by (rule complex_mult_one_left)
show "0 \<noteq> (1::complex)"
by (simp add: complex_zero_def complex_one_def)
show "(u + v) * w = u * w + v * w"
by (simp add: complex_mult_def complex_add_def left_distrib real_diff_def add_ac)
show "z+u = z+v ==> u=v"
proof -
assume eq: "z+u = z+v"
hence "(-z + z) + u = (-z + z) + v" by (simp only: eq complex_add_assoc)
thus "u = v" by (simp add: complex_add_minus_left complex_add_zero_left)
qed
assume neq: "w \<noteq> 0"
thus "z / w = z * inverse w"
by (simp add: complex_divide_def)
show "inverse w * w = 1"
by (simp add: neq complex_mult_inv_left)
qed
instance complex :: division_by_zero
proof
show inv: "inverse 0 = (0::complex)"
by (simp add: complex_inverse_def complex_zero_def)
fix x :: complex
show "x/0 = 0"
by (simp add: complex_divide_def inv)
qed
subsection{*Embedding Properties for @{term complex_of_real} Map*}
lemma complex_of_real_one: "complex_of_real 1 = 1"
by (simp add: complex_one_def complex_of_real_def)
declare complex_of_real_one [simp]
lemma complex_of_real_zero: "complex_of_real 0 = 0"
by (simp add: complex_zero_def complex_of_real_def)
declare complex_of_real_zero [simp]
lemma complex_of_real_eq_iff:
"(complex_of_real x = complex_of_real y) = (x = y)"
by (simp add: complex_of_real_def)
declare complex_of_real_eq_iff [iff]
lemma complex_of_real_minus: "complex_of_real(-x) = - complex_of_real x"
by (simp add: complex_of_real_def complex_minus)
lemma complex_of_real_inverse:
"complex_of_real(inverse x) = inverse(complex_of_real x)"
apply (case_tac "x=0", simp)
apply (simp add: complex_inverse complex_of_real_def real_divide_def
inverse_mult_distrib power2_eq_square)
done
lemma complex_of_real_add:
"complex_of_real x + complex_of_real y = complex_of_real (x + y)"
by (simp add: complex_add complex_of_real_def)
lemma complex_of_real_diff:
"complex_of_real x - complex_of_real y = complex_of_real (x - y)"
by (simp add: complex_of_real_minus [symmetric] complex_diff_def complex_of_real_add)
lemma complex_of_real_mult:
"complex_of_real x * complex_of_real y = complex_of_real (x * y)"
by (simp add: complex_mult complex_of_real_def)
lemma complex_of_real_divide:
"complex_of_real x / complex_of_real y = complex_of_real(x/y)"
apply (simp add: complex_divide_def)
apply (case_tac "y=0", simp)
apply (simp add: complex_of_real_mult [symmetric] complex_of_real_inverse real_divide_def)
done
lemma complex_mod: "cmod (Complex x y) = sqrt(x ^ 2 + y ^ 2)"
by (simp add: cmod_def)
lemma complex_mod_zero: "cmod(0) = 0"
by (simp add: cmod_def)
declare complex_mod_zero [simp]
lemma complex_mod_one [simp]: "cmod(1) = 1"
by (simp add: cmod_def power2_eq_square)
lemma complex_mod_complex_of_real: "cmod(complex_of_real x) = abs x"
by (simp add: complex_of_real_def power2_eq_square complex_mod)
declare complex_mod_complex_of_real [simp]
lemma complex_of_real_abs:
"complex_of_real (abs x) = complex_of_real(cmod(complex_of_real x))"
by simp
subsection{*Conjugation is an Automorphism*}
lemma complex_cnj: "cnj (Complex x y) = Complex x (-y)"
by (simp add: cnj_def)
lemma complex_cnj_cancel_iff: "(cnj x = cnj y) = (x = y)"
by (simp add: cnj_def complex_Re_Im_cancel_iff)
declare complex_cnj_cancel_iff [simp]
lemma complex_cnj_cnj: "cnj (cnj z) = z"
by (simp add: cnj_def)
declare complex_cnj_cnj [simp]
lemma complex_cnj_complex_of_real:
"cnj (complex_of_real x) = complex_of_real x"
by (simp add: complex_of_real_def complex_cnj)
declare complex_cnj_complex_of_real [simp]
lemma complex_mod_cnj: "cmod (cnj z) = cmod z"
by (induct z, simp add: complex_cnj complex_mod power2_eq_square)
declare complex_mod_cnj [simp]
lemma complex_cnj_minus: "cnj (-z) = - cnj z"
by (simp add: cnj_def complex_minus complex_Re_minus complex_Im_minus)
lemma complex_cnj_inverse: "cnj(inverse z) = inverse(cnj z)"
by (induct z, simp add: complex_cnj complex_inverse power2_eq_square)
lemma complex_cnj_add: "cnj(w + z) = cnj(w) + cnj(z)"
by (induct w, induct z, simp add: complex_cnj complex_add)
lemma complex_cnj_diff: "cnj(w - z) = cnj(w) - cnj(z)"
by (simp add: complex_diff_def complex_cnj_add complex_cnj_minus)
lemma complex_cnj_mult: "cnj(w * z) = cnj(w) * cnj(z)"
by (induct w, induct z, simp add: complex_cnj complex_mult)
lemma complex_cnj_divide: "cnj(w / z) = (cnj w)/(cnj z)"
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
lemma complex_cnj_one: "cnj 1 = 1"
by (simp add: cnj_def complex_one_def)
declare complex_cnj_one [simp]
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re(z))"
by (induct z, simp add: complex_add complex_cnj complex_of_real_def)
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im(z)) * ii"
apply (induct z)
apply (simp add: complex_add complex_cnj complex_of_real_def complex_diff_def
complex_minus i_def complex_mult)
done
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
by (simp add: cnj_def complex_zero_def)
lemma complex_cnj_zero_iff: "(cnj z = 0) = (z = 0)"
by (induct z, simp add: complex_zero_def complex_cnj)
declare complex_cnj_zero_iff [iff]
lemma complex_mult_cnj: "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)"
by (induct z, simp add: complex_cnj complex_mult complex_of_real_def power2_eq_square)
subsection{*Algebra*}
lemma complex_add_left_cancel_zero: "(x + y = x) = (y = (0::complex))"
by (induct x, induct y, simp add: complex_zero_def complex_add)
declare complex_add_left_cancel_zero [simp]
lemma complex_diff_mult_distrib: "((z1::complex) - z2) * w = (z1 * w) - (z2 * w)"
by (simp add: complex_diff_def left_distrib)
lemma complex_diff_mult_distrib2: "(w::complex) * (z1 - z2) = (w * z1) - (w * z2)"
by (simp add: complex_diff_def right_distrib)
subsection{*Modulus*}
lemma complex_mod_eq_zero_cancel: "(cmod x = 0) = (x = 0)"
apply (induct x)
apply (auto intro: real_sum_squares_cancel real_sum_squares_cancel2
simp add: complex_mod complex_zero_def power2_eq_square)
done
declare complex_mod_eq_zero_cancel [simp]
lemma complex_mod_complex_of_real_of_nat:
"cmod (complex_of_real(real (n::nat))) = real n"
by simp
declare complex_mod_complex_of_real_of_nat [simp]
lemma complex_mod_minus: "cmod (-x) = cmod(x)"
by (induct x, simp add: complex_mod complex_minus power2_eq_square)
declare complex_mod_minus [simp]
lemma complex_mod_mult_cnj: "cmod(z * cnj(z)) = cmod(z) ^ 2"
apply (induct z, simp add: complex_mod complex_cnj complex_mult)
apply (simp add: power2_eq_square real_abs_def)
done
lemma complex_mod_squared: "cmod(Complex x y) ^ 2 = x ^ 2 + y ^ 2"
by (simp add: cmod_def)
lemma complex_mod_ge_zero: "0 \<le> cmod x"
by (simp add: cmod_def)
declare complex_mod_ge_zero [simp]
lemma abs_cmod_cancel: "abs(cmod x) = cmod x"
by (simp add: abs_if linorder_not_less)
declare abs_cmod_cancel [simp]
lemma complex_mod_mult: "cmod(x*y) = cmod(x) * cmod(y)"
apply (induct x, induct y)
apply (auto simp add: complex_mult complex_mod real_sqrt_mult_distrib2 [symmetric] simp del: realpow_Suc)
apply (rule_tac n = 1 in power_inject_base)
apply (auto simp add: power2_eq_square [symmetric] simp del: realpow_Suc)
apply (auto simp add: real_diff_def power2_eq_square right_distrib left_distrib add_ac mult_ac)
done
lemma complex_mod_add_squared_eq: "cmod(x + y) ^ 2 = cmod(x) ^ 2 + cmod(y) ^ 2 + 2 * Re(x * cnj y)"
apply (induct x, induct y)
apply (auto simp add: complex_add complex_mod_squared complex_mult complex_cnj real_diff_def simp del: realpow_Suc)
apply (auto simp add: right_distrib left_distrib power2_eq_square mult_ac add_ac)
done
lemma complex_Re_mult_cnj_le_cmod: "Re(x * cnj y) \<le> cmod(x * cnj y)"
apply (induct x, induct y)
apply (auto simp add: complex_mod complex_mult complex_cnj real_diff_def simp del: realpow_Suc)
done
declare complex_Re_mult_cnj_le_cmod [simp]
lemma complex_Re_mult_cnj_le_cmod2: "Re(x * cnj y) \<le> cmod(x * y)"
by (insert complex_Re_mult_cnj_le_cmod [of x y], simp add: complex_mod_mult)
declare complex_Re_mult_cnj_le_cmod2 [simp]
lemma real_sum_squared_expand: "((x::real) + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y"
by (simp add: left_distrib right_distrib power2_eq_square)
lemma complex_mod_triangle_squared: "cmod (x + y) ^ 2 \<le> (cmod(x) + cmod(y)) ^ 2"
by (simp add: real_sum_squared_expand complex_mod_add_squared_eq real_mult_assoc complex_mod_mult [symmetric])
declare complex_mod_triangle_squared [simp]
lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
by (rule order_trans [OF _ complex_mod_ge_zero], simp)
declare complex_mod_minus_le_complex_mod [simp]
lemma complex_mod_triangle_ineq: "cmod (x + y) \<le> cmod(x) + cmod(y)"
apply (rule_tac n = 1 in realpow_increasing)
apply (auto intro: order_trans [OF _ complex_mod_ge_zero]
simp add: power2_eq_square [symmetric])
done
declare complex_mod_triangle_ineq [simp]
lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \<le> cmod a"
by (insert complex_mod_triangle_ineq [THEN add_right_mono, of b a"-cmod b"], simp)
declare complex_mod_triangle_ineq2 [simp]
lemma complex_mod_diff_commute: "cmod (x - y) = cmod (y - x)"
apply (induct x, induct y)
apply (auto simp add: complex_diff complex_mod right_diff_distrib power2_eq_square left_diff_distrib add_ac mult_ac)
done
lemma complex_mod_add_less: "[| cmod x < r; cmod y < s |] ==> cmod (x + y) < r + s"
by (auto intro: order_le_less_trans complex_mod_triangle_ineq)
lemma complex_mod_mult_less: "[| cmod x < r; cmod y < s |] ==> cmod (x * y) < r * s"
by (auto intro: real_mult_less_mono' simp add: complex_mod_mult)
lemma complex_mod_diff_ineq: "cmod(a) - cmod(b) \<le> cmod(a + b)"
apply (rule linorder_cases [of "cmod(a)" "cmod (b)"])
apply auto
apply (rule order_trans [of _ 0], rule order_less_imp_le)
apply (simp add: compare_rls, simp)
apply (simp add: compare_rls)
apply (rule complex_mod_minus [THEN subst])
apply (rule order_trans)
apply (rule_tac [2] complex_mod_triangle_ineq)
apply (auto simp add: add_ac)
done
declare complex_mod_diff_ineq [simp]
lemma complex_Re_le_cmod: "Re z \<le> cmod z"
by (induct z, simp add: complex_mod del: realpow_Suc)
declare complex_Re_le_cmod [simp]
lemma complex_mod_gt_zero: "z \<noteq> 0 ==> 0 < cmod z"
apply (insert complex_mod_ge_zero [of z])
apply (drule order_le_imp_less_or_eq, auto)
done
subsection{*A Few More Theorems*}
lemma complex_mod_inverse: "cmod(inverse x) = inverse(cmod x)"
apply (case_tac "x=0", simp)
apply (rule_tac c1 = "cmod x" in real_mult_left_cancel [THEN iffD1])
apply (auto simp add: complex_mod_mult [symmetric])
done
lemma complex_mod_divide: "cmod(x/y) = cmod(x)/(cmod y)"
by (simp add: complex_divide_def real_divide_def, simp add: complex_mod_mult complex_mod_inverse)
lemma complex_inverse_divide: "inverse(x/y) = y/(x::complex)"
by (simp add: complex_divide_def inverse_mult_distrib mult_commute)
declare complex_inverse_divide [simp]
subsection{*Exponentiation*}
primrec
complexpow_0: "z ^ 0 = 1"
complexpow_Suc: "z ^ (Suc n) = (z::complex) * (z ^ n)"
instance complex :: ringpower
proof
fix z :: complex
fix n :: nat
show "z^0 = 1" by simp
show "z^(Suc n) = z * (z^n)" by simp
qed
lemma complex_of_real_pow: "complex_of_real (x ^ n) = (complex_of_real x) ^ n"
apply (induct_tac "n")
apply (auto simp add: complex_of_real_mult [symmetric])
done
lemma complex_cnj_pow: "cnj(z ^ n) = cnj(z) ^ n"
apply (induct_tac "n")
apply (auto simp add: complex_cnj_mult)
done
lemma complex_mod_complexpow: "cmod(x ^ n) = cmod(x) ^ n"
apply (induct_tac "n")
apply (auto simp add: complex_mod_mult)
done
lemma complexpow_minus: "(-x::complex) ^ n = (if even n then (x ^ n) else -(x ^ n))"
by (induct_tac "n", auto)
lemma complexpow_i_squared [simp]: "ii ^ 2 = -(1::complex)"
by (simp add: i_def complex_mult complex_one_def complex_minus numeral_2_eq_2)
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
by (simp add: i_def complex_zero_def)
subsection{*The Function @{term sgn}*}
lemma sgn_zero: "sgn 0 = 0"
by (simp add: sgn_def)
declare sgn_zero [simp]
lemma sgn_one: "sgn 1 = 1"
by (simp add: sgn_def)
declare sgn_one [simp]
lemma sgn_minus: "sgn (-z) = - sgn(z)"
by (simp add: sgn_def)
lemma sgn_eq:
"sgn z = z / complex_of_real (cmod z)"
apply (simp add: sgn_def)
done
lemma complex_split: "\<exists>x y. z = complex_of_real(x) + ii * complex_of_real(y)"
apply (induct z)
apply (auto simp add: complex_of_real_def i_def complex_mult complex_add)
done
lemma Re_complex_i: "Re(complex_of_real(x) + ii * complex_of_real(y)) = x"
by (auto simp add: complex_of_real_def i_def complex_mult complex_add)
declare Re_complex_i [simp]
lemma Im_complex_i: "Im(complex_of_real(x) + ii * complex_of_real(y)) = y"
by (auto simp add: complex_of_real_def i_def complex_mult complex_add)
declare Im_complex_i [simp]
lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
by (simp add: i_def complex_of_real_def complex_mult complex_add)
lemma i_mult_eq2: "ii * ii = -(1::complex)"
by (simp add: i_def complex_one_def complex_mult complex_minus)
declare i_mult_eq2 [simp]
lemma cmod_i: "cmod (complex_of_real(x) + ii * complex_of_real(y)) =
sqrt (x ^ 2 + y ^ 2)"
by (simp add: complex_mult complex_add i_def complex_of_real_def cmod_def)
lemma complex_eq_Re_eq:
"complex_of_real xa + ii * complex_of_real ya =
complex_of_real xb + ii * complex_of_real yb
==> xa = xb"
by (simp add: complex_of_real_def i_def complex_mult complex_add)
lemma complex_eq_Im_eq:
"complex_of_real xa + ii * complex_of_real ya =
complex_of_real xb + ii * complex_of_real yb
==> ya = yb"
by (simp add: complex_of_real_def i_def complex_mult complex_add)
lemma complex_eq_cancel_iff: "(complex_of_real xa + ii * complex_of_real ya =
complex_of_real xb + ii * complex_of_real yb) = ((xa = xb) & (ya = yb))"
by (auto intro: complex_eq_Im_eq complex_eq_Re_eq)
declare complex_eq_cancel_iff [iff]
lemma complex_eq_cancel_iffA: "(complex_of_real xa + complex_of_real ya * ii =
complex_of_real xb + complex_of_real yb * ii) = ((xa = xb) & (ya = yb))"
by (simp add: mult_commute)
declare complex_eq_cancel_iffA [iff]
lemma complex_eq_cancel_iffB: "(complex_of_real xa + complex_of_real ya * ii =
complex_of_real xb + ii * complex_of_real yb) = ((xa = xb) & (ya = yb))"
by (auto simp add: mult_commute)
declare complex_eq_cancel_iffB [iff]
lemma complex_eq_cancel_iffC: "(complex_of_real xa + ii * complex_of_real ya =
complex_of_real xb + complex_of_real yb * ii) = ((xa = xb) & (ya = yb))"
by (auto simp add: mult_commute)
declare complex_eq_cancel_iffC [iff]
lemma complex_eq_cancel_iff2: "(complex_of_real x + ii * complex_of_real y =
complex_of_real xa) = (x = xa & y = 0)"
apply (cut_tac xa = x and ya = y and xb = xa and yb = 0 in complex_eq_cancel_iff)
apply (simp del: complex_eq_cancel_iff)
done
declare complex_eq_cancel_iff2 [simp]
lemma complex_eq_cancel_iff2a: "(complex_of_real x + complex_of_real y * ii =
complex_of_real xa) = (x = xa & y = 0)"
by (auto simp add: mult_commute)
declare complex_eq_cancel_iff2a [simp]
lemma complex_eq_cancel_iff3: "(complex_of_real x + ii * complex_of_real y =
ii * complex_of_real ya) = (x = 0 & y = ya)"
apply (cut_tac xa = x and ya = y and xb = 0 and yb = ya in complex_eq_cancel_iff)
apply (simp del: complex_eq_cancel_iff)
done
declare complex_eq_cancel_iff3 [simp]
lemma complex_eq_cancel_iff3a: "(complex_of_real x + complex_of_real y * ii =
ii * complex_of_real ya) = (x = 0 & y = ya)"
by (auto simp add: mult_commute)
declare complex_eq_cancel_iff3a [simp]
lemma complex_split_Re_zero:
"complex_of_real x + ii * complex_of_real y = 0
==> x = 0"
by (simp add: complex_of_real_def i_def complex_zero_def complex_mult complex_add)
lemma complex_split_Im_zero:
"complex_of_real x + ii * complex_of_real y = 0
==> y = 0"
by (simp add: complex_of_real_def i_def complex_zero_def complex_mult complex_add)
lemma Re_sgn: "Re(sgn z) = Re(z)/cmod z"
apply (induct z)
apply (simp add: sgn_def complex_divide_def complex_of_real_inverse [symmetric])
apply (simp add: complex_of_real_def complex_mult real_divide_def)
done
declare Re_sgn [simp]
lemma Im_sgn:
"Im(sgn z) = Im(z)/cmod z"
apply (induct z)
apply (simp add: sgn_def complex_divide_def complex_of_real_inverse [symmetric])
apply (simp add: complex_of_real_def complex_mult real_divide_def)
done
declare Im_sgn [simp]
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 complex_mult complex_add
complex_diff_def complex_minus complex_inverse real_divide_def)
(*----------------------------------------------------------------------------*)
(* 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 complex_of_real_zero_iff [simp]: "(complex_of_real y = 0) = (y = 0)"
by (auto simp add: complex_zero_def complex_of_real_def)
lemma Re_mult_i_eq: "Re (ii * complex_of_real y) = 0"
by (simp add: i_def complex_of_real_def complex_mult)
declare Re_mult_i_eq [simp]
lemma Im_mult_i_eq: "Im (ii * complex_of_real y) = y"
by (simp add: i_def complex_of_real_def complex_mult)
declare Im_mult_i_eq [simp]
lemma complex_mod_mult_i: "cmod (ii * complex_of_real y) = abs y"
by (simp add: i_def complex_of_real_def complex_mult complex_mod power2_eq_square)
declare complex_mod_mult_i [simp]
lemma cos_arg_i_mult_zero_pos:
"0 < y ==> cos (arg(ii * complex_of_real 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(ii * complex_of_real 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 \<noteq> 0 ==> cos (arg(ii * complex_of_real y)) = 0"
apply (insert linorder_less_linear [of y 0])
apply (auto simp add: cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
done
subsection{*Finally! Polar Form for Complex Numbers*}
lemma complex_split_polar: "\<exists>r a. z = complex_of_real r *
(complex_of_real(cos a) + ii * complex_of_real(sin a))"
apply (cut_tac z = z in complex_split)
apply (auto simp add: polar_Ex right_distrib complex_of_real_mult mult_ac)
done
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
apply (simp add: rcis_def cis_def)
apply (rule complex_split_polar)
done
lemma Re_complex_polar: "Re(complex_of_real r *
(complex_of_real(cos a) + ii * complex_of_real(sin a))) = r * cos a"
by (auto simp add: right_distrib complex_of_real_mult mult_ac)
declare Re_complex_polar [simp]
lemma Re_rcis: "Re(rcis r a) = r * cos a"
by (simp add: rcis_def cis_def)
declare Re_rcis [simp]
lemma Im_complex_polar [simp]:
"Im(complex_of_real r *
(complex_of_real(cos a) + ii * complex_of_real(sin a))) =
r * sin a"
by (auto simp add: right_distrib complex_of_real_mult mult_ac)
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
by (simp add: rcis_def cis_def)
lemma complex_mod_complex_polar [simp]:
"cmod (complex_of_real r *
(complex_of_real(cos a) + ii * complex_of_real(sin a))) =
abs r"
by (auto simp add: right_distrib cmod_i complex_of_real_mult
right_distrib [symmetric] power_mult_distrib mult_ac
simp del: realpow_Suc)
lemma complex_mod_rcis: "cmod(rcis r a) = abs r"
by (simp add: rcis_def cis_def)
declare complex_mod_rcis [simp]
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
apply (simp add: cmod_def)
apply (rule real_sqrt_eq_iff [THEN iffD2])
apply (auto simp add: complex_mult_cnj)
done
lemma complex_Re_cnj: "Re(cnj z) = Re z"
by (induct z, simp add: complex_cnj)
declare complex_Re_cnj [simp]
lemma complex_Im_cnj: "Im(cnj z) = - Im z"
by (induct z, simp add: complex_cnj)
declare complex_Im_cnj [simp]
lemma complex_In_mult_cnj_zero: "Im (z * cnj z) = 0"
by (induct z, simp add: complex_cnj complex_mult)
declare complex_In_mult_cnj_zero [simp]
lemma complex_Re_mult: "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)"
by (induct z, induct w, simp add: complex_mult)
lemma complex_Re_mult_complex_of_real: "Re (z * complex_of_real c) = Re(z) * c"
by (induct z, simp add: complex_of_real_def complex_mult)
declare complex_Re_mult_complex_of_real [simp]
lemma complex_Im_mult_complex_of_real: "Im (z * complex_of_real c) = Im(z) * c"
by (induct z, simp add: complex_of_real_def complex_mult)
declare complex_Im_mult_complex_of_real [simp]
lemma complex_Re_mult_complex_of_real2: "Re (complex_of_real c * z) = c * Re(z)"
by (induct z, simp add: complex_of_real_def complex_mult)
declare complex_Re_mult_complex_of_real2 [simp]
lemma complex_Im_mult_complex_of_real2: "Im (complex_of_real c * z) = c * Im(z)"
by (induct z, simp add: complex_of_real_def complex_mult)
declare complex_Im_mult_complex_of_real2 [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)"
apply (simp add: rcis_def cis_def cos_add sin_add right_distrib left_distrib
mult_ac add_ac)
apply (auto simp add: right_distrib [symmetric] complex_mult_assoc [symmetric] complex_of_real_mult complex_of_real_add complex_add_assoc [symmetric] i_mult_eq simp del: i_mult_eq2)
apply (auto simp add: add_ac)
apply (auto simp add: complex_add_assoc [symmetric] complex_of_real_add right_distrib real_diff_def mult_ac add_ac)
done
lemma cis_mult: "cis a * cis b = cis (a + b)"
by (simp add: cis_rcis_eq rcis_mult)
lemma cis_zero: "cis 0 = 1"
by (simp add: cis_def)
declare cis_zero [simp]
lemma cis_zero2: "cis 0 = complex_of_real 1"
by (simp add: cis_def)
declare cis_zero2 [simp]
lemma rcis_zero_mod: "rcis 0 a = 0"
by (simp add: rcis_def)
declare rcis_zero_mod [simp]
lemma rcis_zero_arg: "rcis r 0 = complex_of_real r"
by (simp add: rcis_def)
declare rcis_zero_arg [simp]
lemma complex_of_real_minus_one:
"complex_of_real (-(1::real)) = -(1::complex)"
apply (simp add: complex_of_real_def complex_one_def complex_minus)
done
lemma complex_i_mult_minus: "ii * (ii * x) = - x"
by (simp add: complex_mult_assoc [symmetric])
declare complex_i_mult_minus [simp]
lemma cis_real_of_nat_Suc_mult:
"cis (real (Suc n) * a) = cis a * cis (real n * a)"
apply (simp add: cis_def)
apply (auto simp add: real_of_nat_Suc left_distrib cos_add sin_add left_distrib right_distrib complex_of_real_add complex_of_real_mult mult_ac add_ac)
apply (auto simp add: right_distrib [symmetric] complex_mult_assoc [symmetric] i_mult_eq complex_of_real_mult complex_of_real_add complex_add_assoc [symmetric] complex_of_real_minus [symmetric] real_diff_def mult_ac simp del: i_mult_eq2)
done
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)"
apply (simp add: rcis_def power_mult_distrib DeMoivre complex_of_real_pow)
done
lemma cis_inverse: "inverse(cis a) = cis (-a)"
by (simp add: cis_def complex_inverse_complex_split complex_of_real_minus complex_diff_def)
declare cis_inverse [simp]
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
apply (case_tac "r=0", simp)
apply (auto simp add: complex_inverse_complex_split right_distrib
complex_of_real_mult rcis_def cis_def power2_eq_square mult_ac)
apply (auto simp add: right_distrib [symmetric] complex_of_real_minus complex_diff_def)
done
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: "Re(cis a) = cos a"
by (simp add: cis_def)
declare Re_cis [simp]
lemma Im_cis: "Im(cis a) = sin a"
by (simp add: cis_def)
declare Im_cis [simp]
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_Im_split:
"expi (ii * complex_of_real y) =
complex_of_real (cos y) + ii * complex_of_real (sin y)"
by (simp add: expi_def cis_def)
lemma expi_Im_cis:
"expi (ii * complex_of_real y) = cis y"
by (simp add: expi_def)
lemma expi_add: "expi(a + b) = expi(a) * expi(b)"
by (simp add: expi_def complex_Re_add exp_add complex_Im_add cis_mult [symmetric] complex_of_real_mult mult_ac)
lemma expi_complex_split:
"expi(complex_of_real x + ii * complex_of_real y) =
complex_of_real (exp(x)) * cis y"
by (simp add: expi_def)
lemma expi_zero: "expi (0::complex) = 1"
by (simp add: expi_def)
declare expi_zero [simp]
lemma complex_Re_mult_eq: "Re (w * z) = Re w * Re z - Im w * Im z"
by (induct z, induct w, simp add: complex_mult)
lemma complex_Im_mult_eq:
"Im (w * z) = Re w * Im z + Im w * Re z"
apply (induct z, induct w, simp add: complex_mult)
done
lemma complex_expi_Ex:
"\<exists>a r. z = complex_of_real r * expi a"
apply (insert rcis_Ex [of z])
apply (auto simp add: expi_def rcis_def complex_mult_assoc [symmetric] complex_of_real_mult)
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
done
ML
{*
val complex_zero_def = thm"complex_zero_def";
val complex_one_def = thm"complex_one_def";
val complex_minus_def = thm"complex_minus_def";
val complex_diff_def = thm"complex_diff_def";
val complex_divide_def = thm"complex_divide_def";
val complex_mult_def = thm"complex_mult_def";
val complex_add_def = thm"complex_add_def";
val complex_of_real_def = thm"complex_of_real_def";
val i_def = thm"i_def";
val expi_def = thm"expi_def";
val cis_def = thm"cis_def";
val rcis_def = thm"rcis_def";
val cmod_def = thm"cmod_def";
val cnj_def = thm"cnj_def";
val sgn_def = thm"sgn_def";
val arg_def = thm"arg_def";
val complexpow_0 = thm"complexpow_0";
val complexpow_Suc = thm"complexpow_Suc";
val Re = thm"Re";
val Im = thm"Im";
val complex_Re_Im_cancel_iff = thm"complex_Re_Im_cancel_iff";
val complex_Re_zero = thm"complex_Re_zero";
val complex_Im_zero = thm"complex_Im_zero";
val complex_Re_one = thm"complex_Re_one";
val complex_Im_one = thm"complex_Im_one";
val complex_Re_i = thm"complex_Re_i";
val complex_Im_i = thm"complex_Im_i";
val Re_complex_of_real_zero = thm"Re_complex_of_real_zero";
val Im_complex_of_real_zero = thm"Im_complex_of_real_zero";
val Re_complex_of_real_one = thm"Re_complex_of_real_one";
val Im_complex_of_real_one = thm"Im_complex_of_real_one";
val Re_complex_of_real = thm"Re_complex_of_real";
val Im_complex_of_real = thm"Im_complex_of_real";
val complex_minus = thm"complex_minus";
val complex_Re_minus = thm"complex_Re_minus";
val complex_Im_minus = thm"complex_Im_minus";
val complex_minus_zero = thm"complex_minus_zero";
val complex_minus_zero_iff = thm"complex_minus_zero_iff";
val complex_add = thm"complex_add";
val complex_Re_add = thm"complex_Re_add";
val complex_Im_add = thm"complex_Im_add";
val complex_add_commute = thm"complex_add_commute";
val complex_add_assoc = thm"complex_add_assoc";
val complex_add_zero_left = thm"complex_add_zero_left";
val complex_add_zero_right = thm"complex_add_zero_right";
val complex_diff = thm"complex_diff";
val complex_mult = thm"complex_mult";
val complex_mult_one_left = thm"complex_mult_one_left";
val complex_mult_one_right = thm"complex_mult_one_right";
val complex_inverse = thm"complex_inverse";
val complex_of_real_one = thm"complex_of_real_one";
val complex_of_real_zero = thm"complex_of_real_zero";
val complex_of_real_eq_iff = thm"complex_of_real_eq_iff";
val complex_of_real_minus = thm"complex_of_real_minus";
val complex_of_real_inverse = thm"complex_of_real_inverse";
val complex_of_real_add = thm"complex_of_real_add";
val complex_of_real_diff = thm"complex_of_real_diff";
val complex_of_real_mult = thm"complex_of_real_mult";
val complex_of_real_divide = thm"complex_of_real_divide";
val complex_of_real_pow = thm"complex_of_real_pow";
val complex_mod = thm"complex_mod";
val complex_mod_zero = thm"complex_mod_zero";
val complex_mod_one = thm"complex_mod_one";
val complex_mod_complex_of_real = thm"complex_mod_complex_of_real";
val complex_of_real_abs = thm"complex_of_real_abs";
val complex_cnj = thm"complex_cnj";
val complex_cnj_cancel_iff = thm"complex_cnj_cancel_iff";
val complex_cnj_cnj = thm"complex_cnj_cnj";
val complex_cnj_complex_of_real = thm"complex_cnj_complex_of_real";
val complex_mod_cnj = thm"complex_mod_cnj";
val complex_cnj_minus = thm"complex_cnj_minus";
val complex_cnj_inverse = thm"complex_cnj_inverse";
val complex_cnj_add = thm"complex_cnj_add";
val complex_cnj_diff = thm"complex_cnj_diff";
val complex_cnj_mult = thm"complex_cnj_mult";
val complex_cnj_divide = thm"complex_cnj_divide";
val complex_cnj_one = thm"complex_cnj_one";
val complex_cnj_pow = thm"complex_cnj_pow";
val complex_add_cnj = thm"complex_add_cnj";
val complex_diff_cnj = thm"complex_diff_cnj";
val complex_cnj_zero = thm"complex_cnj_zero";
val complex_cnj_zero_iff = thm"complex_cnj_zero_iff";
val complex_mult_cnj = thm"complex_mult_cnj";
val complex_add_left_cancel_zero = thm"complex_add_left_cancel_zero";
val complex_diff_mult_distrib = thm"complex_diff_mult_distrib";
val complex_diff_mult_distrib2 = thm"complex_diff_mult_distrib2";
val complex_mod_eq_zero_cancel = thm"complex_mod_eq_zero_cancel";
val complex_mod_complex_of_real_of_nat = thm"complex_mod_complex_of_real_of_nat";
val complex_mod_minus = thm"complex_mod_minus";
val complex_mod_mult_cnj = thm"complex_mod_mult_cnj";
val complex_mod_squared = thm"complex_mod_squared";
val complex_mod_ge_zero = thm"complex_mod_ge_zero";
val abs_cmod_cancel = thm"abs_cmod_cancel";
val complex_mod_mult = thm"complex_mod_mult";
val complex_mod_add_squared_eq = thm"complex_mod_add_squared_eq";
val complex_Re_mult_cnj_le_cmod = thm"complex_Re_mult_cnj_le_cmod";
val complex_Re_mult_cnj_le_cmod2 = thm"complex_Re_mult_cnj_le_cmod2";
val real_sum_squared_expand = thm"real_sum_squared_expand";
val complex_mod_triangle_squared = thm"complex_mod_triangle_squared";
val complex_mod_minus_le_complex_mod = thm"complex_mod_minus_le_complex_mod";
val complex_mod_triangle_ineq = thm"complex_mod_triangle_ineq";
val complex_mod_triangle_ineq2 = thm"complex_mod_triangle_ineq2";
val complex_mod_diff_commute = thm"complex_mod_diff_commute";
val complex_mod_add_less = thm"complex_mod_add_less";
val complex_mod_mult_less = thm"complex_mod_mult_less";
val complex_mod_diff_ineq = thm"complex_mod_diff_ineq";
val complex_Re_le_cmod = thm"complex_Re_le_cmod";
val complex_mod_gt_zero = thm"complex_mod_gt_zero";
val complex_mod_complexpow = thm"complex_mod_complexpow";
val complexpow_minus = thm"complexpow_minus";
val complex_mod_inverse = thm"complex_mod_inverse";
val complex_mod_divide = thm"complex_mod_divide";
val complex_inverse_divide = thm"complex_inverse_divide";
val complexpow_i_squared = thm"complexpow_i_squared";
val complex_i_not_zero = thm"complex_i_not_zero";
val sgn_zero = thm"sgn_zero";
val sgn_one = thm"sgn_one";
val sgn_minus = thm"sgn_minus";
val sgn_eq = thm"sgn_eq";
val complex_split = thm"complex_split";
val Re_complex_i = thm"Re_complex_i";
val Im_complex_i = thm"Im_complex_i";
val i_mult_eq = thm"i_mult_eq";
val i_mult_eq2 = thm"i_mult_eq2";
val cmod_i = thm"cmod_i";
val complex_eq_Re_eq = thm"complex_eq_Re_eq";
val complex_eq_Im_eq = thm"complex_eq_Im_eq";
val complex_eq_cancel_iff = thm"complex_eq_cancel_iff";
val complex_eq_cancel_iffA = thm"complex_eq_cancel_iffA";
val complex_eq_cancel_iffB = thm"complex_eq_cancel_iffB";
val complex_eq_cancel_iffC = thm"complex_eq_cancel_iffC";
val complex_eq_cancel_iff2 = thm"complex_eq_cancel_iff2";
val complex_eq_cancel_iff2a = thm"complex_eq_cancel_iff2a";
val complex_eq_cancel_iff3 = thm"complex_eq_cancel_iff3";
val complex_eq_cancel_iff3a = thm"complex_eq_cancel_iff3a";
val complex_split_Re_zero = thm"complex_split_Re_zero";
val complex_split_Im_zero = thm"complex_split_Im_zero";
val Re_sgn = thm"Re_sgn";
val Im_sgn = thm"Im_sgn";
val complex_inverse_complex_split = thm"complex_inverse_complex_split";
val Re_mult_i_eq = thm"Re_mult_i_eq";
val Im_mult_i_eq = thm"Im_mult_i_eq";
val complex_mod_mult_i = thm"complex_mod_mult_i";
val cos_arg_i_mult_zero = thm"cos_arg_i_mult_zero";
val complex_of_real_zero_iff = thm"complex_of_real_zero_iff";
val complex_split_polar = thm"complex_split_polar";
val rcis_Ex = thm"rcis_Ex";
val Re_complex_polar = thm"Re_complex_polar";
val Re_rcis = thm"Re_rcis";
val Im_complex_polar = thm"Im_complex_polar";
val Im_rcis = thm"Im_rcis";
val complex_mod_complex_polar = thm"complex_mod_complex_polar";
val complex_mod_rcis = thm"complex_mod_rcis";
val complex_mod_sqrt_Re_mult_cnj = thm"complex_mod_sqrt_Re_mult_cnj";
val complex_Re_cnj = thm"complex_Re_cnj";
val complex_Im_cnj = thm"complex_Im_cnj";
val complex_In_mult_cnj_zero = thm"complex_In_mult_cnj_zero";
val complex_Re_mult = thm"complex_Re_mult";
val complex_Re_mult_complex_of_real = thm"complex_Re_mult_complex_of_real";
val complex_Im_mult_complex_of_real = thm"complex_Im_mult_complex_of_real";
val complex_Re_mult_complex_of_real2 = thm"complex_Re_mult_complex_of_real2";
val complex_Im_mult_complex_of_real2 = thm"complex_Im_mult_complex_of_real2";
val cis_rcis_eq = thm"cis_rcis_eq";
val rcis_mult = thm"rcis_mult";
val cis_mult = thm"cis_mult";
val cis_zero = thm"cis_zero";
val cis_zero2 = thm"cis_zero2";
val rcis_zero_mod = thm"rcis_zero_mod";
val rcis_zero_arg = thm"rcis_zero_arg";
val complex_of_real_minus_one = thm"complex_of_real_minus_one";
val complex_i_mult_minus = thm"complex_i_mult_minus";
val cis_real_of_nat_Suc_mult = thm"cis_real_of_nat_Suc_mult";
val DeMoivre = thm"DeMoivre";
val DeMoivre2 = thm"DeMoivre2";
val cis_inverse = thm"cis_inverse";
val rcis_inverse = thm"rcis_inverse";
val cis_divide = thm"cis_divide";
val rcis_divide = thm"rcis_divide";
val Re_cis = thm"Re_cis";
val Im_cis = thm"Im_cis";
val cos_n_Re_cis_pow_n = thm"cos_n_Re_cis_pow_n";
val sin_n_Im_cis_pow_n = thm"sin_n_Im_cis_pow_n";
val expi_Im_split = thm"expi_Im_split";
val expi_Im_cis = thm"expi_Im_cis";
val expi_add = thm"expi_add";
val expi_complex_split = thm"expi_complex_split";
val expi_zero = thm"expi_zero";
val complex_Re_mult_eq = thm"complex_Re_mult_eq";
val complex_Im_mult_eq = thm"complex_Im_mult_eq";
val complex_expi_Ex = thm"complex_expi_Ex";
*}
end