wenzelm@41959: (*  Title:       HOL/Complex.thy
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
haftmann@28952: imports Transcendental
nipkow@15131: begin
paulson@13957: 
paulson@14373: datatype complex = Complex real real
paulson@13957: 
huffman@44724: primrec Re :: "complex \<Rightarrow> real"
huffman@44724:   where Re: "Re (Complex x y) = x"
paulson@14373: 
huffman@44724: primrec Im :: "complex \<Rightarrow> real"
huffman@44724:   where 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: 
huffman@44065: lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"
haftmann@25712:   by (induct x, induct y) simp
huffman@23125: 
huffman@44065: lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"
haftmann@25712:   by (induct x, induct y) simp
huffman@23125: 
huffman@23125: 
huffman@23125: subsection {* Addition and Subtraction *}
huffman@23125: 
haftmann@25599: instantiation complex :: ab_group_add
haftmann@25571: begin
haftmann@25571: 
huffman@44724: definition complex_zero_def:
huffman@44724:   "0 = Complex 0 0"
haftmann@25571: 
huffman@44724: definition complex_add_def:
huffman@44724:   "x + y = Complex (Re x + Re y) (Im x + Im y)"
huffman@23124: 
huffman@44724: definition complex_minus_def:
huffman@44724:   "- x = Complex (- Re x) (- Im x)"
paulson@14323: 
huffman@44724: definition complex_diff_def:
huffman@44724:   "x - (y\<Colon>complex) = x + - y"
haftmann@25571: 
haftmann@25599: lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
haftmann@25599:   by (simp add: complex_zero_def)
paulson@14323: 
paulson@14374: lemma complex_Re_zero [simp]: "Re 0 = 0"
haftmann@25599:   by (simp add: complex_zero_def)
paulson@14374: 
paulson@14374: lemma complex_Im_zero [simp]: "Im 0 = 0"
haftmann@25599:   by (simp add: complex_zero_def)
haftmann@25599: 
haftmann@25712: lemma complex_add [simp]:
haftmann@25712:   "Complex a b + Complex c d = Complex (a + c) (b + d)"
haftmann@25712:   by (simp add: complex_add_def)
haftmann@25712: 
haftmann@25599: lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y"
haftmann@25599:   by (simp add: complex_add_def)
haftmann@25599: 
haftmann@25599: lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y"
haftmann@25599:   by (simp add: complex_add_def)
paulson@14323: 
haftmann@25712: lemma complex_minus [simp]:
haftmann@25712:   "- (Complex a b) = Complex (- a) (- b)"
haftmann@25599:   by (simp add: complex_minus_def)
huffman@23125: 
huffman@23125: lemma complex_Re_minus [simp]: "Re (- x) = - Re x"
haftmann@25599:   by (simp add: complex_minus_def)
huffman@23125: 
huffman@23125: lemma complex_Im_minus [simp]: "Im (- x) = - Im x"
haftmann@25599:   by (simp add: complex_minus_def)
huffman@23125: 
huffman@23275: lemma complex_diff [simp]:
huffman@23125:   "Complex a b - Complex c d = Complex (a - c) (b - d)"
haftmann@25599:   by (simp add: complex_diff_def)
huffman@23125: 
huffman@23125: lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y"
haftmann@25599:   by (simp add: complex_diff_def)
huffman@23125: 
huffman@23125: lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y"
haftmann@25599:   by (simp add: complex_diff_def)
huffman@23125: 
haftmann@25712: instance
haftmann@25712:   by intro_classes (simp_all add: complex_add_def complex_diff_def)
haftmann@25712: 
haftmann@25712: end
haftmann@25712: 
haftmann@25712: 
huffman@23125: subsection {* Multiplication and Division *}
huffman@23125: 
haftmann@36409: instantiation complex :: field_inverse_zero
haftmann@25571: begin
haftmann@25571: 
huffman@44724: definition complex_one_def:
huffman@44724:   "1 = Complex 1 0"
haftmann@25571: 
huffman@44724: definition complex_mult_def:
huffman@44724:   "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
huffman@23125: 
huffman@44724: definition complex_inverse_def:
huffman@44724:   "inverse x =
haftmann@25571:     Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))"
huffman@23125: 
huffman@44724: definition complex_divide_def:
huffman@44724:   "x / (y\<Colon>complex) = x * inverse y"
haftmann@25571: 
huffman@23125: lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"
haftmann@25712:   by (simp add: complex_one_def)
huffman@22861: 
paulson@14374: lemma complex_Re_one [simp]: "Re 1 = 1"
haftmann@25712:   by (simp add: complex_one_def)
paulson@14323: 
paulson@14374: lemma complex_Im_one [simp]: "Im 1 = 0"
haftmann@25712:   by (simp add: complex_one_def)
paulson@14323: 
huffman@23125: lemma complex_mult [simp]:
huffman@23125:   "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
haftmann@25712:   by (simp add: complex_mult_def)
paulson@14323: 
huffman@23125: lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"
haftmann@25712:   by (simp add: complex_mult_def)
paulson@14323: 
huffman@23125: lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"
haftmann@25712:   by (simp add: complex_mult_def)
paulson@14323: 
paulson@14377: lemma complex_inverse [simp]:
huffman@23125:   "inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))"
haftmann@25712:   by (simp add: complex_inverse_def)
paulson@14335: 
huffman@23125: lemma complex_Re_inverse:
huffman@23125:   "Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
haftmann@25712:   by (simp add: complex_inverse_def)
paulson@14323: 
huffman@23125: lemma complex_Im_inverse:
huffman@23125:   "Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
haftmann@25712:   by (simp add: complex_inverse_def)
paulson@14335: 
haftmann@25712: instance
haftmann@25712:   by intro_classes (simp_all add: complex_mult_def
huffman@44724:     right_distrib left_distrib right_diff_distrib left_diff_distrib
huffman@44724:     complex_inverse_def complex_divide_def
huffman@44724:     power2_eq_square add_divide_distrib [symmetric]
huffman@44724:     complex_eq_iff)
paulson@14335: 
haftmann@25712: end
huffman@23125: 
huffman@23125: 
huffman@23125: subsection {* Numerals and Arithmetic *}
huffman@23125: 
haftmann@25571: instantiation complex :: number_ring
haftmann@25571: begin
huffman@23125: 
huffman@44724: definition complex_number_of_def:
huffman@44724:   "number_of w = (of_int w \<Colon> complex)"
haftmann@25571: 
haftmann@25571: instance
haftmann@25712:   by intro_classes (simp only: complex_number_of_def)
haftmann@25571: 
haftmann@25571: end
huffman@23125: 
huffman@23125: lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
huffman@44724:   by (induct n) simp_all
huffman@20556: 
huffman@23125: lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
huffman@44724:   by (induct n) simp_all
huffman@23125: 
huffman@23125: lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
huffman@44724:   by (cases z rule: int_diff_cases) simp
huffman@23125: 
huffman@23125: lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
huffman@44724:   by (cases z rule: int_diff_cases) simp
huffman@23125: 
huffman@23125: lemma complex_Re_number_of [simp]: "Re (number_of v) = number_of v"
huffman@44724:   unfolding number_of_eq by (rule complex_Re_of_int)
huffman@20556: 
huffman@23125: lemma complex_Im_number_of [simp]: "Im (number_of v) = 0"
huffman@44724:   unfolding number_of_eq by (rule complex_Im_of_int)
huffman@23125: 
huffman@23125: lemma Complex_eq_number_of [simp]:
huffman@23125:   "(Complex a b = number_of w) = (a = number_of w \<and> b = 0)"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: 
huffman@23125: subsection {* Scalar Multiplication *}
huffman@20556: 
haftmann@25712: instantiation complex :: real_field
haftmann@25571: begin
haftmann@25571: 
huffman@44724: definition complex_scaleR_def:
huffman@44724:   "scaleR r x = Complex (r * Re x) (r * Im x)"
haftmann@25571: 
huffman@23125: lemma complex_scaleR [simp]:
huffman@23125:   "scaleR r (Complex a b) = Complex (r * a) (r * b)"
haftmann@25712:   unfolding complex_scaleR_def by simp
huffman@23125: 
huffman@23125: lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"
haftmann@25712:   unfolding complex_scaleR_def by simp
huffman@23125: 
huffman@23125: lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"
haftmann@25712:   unfolding complex_scaleR_def by simp
huffman@22972: 
haftmann@25712: instance
huffman@20556: proof
huffman@23125:   fix a b :: real and x y :: complex
huffman@23125:   show "scaleR a (x + y) = scaleR a x + scaleR a y"
huffman@44065:     by (simp add: complex_eq_iff right_distrib)
huffman@23125:   show "scaleR (a + b) x = scaleR a x + scaleR b x"
huffman@44065:     by (simp add: complex_eq_iff left_distrib)
huffman@23125:   show "scaleR a (scaleR b x) = scaleR (a * b) x"
huffman@44065:     by (simp add: complex_eq_iff mult_assoc)
huffman@23125:   show "scaleR 1 x = x"
huffman@44065:     by (simp add: complex_eq_iff)
huffman@23125:   show "scaleR a x * y = scaleR a (x * y)"
huffman@44065:     by (simp add: complex_eq_iff algebra_simps)
huffman@23125:   show "x * scaleR a y = scaleR a (x * y)"
huffman@44065:     by (simp add: complex_eq_iff algebra_simps)
huffman@20556: qed
huffman@20556: 
haftmann@25712: end
haftmann@25712: 
huffman@20556: 
huffman@23125: subsection{* Properties of Embedding from Reals *}
paulson@14323: 
huffman@44724: abbreviation complex_of_real :: "real \<Rightarrow> complex"
huffman@44724:   where "complex_of_real \<equiv> of_real"
huffman@20557: 
huffman@20557: lemma complex_of_real_def: "complex_of_real r = Complex r 0"
huffman@44724:   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@44724:   by (simp add: complex_of_real_def)
huffman@20557: 
huffman@20557: lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
huffman@44724:   by (simp add: complex_of_real_def)
huffman@20557: 
paulson@14377: lemma Complex_add_complex_of_real [simp]:
huffman@44724:   shows "Complex x y + complex_of_real r = Complex (x+r) y"
huffman@44724:   by (simp add: complex_of_real_def)
paulson@14377: 
paulson@14377: lemma complex_of_real_add_Complex [simp]:
huffman@44724:   shows "complex_of_real r + Complex x y = Complex (r+x) y"
huffman@44724:   by (simp add: complex_of_real_def)
paulson@14377: 
paulson@14377: lemma Complex_mult_complex_of_real:
huffman@44724:   shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
huffman@44724:   by (simp add: complex_of_real_def)
paulson@14377: 
paulson@14377: lemma complex_of_real_mult_Complex:
huffman@44724:   shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
huffman@44724:   by (simp add: complex_of_real_def)
huffman@20557: 
paulson@14377: 
huffman@23125: subsection {* Vector Norm *}
paulson@14323: 
haftmann@25712: instantiation complex :: real_normed_field
haftmann@25571: begin
haftmann@25571: 
huffman@31413: definition complex_norm_def:
huffman@31413:   "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
haftmann@25571: 
huffman@44724: abbreviation cmod :: "complex \<Rightarrow> real"
huffman@44724:   where "cmod \<equiv> norm"
haftmann@25571: 
huffman@31413: definition complex_sgn_def:
huffman@31413:   "sgn x = x /\<^sub>R cmod x"
haftmann@25571: 
huffman@31413: definition dist_complex_def:
huffman@31413:   "dist x y = cmod (x - y)"
huffman@31413: 
haftmann@37767: definition open_complex_def:
huffman@31492:   "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31292: 
huffman@20557: lemmas cmod_def = complex_norm_def
huffman@20557: 
huffman@23125: lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)"
haftmann@25712:   by (simp add: complex_norm_def)
huffman@22852: 
huffman@31413: instance proof
huffman@31492:   fix r :: real and x y :: complex and S :: "complex set"
huffman@23125:   show "0 \<le> norm x"
huffman@22861:     by (induct x) simp
huffman@23125:   show "(norm x = 0) = (x = 0)"
huffman@22861:     by (induct x) simp
huffman@23125:   show "norm (x + y) \<le> norm x + norm y"
huffman@23125:     by (induct x, induct y)
huffman@23125:        (simp add: real_sqrt_sum_squares_triangle_ineq)
huffman@23125:   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
huffman@23125:     by (induct x)
huffman@23125:        (simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult)
huffman@23125:   show "norm (x * y) = norm x * norm y"
huffman@23125:     by (induct x, induct y)
nipkow@29667:        (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
huffman@31292:   show "sgn x = x /\<^sub>R cmod x"
huffman@31292:     by (rule complex_sgn_def)
huffman@31292:   show "dist x y = cmod (x - y)"
huffman@31292:     by (rule dist_complex_def)
huffman@31492:   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31492:     by (rule open_complex_def)
huffman@24520: qed
huffman@20557: 
haftmann@25712: end
haftmann@25712: 
huffman@44761: lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1"
huffman@44724:   by simp
paulson@14323: 
huffman@44761: lemma cmod_complex_polar:
huffman@44724:   "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
huffman@44724:   by (simp add: norm_mult)
huffman@22861: 
huffman@22861: lemma complex_Re_le_cmod: "Re x \<le> cmod x"
huffman@44724:   unfolding complex_norm_def
huffman@44724:   by (rule real_sqrt_sum_squares_ge1)
huffman@22861: 
huffman@44761: lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
huffman@44724:   by (rule order_trans [OF _ norm_ge_zero], simp)
huffman@22861: 
huffman@44761: lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \<le> cmod a"
huffman@44724:   by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
paulson@14323: 
chaieb@26117: lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
huffman@44724:   by (cases x) simp
chaieb@26117: 
chaieb@26117: lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
huffman@44724:   by (cases x) simp
huffman@44724: 
paulson@14354: 
huffman@23123: subsection {* Completeness of the Complexes *}
huffman@23123: 
huffman@44290: lemma bounded_linear_Re: "bounded_linear Re"
huffman@44290:   by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@44290: 
huffman@44290: lemma bounded_linear_Im: "bounded_linear Im"
huffman@44127:   by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@23123: 
huffman@44290: lemmas tendsto_Re [tendsto_intros] =
huffman@44290:   bounded_linear.tendsto [OF bounded_linear_Re]
huffman@44290: 
huffman@44290: lemmas tendsto_Im [tendsto_intros] =
huffman@44290:   bounded_linear.tendsto [OF bounded_linear_Im]
huffman@44290: 
huffman@44290: lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
huffman@44290: lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
huffman@44290: lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
huffman@44290: lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
huffman@23123: 
huffman@36825: lemma tendsto_Complex [tendsto_intros]:
huffman@44724:   assumes "(f ---> a) F" and "(g ---> b) F"
huffman@44724:   shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
huffman@36825: proof (rule tendstoI)
huffman@36825:   fix r :: real assume "0 < r"
huffman@36825:   hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)
huffman@44724:   have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F"
huffman@44724:     using `(f ---> a) F` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825:   moreover
huffman@44724:   have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F"
huffman@44724:     using `(g ---> b) F` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825:   ultimately
huffman@44724:   show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F"
huffman@36825:     by (rule eventually_elim2)
huffman@36825:        (simp add: dist_norm real_sqrt_sum_squares_less)
huffman@36825: qed
huffman@36825: 
huffman@23123: instance complex :: banach
huffman@23123: proof
huffman@23123:   fix X :: "nat \<Rightarrow> complex"
huffman@23123:   assume X: "Cauchy X"
huffman@44290:   from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"
huffman@23123:     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@44290:   from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
huffman@23123:     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@23123:   have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
huffman@44748:     using tendsto_Complex [OF 1 2] by simp
huffman@23123:   thus "convergent X"
huffman@23123:     by (rule convergentI)
huffman@23123: qed
huffman@23123: 
huffman@23123: 
huffman@23125: subsection {* The Complex Number @{term "\<i>"} *}
huffman@23125: 
huffman@44724: definition "ii" :: complex  ("\<i>")
huffman@44724:   where i_def: "ii \<equiv> Complex 0 1"
huffman@23125: 
huffman@23125: lemma complex_Re_i [simp]: "Re ii = 0"
huffman@44724:   by (simp add: i_def)
paulson@14354: 
huffman@23125: lemma complex_Im_i [simp]: "Im ii = 1"
huffman@44724:   by (simp add: i_def)
huffman@23125: 
huffman@23125: lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
huffman@44724:   by (simp add: i_def)
huffman@23125: 
huffman@23125: lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_i_not_one [simp]: "ii \<noteq> 1"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23124: 
huffman@23125: lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
huffman@44724:   by (simp add: i_def complex_of_real_def)
huffman@23125: 
huffman@23125: lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
huffman@44724:   by (simp add: i_def complex_of_real_def)
huffman@23125: 
huffman@23125: lemma i_squared [simp]: "ii * ii = -1"
huffman@44724:   by (simp add: i_def)
huffman@23125: 
huffman@23125: lemma power2_i [simp]: "ii\<twosuperior> = -1"
huffman@44724:   by (simp add: power2_eq_square)
huffman@23125: 
huffman@23125: lemma inverse_i [simp]: "inverse ii = - ii"
huffman@44724:   by (rule inverse_unique, simp)
paulson@14354: 
paulson@14354: 
huffman@23125: subsection {* Complex Conjugation *}
huffman@23125: 
huffman@44724: definition cnj :: "complex \<Rightarrow> complex" where
huffman@23125:   "cnj z = Complex (Re z) (- Im z)"
huffman@23125: 
huffman@23125: lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
huffman@44724:   by (simp add: cnj_def)
huffman@23125: 
huffman@23125: lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
huffman@44724:   by (simp add: cnj_def)
huffman@23125: 
huffman@23125: lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
huffman@44724:   by (simp add: cnj_def)
huffman@23125: 
huffman@23125: lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
huffman@44724:   by (simp add: cnj_def)
huffman@23125: 
huffman@23125: lemma complex_cnj_zero [simp]: "cnj 0 = 0"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_cnj_minus: "cnj (- x) = - cnj x"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_cnj_one [simp]: "cnj 1 = 1"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
huffman@44724:   by (simp add: complex_inverse_def)
paulson@14323: 
huffman@23125: lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
huffman@44724:   by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
huffman@23125: 
huffman@23125: lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
huffman@44724:   by (induct n, simp_all add: complex_cnj_mult)
huffman@23125: 
huffman@23125: lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
huffman@44724:   by (simp add: complex_norm_def)
paulson@14323: 
huffman@23125: lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_cnj_i [simp]: "cnj ii = - ii"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
huffman@44724:   by (simp add: complex_eq_iff)
huffman@23125: 
huffman@23125: lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
huffman@44724:   by (simp add: complex_eq_iff)
paulson@14354: 
huffman@23125: lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
huffman@44724:   by (simp add: complex_eq_iff power2_eq_square)
huffman@23125: 
huffman@23125: lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
huffman@44724:   by (simp add: norm_mult power2_eq_square)
huffman@23125: 
huffman@44290: lemma bounded_linear_cnj: "bounded_linear cnj"
huffman@44127:   using complex_cnj_add complex_cnj_scaleR
huffman@44127:   by (rule bounded_linear_intro [where K=1], simp)
paulson@14354: 
huffman@44290: lemmas tendsto_cnj [tendsto_intros] =
huffman@44290:   bounded_linear.tendsto [OF bounded_linear_cnj]
huffman@44290: 
huffman@44290: lemmas isCont_cnj [simp] =
huffman@44290:   bounded_linear.isCont [OF bounded_linear_cnj]
huffman@44290: 
paulson@14354: 
huffman@22972: subsection{*The Functions @{term sgn} and @{term arg}*}
paulson@14323: 
huffman@22972: text {*------------ Argand -------------*}
huffman@20557: 
huffman@44724: definition arg :: "complex => real" where
huffman@20557:   "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)"
huffman@20557: 
paulson@14374: lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
huffman@44724:   by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
paulson@14323: 
paulson@14374: lemma complex_eq_cancel_iff2 [simp]:
huffman@44724:   shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
huffman@44724:   by (simp add: complex_of_real_def)
paulson@14323: 
paulson@14374: lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
huffman@44724:   by (simp add: complex_sgn_def divide_inverse)
paulson@14323: 
paulson@14374: lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
huffman@44724:   by (simp add: complex_sgn_def divide_inverse)
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@44724:   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: 
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 \<noteq> 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@44715: text {* An abbreviation for @{text "cos a + i sin a"}. *}
huffman@20557: 
huffman@44715: definition cis :: "real \<Rightarrow> complex" where
huffman@20557:   "cis a = Complex (cos a) (sin a)"
huffman@20557: 
huffman@44715: text {* An abbreviation for @{text "r(cos a + i sin a)"}. *}
huffman@44715: 
huffman@44715: definition rcis :: "[real, real] \<Rightarrow> complex" where
huffman@20557:   "rcis r a = complex_of_real r * cis a"
huffman@20557: 
huffman@44291: abbreviation expi :: "complex \<Rightarrow> complex"
huffman@44291:   where "expi \<equiv> exp"
huffman@44291: 
huffman@44712: lemma cis_conv_exp: "cis b = exp (Complex 0 b)"
huffman@44291: proof (rule complex_eqI)
huffman@44291:   { fix n have "Complex 0 b ^ n =
huffman@44291:     real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"
huffman@44291:       apply (induct n)
huffman@44291:       apply (simp add: cos_coeff_def sin_coeff_def)
huffman@44291:       apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)
huffman@44291:       done } note * = this
huffman@44712:   show "Re (cis b) = Re (exp (Complex 0 b))"
huffman@44291:     unfolding exp_def cis_def cos_def
huffman@44291:     by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],
huffman@44291:       simp add: * mult_assoc [symmetric])
huffman@44712:   show "Im (cis b) = Im (exp (Complex 0 b))"
huffman@44291:     unfolding exp_def cis_def sin_def
huffman@44291:     by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],
huffman@44291:       simp add: * mult_assoc [symmetric])
huffman@44291: qed
huffman@44291: 
huffman@44291: lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"
huffman@44712:   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp
huffman@20557: 
paulson@14374: lemma complex_split_polar:
paulson@14377:      "\<exists>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: "\<exists>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"
huffman@44724:   by (simp add: rcis_def cis_def)
paulson@14323: 
paulson@14348: lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
huffman@44724:   by (simp add: rcis_def cis_def)
paulson@14323: 
paulson@14374: lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
huffman@44761:   by (simp add: rcis_def cis_def norm_mult)
paulson@14323: 
huffman@23125: lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
huffman@44724:   by (simp add: cmod_def power2_eq_square)
huffman@23125: 
paulson@14374: lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
huffman@44724:   by simp
paulson@14323: 
paulson@14323: lemma cis_rcis_eq: "cis a = rcis 1 a"
huffman@44724:   by (simp add: rcis_def)
paulson@14323: 
paulson@14374: lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
huffman@44724:   by (simp add: rcis_def cis_def cos_add sin_add right_distrib
huffman@44724:     right_diff_distrib complex_of_real_def)
paulson@14323: 
paulson@14323: lemma cis_mult: "cis a * cis b = cis (a + b)"
huffman@44724:   by (simp add: cis_rcis_eq rcis_mult)
paulson@14323: 
paulson@14374: lemma cis_zero [simp]: "cis 0 = 1"
huffman@44724:   by (simp add: cis_def complex_one_def)
paulson@14323: 
paulson@14374: lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
huffman@44724:   by (simp add: rcis_def)
paulson@14323: 
paulson@14374: lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
huffman@44724:   by (simp add: rcis_def)
paulson@14323: 
paulson@14374: lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
huffman@44724:   by (simp add: mult_assoc [symmetric])
paulson@14323: 
paulson@14323: lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
huffman@44824:   by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
paulson@14323: 
paulson@14374: lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
huffman@44724:   by (simp add: rcis_def power_mult_distrib DeMoivre)
paulson@14323: 
paulson@14374: lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
huffman@44724:   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)"
huffman@44724:   by (simp add: divide_inverse rcis_def)
paulson@14323: 
paulson@14323: lemma cis_divide: "cis a / cis b = cis (a - b)"
huffman@44724:   by (simp add: complex_divide_def cis_mult diff_minus)
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)
haftmann@37887: apply (simp add: rcis_inverse rcis_mult diff_minus)
paulson@14323: done
paulson@14323: 
paulson@14374: lemma Re_cis [simp]: "Re(cis a) = cos a"
huffman@44724:   by (simp add: cis_def)
paulson@14323: 
paulson@14374: lemma Im_cis [simp]: "Im(cis a) = sin a"
huffman@44724:   by (simp add: cis_def)
paulson@14323: 
paulson@14323: lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
huffman@44724:   by (auto simp add: DeMoivre)
paulson@14323: 
paulson@14323: lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
huffman@44724:   by (auto simp add: DeMoivre)
paulson@14323: 
paulson@14374: lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
paulson@14373: apply (insert rcis_Ex [of z])
huffman@23125: apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
paulson@14334: apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
paulson@14323: done
paulson@14323: 
paulson@14387: lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
huffman@44724:   by (simp add: expi_def cis_def)
paulson@14387: 
huffman@44065: text {* Legacy theorem names *}
huffman@44065: 
huffman@44065: lemmas expand_complex_eq = complex_eq_iff
huffman@44065: lemmas complex_Re_Im_cancel_iff = complex_eq_iff
huffman@44065: lemmas complex_equality = complex_eqI
huffman@44065: 
paulson@13957: end