src/HOL/Complex.thy
author hoelzl
Mon Mar 31 12:32:15 2014 +0200 (2014-03-31)
changeset 56331 bea2196627cb
parent 56238 5d147e1e18d1
child 56369 2704ca85be98
permissions -rw-r--r--
add complex_of_real coercion
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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)\<^sup>2 + (Im x)\<^sup>2)) (- Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2))"
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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]:
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  "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"
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  by (simp add: complex_one_def)
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lemma Complex_eq_neg_1 [simp]:
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  "Complex a b = - 1 \<longleftrightarrow> 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\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"
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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)\<^sup>2 + (Im x)\<^sup>2)"
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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)\<^sup>2 + (Im x)\<^sup>2)"
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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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    distrib_left distrib_right 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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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_numeral [simp]: "Re (numeral v) = numeral v"
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  using complex_Re_of_int [of "numeral v"] by simp
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lemma complex_Re_neg_numeral [simp]: "Re (- numeral v) = - numeral v"
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  using complex_Re_of_int [of "- numeral v"] by simp
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lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"
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  using complex_Im_of_int [of "numeral v"] by simp
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lemma complex_Im_neg_numeral [simp]: "Im (- numeral v) = 0"
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  using complex_Im_of_int [of "- numeral v"] by simp
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lemma Complex_eq_numeral [simp]:
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  "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"
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  by (simp add: complex_eq_iff)
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lemma Complex_eq_neg_numeral [simp]:
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  "Complex a b = - numeral w \<longleftrightarrow> a = - numeral 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 distrib_left)
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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 distrib_right)
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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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declare [[coercion complex_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)\<^sup>2 + (Im z)\<^sup>2)"
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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\<^sup>2 + y\<^sup>2)"
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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 "(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 distrib_left [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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chaieb@26117
   336
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
huffman@44724
   337
  by (cases x) simp
chaieb@26117
   338
chaieb@26117
   339
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
huffman@44724
   340
  by (cases x) simp
huffman@44724
   341
huffman@44843
   342
text {* Properties of complex signum. *}
huffman@44843
   343
huffman@44843
   344
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
huffman@44843
   345
  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
huffman@44843
   346
huffman@44843
   347
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
huffman@44843
   348
  by (simp add: complex_sgn_def divide_inverse)
huffman@44843
   349
huffman@44843
   350
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
huffman@44843
   351
  by (simp add: complex_sgn_def divide_inverse)
huffman@44843
   352
paulson@14354
   353
huffman@23123
   354
subsection {* Completeness of the Complexes *}
huffman@23123
   355
huffman@44290
   356
lemma bounded_linear_Re: "bounded_linear Re"
huffman@44290
   357
  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@44290
   358
huffman@44290
   359
lemma bounded_linear_Im: "bounded_linear Im"
huffman@44127
   360
  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@23123
   361
huffman@44290
   362
lemmas tendsto_Re [tendsto_intros] =
huffman@44290
   363
  bounded_linear.tendsto [OF bounded_linear_Re]
huffman@44290
   364
huffman@44290
   365
lemmas tendsto_Im [tendsto_intros] =
huffman@44290
   366
  bounded_linear.tendsto [OF bounded_linear_Im]
huffman@44290
   367
huffman@44290
   368
lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
huffman@44290
   369
lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
huffman@44290
   370
lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
huffman@44290
   371
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
huffman@23123
   372
huffman@36825
   373
lemma tendsto_Complex [tendsto_intros]:
huffman@44724
   374
  assumes "(f ---> a) F" and "(g ---> b) F"
huffman@44724
   375
  shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
huffman@36825
   376
proof (rule tendstoI)
huffman@36825
   377
  fix r :: real assume "0 < r"
huffman@36825
   378
  hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)
huffman@44724
   379
  have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F"
huffman@44724
   380
    using `(f ---> a) F` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   381
  moreover
huffman@44724
   382
  have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F"
huffman@44724
   383
    using `(g ---> b) F` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   384
  ultimately
huffman@44724
   385
  show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F"
huffman@36825
   386
    by (rule eventually_elim2)
huffman@36825
   387
       (simp add: dist_norm real_sqrt_sum_squares_less)
huffman@36825
   388
qed
huffman@36825
   389
huffman@23123
   390
instance complex :: banach
huffman@23123
   391
proof
huffman@23123
   392
  fix X :: "nat \<Rightarrow> complex"
huffman@23123
   393
  assume X: "Cauchy X"
huffman@44290
   394
  from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"
huffman@23123
   395
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@44290
   396
  from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
huffman@23123
   397
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@23123
   398
  have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
huffman@44748
   399
    using tendsto_Complex [OF 1 2] by simp
huffman@23123
   400
  thus "convergent X"
huffman@23123
   401
    by (rule convergentI)
huffman@23123
   402
qed
huffman@23123
   403
lp15@56238
   404
declare
lp15@56238
   405
  DERIV_power[where 'a=complex, THEN DERIV_cong,
lp15@56238
   406
              unfolded of_nat_def[symmetric], DERIV_intros]
lp15@56238
   407
huffman@23123
   408
huffman@44827
   409
subsection {* The Complex Number $i$ *}
huffman@23125
   410
huffman@44724
   411
definition "ii" :: complex  ("\<i>")
huffman@44724
   412
  where i_def: "ii \<equiv> Complex 0 1"
huffman@23125
   413
huffman@23125
   414
lemma complex_Re_i [simp]: "Re ii = 0"
huffman@44724
   415
  by (simp add: i_def)
paulson@14354
   416
huffman@23125
   417
lemma complex_Im_i [simp]: "Im ii = 1"
huffman@44724
   418
  by (simp add: i_def)
huffman@23125
   419
huffman@23125
   420
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
huffman@44724
   421
  by (simp add: i_def)
huffman@23125
   422
huffman@44902
   423
lemma norm_ii [simp]: "norm ii = 1"
huffman@44902
   424
  by (simp add: i_def)
huffman@44902
   425
huffman@23125
   426
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
huffman@44724
   427
  by (simp add: complex_eq_iff)
huffman@23125
   428
huffman@23125
   429
lemma complex_i_not_one [simp]: "ii \<noteq> 1"
huffman@44724
   430
  by (simp add: complex_eq_iff)
huffman@23124
   431
huffman@47108
   432
lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"
huffman@47108
   433
  by (simp add: complex_eq_iff)
huffman@47108
   434
haftmann@54489
   435
lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> - numeral w"
huffman@44724
   436
  by (simp add: complex_eq_iff)
huffman@23125
   437
huffman@23125
   438
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
huffman@44724
   439
  by (simp add: complex_eq_iff)
huffman@23125
   440
huffman@23125
   441
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
huffman@44724
   442
  by (simp add: complex_eq_iff)
huffman@23125
   443
huffman@23125
   444
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
huffman@44724
   445
  by (simp add: i_def complex_of_real_def)
huffman@23125
   446
huffman@23125
   447
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
huffman@44724
   448
  by (simp add: i_def complex_of_real_def)
huffman@23125
   449
huffman@23125
   450
lemma i_squared [simp]: "ii * ii = -1"
huffman@44724
   451
  by (simp add: i_def)
huffman@23125
   452
wenzelm@53015
   453
lemma power2_i [simp]: "ii\<^sup>2 = -1"
huffman@44724
   454
  by (simp add: power2_eq_square)
huffman@23125
   455
huffman@23125
   456
lemma inverse_i [simp]: "inverse ii = - ii"
huffman@44724
   457
  by (rule inverse_unique, simp)
paulson@14354
   458
huffman@44827
   459
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
huffman@44827
   460
  by (simp add: mult_assoc [symmetric])
huffman@44827
   461
paulson@14354
   462
huffman@23125
   463
subsection {* Complex Conjugation *}
huffman@23125
   464
huffman@44724
   465
definition cnj :: "complex \<Rightarrow> complex" where
huffman@23125
   466
  "cnj z = Complex (Re z) (- Im z)"
huffman@23125
   467
huffman@23125
   468
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
huffman@44724
   469
  by (simp add: cnj_def)
huffman@23125
   470
huffman@23125
   471
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
huffman@44724
   472
  by (simp add: cnj_def)
huffman@23125
   473
huffman@23125
   474
lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
huffman@44724
   475
  by (simp add: cnj_def)
huffman@23125
   476
huffman@23125
   477
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
huffman@44724
   478
  by (simp add: complex_eq_iff)
huffman@23125
   479
huffman@23125
   480
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
huffman@44724
   481
  by (simp add: cnj_def)
huffman@23125
   482
huffman@23125
   483
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
huffman@44724
   484
  by (simp add: complex_eq_iff)
huffman@23125
   485
huffman@23125
   486
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
huffman@44724
   487
  by (simp add: complex_eq_iff)
huffman@23125
   488
huffman@23125
   489
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
huffman@44724
   490
  by (simp add: complex_eq_iff)
huffman@23125
   491
huffman@23125
   492
lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
huffman@44724
   493
  by (simp add: complex_eq_iff)
huffman@23125
   494
huffman@23125
   495
lemma complex_cnj_minus: "cnj (- x) = - cnj x"
huffman@44724
   496
  by (simp add: complex_eq_iff)
huffman@23125
   497
huffman@23125
   498
lemma complex_cnj_one [simp]: "cnj 1 = 1"
huffman@44724
   499
  by (simp add: complex_eq_iff)
huffman@23125
   500
huffman@23125
   501
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
huffman@44724
   502
  by (simp add: complex_eq_iff)
huffman@23125
   503
huffman@23125
   504
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
huffman@44724
   505
  by (simp add: complex_inverse_def)
paulson@14323
   506
huffman@23125
   507
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
huffman@44724
   508
  by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
huffman@23125
   509
huffman@23125
   510
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
huffman@44724
   511
  by (induct n, simp_all add: complex_cnj_mult)
huffman@23125
   512
huffman@23125
   513
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
huffman@44724
   514
  by (simp add: complex_eq_iff)
huffman@23125
   515
huffman@23125
   516
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
huffman@44724
   517
  by (simp add: complex_eq_iff)
huffman@23125
   518
huffman@47108
   519
lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
huffman@47108
   520
  by (simp add: complex_eq_iff)
huffman@47108
   521
haftmann@54489
   522
lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
huffman@44724
   523
  by (simp add: complex_eq_iff)
huffman@23125
   524
huffman@23125
   525
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
huffman@44724
   526
  by (simp add: complex_eq_iff)
huffman@23125
   527
huffman@23125
   528
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
huffman@44724
   529
  by (simp add: complex_norm_def)
paulson@14323
   530
huffman@23125
   531
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
huffman@44724
   532
  by (simp add: complex_eq_iff)
huffman@23125
   533
huffman@23125
   534
lemma complex_cnj_i [simp]: "cnj ii = - ii"
huffman@44724
   535
  by (simp add: complex_eq_iff)
huffman@23125
   536
huffman@23125
   537
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
huffman@44724
   538
  by (simp add: complex_eq_iff)
huffman@23125
   539
huffman@23125
   540
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
huffman@44724
   541
  by (simp add: complex_eq_iff)
paulson@14354
   542
wenzelm@53015
   543
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
huffman@44724
   544
  by (simp add: complex_eq_iff power2_eq_square)
huffman@23125
   545
wenzelm@53015
   546
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"
huffman@44724
   547
  by (simp add: norm_mult power2_eq_square)
huffman@23125
   548
huffman@44827
   549
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
huffman@44827
   550
  by (simp add: cmod_def power2_eq_square)
huffman@44827
   551
huffman@44827
   552
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
huffman@44827
   553
  by simp
huffman@44827
   554
huffman@44290
   555
lemma bounded_linear_cnj: "bounded_linear cnj"
huffman@44127
   556
  using complex_cnj_add complex_cnj_scaleR
huffman@44127
   557
  by (rule bounded_linear_intro [where K=1], simp)
paulson@14354
   558
huffman@44290
   559
lemmas tendsto_cnj [tendsto_intros] =
huffman@44290
   560
  bounded_linear.tendsto [OF bounded_linear_cnj]
huffman@44290
   561
huffman@44290
   562
lemmas isCont_cnj [simp] =
huffman@44290
   563
  bounded_linear.isCont [OF bounded_linear_cnj]
huffman@44290
   564
paulson@14354
   565
lp15@55734
   566
subsection{*Basic Lemmas*}
lp15@55734
   567
lp15@55734
   568
lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"
lp15@55734
   569
  by (metis complex_Im_zero complex_Re_zero complex_eqI sum_power2_eq_zero_iff)
lp15@55734
   570
lp15@55734
   571
lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"
lp15@55734
   572
by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)
lp15@55734
   573
lp15@55734
   574
lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"
lp15@55734
   575
apply (cases z, auto)
lp15@55734
   576
by (metis complex_of_real_def of_real_add of_real_power power2_eq_square)
lp15@55734
   577
lp15@55734
   578
lemma complex_div_eq_0: 
lp15@55734
   579
    "(Re(a / b) = 0 \<longleftrightarrow> Re(a * cnj b) = 0) & (Im(a / b) = 0 \<longleftrightarrow> Im(a * cnj b) = 0)"
lp15@55734
   580
proof (cases "b=0")
lp15@55734
   581
  case True then show ?thesis by auto
lp15@55734
   582
next
lp15@55734
   583
  case False
lp15@55734
   584
  show ?thesis
lp15@55734
   585
  proof (cases b)
lp15@55734
   586
    case (Complex x y)
lp15@55734
   587
    then have "x\<^sup>2 + y\<^sup>2 > 0"
lp15@55734
   588
      by (metis Complex_eq_0 False sum_power2_gt_zero_iff)
lp15@55734
   589
    then have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)"
lp15@55734
   590
      by (metis add_divide_distrib)
lp15@55734
   591
    with Complex False show ?thesis
lp15@55734
   592
      by (auto simp: complex_divide_def)
lp15@55734
   593
  qed
lp15@55734
   594
qed
lp15@55734
   595
lp15@55734
   596
lemma re_complex_div_eq_0: "Re(a / b) = 0 \<longleftrightarrow> Re(a * cnj b) = 0"
lp15@55734
   597
  and im_complex_div_eq_0: "Im(a / b) = 0 \<longleftrightarrow> Im(a * cnj b) = 0"
lp15@55734
   598
using complex_div_eq_0 by auto
lp15@55734
   599
lp15@55734
   600
lp15@55734
   601
lemma complex_div_gt_0: 
lp15@55734
   602
    "(Re(a / b) > 0 \<longleftrightarrow> Re(a * cnj b) > 0) & (Im(a / b) > 0 \<longleftrightarrow> Im(a * cnj b) > 0)"
lp15@55734
   603
proof (cases "b=0")
lp15@55734
   604
  case True then show ?thesis by auto
lp15@55734
   605
next
lp15@55734
   606
  case False
lp15@55734
   607
  show ?thesis
lp15@55734
   608
  proof (cases b)
lp15@55734
   609
    case (Complex x y)
lp15@55734
   610
    then have "x\<^sup>2 + y\<^sup>2 > 0"
lp15@55734
   611
      by (metis Complex_eq_0 False sum_power2_gt_zero_iff)
lp15@55734
   612
    moreover have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)"
lp15@55734
   613
      by (metis add_divide_distrib)
lp15@55734
   614
    ultimately show ?thesis using Complex False `0 < x\<^sup>2 + y\<^sup>2`
lp15@55734
   615
      apply (simp add: complex_divide_def  zero_less_divide_iff less_divide_eq)
lp15@55734
   616
      apply (metis less_divide_eq mult_zero_left diff_conv_add_uminus minus_divide_left)
lp15@55734
   617
      done
lp15@55734
   618
  qed
lp15@55734
   619
qed
lp15@55734
   620
lp15@55734
   621
lemma re_complex_div_gt_0: "Re(a / b) > 0 \<longleftrightarrow> Re(a * cnj b) > 0"
lp15@55734
   622
  and im_complex_div_gt_0: "Im(a / b) > 0 \<longleftrightarrow> Im(a * cnj b) > 0"
lp15@55734
   623
using complex_div_gt_0 by auto
lp15@55734
   624
lp15@55734
   625
lemma re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0"
lp15@55734
   626
  by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0)
lp15@55734
   627
lp15@55734
   628
lemma im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0"
lp15@55734
   629
  by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less)
lp15@55734
   630
lp15@55734
   631
lemma re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0"
boehmes@55759
   632
  by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0)
lp15@55734
   633
lp15@55734
   634
lemma im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0"
lp15@55734
   635
  by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff)
lp15@55734
   636
lp15@55734
   637
lemma re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0"
lp15@55734
   638
  by (metis not_le re_complex_div_gt_0)
lp15@55734
   639
lp15@55734
   640
lemma im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0"
lp15@55734
   641
  by (metis im_complex_div_gt_0 not_le)
lp15@55734
   642
lp15@56217
   643
lemma Re_setsum: "Re(setsum f s) = setsum (%x. Re(f x)) s"
lp15@56217
   644
apply (cases "finite s")
lp15@55734
   645
  by (induct s rule: finite_induct) auto
lp15@55734
   646
lp15@56217
   647
lemma Im_setsum: "Im(setsum f s) = setsum (%x. Im(f x)) s"
lp15@56217
   648
apply (cases "finite s")
lp15@56217
   649
  by (induct s rule: finite_induct) auto
lp15@56217
   650
lp15@56217
   651
lemma Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"
lp15@56217
   652
apply (cases "finite s")
lp15@55734
   653
  by (induct s rule: finite_induct) auto
lp15@55734
   654
lp15@56217
   655
lemma Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"
lp15@56217
   656
  by (metis Complex_setsum')
lp15@56217
   657
lp15@56217
   658
lemma cnj_setsum: "cnj (setsum f s) = setsum (%x. cnj (f x)) s"
lp15@56217
   659
apply (cases "finite s")
lp15@56217
   660
  by (induct s rule: finite_induct) (auto simp: complex_cnj_add)
lp15@56217
   661
lp15@56217
   662
lemma of_real_setsum: "of_real (setsum f s) = setsum (%x. of_real(f x)) s"
lp15@56217
   663
apply (cases "finite s")
lp15@55734
   664
  by (induct s rule: finite_induct) auto
lp15@55734
   665
lp15@56217
   666
lemma of_real_setprod: "of_real (setprod f s) = setprod (%x. of_real(f x)) s"
lp15@56217
   667
apply (cases "finite s")
lp15@56217
   668
  by (induct s rule: finite_induct) auto
lp15@55734
   669
lp15@55734
   670
lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z"
lp15@55734
   671
by (metis Reals_cases Reals_of_real complex.exhaust complex.inject complex_cnj 
lp15@55734
   672
          complex_of_real_def equal_neg_zero)
lp15@55734
   673
lp15@55734
   674
lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"
lp15@55734
   675
  by (metis Reals_of_real complex_of_real_def)
lp15@55734
   676
lp15@55734
   677
lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm(z) = abs(Re z)"
lp15@55734
   678
  by (metis Re_complex_of_real Reals_cases norm_of_real)
lp15@55734
   679
lp15@55734
   680
paulson@14323
   681
subsection{*Finally! Polar Form for Complex Numbers*}
paulson@14323
   682
huffman@44827
   683
subsubsection {* $\cos \theta + i \sin \theta$ *}
huffman@20557
   684
huffman@44715
   685
definition cis :: "real \<Rightarrow> complex" where
huffman@20557
   686
  "cis a = Complex (cos a) (sin a)"
huffman@20557
   687
huffman@44827
   688
lemma Re_cis [simp]: "Re (cis a) = cos a"
huffman@44827
   689
  by (simp add: cis_def)
huffman@44827
   690
huffman@44827
   691
lemma Im_cis [simp]: "Im (cis a) = sin a"
huffman@44827
   692
  by (simp add: cis_def)
huffman@44827
   693
huffman@44827
   694
lemma cis_zero [simp]: "cis 0 = 1"
huffman@44827
   695
  by (simp add: cis_def)
huffman@44827
   696
huffman@44828
   697
lemma norm_cis [simp]: "norm (cis a) = 1"
huffman@44828
   698
  by (simp add: cis_def)
huffman@44828
   699
huffman@44828
   700
lemma sgn_cis [simp]: "sgn (cis a) = cis a"
huffman@44828
   701
  by (simp add: sgn_div_norm)
huffman@44828
   702
huffman@44828
   703
lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
huffman@44828
   704
  by (metis norm_cis norm_zero zero_neq_one)
huffman@44828
   705
huffman@44827
   706
lemma cis_mult: "cis a * cis b = cis (a + b)"
huffman@44827
   707
  by (simp add: cis_def cos_add sin_add)
huffman@44827
   708
huffman@44827
   709
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
huffman@44827
   710
  by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
huffman@44827
   711
huffman@44827
   712
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
huffman@44827
   713
  by (simp add: cis_def)
huffman@44827
   714
huffman@44827
   715
lemma cis_divide: "cis a / cis b = cis (a - b)"
haftmann@54230
   716
  by (simp add: complex_divide_def cis_mult)
huffman@44827
   717
huffman@44827
   718
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
huffman@44827
   719
  by (auto simp add: DeMoivre)
huffman@44827
   720
huffman@44827
   721
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
huffman@44827
   722
  by (auto simp add: DeMoivre)
huffman@44827
   723
huffman@44827
   724
subsubsection {* $r(\cos \theta + i \sin \theta)$ *}
huffman@44715
   725
huffman@44715
   726
definition rcis :: "[real, real] \<Rightarrow> complex" where
huffman@20557
   727
  "rcis r a = complex_of_real r * cis a"
huffman@20557
   728
huffman@44827
   729
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
huffman@44828
   730
  by (simp add: rcis_def)
huffman@44827
   731
huffman@44827
   732
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
huffman@44828
   733
  by (simp add: rcis_def)
huffman@44827
   734
huffman@44827
   735
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
huffman@44828
   736
  by (simp add: complex_eq_iff polar_Ex)
huffman@44827
   737
huffman@44827
   738
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
huffman@44828
   739
  by (simp add: rcis_def norm_mult)
huffman@44827
   740
huffman@44827
   741
lemma cis_rcis_eq: "cis a = rcis 1 a"
huffman@44827
   742
  by (simp add: rcis_def)
huffman@44827
   743
huffman@44827
   744
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
huffman@44828
   745
  by (simp add: rcis_def cis_mult)
huffman@44827
   746
huffman@44827
   747
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
huffman@44827
   748
  by (simp add: rcis_def)
huffman@44827
   749
huffman@44827
   750
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
huffman@44827
   751
  by (simp add: rcis_def)
huffman@44827
   752
huffman@44828
   753
lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
huffman@44828
   754
  by (simp add: rcis_def)
huffman@44828
   755
huffman@44827
   756
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
huffman@44827
   757
  by (simp add: rcis_def power_mult_distrib DeMoivre)
huffman@44827
   758
huffman@44827
   759
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
huffman@44827
   760
  by (simp add: divide_inverse rcis_def)
huffman@44827
   761
huffman@44827
   762
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
huffman@44828
   763
  by (simp add: rcis_def cis_divide [symmetric])
huffman@44827
   764
huffman@44827
   765
subsubsection {* Complex exponential *}
huffman@44827
   766
huffman@44291
   767
abbreviation expi :: "complex \<Rightarrow> complex"
huffman@44291
   768
  where "expi \<equiv> exp"
huffman@44291
   769
huffman@44712
   770
lemma cis_conv_exp: "cis b = exp (Complex 0 b)"
huffman@44291
   771
proof (rule complex_eqI)
huffman@44291
   772
  { fix n have "Complex 0 b ^ n =
huffman@44291
   773
    real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"
huffman@44291
   774
      apply (induct n)
huffman@44291
   775
      apply (simp add: cos_coeff_def sin_coeff_def)
huffman@44291
   776
      apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)
huffman@44291
   777
      done } note * = this
huffman@44712
   778
  show "Re (cis b) = Re (exp (Complex 0 b))"
huffman@44291
   779
    unfolding exp_def cis_def cos_def
huffman@44291
   780
    by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],
huffman@44291
   781
      simp add: * mult_assoc [symmetric])
huffman@44712
   782
  show "Im (cis b) = Im (exp (Complex 0 b))"
huffman@44291
   783
    unfolding exp_def cis_def sin_def
huffman@44291
   784
    by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],
huffman@44291
   785
      simp add: * mult_assoc [symmetric])
huffman@44291
   786
qed
huffman@44291
   787
huffman@44291
   788
lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"
huffman@44712
   789
  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp
huffman@20557
   790
huffman@44828
   791
lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
huffman@44828
   792
  unfolding expi_def by simp
huffman@44828
   793
huffman@44828
   794
lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
huffman@44828
   795
  unfolding expi_def by simp
huffman@44828
   796
paulson@14374
   797
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
paulson@14373
   798
apply (insert rcis_Ex [of z])
huffman@23125
   799
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
paulson@14334
   800
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
paulson@14323
   801
done
paulson@14323
   802
paulson@14387
   803
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
huffman@44724
   804
  by (simp add: expi_def cis_def)
paulson@14387
   805
huffman@44844
   806
subsubsection {* Complex argument *}
huffman@44844
   807
huffman@44844
   808
definition arg :: "complex \<Rightarrow> real" where
huffman@44844
   809
  "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"
huffman@44844
   810
huffman@44844
   811
lemma arg_zero: "arg 0 = 0"
huffman@44844
   812
  by (simp add: arg_def)
huffman@44844
   813
huffman@44844
   814
lemma of_nat_less_of_int_iff: (* TODO: move *)
huffman@44844
   815
  "(of_nat n :: 'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
huffman@44844
   816
  by (metis of_int_of_nat_eq of_int_less_iff)
huffman@44844
   817
huffman@47108
   818
lemma real_of_nat_less_numeral_iff [simp]: (* TODO: move *)
huffman@47108
   819
  "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"
huffman@47108
   820
  using of_nat_less_of_int_iff [of n "numeral w", where 'a=real]
huffman@47108
   821
  by (simp add: real_of_nat_def zless_nat_eq_int_zless [symmetric])
huffman@44844
   822
huffman@44844
   823
lemma arg_unique:
huffman@44844
   824
  assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
huffman@44844
   825
  shows "arg z = x"
huffman@44844
   826
proof -
huffman@44844
   827
  from assms have "z \<noteq> 0" by auto
huffman@44844
   828
  have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
huffman@44844
   829
  proof
huffman@44844
   830
    fix a def d \<equiv> "a - x"
huffman@44844
   831
    assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
huffman@44844
   832
    from a assms have "- (2*pi) < d \<and> d < 2*pi"
huffman@44844
   833
      unfolding d_def by simp
huffman@44844
   834
    moreover from a assms have "cos a = cos x" and "sin a = sin x"
huffman@44844
   835
      by (simp_all add: complex_eq_iff)
wenzelm@53374
   836
    hence cos: "cos d = 1" unfolding d_def cos_diff by simp
wenzelm@53374
   837
    moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)
huffman@44844
   838
    ultimately have "d = 0"
huffman@44844
   839
      unfolding sin_zero_iff even_mult_two_ex
wenzelm@53374
   840
      by (auto simp add: numeral_2_eq_2 less_Suc_eq)
huffman@44844
   841
    thus "a = x" unfolding d_def by simp
huffman@44844
   842
  qed (simp add: assms del: Re_sgn Im_sgn)
huffman@44844
   843
  with `z \<noteq> 0` show "arg z = x"
huffman@44844
   844
    unfolding arg_def by simp
huffman@44844
   845
qed
huffman@44844
   846
huffman@44844
   847
lemma arg_correct:
huffman@44844
   848
  assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
huffman@44844
   849
proof (simp add: arg_def assms, rule someI_ex)
huffman@44844
   850
  obtain r a where z: "z = rcis r a" using rcis_Ex by fast
huffman@44844
   851
  with assms have "r \<noteq> 0" by auto
huffman@44844
   852
  def b \<equiv> "if 0 < r then a else a + pi"
huffman@44844
   853
  have b: "sgn z = cis b"
huffman@44844
   854
    unfolding z b_def rcis_def using `r \<noteq> 0`
huffman@44844
   855
    by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def)
huffman@44844
   856
  have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
webertj@49962
   857
    by (induct_tac n, simp_all add: distrib_left cis_mult [symmetric],
huffman@44844
   858
      simp add: cis_def)
huffman@44844
   859
  have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
huffman@44844
   860
    by (case_tac x rule: int_diff_cases,
huffman@44844
   861
      simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
huffman@44844
   862
  def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
huffman@44844
   863
  have "sgn z = cis c"
huffman@44844
   864
    unfolding b c_def
huffman@44844
   865
    by (simp add: cis_divide [symmetric] cis_2pi_int)
huffman@44844
   866
  moreover have "- pi < c \<and> c \<le> pi"
huffman@44844
   867
    using ceiling_correct [of "(b - pi) / (2*pi)"]
huffman@44844
   868
    by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)
huffman@44844
   869
  ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
huffman@44844
   870
qed
huffman@44844
   871
huffman@44844
   872
lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
huffman@44844
   873
  by (cases "z = 0", simp_all add: arg_zero arg_correct)
huffman@44844
   874
huffman@44844
   875
lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
huffman@44844
   876
  by (simp add: arg_correct)
huffman@44844
   877
huffman@44844
   878
lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
huffman@44844
   879
  by (cases "z = 0", simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
huffman@44844
   880
huffman@44844
   881
lemma cos_arg_i_mult_zero [simp]:
huffman@44844
   882
     "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
huffman@44844
   883
  using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff)
huffman@44844
   884
huffman@44065
   885
text {* Legacy theorem names *}
huffman@44065
   886
huffman@44065
   887
lemmas expand_complex_eq = complex_eq_iff
huffman@44065
   888
lemmas complex_Re_Im_cancel_iff = complex_eq_iff
huffman@44065
   889
lemmas complex_equality = complex_eqI
huffman@44065
   890
paulson@13957
   891
end