src/HOL/Complex.thy
author hoelzl
Wed Apr 02 18:35:01 2014 +0200 (2014-04-02)
changeset 56369 2704ca85be98
parent 56331 bea2196627cb
child 56381 0556204bc230
permissions -rw-r--r--
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
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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
hoelzl@56369
   342
hoelzl@56369
   343
lemma abs_sqrt_wlog:
hoelzl@56369
   344
  fixes x::"'a::linordered_idom"
hoelzl@56369
   345
  assumes "\<And>x::'a. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"
hoelzl@56369
   346
by (metis abs_ge_zero assms power2_abs)
hoelzl@56369
   347
hoelzl@56369
   348
lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z"
hoelzl@56369
   349
  unfolding complex_norm_def
hoelzl@56369
   350
  apply (rule abs_sqrt_wlog [where x="Re z"])
hoelzl@56369
   351
  apply (rule abs_sqrt_wlog [where x="Im z"])
hoelzl@56369
   352
  apply (rule power2_le_imp_le)
hoelzl@56369
   353
  apply (simp_all add: power2_sum add_commute sum_squares_bound real_sqrt_mult [symmetric])
hoelzl@56369
   354
  done
hoelzl@56369
   355
hoelzl@56369
   356
huffman@44843
   357
text {* Properties of complex signum. *}
huffman@44843
   358
huffman@44843
   359
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
huffman@44843
   360
  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
huffman@44843
   361
huffman@44843
   362
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
huffman@44843
   363
  by (simp add: complex_sgn_def divide_inverse)
huffman@44843
   364
huffman@44843
   365
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
huffman@44843
   366
  by (simp add: complex_sgn_def divide_inverse)
huffman@44843
   367
paulson@14354
   368
huffman@23123
   369
subsection {* Completeness of the Complexes *}
huffman@23123
   370
huffman@44290
   371
lemma bounded_linear_Re: "bounded_linear Re"
huffman@44290
   372
  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@44290
   373
huffman@44290
   374
lemma bounded_linear_Im: "bounded_linear Im"
huffman@44127
   375
  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@23123
   376
huffman@44290
   377
lemmas tendsto_Re [tendsto_intros] =
huffman@44290
   378
  bounded_linear.tendsto [OF bounded_linear_Re]
huffman@44290
   379
huffman@44290
   380
lemmas tendsto_Im [tendsto_intros] =
huffman@44290
   381
  bounded_linear.tendsto [OF bounded_linear_Im]
huffman@44290
   382
huffman@44290
   383
lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
huffman@44290
   384
lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
huffman@44290
   385
lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
huffman@44290
   386
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
huffman@23123
   387
hoelzl@56369
   388
lemma continuous_Re: "continuous_on X Re"
hoelzl@56369
   389
  by (metis (poly_guards_query) bounded_linear.continuous_on bounded_linear_Re 
hoelzl@56369
   390
            continuous_on_cong continuous_on_id)
hoelzl@56369
   391
hoelzl@56369
   392
lemma continuous_Im: "continuous_on X Im"
hoelzl@56369
   393
  by (metis (poly_guards_query) bounded_linear.continuous_on bounded_linear_Im 
hoelzl@56369
   394
            continuous_on_cong continuous_on_id)
hoelzl@56369
   395
hoelzl@56369
   396
lemma has_derivative_Re [has_derivative_intros] : "(Re has_derivative Re) F"
hoelzl@56369
   397
  by (auto simp add: has_derivative_def bounded_linear_Re tendsto_const)
hoelzl@56369
   398
hoelzl@56369
   399
lemma has_derivative_Im [has_derivative_intros] : "(Im has_derivative Im) F"
hoelzl@56369
   400
  by (auto simp add: has_derivative_def bounded_linear_Im tendsto_const)
hoelzl@56369
   401
huffman@36825
   402
lemma tendsto_Complex [tendsto_intros]:
huffman@44724
   403
  assumes "(f ---> a) F" and "(g ---> b) F"
huffman@44724
   404
  shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
huffman@36825
   405
proof (rule tendstoI)
huffman@36825
   406
  fix r :: real assume "0 < r"
huffman@36825
   407
  hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)
huffman@44724
   408
  have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F"
huffman@44724
   409
    using `(f ---> a) F` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   410
  moreover
huffman@44724
   411
  have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F"
huffman@44724
   412
    using `(g ---> b) F` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   413
  ultimately
huffman@44724
   414
  show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F"
huffman@36825
   415
    by (rule eventually_elim2)
huffman@36825
   416
       (simp add: dist_norm real_sqrt_sum_squares_less)
huffman@36825
   417
qed
huffman@36825
   418
hoelzl@56369
   419
hoelzl@56369
   420
lemma tendsto_complex_iff:
hoelzl@56369
   421
  "(f ---> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) ---> Re x) F \<and> ((\<lambda>x. Im (f x)) ---> Im x) F)"
hoelzl@56369
   422
proof -
hoelzl@56369
   423
  have f: "f = (\<lambda>x. Complex (Re (f x)) (Im (f x)))" and x: "x = Complex (Re x) (Im x)"
hoelzl@56369
   424
    by simp_all
hoelzl@56369
   425
  show ?thesis
hoelzl@56369
   426
    apply (subst f)
hoelzl@56369
   427
    apply (subst x)
hoelzl@56369
   428
    apply (intro iffI tendsto_Complex conjI)
hoelzl@56369
   429
    apply (simp_all add: tendsto_Re tendsto_Im)
hoelzl@56369
   430
    done
hoelzl@56369
   431
qed
hoelzl@56369
   432
huffman@23123
   433
instance complex :: banach
huffman@23123
   434
proof
huffman@23123
   435
  fix X :: "nat \<Rightarrow> complex"
huffman@23123
   436
  assume X: "Cauchy X"
huffman@44290
   437
  from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"
huffman@23123
   438
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@44290
   439
  from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
huffman@23123
   440
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@23123
   441
  have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
huffman@44748
   442
    using tendsto_Complex [OF 1 2] by simp
huffman@23123
   443
  thus "convergent X"
huffman@23123
   444
    by (rule convergentI)
huffman@23123
   445
qed
huffman@23123
   446
lp15@56238
   447
declare
lp15@56238
   448
  DERIV_power[where 'a=complex, THEN DERIV_cong,
lp15@56238
   449
              unfolded of_nat_def[symmetric], DERIV_intros]
lp15@56238
   450
huffman@23123
   451
huffman@44827
   452
subsection {* The Complex Number $i$ *}
huffman@23125
   453
huffman@44724
   454
definition "ii" :: complex  ("\<i>")
huffman@44724
   455
  where i_def: "ii \<equiv> Complex 0 1"
huffman@23125
   456
huffman@23125
   457
lemma complex_Re_i [simp]: "Re ii = 0"
huffman@44724
   458
  by (simp add: i_def)
paulson@14354
   459
huffman@23125
   460
lemma complex_Im_i [simp]: "Im ii = 1"
huffman@44724
   461
  by (simp add: i_def)
huffman@23125
   462
huffman@23125
   463
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
huffman@44724
   464
  by (simp add: i_def)
huffman@23125
   465
huffman@44902
   466
lemma norm_ii [simp]: "norm ii = 1"
huffman@44902
   467
  by (simp add: i_def)
huffman@44902
   468
huffman@23125
   469
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
huffman@44724
   470
  by (simp add: complex_eq_iff)
huffman@23125
   471
huffman@23125
   472
lemma complex_i_not_one [simp]: "ii \<noteq> 1"
huffman@44724
   473
  by (simp add: complex_eq_iff)
huffman@23124
   474
huffman@47108
   475
lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"
huffman@47108
   476
  by (simp add: complex_eq_iff)
huffman@47108
   477
haftmann@54489
   478
lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> - numeral w"
huffman@44724
   479
  by (simp add: complex_eq_iff)
huffman@23125
   480
huffman@23125
   481
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
huffman@44724
   482
  by (simp add: complex_eq_iff)
huffman@23125
   483
huffman@23125
   484
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
huffman@44724
   485
  by (simp add: complex_eq_iff)
huffman@23125
   486
huffman@23125
   487
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
huffman@44724
   488
  by (simp add: i_def complex_of_real_def)
huffman@23125
   489
huffman@23125
   490
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
huffman@44724
   491
  by (simp add: i_def complex_of_real_def)
huffman@23125
   492
huffman@23125
   493
lemma i_squared [simp]: "ii * ii = -1"
huffman@44724
   494
  by (simp add: i_def)
huffman@23125
   495
wenzelm@53015
   496
lemma power2_i [simp]: "ii\<^sup>2 = -1"
huffman@44724
   497
  by (simp add: power2_eq_square)
huffman@23125
   498
huffman@23125
   499
lemma inverse_i [simp]: "inverse ii = - ii"
huffman@44724
   500
  by (rule inverse_unique, simp)
paulson@14354
   501
huffman@44827
   502
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
huffman@44827
   503
  by (simp add: mult_assoc [symmetric])
huffman@44827
   504
paulson@14354
   505
huffman@23125
   506
subsection {* Complex Conjugation *}
huffman@23125
   507
huffman@44724
   508
definition cnj :: "complex \<Rightarrow> complex" where
huffman@23125
   509
  "cnj z = Complex (Re z) (- Im z)"
huffman@23125
   510
huffman@23125
   511
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
huffman@44724
   512
  by (simp add: cnj_def)
huffman@23125
   513
huffman@23125
   514
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
huffman@44724
   515
  by (simp add: cnj_def)
huffman@23125
   516
huffman@23125
   517
lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
huffman@44724
   518
  by (simp add: cnj_def)
huffman@23125
   519
huffman@23125
   520
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
huffman@44724
   521
  by (simp add: complex_eq_iff)
huffman@23125
   522
huffman@23125
   523
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
huffman@44724
   524
  by (simp add: cnj_def)
huffman@23125
   525
huffman@23125
   526
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
huffman@44724
   527
  by (simp add: complex_eq_iff)
huffman@23125
   528
huffman@23125
   529
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
huffman@44724
   530
  by (simp add: complex_eq_iff)
huffman@23125
   531
huffman@23125
   532
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
huffman@44724
   533
  by (simp add: complex_eq_iff)
huffman@23125
   534
hoelzl@56369
   535
lemma cnj_setsum: "cnj (setsum f s) = (\<Sum>x\<in>s. cnj (f x))"
hoelzl@56369
   536
  by (induct s rule: infinite_finite_induct) (auto simp: complex_cnj_add)
hoelzl@56369
   537
huffman@23125
   538
lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
huffman@44724
   539
  by (simp add: complex_eq_iff)
huffman@23125
   540
huffman@23125
   541
lemma complex_cnj_minus: "cnj (- x) = - cnj x"
huffman@44724
   542
  by (simp add: complex_eq_iff)
huffman@23125
   543
huffman@23125
   544
lemma complex_cnj_one [simp]: "cnj 1 = 1"
huffman@44724
   545
  by (simp add: complex_eq_iff)
huffman@23125
   546
huffman@23125
   547
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
huffman@44724
   548
  by (simp add: complex_eq_iff)
huffman@23125
   549
hoelzl@56369
   550
lemma cnj_setprod: "cnj (setprod f s) = (\<Prod>x\<in>s. cnj (f x))"
hoelzl@56369
   551
  by (induct s rule: infinite_finite_induct) (auto simp: complex_cnj_mult)
hoelzl@56369
   552
huffman@23125
   553
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
huffman@44724
   554
  by (simp add: complex_inverse_def)
paulson@14323
   555
huffman@23125
   556
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
huffman@44724
   557
  by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
huffman@23125
   558
huffman@23125
   559
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
huffman@44724
   560
  by (induct n, simp_all add: complex_cnj_mult)
huffman@23125
   561
huffman@23125
   562
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
huffman@44724
   563
  by (simp add: complex_eq_iff)
huffman@23125
   564
huffman@23125
   565
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
huffman@44724
   566
  by (simp add: complex_eq_iff)
huffman@23125
   567
huffman@47108
   568
lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
huffman@47108
   569
  by (simp add: complex_eq_iff)
huffman@47108
   570
haftmann@54489
   571
lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
huffman@44724
   572
  by (simp add: complex_eq_iff)
huffman@23125
   573
huffman@23125
   574
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
huffman@44724
   575
  by (simp add: complex_eq_iff)
huffman@23125
   576
huffman@23125
   577
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
huffman@44724
   578
  by (simp add: complex_norm_def)
paulson@14323
   579
huffman@23125
   580
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
huffman@44724
   581
  by (simp add: complex_eq_iff)
huffman@23125
   582
huffman@23125
   583
lemma complex_cnj_i [simp]: "cnj ii = - ii"
huffman@44724
   584
  by (simp add: complex_eq_iff)
huffman@23125
   585
huffman@23125
   586
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
huffman@44724
   587
  by (simp add: complex_eq_iff)
huffman@23125
   588
huffman@23125
   589
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
huffman@44724
   590
  by (simp add: complex_eq_iff)
paulson@14354
   591
wenzelm@53015
   592
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
huffman@44724
   593
  by (simp add: complex_eq_iff power2_eq_square)
huffman@23125
   594
wenzelm@53015
   595
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"
huffman@44724
   596
  by (simp add: norm_mult power2_eq_square)
huffman@23125
   597
huffman@44827
   598
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
huffman@44827
   599
  by (simp add: cmod_def power2_eq_square)
huffman@44827
   600
huffman@44827
   601
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
huffman@44827
   602
  by simp
huffman@44827
   603
huffman@44290
   604
lemma bounded_linear_cnj: "bounded_linear cnj"
huffman@44127
   605
  using complex_cnj_add complex_cnj_scaleR
huffman@44127
   606
  by (rule bounded_linear_intro [where K=1], simp)
paulson@14354
   607
huffman@44290
   608
lemmas tendsto_cnj [tendsto_intros] =
huffman@44290
   609
  bounded_linear.tendsto [OF bounded_linear_cnj]
huffman@44290
   610
huffman@44290
   611
lemmas isCont_cnj [simp] =
huffman@44290
   612
  bounded_linear.isCont [OF bounded_linear_cnj]
huffman@44290
   613
hoelzl@56369
   614
lemma lim_cnj: "((\<lambda>x. cnj(f x)) ---> cnj l) F \<longleftrightarrow> (f ---> l) F"
hoelzl@56369
   615
  by (simp add: tendsto_iff dist_complex_def Complex.complex_cnj_diff [symmetric])
hoelzl@56369
   616
hoelzl@56369
   617
lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
hoelzl@56369
   618
  by (simp add: sums_def lim_cnj cnj_setsum [symmetric])
hoelzl@56369
   619
paulson@14354
   620
lp15@55734
   621
subsection{*Basic Lemmas*}
lp15@55734
   622
lp15@55734
   623
lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"
lp15@55734
   624
  by (metis complex_Im_zero complex_Re_zero complex_eqI sum_power2_eq_zero_iff)
lp15@55734
   625
lp15@55734
   626
lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"
lp15@55734
   627
by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)
lp15@55734
   628
lp15@55734
   629
lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"
lp15@55734
   630
apply (cases z, auto)
lp15@55734
   631
by (metis complex_of_real_def of_real_add of_real_power power2_eq_square)
lp15@55734
   632
lp15@55734
   633
lemma complex_div_eq_0: 
lp15@55734
   634
    "(Re(a / b) = 0 \<longleftrightarrow> Re(a * cnj b) = 0) & (Im(a / b) = 0 \<longleftrightarrow> Im(a * cnj b) = 0)"
lp15@55734
   635
proof (cases "b=0")
lp15@55734
   636
  case True then show ?thesis by auto
lp15@55734
   637
next
lp15@55734
   638
  case False
lp15@55734
   639
  show ?thesis
lp15@55734
   640
  proof (cases b)
lp15@55734
   641
    case (Complex x y)
lp15@55734
   642
    then have "x\<^sup>2 + y\<^sup>2 > 0"
lp15@55734
   643
      by (metis Complex_eq_0 False sum_power2_gt_zero_iff)
lp15@55734
   644
    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
   645
      by (metis add_divide_distrib)
lp15@55734
   646
    with Complex False show ?thesis
lp15@55734
   647
      by (auto simp: complex_divide_def)
lp15@55734
   648
  qed
lp15@55734
   649
qed
lp15@55734
   650
lp15@55734
   651
lemma re_complex_div_eq_0: "Re(a / b) = 0 \<longleftrightarrow> Re(a * cnj b) = 0"
lp15@55734
   652
  and im_complex_div_eq_0: "Im(a / b) = 0 \<longleftrightarrow> Im(a * cnj b) = 0"
lp15@55734
   653
using complex_div_eq_0 by auto
lp15@55734
   654
lp15@55734
   655
lp15@55734
   656
lemma complex_div_gt_0: 
lp15@55734
   657
    "(Re(a / b) > 0 \<longleftrightarrow> Re(a * cnj b) > 0) & (Im(a / b) > 0 \<longleftrightarrow> Im(a * cnj b) > 0)"
lp15@55734
   658
proof (cases "b=0")
lp15@55734
   659
  case True then show ?thesis by auto
lp15@55734
   660
next
lp15@55734
   661
  case False
lp15@55734
   662
  show ?thesis
lp15@55734
   663
  proof (cases b)
lp15@55734
   664
    case (Complex x y)
lp15@55734
   665
    then have "x\<^sup>2 + y\<^sup>2 > 0"
lp15@55734
   666
      by (metis Complex_eq_0 False sum_power2_gt_zero_iff)
lp15@55734
   667
    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
   668
      by (metis add_divide_distrib)
lp15@55734
   669
    ultimately show ?thesis using Complex False `0 < x\<^sup>2 + y\<^sup>2`
lp15@55734
   670
      apply (simp add: complex_divide_def  zero_less_divide_iff less_divide_eq)
lp15@55734
   671
      apply (metis less_divide_eq mult_zero_left diff_conv_add_uminus minus_divide_left)
lp15@55734
   672
      done
lp15@55734
   673
  qed
lp15@55734
   674
qed
lp15@55734
   675
lp15@55734
   676
lemma re_complex_div_gt_0: "Re(a / b) > 0 \<longleftrightarrow> Re(a * cnj b) > 0"
lp15@55734
   677
  and im_complex_div_gt_0: "Im(a / b) > 0 \<longleftrightarrow> Im(a * cnj b) > 0"
lp15@55734
   678
using complex_div_gt_0 by auto
lp15@55734
   679
lp15@55734
   680
lemma re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0"
lp15@55734
   681
  by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0)
lp15@55734
   682
lp15@55734
   683
lemma im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0"
lp15@55734
   684
  by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less)
lp15@55734
   685
lp15@55734
   686
lemma re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0"
boehmes@55759
   687
  by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0)
lp15@55734
   688
lp15@55734
   689
lemma im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0"
lp15@55734
   690
  by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff)
lp15@55734
   691
lp15@55734
   692
lemma re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0"
lp15@55734
   693
  by (metis not_le re_complex_div_gt_0)
lp15@55734
   694
lp15@55734
   695
lemma im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0"
lp15@55734
   696
  by (metis im_complex_div_gt_0 not_le)
lp15@55734
   697
lp15@56217
   698
lemma Re_setsum: "Re(setsum f s) = setsum (%x. Re(f x)) s"
hoelzl@56369
   699
  by (induct s rule: infinite_finite_induct) auto
lp15@55734
   700
lp15@56217
   701
lemma Im_setsum: "Im(setsum f s) = setsum (%x. Im(f x)) s"
hoelzl@56369
   702
  by (induct s rule: infinite_finite_induct) auto
hoelzl@56369
   703
hoelzl@56369
   704
lemma sums_complex_iff: "f sums x \<longleftrightarrow> ((\<lambda>x. Re (f x)) sums Re x) \<and> ((\<lambda>x. Im (f x)) sums Im x)"
hoelzl@56369
   705
  unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..
hoelzl@56369
   706
  
hoelzl@56369
   707
lemma sums_Re: "f sums a \<Longrightarrow> (\<lambda>x. Re (f x)) sums Re a"
hoelzl@56369
   708
  unfolding sums_complex_iff by blast
hoelzl@56369
   709
hoelzl@56369
   710
lemma sums_Im: "f sums a \<Longrightarrow> (\<lambda>x. Im (f x)) sums Im a"
hoelzl@56369
   711
  unfolding sums_complex_iff by blast
hoelzl@56369
   712
hoelzl@56369
   713
lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and>  summable (\<lambda>x. Im (f x))"
hoelzl@56369
   714
  unfolding summable_def sums_complex_iff[abs_def] by (metis Im.simps Re.simps)
hoelzl@56369
   715
hoelzl@56369
   716
lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f"
hoelzl@56369
   717
  unfolding summable_complex_iff by simp
hoelzl@56369
   718
hoelzl@56369
   719
lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))"
hoelzl@56369
   720
  unfolding summable_complex_iff by blast
hoelzl@56369
   721
hoelzl@56369
   722
lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))"
hoelzl@56369
   723
  unfolding summable_complex_iff by blast
lp15@56217
   724
lp15@56217
   725
lemma Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"
hoelzl@56369
   726
  by (induct s rule: infinite_finite_induct) auto
lp15@55734
   727
lp15@56217
   728
lemma Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"
lp15@56217
   729
  by (metis Complex_setsum')
lp15@56217
   730
lp15@56217
   731
lemma of_real_setsum: "of_real (setsum f s) = setsum (%x. of_real(f x)) s"
hoelzl@56369
   732
  by (induct s rule: infinite_finite_induct) auto
lp15@55734
   733
lp15@56217
   734
lemma of_real_setprod: "of_real (setprod f s) = setprod (%x. of_real(f x)) s"
hoelzl@56369
   735
  by (induct s rule: infinite_finite_induct) auto
lp15@55734
   736
lp15@55734
   737
lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z"
lp15@55734
   738
by (metis Reals_cases Reals_of_real complex.exhaust complex.inject complex_cnj 
lp15@55734
   739
          complex_of_real_def equal_neg_zero)
lp15@55734
   740
lp15@55734
   741
lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"
lp15@55734
   742
  by (metis Reals_of_real complex_of_real_def)
lp15@55734
   743
lp15@55734
   744
lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm(z) = abs(Re z)"
lp15@55734
   745
  by (metis Re_complex_of_real Reals_cases norm_of_real)
lp15@55734
   746
hoelzl@56369
   747
lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
hoelzl@56369
   748
  by (metis Complex_in_Reals Im_complex_of_real Reals_cases complex_surj)
hoelzl@56369
   749
hoelzl@56369
   750
lemma series_comparison_complex:
hoelzl@56369
   751
  fixes f:: "nat \<Rightarrow> 'a::banach"
hoelzl@56369
   752
  assumes sg: "summable g"
hoelzl@56369
   753
     and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"
hoelzl@56369
   754
     and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"
hoelzl@56369
   755
  shows "summable f"
hoelzl@56369
   756
proof -
hoelzl@56369
   757
  have g: "\<And>n. cmod (g n) = Re (g n)" using assms
hoelzl@56369
   758
    by (metis abs_of_nonneg in_Reals_norm)
hoelzl@56369
   759
  show ?thesis
hoelzl@56369
   760
    apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])
hoelzl@56369
   761
    using sg
hoelzl@56369
   762
    apply (auto simp: summable_def)
hoelzl@56369
   763
    apply (rule_tac x="Re s" in exI)
hoelzl@56369
   764
    apply (auto simp: g sums_Re)
hoelzl@56369
   765
    apply (metis fg g)
hoelzl@56369
   766
    done
hoelzl@56369
   767
qed
lp15@55734
   768
paulson@14323
   769
subsection{*Finally! Polar Form for Complex Numbers*}
paulson@14323
   770
huffman@44827
   771
subsubsection {* $\cos \theta + i \sin \theta$ *}
huffman@20557
   772
huffman@44715
   773
definition cis :: "real \<Rightarrow> complex" where
huffman@20557
   774
  "cis a = Complex (cos a) (sin a)"
huffman@20557
   775
huffman@44827
   776
lemma Re_cis [simp]: "Re (cis a) = cos a"
huffman@44827
   777
  by (simp add: cis_def)
huffman@44827
   778
huffman@44827
   779
lemma Im_cis [simp]: "Im (cis a) = sin a"
huffman@44827
   780
  by (simp add: cis_def)
huffman@44827
   781
huffman@44827
   782
lemma cis_zero [simp]: "cis 0 = 1"
huffman@44827
   783
  by (simp add: cis_def)
huffman@44827
   784
huffman@44828
   785
lemma norm_cis [simp]: "norm (cis a) = 1"
huffman@44828
   786
  by (simp add: cis_def)
huffman@44828
   787
huffman@44828
   788
lemma sgn_cis [simp]: "sgn (cis a) = cis a"
huffman@44828
   789
  by (simp add: sgn_div_norm)
huffman@44828
   790
huffman@44828
   791
lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
huffman@44828
   792
  by (metis norm_cis norm_zero zero_neq_one)
huffman@44828
   793
huffman@44827
   794
lemma cis_mult: "cis a * cis b = cis (a + b)"
huffman@44827
   795
  by (simp add: cis_def cos_add sin_add)
huffman@44827
   796
huffman@44827
   797
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
huffman@44827
   798
  by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
huffman@44827
   799
huffman@44827
   800
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
huffman@44827
   801
  by (simp add: cis_def)
huffman@44827
   802
huffman@44827
   803
lemma cis_divide: "cis a / cis b = cis (a - b)"
haftmann@54230
   804
  by (simp add: complex_divide_def cis_mult)
huffman@44827
   805
huffman@44827
   806
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
huffman@44827
   807
  by (auto simp add: DeMoivre)
huffman@44827
   808
huffman@44827
   809
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
huffman@44827
   810
  by (auto simp add: DeMoivre)
huffman@44827
   811
huffman@44827
   812
subsubsection {* $r(\cos \theta + i \sin \theta)$ *}
huffman@44715
   813
huffman@44715
   814
definition rcis :: "[real, real] \<Rightarrow> complex" where
huffman@20557
   815
  "rcis r a = complex_of_real r * cis a"
huffman@20557
   816
huffman@44827
   817
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
huffman@44828
   818
  by (simp add: rcis_def)
huffman@44827
   819
huffman@44827
   820
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
huffman@44828
   821
  by (simp add: rcis_def)
huffman@44827
   822
huffman@44827
   823
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
huffman@44828
   824
  by (simp add: complex_eq_iff polar_Ex)
huffman@44827
   825
huffman@44827
   826
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
huffman@44828
   827
  by (simp add: rcis_def norm_mult)
huffman@44827
   828
huffman@44827
   829
lemma cis_rcis_eq: "cis a = rcis 1 a"
huffman@44827
   830
  by (simp add: rcis_def)
huffman@44827
   831
huffman@44827
   832
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
huffman@44828
   833
  by (simp add: rcis_def cis_mult)
huffman@44827
   834
huffman@44827
   835
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
huffman@44827
   836
  by (simp add: rcis_def)
huffman@44827
   837
huffman@44827
   838
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
huffman@44827
   839
  by (simp add: rcis_def)
huffman@44827
   840
huffman@44828
   841
lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
huffman@44828
   842
  by (simp add: rcis_def)
huffman@44828
   843
huffman@44827
   844
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
huffman@44827
   845
  by (simp add: rcis_def power_mult_distrib DeMoivre)
huffman@44827
   846
huffman@44827
   847
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
huffman@44827
   848
  by (simp add: divide_inverse rcis_def)
huffman@44827
   849
huffman@44827
   850
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
huffman@44828
   851
  by (simp add: rcis_def cis_divide [symmetric])
huffman@44827
   852
huffman@44827
   853
subsubsection {* Complex exponential *}
huffman@44827
   854
huffman@44291
   855
abbreviation expi :: "complex \<Rightarrow> complex"
huffman@44291
   856
  where "expi \<equiv> exp"
huffman@44291
   857
huffman@44712
   858
lemma cis_conv_exp: "cis b = exp (Complex 0 b)"
huffman@44291
   859
proof (rule complex_eqI)
huffman@44291
   860
  { fix n have "Complex 0 b ^ n =
huffman@44291
   861
    real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"
huffman@44291
   862
      apply (induct n)
huffman@44291
   863
      apply (simp add: cos_coeff_def sin_coeff_def)
huffman@44291
   864
      apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)
huffman@44291
   865
      done } note * = this
huffman@44712
   866
  show "Re (cis b) = Re (exp (Complex 0 b))"
huffman@44291
   867
    unfolding exp_def cis_def cos_def
huffman@44291
   868
    by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],
huffman@44291
   869
      simp add: * mult_assoc [symmetric])
huffman@44712
   870
  show "Im (cis b) = Im (exp (Complex 0 b))"
huffman@44291
   871
    unfolding exp_def cis_def sin_def
huffman@44291
   872
    by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],
huffman@44291
   873
      simp add: * mult_assoc [symmetric])
huffman@44291
   874
qed
huffman@44291
   875
huffman@44291
   876
lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"
huffman@44712
   877
  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp
huffman@20557
   878
huffman@44828
   879
lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
huffman@44828
   880
  unfolding expi_def by simp
huffman@44828
   881
huffman@44828
   882
lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
huffman@44828
   883
  unfolding expi_def by simp
huffman@44828
   884
paulson@14374
   885
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
paulson@14373
   886
apply (insert rcis_Ex [of z])
huffman@23125
   887
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
paulson@14334
   888
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
paulson@14323
   889
done
paulson@14323
   890
paulson@14387
   891
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
huffman@44724
   892
  by (simp add: expi_def cis_def)
paulson@14387
   893
huffman@44844
   894
subsubsection {* Complex argument *}
huffman@44844
   895
huffman@44844
   896
definition arg :: "complex \<Rightarrow> real" where
huffman@44844
   897
  "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"
huffman@44844
   898
huffman@44844
   899
lemma arg_zero: "arg 0 = 0"
huffman@44844
   900
  by (simp add: arg_def)
huffman@44844
   901
huffman@44844
   902
lemma of_nat_less_of_int_iff: (* TODO: move *)
huffman@44844
   903
  "(of_nat n :: 'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
huffman@44844
   904
  by (metis of_int_of_nat_eq of_int_less_iff)
huffman@44844
   905
huffman@47108
   906
lemma real_of_nat_less_numeral_iff [simp]: (* TODO: move *)
huffman@47108
   907
  "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"
huffman@47108
   908
  using of_nat_less_of_int_iff [of n "numeral w", where 'a=real]
huffman@47108
   909
  by (simp add: real_of_nat_def zless_nat_eq_int_zless [symmetric])
huffman@44844
   910
huffman@44844
   911
lemma arg_unique:
huffman@44844
   912
  assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
huffman@44844
   913
  shows "arg z = x"
huffman@44844
   914
proof -
huffman@44844
   915
  from assms have "z \<noteq> 0" by auto
huffman@44844
   916
  have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
huffman@44844
   917
  proof
huffman@44844
   918
    fix a def d \<equiv> "a - x"
huffman@44844
   919
    assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
huffman@44844
   920
    from a assms have "- (2*pi) < d \<and> d < 2*pi"
huffman@44844
   921
      unfolding d_def by simp
huffman@44844
   922
    moreover from a assms have "cos a = cos x" and "sin a = sin x"
huffman@44844
   923
      by (simp_all add: complex_eq_iff)
wenzelm@53374
   924
    hence cos: "cos d = 1" unfolding d_def cos_diff by simp
wenzelm@53374
   925
    moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)
huffman@44844
   926
    ultimately have "d = 0"
huffman@44844
   927
      unfolding sin_zero_iff even_mult_two_ex
wenzelm@53374
   928
      by (auto simp add: numeral_2_eq_2 less_Suc_eq)
huffman@44844
   929
    thus "a = x" unfolding d_def by simp
huffman@44844
   930
  qed (simp add: assms del: Re_sgn Im_sgn)
huffman@44844
   931
  with `z \<noteq> 0` show "arg z = x"
huffman@44844
   932
    unfolding arg_def by simp
huffman@44844
   933
qed
huffman@44844
   934
huffman@44844
   935
lemma arg_correct:
huffman@44844
   936
  assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
huffman@44844
   937
proof (simp add: arg_def assms, rule someI_ex)
huffman@44844
   938
  obtain r a where z: "z = rcis r a" using rcis_Ex by fast
huffman@44844
   939
  with assms have "r \<noteq> 0" by auto
huffman@44844
   940
  def b \<equiv> "if 0 < r then a else a + pi"
huffman@44844
   941
  have b: "sgn z = cis b"
huffman@44844
   942
    unfolding z b_def rcis_def using `r \<noteq> 0`
huffman@44844
   943
    by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def)
huffman@44844
   944
  have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
webertj@49962
   945
    by (induct_tac n, simp_all add: distrib_left cis_mult [symmetric],
huffman@44844
   946
      simp add: cis_def)
huffman@44844
   947
  have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
huffman@44844
   948
    by (case_tac x rule: int_diff_cases,
huffman@44844
   949
      simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
huffman@44844
   950
  def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
huffman@44844
   951
  have "sgn z = cis c"
huffman@44844
   952
    unfolding b c_def
huffman@44844
   953
    by (simp add: cis_divide [symmetric] cis_2pi_int)
huffman@44844
   954
  moreover have "- pi < c \<and> c \<le> pi"
huffman@44844
   955
    using ceiling_correct [of "(b - pi) / (2*pi)"]
huffman@44844
   956
    by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)
huffman@44844
   957
  ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
huffman@44844
   958
qed
huffman@44844
   959
huffman@44844
   960
lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
huffman@44844
   961
  by (cases "z = 0", simp_all add: arg_zero arg_correct)
huffman@44844
   962
huffman@44844
   963
lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
huffman@44844
   964
  by (simp add: arg_correct)
huffman@44844
   965
huffman@44844
   966
lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
huffman@44844
   967
  by (cases "z = 0", simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
huffman@44844
   968
huffman@44844
   969
lemma cos_arg_i_mult_zero [simp]:
huffman@44844
   970
     "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
huffman@44844
   971
  using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff)
huffman@44844
   972
huffman@44065
   973
text {* Legacy theorem names *}
huffman@44065
   974
huffman@44065
   975
lemmas expand_complex_eq = complex_eq_iff
huffman@44065
   976
lemmas complex_Re_Im_cancel_iff = complex_eq_iff
huffman@44065
   977
lemmas complex_equality = complex_eqI
huffman@44065
   978
hoelzl@56369
   979
paulson@13957
   980
end