src/HOL/Complex.thy
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(*  Title:       HOL/Complex.thy
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    Author:      Jacques D. Fleuriot
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    Copyright:   2001 University of Edinburgh
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
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*)
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header {* Complex Numbers: Rectangular and Polar Representations *}
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theory Complex
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imports Transcendental
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begin
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datatype complex = Complex real real
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primrec Re :: "complex \<Rightarrow> real"
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  where Re: "Re (Complex x y) = x"
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primrec Im :: "complex \<Rightarrow> real"
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  where Im: "Im (Complex x y) = y"
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lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"
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  by (induct z) simp
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lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"
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  by (induct x, induct y) simp
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lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"
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  by (induct x, induct y) simp
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subsection {* Addition and Subtraction *}
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instantiation complex :: ab_group_add
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begin
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definition complex_zero_def:
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  "0 = Complex 0 0"
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definition complex_add_def:
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  "x + y = Complex (Re x + Re y) (Im x + Im y)"
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definition complex_minus_def:
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  "- x = Complex (- Re x) (- Im x)"
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definition complex_diff_def:
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  "x - (y\<Colon>complex) = x + - y"
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lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
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  by (simp add: complex_zero_def)
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lemma complex_Re_zero [simp]: "Re 0 = 0"
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  by (simp add: complex_zero_def)
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lemma complex_Im_zero [simp]: "Im 0 = 0"
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  by (simp add: complex_zero_def)
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lemma complex_add [simp]:
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  "Complex a b + Complex c d = Complex (a + c) (b + d)"
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  by (simp add: complex_add_def)
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lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y"
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  by (simp add: complex_add_def)
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lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y"
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  by (simp add: complex_add_def)
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lemma complex_minus [simp]:
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  "- (Complex a b) = Complex (- a) (- b)"
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  by (simp add: complex_minus_def)
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lemma complex_Re_minus [simp]: "Re (- x) = - Re x"
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  by (simp add: complex_minus_def)
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lemma complex_Im_minus [simp]: "Im (- x) = - Im x"
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  by (simp add: complex_minus_def)
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lemma complex_diff [simp]:
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  "Complex a b - Complex c d = Complex (a - c) (b - d)"
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  by (simp add: complex_diff_def)
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lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y"
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  by (simp add: complex_diff_def)
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lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y"
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  by (simp add: complex_diff_def)
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instance
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  by intro_classes (simp_all add: complex_add_def complex_diff_def)
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end
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subsection {* Multiplication and Division *}
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instantiation complex :: field_inverse_zero
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begin
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definition complex_one_def:
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  "1 = Complex 1 0"
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definition complex_mult_def:
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  "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
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definition complex_inverse_def:
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  "inverse x =
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    Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))"
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definition complex_divide_def:
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  "x / (y\<Colon>complex) = x * inverse y"
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lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"
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  by (simp add: complex_one_def)
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lemma complex_Re_one [simp]: "Re 1 = 1"
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  by (simp add: complex_one_def)
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lemma complex_Im_one [simp]: "Im 1 = 0"
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  by (simp add: complex_one_def)
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lemma complex_mult [simp]:
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  "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
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  by (simp add: complex_mult_def)
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lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"
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  by (simp add: complex_mult_def)
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lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"
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  by (simp add: complex_mult_def)
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lemma complex_inverse [simp]:
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  "inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))"
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  by (simp add: complex_inverse_def)
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lemma complex_Re_inverse:
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  "Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
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  by (simp add: complex_inverse_def)
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lemma complex_Im_inverse:
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  "Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
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  by (simp add: complex_inverse_def)
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instance
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  by intro_classes (simp_all add: complex_mult_def
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    right_distrib left_distrib right_diff_distrib left_diff_distrib
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    complex_inverse_def complex_divide_def
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    power2_eq_square add_divide_distrib [symmetric]
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    complex_eq_iff)
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end
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subsection {* Numerals and Arithmetic *}
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instantiation complex :: number_ring
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begin
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definition complex_number_of_def:
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  "number_of w = (of_int w \<Colon> complex)"
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instance
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  by intro_classes (simp only: complex_number_of_def)
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end
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lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
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  by (induct n) simp_all
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lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
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  by (induct n) simp_all
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lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
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  by (cases z rule: int_diff_cases) simp
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lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
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  by (cases z rule: int_diff_cases) simp
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lemma complex_Re_number_of [simp]: "Re (number_of v) = number_of v"
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  unfolding number_of_eq by (rule complex_Re_of_int)
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lemma complex_Im_number_of [simp]: "Im (number_of v) = 0"
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  unfolding number_of_eq by (rule complex_Im_of_int)
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lemma Complex_eq_number_of [simp]:
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  "(Complex a b = number_of w) = (a = number_of w \<and> b = 0)"
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  by (simp add: complex_eq_iff)
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subsection {* Scalar Multiplication *}
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instantiation complex :: real_field
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begin
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definition complex_scaleR_def:
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  "scaleR r x = Complex (r * Re x) (r * Im x)"
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lemma complex_scaleR [simp]:
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  "scaleR r (Complex a b) = Complex (r * a) (r * b)"
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  unfolding complex_scaleR_def by simp
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lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"
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  unfolding complex_scaleR_def by simp
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lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"
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  unfolding complex_scaleR_def by simp
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instance
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proof
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  fix a b :: real and x y :: complex
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  show "scaleR a (x + y) = scaleR a x + scaleR a y"
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    by (simp add: complex_eq_iff right_distrib)
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  show "scaleR (a + b) x = scaleR a x + scaleR b x"
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    by (simp add: complex_eq_iff left_distrib)
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  show "scaleR a (scaleR b x) = scaleR (a * b) x"
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    by (simp add: complex_eq_iff mult_assoc)
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  show "scaleR 1 x = x"
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    by (simp add: complex_eq_iff)
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  show "scaleR a x * y = scaleR a (x * y)"
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    by (simp add: complex_eq_iff algebra_simps)
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  show "x * scaleR a y = scaleR a (x * y)"
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    by (simp add: complex_eq_iff algebra_simps)
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qed
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end
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subsection{* Properties of Embedding from Reals *}
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abbreviation complex_of_real :: "real \<Rightarrow> complex"
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  where "complex_of_real \<equiv> of_real"
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lemma complex_of_real_def: "complex_of_real r = Complex r 0"
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  by (simp add: of_real_def complex_scaleR_def)
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lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
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  by (simp add: complex_of_real_def)
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lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
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  by (simp add: complex_of_real_def)
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lemma Complex_add_complex_of_real [simp]:
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  shows "Complex x y + complex_of_real r = Complex (x+r) y"
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  by (simp add: complex_of_real_def)
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lemma complex_of_real_add_Complex [simp]:
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  shows "complex_of_real r + Complex x y = Complex (r+x) y"
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  by (simp add: complex_of_real_def)
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lemma Complex_mult_complex_of_real:
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  shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
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  by (simp add: complex_of_real_def)
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lemma complex_of_real_mult_Complex:
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  shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
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  by (simp add: complex_of_real_def)
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lemma complex_eq_cancel_iff2 [simp]:
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  shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
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  by (simp add: complex_of_real_def)
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lemma complex_split_polar:
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     "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
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  by (simp add: complex_eq_iff polar_Ex)
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subsection {* Vector Norm *}
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instantiation complex :: real_normed_field
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begin
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definition complex_norm_def:
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  "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
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abbreviation cmod :: "complex \<Rightarrow> real"
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  where "cmod \<equiv> norm"
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definition complex_sgn_def:
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  "sgn x = x /\<^sub>R cmod x"
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definition dist_complex_def:
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  "dist x y = cmod (x - y)"
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definition open_complex_def:
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  "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
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lemmas cmod_def = complex_norm_def
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lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)"
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  by (simp add: complex_norm_def)
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instance proof
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  fix r :: real and x y :: complex and S :: "complex set"
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  show "0 \<le> norm x"
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    by (induct x) simp
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  show "(norm x = 0) = (x = 0)"
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    by (induct x) simp
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  show "norm (x + y) \<le> norm x + norm y"
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    by (induct x, induct y)
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       (simp add: real_sqrt_sum_squares_triangle_ineq)
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  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
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    by (induct x)
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       (simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult)
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  show "norm (x * y) = norm x * norm y"
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    by (induct x, induct y)
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       (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
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  show "sgn x = x /\<^sub>R cmod x"
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    by (rule complex_sgn_def)
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  show "dist x y = cmod (x - y)"
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    by (rule dist_complex_def)
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  show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
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    by (rule open_complex_def)
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qed
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end
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lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1"
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  by simp
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lemma cmod_complex_polar:
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  "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
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  by (simp add: norm_mult)
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lemma complex_Re_le_cmod: "Re x \<le> cmod x"
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  unfolding complex_norm_def
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  by (rule real_sqrt_sum_squares_ge1)
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lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
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  by (rule order_trans [OF _ norm_ge_zero], simp)
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lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \<le> cmod a"
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  by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
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lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
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  by (cases x) simp
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lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
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  by (cases x) simp
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text {* Properties of complex signum. *}
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huffman@44843
   340
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
huffman@44843
   341
  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
huffman@44843
   342
huffman@44843
   343
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
huffman@44843
   344
  by (simp add: complex_sgn_def divide_inverse)
huffman@44843
   345
huffman@44843
   346
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
huffman@44843
   347
  by (simp add: complex_sgn_def divide_inverse)
huffman@44843
   348
paulson@14354
   349
huffman@23123
   350
subsection {* Completeness of the Complexes *}
huffman@23123
   351
huffman@44290
   352
lemma bounded_linear_Re: "bounded_linear Re"
huffman@44290
   353
  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@44290
   354
huffman@44290
   355
lemma bounded_linear_Im: "bounded_linear Im"
huffman@44127
   356
  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@23123
   357
huffman@44290
   358
lemmas tendsto_Re [tendsto_intros] =
huffman@44290
   359
  bounded_linear.tendsto [OF bounded_linear_Re]
huffman@44290
   360
huffman@44290
   361
lemmas tendsto_Im [tendsto_intros] =
huffman@44290
   362
  bounded_linear.tendsto [OF bounded_linear_Im]
huffman@44290
   363
huffman@44290
   364
lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
huffman@44290
   365
lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
huffman@44290
   366
lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
huffman@44290
   367
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
huffman@23123
   368
huffman@36825
   369
lemma tendsto_Complex [tendsto_intros]:
huffman@44724
   370
  assumes "(f ---> a) F" and "(g ---> b) F"
huffman@44724
   371
  shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
huffman@36825
   372
proof (rule tendstoI)
huffman@36825
   373
  fix r :: real assume "0 < r"
huffman@36825
   374
  hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)
huffman@44724
   375
  have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F"
huffman@44724
   376
    using `(f ---> a) F` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   377
  moreover
huffman@44724
   378
  have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F"
huffman@44724
   379
    using `(g ---> b) F` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   380
  ultimately
huffman@44724
   381
  show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F"
huffman@36825
   382
    by (rule eventually_elim2)
huffman@36825
   383
       (simp add: dist_norm real_sqrt_sum_squares_less)
huffman@36825
   384
qed
huffman@36825
   385
huffman@23123
   386
instance complex :: banach
huffman@23123
   387
proof
huffman@23123
   388
  fix X :: "nat \<Rightarrow> complex"
huffman@23123
   389
  assume X: "Cauchy X"
huffman@44290
   390
  from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"
huffman@23123
   391
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@44290
   392
  from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
huffman@23123
   393
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@23123
   394
  have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
huffman@44748
   395
    using tendsto_Complex [OF 1 2] by simp
huffman@23123
   396
  thus "convergent X"
huffman@23123
   397
    by (rule convergentI)
huffman@23123
   398
qed
huffman@23123
   399
huffman@23123
   400
huffman@44827
   401
subsection {* The Complex Number $i$ *}
huffman@23125
   402
huffman@44724
   403
definition "ii" :: complex  ("\<i>")
huffman@44724
   404
  where i_def: "ii \<equiv> Complex 0 1"
huffman@23125
   405
huffman@23125
   406
lemma complex_Re_i [simp]: "Re ii = 0"
huffman@44724
   407
  by (simp add: i_def)
paulson@14354
   408
huffman@23125
   409
lemma complex_Im_i [simp]: "Im ii = 1"
huffman@44724
   410
  by (simp add: i_def)
huffman@23125
   411
huffman@23125
   412
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
huffman@44724
   413
  by (simp add: i_def)
huffman@23125
   414
huffman@44902
   415
lemma norm_ii [simp]: "norm ii = 1"
huffman@44902
   416
  by (simp add: i_def)
huffman@44902
   417
huffman@23125
   418
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
huffman@44724
   419
  by (simp add: complex_eq_iff)
huffman@23125
   420
huffman@23125
   421
lemma complex_i_not_one [simp]: "ii \<noteq> 1"
huffman@44724
   422
  by (simp add: complex_eq_iff)
huffman@23124
   423
huffman@23125
   424
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w"
huffman@44724
   425
  by (simp add: complex_eq_iff)
huffman@23125
   426
huffman@23125
   427
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
huffman@44724
   428
  by (simp add: complex_eq_iff)
huffman@23125
   429
huffman@23125
   430
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
huffman@44724
   431
  by (simp add: complex_eq_iff)
huffman@23125
   432
huffman@23125
   433
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
huffman@44724
   434
  by (simp add: i_def complex_of_real_def)
huffman@23125
   435
huffman@23125
   436
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
huffman@44724
   437
  by (simp add: i_def complex_of_real_def)
huffman@23125
   438
huffman@23125
   439
lemma i_squared [simp]: "ii * ii = -1"
huffman@44724
   440
  by (simp add: i_def)
huffman@23125
   441
huffman@23125
   442
lemma power2_i [simp]: "ii\<twosuperior> = -1"
huffman@44724
   443
  by (simp add: power2_eq_square)
huffman@23125
   444
huffman@23125
   445
lemma inverse_i [simp]: "inverse ii = - ii"
huffman@44724
   446
  by (rule inverse_unique, simp)
paulson@14354
   447
huffman@44827
   448
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
huffman@44827
   449
  by (simp add: mult_assoc [symmetric])
huffman@44827
   450
paulson@14354
   451
huffman@23125
   452
subsection {* Complex Conjugation *}
huffman@23125
   453
huffman@44724
   454
definition cnj :: "complex \<Rightarrow> complex" where
huffman@23125
   455
  "cnj z = Complex (Re z) (- Im z)"
huffman@23125
   456
huffman@23125
   457
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
huffman@44724
   458
  by (simp add: cnj_def)
huffman@23125
   459
huffman@23125
   460
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
huffman@44724
   461
  by (simp add: cnj_def)
huffman@23125
   462
huffman@23125
   463
lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
huffman@44724
   464
  by (simp add: cnj_def)
huffman@23125
   465
huffman@23125
   466
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
huffman@44724
   467
  by (simp add: complex_eq_iff)
huffman@23125
   468
huffman@23125
   469
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
huffman@44724
   470
  by (simp add: cnj_def)
huffman@23125
   471
huffman@23125
   472
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
huffman@44724
   473
  by (simp add: complex_eq_iff)
huffman@23125
   474
huffman@23125
   475
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
huffman@44724
   476
  by (simp add: complex_eq_iff)
huffman@23125
   477
huffman@23125
   478
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
huffman@44724
   479
  by (simp add: complex_eq_iff)
huffman@23125
   480
huffman@23125
   481
lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
huffman@44724
   482
  by (simp add: complex_eq_iff)
huffman@23125
   483
huffman@23125
   484
lemma complex_cnj_minus: "cnj (- x) = - cnj x"
huffman@44724
   485
  by (simp add: complex_eq_iff)
huffman@23125
   486
huffman@23125
   487
lemma complex_cnj_one [simp]: "cnj 1 = 1"
huffman@44724
   488
  by (simp add: complex_eq_iff)
huffman@23125
   489
huffman@23125
   490
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
huffman@44724
   491
  by (simp add: complex_eq_iff)
huffman@23125
   492
huffman@23125
   493
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
huffman@44724
   494
  by (simp add: complex_inverse_def)
paulson@14323
   495
huffman@23125
   496
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
huffman@44724
   497
  by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
huffman@23125
   498
huffman@23125
   499
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
huffman@44724
   500
  by (induct n, simp_all add: complex_cnj_mult)
huffman@23125
   501
huffman@23125
   502
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
huffman@44724
   503
  by (simp add: complex_eq_iff)
huffman@23125
   504
huffman@23125
   505
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
huffman@44724
   506
  by (simp add: complex_eq_iff)
huffman@23125
   507
huffman@23125
   508
lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w"
huffman@44724
   509
  by (simp add: complex_eq_iff)
huffman@23125
   510
huffman@23125
   511
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
huffman@44724
   512
  by (simp add: complex_eq_iff)
huffman@23125
   513
huffman@23125
   514
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
huffman@44724
   515
  by (simp add: complex_norm_def)
paulson@14323
   516
huffman@23125
   517
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
huffman@44724
   518
  by (simp add: complex_eq_iff)
huffman@23125
   519
huffman@23125
   520
lemma complex_cnj_i [simp]: "cnj ii = - ii"
huffman@44724
   521
  by (simp add: complex_eq_iff)
huffman@23125
   522
huffman@23125
   523
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
huffman@44724
   524
  by (simp add: complex_eq_iff)
huffman@23125
   525
huffman@23125
   526
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
huffman@44724
   527
  by (simp add: complex_eq_iff)
paulson@14354
   528
huffman@23125
   529
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
huffman@44724
   530
  by (simp add: complex_eq_iff power2_eq_square)
huffman@23125
   531
huffman@23125
   532
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
huffman@44724
   533
  by (simp add: norm_mult power2_eq_square)
huffman@23125
   534
huffman@44827
   535
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
huffman@44827
   536
  by (simp add: cmod_def power2_eq_square)
huffman@44827
   537
huffman@44827
   538
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
huffman@44827
   539
  by simp
huffman@44827
   540
huffman@44290
   541
lemma bounded_linear_cnj: "bounded_linear cnj"
huffman@44127
   542
  using complex_cnj_add complex_cnj_scaleR
huffman@44127
   543
  by (rule bounded_linear_intro [where K=1], simp)
paulson@14354
   544
huffman@44290
   545
lemmas tendsto_cnj [tendsto_intros] =
huffman@44290
   546
  bounded_linear.tendsto [OF bounded_linear_cnj]
huffman@44290
   547
huffman@44290
   548
lemmas isCont_cnj [simp] =
huffman@44290
   549
  bounded_linear.isCont [OF bounded_linear_cnj]
huffman@44290
   550
paulson@14354
   551
paulson@14323
   552
subsection{*Finally! Polar Form for Complex Numbers*}
paulson@14323
   553
huffman@44827
   554
subsubsection {* $\cos \theta + i \sin \theta$ *}
huffman@20557
   555
huffman@44715
   556
definition cis :: "real \<Rightarrow> complex" where
huffman@20557
   557
  "cis a = Complex (cos a) (sin a)"
huffman@20557
   558
huffman@44827
   559
lemma Re_cis [simp]: "Re (cis a) = cos a"
huffman@44827
   560
  by (simp add: cis_def)
huffman@44827
   561
huffman@44827
   562
lemma Im_cis [simp]: "Im (cis a) = sin a"
huffman@44827
   563
  by (simp add: cis_def)
huffman@44827
   564
huffman@44827
   565
lemma cis_zero [simp]: "cis 0 = 1"
huffman@44827
   566
  by (simp add: cis_def)
huffman@44827
   567
huffman@44828
   568
lemma norm_cis [simp]: "norm (cis a) = 1"
huffman@44828
   569
  by (simp add: cis_def)
huffman@44828
   570
huffman@44828
   571
lemma sgn_cis [simp]: "sgn (cis a) = cis a"
huffman@44828
   572
  by (simp add: sgn_div_norm)
huffman@44828
   573
huffman@44828
   574
lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
huffman@44828
   575
  by (metis norm_cis norm_zero zero_neq_one)
huffman@44828
   576
huffman@44827
   577
lemma cis_mult: "cis a * cis b = cis (a + b)"
huffman@44827
   578
  by (simp add: cis_def cos_add sin_add)
huffman@44827
   579
huffman@44827
   580
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
huffman@44827
   581
  by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
huffman@44827
   582
huffman@44827
   583
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
huffman@44827
   584
  by (simp add: cis_def)
huffman@44827
   585
huffman@44827
   586
lemma cis_divide: "cis a / cis b = cis (a - b)"
huffman@44827
   587
  by (simp add: complex_divide_def cis_mult diff_minus)
huffman@44827
   588
huffman@44827
   589
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
huffman@44827
   590
  by (auto simp add: DeMoivre)
huffman@44827
   591
huffman@44827
   592
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
huffman@44827
   593
  by (auto simp add: DeMoivre)
huffman@44827
   594
huffman@44827
   595
subsubsection {* $r(\cos \theta + i \sin \theta)$ *}
huffman@44715
   596
huffman@44715
   597
definition rcis :: "[real, real] \<Rightarrow> complex" where
huffman@20557
   598
  "rcis r a = complex_of_real r * cis a"
huffman@20557
   599
huffman@44827
   600
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
huffman@44828
   601
  by (simp add: rcis_def)
huffman@44827
   602
huffman@44827
   603
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
huffman@44828
   604
  by (simp add: rcis_def)
huffman@44827
   605
huffman@44827
   606
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
huffman@44828
   607
  by (simp add: complex_eq_iff polar_Ex)
huffman@44827
   608
huffman@44827
   609
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
huffman@44828
   610
  by (simp add: rcis_def norm_mult)
huffman@44827
   611
huffman@44827
   612
lemma cis_rcis_eq: "cis a = rcis 1 a"
huffman@44827
   613
  by (simp add: rcis_def)
huffman@44827
   614
huffman@44827
   615
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
huffman@44828
   616
  by (simp add: rcis_def cis_mult)
huffman@44827
   617
huffman@44827
   618
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
huffman@44827
   619
  by (simp add: rcis_def)
huffman@44827
   620
huffman@44827
   621
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
huffman@44827
   622
  by (simp add: rcis_def)
huffman@44827
   623
huffman@44828
   624
lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
huffman@44828
   625
  by (simp add: rcis_def)
huffman@44828
   626
huffman@44827
   627
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
huffman@44827
   628
  by (simp add: rcis_def power_mult_distrib DeMoivre)
huffman@44827
   629
huffman@44827
   630
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
huffman@44827
   631
  by (simp add: divide_inverse rcis_def)
huffman@44827
   632
huffman@44827
   633
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
huffman@44828
   634
  by (simp add: rcis_def cis_divide [symmetric])
huffman@44827
   635
huffman@44827
   636
subsubsection {* Complex exponential *}
huffman@44827
   637
huffman@44291
   638
abbreviation expi :: "complex \<Rightarrow> complex"
huffman@44291
   639
  where "expi \<equiv> exp"
huffman@44291
   640
huffman@44712
   641
lemma cis_conv_exp: "cis b = exp (Complex 0 b)"
huffman@44291
   642
proof (rule complex_eqI)
huffman@44291
   643
  { fix n have "Complex 0 b ^ n =
huffman@44291
   644
    real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"
huffman@44291
   645
      apply (induct n)
huffman@44291
   646
      apply (simp add: cos_coeff_def sin_coeff_def)
huffman@44291
   647
      apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)
huffman@44291
   648
      done } note * = this
huffman@44712
   649
  show "Re (cis b) = Re (exp (Complex 0 b))"
huffman@44291
   650
    unfolding exp_def cis_def cos_def
huffman@44291
   651
    by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],
huffman@44291
   652
      simp add: * mult_assoc [symmetric])
huffman@44712
   653
  show "Im (cis b) = Im (exp (Complex 0 b))"
huffman@44291
   654
    unfolding exp_def cis_def sin_def
huffman@44291
   655
    by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],
huffman@44291
   656
      simp add: * mult_assoc [symmetric])
huffman@44291
   657
qed
huffman@44291
   658
huffman@44291
   659
lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"
huffman@44712
   660
  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp
huffman@20557
   661
huffman@44828
   662
lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
huffman@44828
   663
  unfolding expi_def by simp
huffman@44828
   664
huffman@44828
   665
lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
huffman@44828
   666
  unfolding expi_def by simp
huffman@44828
   667
paulson@14374
   668
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
paulson@14373
   669
apply (insert rcis_Ex [of z])
huffman@23125
   670
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
paulson@14334
   671
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
paulson@14323
   672
done
paulson@14323
   673
paulson@14387
   674
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
huffman@44724
   675
  by (simp add: expi_def cis_def)
paulson@14387
   676
huffman@44844
   677
subsubsection {* Complex argument *}
huffman@44844
   678
huffman@44844
   679
definition arg :: "complex \<Rightarrow> real" where
huffman@44844
   680
  "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"
huffman@44844
   681
huffman@44844
   682
lemma arg_zero: "arg 0 = 0"
huffman@44844
   683
  by (simp add: arg_def)
huffman@44844
   684
huffman@44844
   685
lemma of_nat_less_of_int_iff: (* TODO: move *)
huffman@44844
   686
  "(of_nat n :: 'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
huffman@44844
   687
  by (metis of_int_of_nat_eq of_int_less_iff)
huffman@44844
   688
huffman@44844
   689
lemma real_of_nat_less_number_of_iff [simp]: (* TODO: move *)
huffman@44844
   690
  "real (n::nat) < number_of w \<longleftrightarrow> n < number_of w"
huffman@44844
   691
  unfolding real_of_nat_def nat_number_of_def number_of_eq
huffman@44844
   692
  by (simp add: of_nat_less_of_int_iff zless_nat_eq_int_zless)
huffman@44844
   693
huffman@44844
   694
lemma arg_unique:
huffman@44844
   695
  assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
huffman@44844
   696
  shows "arg z = x"
huffman@44844
   697
proof -
huffman@44844
   698
  from assms have "z \<noteq> 0" by auto
huffman@44844
   699
  have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
huffman@44844
   700
  proof
huffman@44844
   701
    fix a def d \<equiv> "a - x"
huffman@44844
   702
    assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
huffman@44844
   703
    from a assms have "- (2*pi) < d \<and> d < 2*pi"
huffman@44844
   704
      unfolding d_def by simp
huffman@44844
   705
    moreover from a assms have "cos a = cos x" and "sin a = sin x"
huffman@44844
   706
      by (simp_all add: complex_eq_iff)
huffman@44844
   707
    hence "cos d = 1" unfolding d_def cos_diff by simp
huffman@44844
   708
    moreover hence "sin d = 0" by (rule cos_one_sin_zero)
huffman@44844
   709
    ultimately have "d = 0"
huffman@44844
   710
      unfolding sin_zero_iff even_mult_two_ex
huffman@44844
   711
      by (safe, auto simp add: numeral_2_eq_2 less_Suc_eq)
huffman@44844
   712
    thus "a = x" unfolding d_def by simp
huffman@44844
   713
  qed (simp add: assms del: Re_sgn Im_sgn)
huffman@44844
   714
  with `z \<noteq> 0` show "arg z = x"
huffman@44844
   715
    unfolding arg_def by simp
huffman@44844
   716
qed
huffman@44844
   717
huffman@44844
   718
lemma arg_correct:
huffman@44844
   719
  assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
huffman@44844
   720
proof (simp add: arg_def assms, rule someI_ex)
huffman@44844
   721
  obtain r a where z: "z = rcis r a" using rcis_Ex by fast
huffman@44844
   722
  with assms have "r \<noteq> 0" by auto
huffman@44844
   723
  def b \<equiv> "if 0 < r then a else a + pi"
huffman@44844
   724
  have b: "sgn z = cis b"
huffman@44844
   725
    unfolding z b_def rcis_def using `r \<noteq> 0`
huffman@44844
   726
    by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def)
huffman@44844
   727
  have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
huffman@44844
   728
    by (induct_tac n, simp_all add: right_distrib cis_mult [symmetric],
huffman@44844
   729
      simp add: cis_def)
huffman@44844
   730
  have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
huffman@44844
   731
    by (case_tac x rule: int_diff_cases,
huffman@44844
   732
      simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
huffman@44844
   733
  def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
huffman@44844
   734
  have "sgn z = cis c"
huffman@44844
   735
    unfolding b c_def
huffman@44844
   736
    by (simp add: cis_divide [symmetric] cis_2pi_int)
huffman@44844
   737
  moreover have "- pi < c \<and> c \<le> pi"
huffman@44844
   738
    using ceiling_correct [of "(b - pi) / (2*pi)"]
huffman@44844
   739
    by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)
huffman@44844
   740
  ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
huffman@44844
   741
qed
huffman@44844
   742
huffman@44844
   743
lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
huffman@44844
   744
  by (cases "z = 0", simp_all add: arg_zero arg_correct)
huffman@44844
   745
huffman@44844
   746
lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
huffman@44844
   747
  by (simp add: arg_correct)
huffman@44844
   748
huffman@44844
   749
lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
huffman@44844
   750
  by (cases "z = 0", simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
huffman@44844
   751
huffman@44844
   752
lemma cos_arg_i_mult_zero [simp]:
huffman@44844
   753
     "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
huffman@44844
   754
  using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff)
huffman@44844
   755
huffman@44065
   756
text {* Legacy theorem names *}
huffman@44065
   757
huffman@44065
   758
lemmas expand_complex_eq = complex_eq_iff
huffman@44065
   759
lemmas complex_Re_Im_cancel_iff = complex_eq_iff
huffman@44065
   760
lemmas complex_equality = complex_eqI
huffman@44065
   761
paulson@13957
   762
end