src/HOL/Complex.thy
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(*  Title:       HOL/Complex.thy
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    Author:      Jacques D. Fleuriot
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    Copyright:   2001 University of Edinburgh
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
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*)
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header {* Complex Numbers: Rectangular and Polar Representations *}
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theory Complex
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imports Transcendental
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begin
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datatype complex = Complex real real
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primrec Re :: "complex \<Rightarrow> real"
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  where Re: "Re (Complex x y) = x"
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primrec Im :: "complex \<Rightarrow> real"
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  where Im: "Im (Complex x y) = y"
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lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"
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  by (induct z) simp
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lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"
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  by (induct x, induct y) simp
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lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"
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  by (induct x, induct y) simp
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subsection {* Addition and Subtraction *}
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instantiation complex :: ab_group_add
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begin
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definition complex_zero_def:
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  "0 = Complex 0 0"
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definition complex_add_def:
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  "x + y = Complex (Re x + Re y) (Im x + Im y)"
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definition complex_minus_def:
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  "- x = Complex (- Re x) (- Im x)"
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definition complex_diff_def:
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  "x - (y\<Colon>complex) = x + - y"
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lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
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  by (simp add: complex_zero_def)
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lemma complex_Re_zero [simp]: "Re 0 = 0"
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  by (simp add: complex_zero_def)
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lemma complex_Im_zero [simp]: "Im 0 = 0"
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  by (simp add: complex_zero_def)
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lemma complex_add [simp]:
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  "Complex a b + Complex c d = Complex (a + c) (b + d)"
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  by (simp add: complex_add_def)
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lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y"
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  by (simp add: complex_add_def)
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lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y"
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  by (simp add: complex_add_def)
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lemma complex_minus [simp]:
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  "- (Complex a b) = Complex (- a) (- b)"
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  by (simp add: complex_minus_def)
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lemma complex_Re_minus [simp]: "Re (- x) = - Re x"
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  by (simp add: complex_minus_def)
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lemma complex_Im_minus [simp]: "Im (- x) = - Im x"
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  by (simp add: complex_minus_def)
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lemma complex_diff [simp]:
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  "Complex a b - Complex c d = Complex (a - c) (b - d)"
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  by (simp add: complex_diff_def)
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lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y"
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  by (simp add: complex_diff_def)
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lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y"
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  by (simp add: complex_diff_def)
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instance
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  by intro_classes (simp_all add: complex_add_def complex_diff_def)
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end
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subsection {* Multiplication and Division *}
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instantiation complex :: field_inverse_zero
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begin
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definition complex_one_def:
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  "1 = Complex 1 0"
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definition complex_mult_def:
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  "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
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definition complex_inverse_def:
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  "inverse x =
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    Complex (Re x / ((Re x)\<^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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81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
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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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81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
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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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   309
  show "sgn x = x /\<^sub>R cmod x"
d24b2692562f definition of dist for complex
huffman
parents: 31021
diff changeset
   310
    by (rule complex_sgn_def)
d24b2692562f definition of dist for complex
huffman
parents: 31021
diff changeset
   311
  show "dist x y = cmod (x - y)"
d24b2692562f definition of dist for complex
huffman
parents: 31021
diff changeset
   312
    by (rule dist_complex_def)
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31419
diff changeset
   313
  show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31419
diff changeset
   314
    by (rule open_complex_def)
24520
40b220403257 fix sgn_div_norm class
huffman
parents: 24506
diff changeset
   315
qed
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   316
25712
f488a37cfad4 instantiation target
haftmann
parents: 25599
diff changeset
   317
end
f488a37cfad4 instantiation target
haftmann
parents: 25599
diff changeset
   318
44761
0694fc3248fd remove some unnecessary simp rules from simpset
huffman
parents: 44749
diff changeset
   319
lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   320
  by simp
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   321
44761
0694fc3248fd remove some unnecessary simp rules from simpset
huffman
parents: 44749
diff changeset
   322
lemma cmod_complex_polar:
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   323
  "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   324
  by (simp add: norm_mult)
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   325
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   326
lemma complex_Re_le_cmod: "Re x \<le> cmod x"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   327
  unfolding complex_norm_def
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   328
  by (rule real_sqrt_sum_squares_ge1)
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   329
44761
0694fc3248fd remove some unnecessary simp rules from simpset
huffman
parents: 44749
diff changeset
   330
lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   331
  by (rule order_trans [OF _ norm_ge_zero], simp)
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   332
44761
0694fc3248fd remove some unnecessary simp rules from simpset
huffman
parents: 44749
diff changeset
   333
lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \<le> cmod a"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   334
  by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   335
26117
ca578d3b9f8c Added trivial theorems aboud cmod
chaieb
parents: 25712
diff changeset
   336
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   337
  by (cases x) simp
26117
ca578d3b9f8c Added trivial theorems aboud cmod
chaieb
parents: 25712
diff changeset
   338
ca578d3b9f8c Added trivial theorems aboud cmod
chaieb
parents: 25712
diff changeset
   339
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   340
  by (cases x) simp
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   341
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   342
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   343
lemma abs_sqrt_wlog:
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   344
  fixes x::"'a::linordered_idom"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   345
  assumes "\<And>x::'a. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   346
by (metis abs_ge_zero assms power2_abs)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   347
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   348
lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   349
  unfolding complex_norm_def
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   350
  apply (rule abs_sqrt_wlog [where x="Re z"])
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   351
  apply (rule abs_sqrt_wlog [where x="Im z"])
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   352
  apply (rule power2_le_imp_le)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   353
  apply (simp_all add: power2_sum add_commute sum_squares_bound real_sqrt_mult [symmetric])
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   354
  done
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   355
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   356
44843
huffman
parents: 44842
diff changeset
   357
text {* Properties of complex signum. *}
huffman
parents: 44842
diff changeset
   358
huffman
parents: 44842
diff changeset
   359
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
huffman
parents: 44842
diff changeset
   360
  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
huffman
parents: 44842
diff changeset
   361
huffman
parents: 44842
diff changeset
   362
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
huffman
parents: 44842
diff changeset
   363
  by (simp add: complex_sgn_def divide_inverse)
huffman
parents: 44842
diff changeset
   364
huffman
parents: 44842
diff changeset
   365
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
huffman
parents: 44842
diff changeset
   366
  by (simp add: complex_sgn_def divide_inverse)
huffman
parents: 44842
diff changeset
   367
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   368
23123
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   369
subsection {* Completeness of the Complexes *}
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   370
44290
23a5137162ea remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents: 44127
diff changeset
   371
lemma bounded_linear_Re: "bounded_linear Re"
23a5137162ea remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents: 44127
diff changeset
   372
  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
23a5137162ea remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents: 44127
diff changeset
   373
23a5137162ea remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents: 44127
diff changeset
   374
lemma bounded_linear_Im: "bounded_linear Im"
44127
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 44065
diff changeset
   375
  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
23123
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   376
44290
23a5137162ea remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents: 44127
diff changeset
   377
lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
23a5137162ea remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents: 44127
diff changeset
   378
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   379
lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   380
lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   381
lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   382
lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   383
lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   384
lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   385
lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   386
lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   387
lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   388
lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   389
lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   390
lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   391
36825
d9320cdcde73 add lemma tendsto_Complex
huffman
parents: 36777
diff changeset
   392
lemma tendsto_Complex [tendsto_intros]:
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   393
  assumes "(f ---> a) F" and "(g ---> b) F"
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   394
  shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
36825
d9320cdcde73 add lemma tendsto_Complex
huffman
parents: 36777
diff changeset
   395
proof (rule tendstoI)
d9320cdcde73 add lemma tendsto_Complex
huffman
parents: 36777
diff changeset
   396
  fix r :: real assume "0 < r"
d9320cdcde73 add lemma tendsto_Complex
huffman
parents: 36777
diff changeset
   397
  hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   398
  have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F"
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   399
    using `(f ---> a) F` and `0 < r / sqrt 2` by (rule tendstoD)
36825
d9320cdcde73 add lemma tendsto_Complex
huffman
parents: 36777
diff changeset
   400
  moreover
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   401
  have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F"
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   402
    using `(g ---> b) F` and `0 < r / sqrt 2` by (rule tendstoD)
36825
d9320cdcde73 add lemma tendsto_Complex
huffman
parents: 36777
diff changeset
   403
  ultimately
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   404
  show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F"
36825
d9320cdcde73 add lemma tendsto_Complex
huffman
parents: 36777
diff changeset
   405
    by (rule eventually_elim2)
d9320cdcde73 add lemma tendsto_Complex
huffman
parents: 36777
diff changeset
   406
       (simp add: dist_norm real_sqrt_sum_squares_less)
d9320cdcde73 add lemma tendsto_Complex
huffman
parents: 36777
diff changeset
   407
qed
d9320cdcde73 add lemma tendsto_Complex
huffman
parents: 36777
diff changeset
   408
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   409
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   410
lemma tendsto_complex_iff:
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   411
  "(f ---> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) ---> Re x) F \<and> ((\<lambda>x. Im (f x)) ---> Im x) F)"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   412
proof -
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   413
  have f: "f = (\<lambda>x. Complex (Re (f x)) (Im (f x)))" and x: "x = Complex (Re x) (Im x)"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   414
    by simp_all
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   415
  show ?thesis
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   416
    apply (subst f)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   417
    apply (subst x)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   418
    apply (intro iffI tendsto_Complex conjI)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   419
    apply (simp_all add: tendsto_Re tendsto_Im)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   420
    done
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   421
qed
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   422
23123
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   423
instance complex :: banach
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   424
proof
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   425
  fix X :: "nat \<Rightarrow> complex"
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   426
  assume X: "Cauchy X"
44290
23a5137162ea remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents: 44127
diff changeset
   427
  from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"
23123
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   428
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
44290
23a5137162ea remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents: 44127
diff changeset
   429
  from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
23123
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   430
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   431
  have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
44748
7f6838b3474a remove redundant lemma LIMSEQ_Complex in favor of tendsto_Complex
huffman
parents: 44724
diff changeset
   432
    using tendsto_Complex [OF 1 2] by simp
23123
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   433
  thus "convergent X"
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   434
    by (rule convergentI)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   435
qed
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   436
56238
5d147e1e18d1 a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents: 56217
diff changeset
   437
declare
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   438
  DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]
56238
5d147e1e18d1 a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents: 56217
diff changeset
   439
23123
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   440
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   441
subsection {* The Complex Number $i$ *}
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   442
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   443
definition "ii" :: complex  ("\<i>")
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   444
  where i_def: "ii \<equiv> Complex 0 1"
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   445
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   446
lemma complex_Re_i [simp]: "Re ii = 0"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   447
  by (simp add: i_def)
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   448
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   449
lemma complex_Im_i [simp]: "Im ii = 1"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   450
  by (simp add: i_def)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   451
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   452
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   453
  by (simp add: i_def)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   454
44902
9ba11d41cd1f move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents: 44846
diff changeset
   455
lemma norm_ii [simp]: "norm ii = 1"
9ba11d41cd1f move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents: 44846
diff changeset
   456
  by (simp add: i_def)
9ba11d41cd1f move lemmas about complex number 'i' to Complex.thy and Library/Inner_Product.thy
huffman
parents: 44846
diff changeset
   457
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   458
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   459
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   460
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   461
lemma complex_i_not_one [simp]: "ii \<noteq> 1"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   462
  by (simp add: complex_eq_iff)
23124
892e0a4551da use new-style instance declarations
huffman
parents: 23123
diff changeset
   463
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 44902
diff changeset
   464
lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 44902
diff changeset
   465
  by (simp add: complex_eq_iff)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 44902
diff changeset
   466
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54230
diff changeset
   467
lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> - numeral w"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   468
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   469
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   470
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   471
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   472
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   473
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   474
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   475
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   476
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   477
  by (simp add: i_def complex_of_real_def)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   478
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   479
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   480
  by (simp add: i_def complex_of_real_def)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   481
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   482
lemma i_squared [simp]: "ii * ii = -1"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   483
  by (simp add: i_def)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   484
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51002
diff changeset
   485
lemma power2_i [simp]: "ii\<^sup>2 = -1"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   486
  by (simp add: power2_eq_square)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   487
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   488
lemma inverse_i [simp]: "inverse ii = - ii"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   489
  by (rule inverse_unique, simp)
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   490
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   491
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   492
  by (simp add: mult_assoc [symmetric])
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   493
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   494
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   495
subsection {* Complex Conjugation *}
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   496
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   497
definition cnj :: "complex \<Rightarrow> complex" where
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   498
  "cnj z = Complex (Re z) (- Im z)"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   499
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   500
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   501
  by (simp add: cnj_def)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   502
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   503
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   504
  by (simp add: cnj_def)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   505
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   506
lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   507
  by (simp add: cnj_def)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   508
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   509
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   510
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   511
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   512
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   513
  by (simp add: cnj_def)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   514
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   515
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   516
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   517
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   518
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   519
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   520
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   521
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   522
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   523
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   524
lemma cnj_setsum: "cnj (setsum f s) = (\<Sum>x\<in>s. cnj (f x))"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   525
  by (induct s rule: infinite_finite_induct) (auto simp: complex_cnj_add)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   526
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   527
lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   528
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   529
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   530
lemma complex_cnj_minus: "cnj (- x) = - cnj x"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   531
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   532
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   533
lemma complex_cnj_one [simp]: "cnj 1 = 1"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   534
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   535
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   536
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   537
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   538
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   539
lemma cnj_setprod: "cnj (setprod f s) = (\<Prod>x\<in>s. cnj (f x))"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   540
  by (induct s rule: infinite_finite_induct) (auto simp: complex_cnj_mult)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   541
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   542
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   543
  by (simp add: complex_inverse_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   544
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   545
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   546
  by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   547
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   548
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   549
  by (induct n, simp_all add: complex_cnj_mult)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   550
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   551
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   552
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   553
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   554
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   555
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   556
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 44902
diff changeset
   557
lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 44902
diff changeset
   558
  by (simp add: complex_eq_iff)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 44902
diff changeset
   559
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54230
diff changeset
   560
lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   561
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   562
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   563
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   564
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   565
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   566
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   567
  by (simp add: complex_norm_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   568
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   569
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   570
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   571
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   572
lemma complex_cnj_i [simp]: "cnj ii = - ii"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   573
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   574
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   575
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   576
  by (simp add: complex_eq_iff)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   577
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   578
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   579
  by (simp add: complex_eq_iff)
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   580
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51002
diff changeset
   581
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   582
  by (simp add: complex_eq_iff power2_eq_square)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   583
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51002
diff changeset
   584
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   585
  by (simp add: norm_mult power2_eq_square)
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   586
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   587
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   588
  by (simp add: cmod_def power2_eq_square)
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   589
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   590
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   591
  by simp
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   592
44290
23a5137162ea remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents: 44127
diff changeset
   593
lemma bounded_linear_cnj: "bounded_linear cnj"
44127
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 44065
diff changeset
   594
  using complex_cnj_add complex_cnj_scaleR
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 44065
diff changeset
   595
  by (rule bounded_linear_intro [where K=1], simp)
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   596
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   597
lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   598
lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   599
lemmas continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   600
lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56369
diff changeset
   601
lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]
44290
23a5137162ea remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents: 44127
diff changeset
   602
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   603
lemma lim_cnj: "((\<lambda>x. cnj(f x)) ---> cnj l) F \<longleftrightarrow> (f ---> l) F"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   604
  by (simp add: tendsto_iff dist_complex_def Complex.complex_cnj_diff [symmetric])
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   605
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   606
lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   607
  by (simp add: sums_def lim_cnj cnj_setsum [symmetric])
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   608
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   609
55734
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   610
subsection{*Basic Lemmas*}
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   611
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   612
lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   613
  by (metis complex_Im_zero complex_Re_zero complex_eqI sum_power2_eq_zero_iff)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   614
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   615
lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   616
by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   617
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   618
lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   619
apply (cases z, auto)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   620
by (metis complex_of_real_def of_real_add of_real_power power2_eq_square)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   621
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   622
lemma complex_div_eq_0: 
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   623
    "(Re(a / b) = 0 \<longleftrightarrow> Re(a * cnj b) = 0) & (Im(a / b) = 0 \<longleftrightarrow> Im(a * cnj b) = 0)"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   624
proof (cases "b=0")
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   625
  case True then show ?thesis by auto
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   626
next
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   627
  case False
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   628
  show ?thesis
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   629
  proof (cases b)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   630
    case (Complex x y)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   631
    then have "x\<^sup>2 + y\<^sup>2 > 0"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   632
      by (metis Complex_eq_0 False sum_power2_gt_zero_iff)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   633
    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)"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   634
      by (metis add_divide_distrib)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   635
    with Complex False show ?thesis
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   636
      by (auto simp: complex_divide_def)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   637
  qed
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   638
qed
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   639
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   640
lemma re_complex_div_eq_0: "Re(a / b) = 0 \<longleftrightarrow> Re(a * cnj b) = 0"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   641
  and im_complex_div_eq_0: "Im(a / b) = 0 \<longleftrightarrow> Im(a * cnj b) = 0"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   642
using complex_div_eq_0 by auto
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   643
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   644
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   645
lemma complex_div_gt_0: 
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   646
    "(Re(a / b) > 0 \<longleftrightarrow> Re(a * cnj b) > 0) & (Im(a / b) > 0 \<longleftrightarrow> Im(a * cnj b) > 0)"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   647
proof (cases "b=0")
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   648
  case True then show ?thesis by auto
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   649
next
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   650
  case False
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   651
  show ?thesis
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   652
  proof (cases b)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   653
    case (Complex x y)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   654
    then have "x\<^sup>2 + y\<^sup>2 > 0"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   655
      by (metis Complex_eq_0 False sum_power2_gt_zero_iff)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   656
    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)"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   657
      by (metis add_divide_distrib)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   658
    ultimately show ?thesis using Complex False `0 < x\<^sup>2 + y\<^sup>2`
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   659
      apply (simp add: complex_divide_def  zero_less_divide_iff less_divide_eq)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   660
      apply (metis less_divide_eq mult_zero_left diff_conv_add_uminus minus_divide_left)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   661
      done
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   662
  qed
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   663
qed
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   664
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   665
lemma re_complex_div_gt_0: "Re(a / b) > 0 \<longleftrightarrow> Re(a * cnj b) > 0"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   666
  and im_complex_div_gt_0: "Im(a / b) > 0 \<longleftrightarrow> Im(a * cnj b) > 0"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   667
using complex_div_gt_0 by auto
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   668
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   669
lemma re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   670
  by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   671
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   672
lemma im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   673
  by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   674
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   675
lemma re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0"
55759
fe3d8f585c20 replaced smt-based proof with metis proof that requires no external tool
boehmes
parents: 55734
diff changeset
   676
  by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0)
55734
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   677
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   678
lemma im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   679
  by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   680
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   681
lemma re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   682
  by (metis not_le re_complex_div_gt_0)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   683
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   684
lemma im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   685
  by (metis im_complex_div_gt_0 not_le)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   686
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55759
diff changeset
   687
lemma Re_setsum: "Re(setsum f s) = setsum (%x. Re(f x)) s"
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   688
  by (induct s rule: infinite_finite_induct) auto
55734
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   689
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55759
diff changeset
   690
lemma Im_setsum: "Im(setsum f s) = setsum (%x. Im(f x)) s"
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   691
  by (induct s rule: infinite_finite_induct) auto
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   692
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   693
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)"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   694
  unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   695
  
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   696
lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and>  summable (\<lambda>x. Im (f x))"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   697
  unfolding summable_def sums_complex_iff[abs_def] by (metis Im.simps Re.simps)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   698
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   699
lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   700
  unfolding summable_complex_iff by simp
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   701
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   702
lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   703
  unfolding summable_complex_iff by blast
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   704
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   705
lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   706
  unfolding summable_complex_iff by blast
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55759
diff changeset
   707
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55759
diff changeset
   708
lemma Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   709
  by (induct s rule: infinite_finite_induct) auto
55734
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   710
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55759
diff changeset
   711
lemma Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55759
diff changeset
   712
  by (metis Complex_setsum')
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55759
diff changeset
   713
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55759
diff changeset
   714
lemma of_real_setsum: "of_real (setsum f s) = setsum (%x. of_real(f x)) s"
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   715
  by (induct s rule: infinite_finite_induct) auto
55734
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   716
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55759
diff changeset
   717
lemma of_real_setprod: "of_real (setprod f s) = setprod (%x. of_real(f x)) s"
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   718
  by (induct s rule: infinite_finite_induct) auto
55734
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   719
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   720
lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   721
by (metis Reals_cases Reals_of_real complex.exhaust complex.inject complex_cnj 
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   722
          complex_of_real_def equal_neg_zero)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   723
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   724
lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   725
  by (metis Reals_of_real complex_of_real_def)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   726
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   727
lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm(z) = abs(Re z)"
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   728
  by (metis Re_complex_of_real Reals_cases norm_of_real)
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   729
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   730
lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   731
  by (metis Complex_in_Reals Im_complex_of_real Reals_cases complex_surj)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   732
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   733
lemma series_comparison_complex:
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   734
  fixes f:: "nat \<Rightarrow> 'a::banach"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   735
  assumes sg: "summable g"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   736
     and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   737
     and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   738
  shows "summable f"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   739
proof -
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   740
  have g: "\<And>n. cmod (g n) = Re (g n)" using assms
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   741
    by (metis abs_of_nonneg in_Reals_norm)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   742
  show ?thesis
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   743
    apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   744
    using sg
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   745
    apply (auto simp: summable_def)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   746
    apply (rule_tac x="Re s" in exI)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   747
    apply (auto simp: g sums_Re)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   748
    apply (metis fg g)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   749
    done
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56331
diff changeset
   750
qed
55734
3f5b2745d659 More complex-related lemmas
paulson <lp15@cam.ac.uk>
parents: 54489
diff changeset
   751
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   752
subsection{*Finally! Polar Form for Complex Numbers*}
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   753
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   754
subsubsection {* $\cos \theta + i \sin \theta$ *}
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   755
44715
1a17d8913976 tuned comments
huffman
parents: 44712
diff changeset
   756
definition cis :: "real \<Rightarrow> complex" where
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   757
  "cis a = Complex (cos a) (sin a)"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   758
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   759
lemma Re_cis [simp]: "Re (cis a) = cos a"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   760
  by (simp add: cis_def)
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   761
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   762
lemma Im_cis [simp]: "Im (cis a) = sin a"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   763
  by (simp add: cis_def)
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   764
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   765
lemma cis_zero [simp]: "cis 0 = 1"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   766
  by (simp add: cis_def)
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   767
44828
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   768
lemma norm_cis [simp]: "norm (cis a) = 1"
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   769
  by (simp add: cis_def)
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   770
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   771
lemma sgn_cis [simp]: "sgn (cis a) = cis a"
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   772
  by (simp add: sgn_div_norm)
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   773
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   774
lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   775
  by (metis norm_cis norm_zero zero_neq_one)
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   776
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   777
lemma cis_mult: "cis a * cis b = cis (a + b)"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   778
  by (simp add: cis_def cos_add sin_add)
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   779
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   780
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   781
  by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   782
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   783
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   784
  by (simp add: cis_def)
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   785
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   786
lemma cis_divide: "cis a / cis b = cis (a - b)"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53374
diff changeset
   787
  by (simp add: complex_divide_def cis_mult)
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   788
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   789
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   790
  by (auto simp add: DeMoivre)
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   791
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   792
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   793
  by (auto simp add: DeMoivre)
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   794
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   795
subsubsection {* $r(\cos \theta + i \sin \theta)$ *}
44715
1a17d8913976 tuned comments
huffman
parents: 44712
diff changeset
   796
1a17d8913976 tuned comments
huffman
parents: 44712
diff changeset
   797
definition rcis :: "[real, real] \<Rightarrow> complex" where
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   798
  "rcis r a = complex_of_real r * cis a"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   799
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   800
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
44828
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   801
  by (simp add: rcis_def)
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   802
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   803
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
44828
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   804
  by (simp add: rcis_def)
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   805
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   806
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
44828
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   807
  by (simp add: complex_eq_iff polar_Ex)
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   808
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   809
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
44828
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   810
  by (simp add: rcis_def norm_mult)
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   811
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   812
lemma cis_rcis_eq: "cis a = rcis 1 a"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   813
  by (simp add: rcis_def)
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   814
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   815
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
44828
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   816
  by (simp add: rcis_def cis_mult)
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   817
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   818
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   819
  by (simp add: rcis_def)
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   820
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   821
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   822
  by (simp add: rcis_def)
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   823
44828
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   824
lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   825
  by (simp add: rcis_def)
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   826
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   827
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   828
  by (simp add: rcis_def power_mult_distrib DeMoivre)
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   829
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   830
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   831
  by (simp add: divide_inverse rcis_def)
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   832
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   833
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
44828
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   834
  by (simp add: rcis_def cis_divide [symmetric])
44827
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   835
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   836
subsubsection {* Complex exponential *}
4d1384a1fc82 Complex.thy: move theorems into appropriate subsections
huffman
parents: 44825
diff changeset
   837
44291
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   838
abbreviation expi :: "complex \<Rightarrow> complex"
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   839
  where "expi \<equiv> exp"
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   840
44712
1e490e891c88 replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents: 44711
diff changeset
   841
lemma cis_conv_exp: "cis b = exp (Complex 0 b)"
44291
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   842
proof (rule complex_eqI)
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   843
  { fix n have "Complex 0 b ^ n =
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   844
    real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   845
      apply (induct n)
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   846
      apply (simp add: cos_coeff_def sin_coeff_def)
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   847
      apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   848
      done } note * = this
44712
1e490e891c88 replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents: 44711
diff changeset
   849
  show "Re (cis b) = Re (exp (Complex 0 b))"
44291
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   850
    unfolding exp_def cis_def cos_def
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   851
    by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   852
      simp add: * mult_assoc [symmetric])
44712
1e490e891c88 replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents: 44711
diff changeset
   853
  show "Im (cis b) = Im (exp (Complex 0 b))"
44291
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   854
    unfolding exp_def cis_def sin_def
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   855
    by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   856
      simp add: * mult_assoc [symmetric])
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   857
qed
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   858
dbd9965745fd define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents: 44290
diff changeset
   859
lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"
44712
1e490e891c88 replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents: 44711
diff changeset
   860
  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   861
44828
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   862
lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   863
  unfolding expi_def by simp
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   864
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   865
lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   866
  unfolding expi_def by simp
3d6a79e0e1d0 add some new lemmas about cis and rcis;
huffman
parents: 44827
diff changeset
   867
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   868
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   869
apply (insert rcis_Ex [of z])
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   870
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   871
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   872
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   873
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   874
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
44724
0b900a9d8023 tuned indentation
huffman
parents: 44715
diff changeset
   875
  by (simp add: expi_def cis_def)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   876
44844
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   877
subsubsection {* Complex argument *}
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   878
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   879
definition arg :: "complex \<Rightarrow> real" where
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   880
  "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   881
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   882
lemma arg_zero: "arg 0 = 0"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   883
  by (simp add: arg_def)
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   884
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   885
lemma of_nat_less_of_int_iff: (* TODO: move *)
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   886
  "(of_nat n :: 'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   887
  by (metis of_int_of_nat_eq of_int_less_iff)
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   888
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 44902
diff changeset
   889
lemma real_of_nat_less_numeral_iff [simp]: (* TODO: move *)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 44902
diff changeset
   890
  "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 44902
diff changeset
   891
  using of_nat_less_of_int_iff [of n "numeral w", where 'a=real]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 44902
diff changeset
   892
  by (simp add: real_of_nat_def zless_nat_eq_int_zless [symmetric])
44844
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   893
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   894
lemma arg_unique:
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   895
  assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   896
  shows "arg z = x"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   897
proof -
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   898
  from assms have "z \<noteq> 0" by auto
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   899
  have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   900
  proof
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   901
    fix a def d \<equiv> "a - x"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   902
    assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   903
    from a assms have "- (2*pi) < d \<and> d < 2*pi"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   904
      unfolding d_def by simp
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   905
    moreover from a assms have "cos a = cos x" and "sin a = sin x"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   906
      by (simp_all add: complex_eq_iff)
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
   907
    hence cos: "cos d = 1" unfolding d_def cos_diff by simp
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
   908
    moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)
44844
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   909
    ultimately have "d = 0"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   910
      unfolding sin_zero_iff even_mult_two_ex
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
   911
      by (auto simp add: numeral_2_eq_2 less_Suc_eq)
44844
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   912
    thus "a = x" unfolding d_def by simp
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   913
  qed (simp add: assms del: Re_sgn Im_sgn)
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   914
  with `z \<noteq> 0` show "arg z = x"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   915
    unfolding arg_def by simp
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   916
qed
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   917
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   918
lemma arg_correct:
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   919
  assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   920
proof (simp add: arg_def assms, rule someI_ex)
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   921
  obtain r a where z: "z = rcis r a" using rcis_Ex by fast
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   922
  with assms have "r \<noteq> 0" by auto
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   923
  def b \<equiv> "if 0 < r then a else a + pi"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   924
  have b: "sgn z = cis b"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   925
    unfolding z b_def rcis_def using `r \<noteq> 0`
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   926
    by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def)
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   927
  have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47108
diff changeset
   928
    by (induct_tac n, simp_all add: distrib_left cis_mult [symmetric],
44844
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   929
      simp add: cis_def)
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   930
  have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   931
    by (case_tac x rule: int_diff_cases,
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   932
      simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   933
  def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   934
  have "sgn z = cis c"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   935
    unfolding b c_def
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   936
    by (simp add: cis_divide [symmetric] cis_2pi_int)
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   937
  moreover have "- pi < c \<and> c \<le> pi"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   938
    using ceiling_correct [of "(b - pi) / (2*pi)"]
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   939
    by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   940
  ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   941
qed
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   942
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   943
lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   944
  by (cases "z = 0", simp_all add: arg_zero arg_correct)
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   945
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   946
lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   947
  by (simp add: arg_correct)
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   948
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   949
lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   950
  by (cases "z = 0", simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   951
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   952
lemma cos_arg_i_mult_zero [simp]:
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   953
     "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   954
  using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff)
f74a4175a3a8 prove existence, uniqueness, and other properties of complex arg function
huffman
parents: 44843
diff changeset
   955
44065
eb64ffccfc75 standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents: 41959
diff changeset
   956
text {* Legacy theorem names *}
eb64ffccfc75 standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents: 41959
diff changeset
   957
eb64ffccfc75 standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents: 41959
diff changeset
   958
lemmas expand_complex_eq = complex_eq_iff
eb64ffccfc75 standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents: 41959
diff changeset
   959
lemmas complex_Re_Im_cancel_iff = complex_eq_iff
eb64ffccfc75 standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents: 41959
diff changeset
   960
lemmas complex_equality = complex_eqI
eb64ffccfc75 standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents: 41959
diff changeset
   961
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
   962
end