src/HOL/Complex.thy
author hoelzl
Thu Jan 31 11:31:27 2013 +0100 (2013-01-31)
changeset 50999 3de230ed0547
parent 49962 a8cc904a6820
child 51002 496013a6eb38
permissions -rw-r--r--
introduce order topology
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(*  Title:       HOL/Complex.thy
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    Author:      Jacques D. Fleuriot
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    Copyright:   2001 University of Edinburgh
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
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*)
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header {* Complex Numbers: Rectangular and Polar Representations *}
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theory Complex
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imports Transcendental
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begin
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datatype complex = Complex real real
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primrec Re :: "complex \<Rightarrow> real"
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  where Re: "Re (Complex x y) = x"
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primrec Im :: "complex \<Rightarrow> real"
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  where Im: "Im (Complex x y) = y"
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lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"
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  by (induct z) simp
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lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"
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  by (induct x, induct y) simp
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lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"
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  by (induct x, induct y) simp
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subsection {* Addition and Subtraction *}
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instantiation complex :: ab_group_add
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begin
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definition complex_zero_def:
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  "0 = Complex 0 0"
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definition complex_add_def:
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  "x + y = Complex (Re x + Re y) (Im x + Im y)"
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definition complex_minus_def:
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  "- x = Complex (- Re x) (- Im x)"
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definition complex_diff_def:
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  "x - (y\<Colon>complex) = x + - y"
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lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
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  by (simp add: complex_zero_def)
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lemma complex_Re_zero [simp]: "Re 0 = 0"
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  by (simp add: complex_zero_def)
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lemma complex_Im_zero [simp]: "Im 0 = 0"
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  by (simp add: complex_zero_def)
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lemma complex_add [simp]:
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  "Complex a b + Complex c d = Complex (a + c) (b + d)"
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  by (simp add: complex_add_def)
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lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y"
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  by (simp add: complex_add_def)
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lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y"
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  by (simp add: complex_add_def)
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lemma complex_minus [simp]:
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  "- (Complex a b) = Complex (- a) (- b)"
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  by (simp add: complex_minus_def)
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lemma complex_Re_minus [simp]: "Re (- x) = - Re x"
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  by (simp add: complex_minus_def)
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lemma complex_Im_minus [simp]: "Im (- x) = - Im x"
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  by (simp add: complex_minus_def)
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lemma complex_diff [simp]:
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  "Complex a b - Complex c d = Complex (a - c) (b - d)"
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  by (simp add: complex_diff_def)
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lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y"
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  by (simp add: complex_diff_def)
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lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y"
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  by (simp add: complex_diff_def)
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instance
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  by intro_classes (simp_all add: complex_add_def complex_diff_def)
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end
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subsection {* Multiplication and Division *}
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instantiation complex :: field_inverse_zero
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begin
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definition complex_one_def:
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  "1 = Complex 1 0"
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definition complex_mult_def:
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  "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
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definition complex_inverse_def:
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  "inverse x =
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    Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))"
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definition complex_divide_def:
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  "x / (y\<Colon>complex) = x * inverse y"
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lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"
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  by (simp add: complex_one_def)
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lemma complex_Re_one [simp]: "Re 1 = 1"
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  by (simp add: complex_one_def)
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lemma complex_Im_one [simp]: "Im 1 = 0"
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  by (simp add: complex_one_def)
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lemma complex_mult [simp]:
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  "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
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  by (simp add: complex_mult_def)
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lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"
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  by (simp add: complex_mult_def)
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lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"
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  by (simp add: complex_mult_def)
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lemma complex_inverse [simp]:
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  "inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))"
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  by (simp add: complex_inverse_def)
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lemma complex_Re_inverse:
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  "Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
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  by (simp add: complex_inverse_def)
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lemma complex_Im_inverse:
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  "Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
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  by (simp add: complex_inverse_def)
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instance
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  by intro_classes (simp_all add: complex_mult_def
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    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 (neg_numeral v) = neg_numeral v"
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  using complex_Re_of_int [of "neg_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 (neg_numeral v) = 0"
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  using complex_Im_of_int [of "neg_numeral v"] by simp
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lemma Complex_eq_numeral [simp]:
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  "(Complex a b = numeral w) = (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 = neg_numeral w) = (a = neg_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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lemma complex_of_real_def: "complex_of_real r = Complex r 0"
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  by (simp add: of_real_def complex_scaleR_def)
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lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
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  by (simp add: complex_of_real_def)
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lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
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  by (simp add: complex_of_real_def)
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lemma Complex_add_complex_of_real [simp]:
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  shows "Complex x y + complex_of_real r = Complex (x+r) y"
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  by (simp add: complex_of_real_def)
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lemma complex_of_real_add_Complex [simp]:
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  shows "complex_of_real r + Complex x y = Complex (r+x) y"
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  by (simp add: complex_of_real_def)
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lemma Complex_mult_complex_of_real:
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  shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
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  by (simp add: complex_of_real_def)
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lemma complex_of_real_mult_Complex:
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  shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
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  by (simp add: complex_of_real_def)
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lemma complex_eq_cancel_iff2 [simp]:
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  shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
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  by (simp add: complex_of_real_def)
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lemma complex_split_polar:
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     "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
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  by (simp add: complex_eq_iff polar_Ex)
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subsection {* Vector Norm *}
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instantiation complex :: real_normed_field
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begin
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definition complex_norm_def:
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  "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
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abbreviation cmod :: "complex \<Rightarrow> real"
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  where "cmod \<equiv> norm"
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definition complex_sgn_def:
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  "sgn x = x /\<^sub>R cmod x"
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definition dist_complex_def:
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  "dist x y = cmod (x - y)"
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definition open_complex_def:
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  "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
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lemmas cmod_def = complex_norm_def
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lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)"
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  by (simp add: complex_norm_def)
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instance proof
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  fix r :: real and x y :: complex and S :: "complex set"
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  show "0 \<le> norm x"
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    by (induct x) simp
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  show "(norm x = 0) = (x = 0)"
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    by (induct x) simp
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  show "norm (x + y) \<le> norm x + norm y"
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    by (induct x, induct y)
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       (simp add: real_sqrt_sum_squares_triangle_ineq)
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  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
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    by (induct x)
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       (simp add: power_mult_distrib 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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lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
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  by (cases x) simp
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lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
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   335
  by (cases x) simp
huffman@44724
   336
huffman@44843
   337
text {* Properties of complex signum. *}
huffman@44843
   338
huffman@44843
   339
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
huffman@44843
   340
  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
huffman@44843
   341
huffman@44843
   342
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
huffman@44843
   343
  by (simp add: complex_sgn_def divide_inverse)
huffman@44843
   344
huffman@44843
   345
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
huffman@44843
   346
  by (simp add: complex_sgn_def divide_inverse)
huffman@44843
   347
paulson@14354
   348
huffman@23123
   349
subsection {* Completeness of the Complexes *}
huffman@23123
   350
huffman@44290
   351
lemma bounded_linear_Re: "bounded_linear Re"
huffman@44290
   352
  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@44290
   353
huffman@44290
   354
lemma bounded_linear_Im: "bounded_linear Im"
huffman@44127
   355
  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@23123
   356
huffman@44290
   357
lemmas tendsto_Re [tendsto_intros] =
huffman@44290
   358
  bounded_linear.tendsto [OF bounded_linear_Re]
huffman@44290
   359
huffman@44290
   360
lemmas tendsto_Im [tendsto_intros] =
huffman@44290
   361
  bounded_linear.tendsto [OF bounded_linear_Im]
huffman@44290
   362
huffman@44290
   363
lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
huffman@44290
   364
lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
huffman@44290
   365
lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
huffman@44290
   366
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
huffman@23123
   367
huffman@36825
   368
lemma tendsto_Complex [tendsto_intros]:
huffman@44724
   369
  assumes "(f ---> a) F" and "(g ---> b) F"
huffman@44724
   370
  shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
huffman@36825
   371
proof (rule tendstoI)
huffman@36825
   372
  fix r :: real assume "0 < r"
huffman@36825
   373
  hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)
huffman@44724
   374
  have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F"
huffman@44724
   375
    using `(f ---> a) F` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   376
  moreover
huffman@44724
   377
  have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F"
huffman@44724
   378
    using `(g ---> b) F` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   379
  ultimately
huffman@44724
   380
  show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F"
huffman@36825
   381
    by (rule eventually_elim2)
huffman@36825
   382
       (simp add: dist_norm real_sqrt_sum_squares_less)
huffman@36825
   383
qed
huffman@36825
   384
huffman@23123
   385
instance complex :: banach
huffman@23123
   386
proof
huffman@23123
   387
  fix X :: "nat \<Rightarrow> complex"
huffman@23123
   388
  assume X: "Cauchy X"
huffman@44290
   389
  from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"
huffman@23123
   390
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@44290
   391
  from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
huffman@23123
   392
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@23123
   393
  have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
huffman@44748
   394
    using tendsto_Complex [OF 1 2] by simp
huffman@23123
   395
  thus "convergent X"
huffman@23123
   396
    by (rule convergentI)
huffman@23123
   397
qed
huffman@23123
   398
huffman@23123
   399
huffman@44827
   400
subsection {* The Complex Number $i$ *}
huffman@23125
   401
huffman@44724
   402
definition "ii" :: complex  ("\<i>")
huffman@44724
   403
  where i_def: "ii \<equiv> Complex 0 1"
huffman@23125
   404
huffman@23125
   405
lemma complex_Re_i [simp]: "Re ii = 0"
huffman@44724
   406
  by (simp add: i_def)
paulson@14354
   407
huffman@23125
   408
lemma complex_Im_i [simp]: "Im ii = 1"
huffman@44724
   409
  by (simp add: i_def)
huffman@23125
   410
huffman@23125
   411
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
huffman@44724
   412
  by (simp add: i_def)
huffman@23125
   413
huffman@44902
   414
lemma norm_ii [simp]: "norm ii = 1"
huffman@44902
   415
  by (simp add: i_def)
huffman@44902
   416
huffman@23125
   417
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
huffman@44724
   418
  by (simp add: complex_eq_iff)
huffman@23125
   419
huffman@23125
   420
lemma complex_i_not_one [simp]: "ii \<noteq> 1"
huffman@44724
   421
  by (simp add: complex_eq_iff)
huffman@23124
   422
huffman@47108
   423
lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"
huffman@47108
   424
  by (simp add: complex_eq_iff)
huffman@47108
   425
huffman@47108
   426
lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> neg_numeral w"
huffman@44724
   427
  by (simp add: complex_eq_iff)
huffman@23125
   428
huffman@23125
   429
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
huffman@44724
   430
  by (simp add: complex_eq_iff)
huffman@23125
   431
huffman@23125
   432
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
huffman@44724
   433
  by (simp add: complex_eq_iff)
huffman@23125
   434
huffman@23125
   435
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
huffman@44724
   436
  by (simp add: i_def complex_of_real_def)
huffman@23125
   437
huffman@23125
   438
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
huffman@44724
   439
  by (simp add: i_def complex_of_real_def)
huffman@23125
   440
huffman@23125
   441
lemma i_squared [simp]: "ii * ii = -1"
huffman@44724
   442
  by (simp add: i_def)
huffman@23125
   443
huffman@23125
   444
lemma power2_i [simp]: "ii\<twosuperior> = -1"
huffman@44724
   445
  by (simp add: power2_eq_square)
huffman@23125
   446
huffman@23125
   447
lemma inverse_i [simp]: "inverse ii = - ii"
huffman@44724
   448
  by (rule inverse_unique, simp)
paulson@14354
   449
huffman@44827
   450
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
huffman@44827
   451
  by (simp add: mult_assoc [symmetric])
huffman@44827
   452
paulson@14354
   453
huffman@23125
   454
subsection {* Complex Conjugation *}
huffman@23125
   455
huffman@44724
   456
definition cnj :: "complex \<Rightarrow> complex" where
huffman@23125
   457
  "cnj z = Complex (Re z) (- Im z)"
huffman@23125
   458
huffman@23125
   459
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
huffman@44724
   460
  by (simp add: cnj_def)
huffman@23125
   461
huffman@23125
   462
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
huffman@44724
   463
  by (simp add: cnj_def)
huffman@23125
   464
huffman@23125
   465
lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
huffman@44724
   466
  by (simp add: cnj_def)
huffman@23125
   467
huffman@23125
   468
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
huffman@44724
   469
  by (simp add: complex_eq_iff)
huffman@23125
   470
huffman@23125
   471
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
huffman@44724
   472
  by (simp add: cnj_def)
huffman@23125
   473
huffman@23125
   474
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
huffman@44724
   475
  by (simp add: complex_eq_iff)
huffman@23125
   476
huffman@23125
   477
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
huffman@44724
   478
  by (simp add: complex_eq_iff)
huffman@23125
   479
huffman@23125
   480
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
huffman@44724
   481
  by (simp add: complex_eq_iff)
huffman@23125
   482
huffman@23125
   483
lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
huffman@44724
   484
  by (simp add: complex_eq_iff)
huffman@23125
   485
huffman@23125
   486
lemma complex_cnj_minus: "cnj (- x) = - cnj x"
huffman@44724
   487
  by (simp add: complex_eq_iff)
huffman@23125
   488
huffman@23125
   489
lemma complex_cnj_one [simp]: "cnj 1 = 1"
huffman@44724
   490
  by (simp add: complex_eq_iff)
huffman@23125
   491
huffman@23125
   492
lemma complex_cnj_mult: "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_inverse: "cnj (inverse x) = inverse (cnj x)"
huffman@44724
   496
  by (simp add: complex_inverse_def)
paulson@14323
   497
huffman@23125
   498
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
huffman@44724
   499
  by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
huffman@23125
   500
huffman@23125
   501
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
huffman@44724
   502
  by (induct n, simp_all add: complex_cnj_mult)
huffman@23125
   503
huffman@23125
   504
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
huffman@44724
   505
  by (simp add: complex_eq_iff)
huffman@23125
   506
huffman@23125
   507
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
huffman@44724
   508
  by (simp add: complex_eq_iff)
huffman@23125
   509
huffman@47108
   510
lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
huffman@47108
   511
  by (simp add: complex_eq_iff)
huffman@47108
   512
huffman@47108
   513
lemma complex_cnj_neg_numeral [simp]: "cnj (neg_numeral w) = neg_numeral w"
huffman@44724
   514
  by (simp add: complex_eq_iff)
huffman@23125
   515
huffman@23125
   516
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
huffman@44724
   517
  by (simp add: complex_eq_iff)
huffman@23125
   518
huffman@23125
   519
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
huffman@44724
   520
  by (simp add: complex_norm_def)
paulson@14323
   521
huffman@23125
   522
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
huffman@44724
   523
  by (simp add: complex_eq_iff)
huffman@23125
   524
huffman@23125
   525
lemma complex_cnj_i [simp]: "cnj ii = - ii"
huffman@44724
   526
  by (simp add: complex_eq_iff)
huffman@23125
   527
huffman@23125
   528
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
huffman@44724
   529
  by (simp add: complex_eq_iff)
huffman@23125
   530
huffman@23125
   531
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
huffman@44724
   532
  by (simp add: complex_eq_iff)
paulson@14354
   533
huffman@23125
   534
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
huffman@44724
   535
  by (simp add: complex_eq_iff power2_eq_square)
huffman@23125
   536
huffman@23125
   537
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
huffman@44724
   538
  by (simp add: norm_mult power2_eq_square)
huffman@23125
   539
huffman@44827
   540
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
huffman@44827
   541
  by (simp add: cmod_def power2_eq_square)
huffman@44827
   542
huffman@44827
   543
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
huffman@44827
   544
  by simp
huffman@44827
   545
huffman@44290
   546
lemma bounded_linear_cnj: "bounded_linear cnj"
huffman@44127
   547
  using complex_cnj_add complex_cnj_scaleR
huffman@44127
   548
  by (rule bounded_linear_intro [where K=1], simp)
paulson@14354
   549
huffman@44290
   550
lemmas tendsto_cnj [tendsto_intros] =
huffman@44290
   551
  bounded_linear.tendsto [OF bounded_linear_cnj]
huffman@44290
   552
huffman@44290
   553
lemmas isCont_cnj [simp] =
huffman@44290
   554
  bounded_linear.isCont [OF bounded_linear_cnj]
huffman@44290
   555
paulson@14354
   556
paulson@14323
   557
subsection{*Finally! Polar Form for Complex Numbers*}
paulson@14323
   558
huffman@44827
   559
subsubsection {* $\cos \theta + i \sin \theta$ *}
huffman@20557
   560
huffman@44715
   561
definition cis :: "real \<Rightarrow> complex" where
huffman@20557
   562
  "cis a = Complex (cos a) (sin a)"
huffman@20557
   563
huffman@44827
   564
lemma Re_cis [simp]: "Re (cis a) = cos a"
huffman@44827
   565
  by (simp add: cis_def)
huffman@44827
   566
huffman@44827
   567
lemma Im_cis [simp]: "Im (cis a) = sin a"
huffman@44827
   568
  by (simp add: cis_def)
huffman@44827
   569
huffman@44827
   570
lemma cis_zero [simp]: "cis 0 = 1"
huffman@44827
   571
  by (simp add: cis_def)
huffman@44827
   572
huffman@44828
   573
lemma norm_cis [simp]: "norm (cis a) = 1"
huffman@44828
   574
  by (simp add: cis_def)
huffman@44828
   575
huffman@44828
   576
lemma sgn_cis [simp]: "sgn (cis a) = cis a"
huffman@44828
   577
  by (simp add: sgn_div_norm)
huffman@44828
   578
huffman@44828
   579
lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
huffman@44828
   580
  by (metis norm_cis norm_zero zero_neq_one)
huffman@44828
   581
huffman@44827
   582
lemma cis_mult: "cis a * cis b = cis (a + b)"
huffman@44827
   583
  by (simp add: cis_def cos_add sin_add)
huffman@44827
   584
huffman@44827
   585
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
huffman@44827
   586
  by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
huffman@44827
   587
huffman@44827
   588
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
huffman@44827
   589
  by (simp add: cis_def)
huffman@44827
   590
huffman@44827
   591
lemma cis_divide: "cis a / cis b = cis (a - b)"
huffman@44827
   592
  by (simp add: complex_divide_def cis_mult diff_minus)
huffman@44827
   593
huffman@44827
   594
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
huffman@44827
   595
  by (auto simp add: DeMoivre)
huffman@44827
   596
huffman@44827
   597
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
huffman@44827
   598
  by (auto simp add: DeMoivre)
huffman@44827
   599
huffman@44827
   600
subsubsection {* $r(\cos \theta + i \sin \theta)$ *}
huffman@44715
   601
huffman@44715
   602
definition rcis :: "[real, real] \<Rightarrow> complex" where
huffman@20557
   603
  "rcis r a = complex_of_real r * cis a"
huffman@20557
   604
huffman@44827
   605
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
huffman@44828
   606
  by (simp add: rcis_def)
huffman@44827
   607
huffman@44827
   608
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
huffman@44828
   609
  by (simp add: rcis_def)
huffman@44827
   610
huffman@44827
   611
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
huffman@44828
   612
  by (simp add: complex_eq_iff polar_Ex)
huffman@44827
   613
huffman@44827
   614
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
huffman@44828
   615
  by (simp add: rcis_def norm_mult)
huffman@44827
   616
huffman@44827
   617
lemma cis_rcis_eq: "cis a = rcis 1 a"
huffman@44827
   618
  by (simp add: rcis_def)
huffman@44827
   619
huffman@44827
   620
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
huffman@44828
   621
  by (simp add: rcis_def cis_mult)
huffman@44827
   622
huffman@44827
   623
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
huffman@44827
   624
  by (simp add: rcis_def)
huffman@44827
   625
huffman@44827
   626
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
huffman@44827
   627
  by (simp add: rcis_def)
huffman@44827
   628
huffman@44828
   629
lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
huffman@44828
   630
  by (simp add: rcis_def)
huffman@44828
   631
huffman@44827
   632
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
huffman@44827
   633
  by (simp add: rcis_def power_mult_distrib DeMoivre)
huffman@44827
   634
huffman@44827
   635
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
huffman@44827
   636
  by (simp add: divide_inverse rcis_def)
huffman@44827
   637
huffman@44827
   638
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
huffman@44828
   639
  by (simp add: rcis_def cis_divide [symmetric])
huffman@44827
   640
huffman@44827
   641
subsubsection {* Complex exponential *}
huffman@44827
   642
huffman@44291
   643
abbreviation expi :: "complex \<Rightarrow> complex"
huffman@44291
   644
  where "expi \<equiv> exp"
huffman@44291
   645
huffman@44712
   646
lemma cis_conv_exp: "cis b = exp (Complex 0 b)"
huffman@44291
   647
proof (rule complex_eqI)
huffman@44291
   648
  { fix n have "Complex 0 b ^ n =
huffman@44291
   649
    real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"
huffman@44291
   650
      apply (induct n)
huffman@44291
   651
      apply (simp add: cos_coeff_def sin_coeff_def)
huffman@44291
   652
      apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)
huffman@44291
   653
      done } note * = this
huffman@44712
   654
  show "Re (cis b) = Re (exp (Complex 0 b))"
huffman@44291
   655
    unfolding exp_def cis_def cos_def
huffman@44291
   656
    by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],
huffman@44291
   657
      simp add: * mult_assoc [symmetric])
huffman@44712
   658
  show "Im (cis b) = Im (exp (Complex 0 b))"
huffman@44291
   659
    unfolding exp_def cis_def sin_def
huffman@44291
   660
    by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],
huffman@44291
   661
      simp add: * mult_assoc [symmetric])
huffman@44291
   662
qed
huffman@44291
   663
huffman@44291
   664
lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"
huffman@44712
   665
  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp
huffman@20557
   666
huffman@44828
   667
lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
huffman@44828
   668
  unfolding expi_def by simp
huffman@44828
   669
huffman@44828
   670
lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
huffman@44828
   671
  unfolding expi_def by simp
huffman@44828
   672
paulson@14374
   673
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
paulson@14373
   674
apply (insert rcis_Ex [of z])
huffman@23125
   675
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
paulson@14334
   676
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
paulson@14323
   677
done
paulson@14323
   678
paulson@14387
   679
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
huffman@44724
   680
  by (simp add: expi_def cis_def)
paulson@14387
   681
huffman@44844
   682
subsubsection {* Complex argument *}
huffman@44844
   683
huffman@44844
   684
definition arg :: "complex \<Rightarrow> real" where
huffman@44844
   685
  "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"
huffman@44844
   686
huffman@44844
   687
lemma arg_zero: "arg 0 = 0"
huffman@44844
   688
  by (simp add: arg_def)
huffman@44844
   689
huffman@44844
   690
lemma of_nat_less_of_int_iff: (* TODO: move *)
huffman@44844
   691
  "(of_nat n :: 'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
huffman@44844
   692
  by (metis of_int_of_nat_eq of_int_less_iff)
huffman@44844
   693
huffman@47108
   694
lemma real_of_nat_less_numeral_iff [simp]: (* TODO: move *)
huffman@47108
   695
  "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"
huffman@47108
   696
  using of_nat_less_of_int_iff [of n "numeral w", where 'a=real]
huffman@47108
   697
  by (simp add: real_of_nat_def zless_nat_eq_int_zless [symmetric])
huffman@44844
   698
huffman@44844
   699
lemma arg_unique:
huffman@44844
   700
  assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
huffman@44844
   701
  shows "arg z = x"
huffman@44844
   702
proof -
huffman@44844
   703
  from assms have "z \<noteq> 0" by auto
huffman@44844
   704
  have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
huffman@44844
   705
  proof
huffman@44844
   706
    fix a def d \<equiv> "a - x"
huffman@44844
   707
    assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
huffman@44844
   708
    from a assms have "- (2*pi) < d \<and> d < 2*pi"
huffman@44844
   709
      unfolding d_def by simp
huffman@44844
   710
    moreover from a assms have "cos a = cos x" and "sin a = sin x"
huffman@44844
   711
      by (simp_all add: complex_eq_iff)
huffman@44844
   712
    hence "cos d = 1" unfolding d_def cos_diff by simp
huffman@44844
   713
    moreover hence "sin d = 0" by (rule cos_one_sin_zero)
huffman@44844
   714
    ultimately have "d = 0"
huffman@44844
   715
      unfolding sin_zero_iff even_mult_two_ex
huffman@44844
   716
      by (safe, auto simp add: numeral_2_eq_2 less_Suc_eq)
huffman@44844
   717
    thus "a = x" unfolding d_def by simp
huffman@44844
   718
  qed (simp add: assms del: Re_sgn Im_sgn)
huffman@44844
   719
  with `z \<noteq> 0` show "arg z = x"
huffman@44844
   720
    unfolding arg_def by simp
huffman@44844
   721
qed
huffman@44844
   722
huffman@44844
   723
lemma arg_correct:
huffman@44844
   724
  assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
huffman@44844
   725
proof (simp add: arg_def assms, rule someI_ex)
huffman@44844
   726
  obtain r a where z: "z = rcis r a" using rcis_Ex by fast
huffman@44844
   727
  with assms have "r \<noteq> 0" by auto
huffman@44844
   728
  def b \<equiv> "if 0 < r then a else a + pi"
huffman@44844
   729
  have b: "sgn z = cis b"
huffman@44844
   730
    unfolding z b_def rcis_def using `r \<noteq> 0`
huffman@44844
   731
    by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def)
huffman@44844
   732
  have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
webertj@49962
   733
    by (induct_tac n, simp_all add: distrib_left cis_mult [symmetric],
huffman@44844
   734
      simp add: cis_def)
huffman@44844
   735
  have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
huffman@44844
   736
    by (case_tac x rule: int_diff_cases,
huffman@44844
   737
      simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
huffman@44844
   738
  def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
huffman@44844
   739
  have "sgn z = cis c"
huffman@44844
   740
    unfolding b c_def
huffman@44844
   741
    by (simp add: cis_divide [symmetric] cis_2pi_int)
huffman@44844
   742
  moreover have "- pi < c \<and> c \<le> pi"
huffman@44844
   743
    using ceiling_correct [of "(b - pi) / (2*pi)"]
huffman@44844
   744
    by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)
huffman@44844
   745
  ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
huffman@44844
   746
qed
huffman@44844
   747
huffman@44844
   748
lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
huffman@44844
   749
  by (cases "z = 0", simp_all add: arg_zero arg_correct)
huffman@44844
   750
huffman@44844
   751
lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
huffman@44844
   752
  by (simp add: arg_correct)
huffman@44844
   753
huffman@44844
   754
lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
huffman@44844
   755
  by (cases "z = 0", simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
huffman@44844
   756
huffman@44844
   757
lemma cos_arg_i_mult_zero [simp]:
huffman@44844
   758
     "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
huffman@44844
   759
  using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff)
huffman@44844
   760
huffman@44065
   761
text {* Legacy theorem names *}
huffman@44065
   762
huffman@44065
   763
lemmas expand_complex_eq = complex_eq_iff
huffman@44065
   764
lemmas complex_Re_Im_cancel_iff = complex_eq_iff
huffman@44065
   765
lemmas complex_equality = complex_eqI
huffman@44065
   766
paulson@13957
   767
end