src/HOL/Complex/Complex.thy
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(*  Title:       Complex.thy
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    ID:      $Id$
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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 "../Real/Real" "../Hyperreal/Transcendental"
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begin
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datatype complex = Complex real real
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primrec
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  Re :: "complex \<Rightarrow> real"
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where
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  Re: "Re (Complex x y) = x"
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primrec
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  Im :: "complex \<Rightarrow> real"
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where
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  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_equality [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 expand_complex_eq: "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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lemmas complex_Re_Im_cancel_iff = expand_complex_eq
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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
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  complex_zero_def: "0 = Complex 0 0"
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definition
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  complex_add_def: "x + y = Complex (Re x + Re y) (Im x + Im y)"
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definition
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  complex_minus_def: "- x = Complex (- Re x) (- Im x)"
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definition
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  complex_diff_def: "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, division_by_zero}"
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begin
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definition
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  complex_one_def: "1 = Complex 1 0"
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definition
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  complex_mult_def: "x * y =
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    Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
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definition
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  complex_inverse_def: "inverse x =
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    Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))"
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definition
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  complex_divide_def: "x / (y\<Colon>complex) = x * inverse y"
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lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"
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  by (simp add: complex_one_def)
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lemma complex_Re_one [simp]: "Re 1 = 1"
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  by (simp add: complex_one_def)
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lemma complex_Im_one [simp]: "Im 1 = 0"
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  by (simp add: complex_one_def)
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lemma complex_mult [simp]:
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  "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
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  by (simp add: complex_mult_def)
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lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"
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  by (simp add: complex_mult_def)
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lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"
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  by (simp add: complex_mult_def)
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lemma complex_inverse [simp]:
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  "inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))"
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  by (simp add: complex_inverse_def)
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lemma complex_Re_inverse:
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  "Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
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  by (simp add: complex_inverse_def)
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lemma complex_Im_inverse:
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  "Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
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  by (simp add: complex_inverse_def)
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instance
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  by intro_classes (simp_all add: complex_mult_def
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  right_distrib left_distrib right_diff_distrib left_diff_distrib
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  complex_inverse_def complex_divide_def
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  power2_eq_square add_divide_distrib [symmetric]
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  expand_complex_eq)
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end
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subsection {* Exponentiation *}
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instantiation complex :: recpower
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begin
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primrec power_complex where
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  complexpow_0:     "z ^ 0     = (1\<Colon>complex)"
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  | complexpow_Suc: "z ^ Suc n = (z\<Colon>complex) * z ^ n"
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instance by intro_classes simp_all
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end
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subsection {* Numerals and Arithmetic *}
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instantiation complex :: number_ring
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begin
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definition number_of_complex where
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  complex_number_of_def: "number_of w = (of_int w \<Colon> complex)"
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instance
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  by intro_classes (simp only: complex_number_of_def)
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end
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lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
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by (induct n) simp_all
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lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
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by (induct n) simp_all
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lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
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by (cases z rule: int_diff_cases) simp
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lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
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by (cases z rule: int_diff_cases) simp
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lemma complex_Re_number_of [simp]: "Re (number_of v) = number_of v"
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unfolding number_of_eq by (rule complex_Re_of_int)
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lemma complex_Im_number_of [simp]: "Im (number_of v) = 0"
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unfolding number_of_eq by (rule complex_Im_of_int)
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lemma Complex_eq_number_of [simp]:
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  "(Complex a b = number_of w) = (a = number_of w \<and> b = 0)"
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by (simp add: expand_complex_eq)
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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
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  complex_scaleR_def: "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: expand_complex_eq right_distrib)
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  show "scaleR (a + b) x = scaleR a x + scaleR b x"
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    by (simp add: expand_complex_eq left_distrib)
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  show "scaleR a (scaleR b x) = scaleR (a * b) x"
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    by (simp add: expand_complex_eq mult_assoc)
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  show "scaleR 1 x = x"
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    by (simp add: expand_complex_eq)
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  show "scaleR a x * y = scaleR a (x * y)"
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    by (simp add: expand_complex_eq ring_simps)
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  show "x * scaleR a y = scaleR a (x * y)"
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    by (simp add: expand_complex_eq ring_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
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  complex_of_real :: "real \<Rightarrow> complex" where
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    "complex_of_real \<equiv> of_real"
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lemma complex_of_real_def: "complex_of_real r = Complex r 0"
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by (simp add: of_real_def complex_scaleR_def)
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lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
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by (simp add: complex_of_real_def)
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lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
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by (simp add: complex_of_real_def)
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lemma Complex_add_complex_of_real [simp]:
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     "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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     "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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     "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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     "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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subsection {* Vector Norm *}
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instantiation complex :: real_normed_field
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begin
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definition
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  complex_norm_def: "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
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abbreviation
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  cmod :: "complex \<Rightarrow> real" where
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  "cmod \<equiv> norm"
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definition
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  complex_sgn_def: "sgn x = x /\<^sub>R cmod x"
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lemmas cmod_def = complex_norm_def
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lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)"
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  by (simp add: complex_norm_def)
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instance
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proof
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  fix r :: real and x y :: complex
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  show "0 \<le> norm x"
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    by (induct x) simp
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  show "(norm x = 0) = (x = 0)"
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    by (induct x) simp
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  show "norm (x + y) \<le> norm x + norm y"
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    by (induct x, induct y)
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       (simp add: real_sqrt_sum_squares_triangle_ineq)
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  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
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    by (induct x)
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       (simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult)
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  show "norm (x * y) = norm x * norm y"
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    by (induct x, induct y)
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       (simp add: real_sqrt_mult [symmetric] power2_eq_square ring_simps)
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  show "sgn x = x /\<^sub>R cmod x" by(simp add: complex_sgn_def)
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parents: 24506
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   318
qed
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   319
25712
f488a37cfad4 instantiation target
haftmann
parents: 25599
diff changeset
   320
end
f488a37cfad4 instantiation target
haftmann
parents: 25599
diff changeset
   321
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   322
lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1"
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   323
by simp
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   324
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   325
lemma cmod_complex_polar [simp]:
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   326
     "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   327
by (simp add: norm_mult)
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   328
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   329
lemma complex_Re_le_cmod: "Re x \<le> cmod x"
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   330
unfolding complex_norm_def
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   331
by (rule real_sqrt_sum_squares_ge1)
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   332
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   333
lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x"
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   334
by (rule order_trans [OF _ norm_ge_zero], simp)
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   335
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   336
lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a"
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   337
by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   338
22861
8ec47039614e clean up complex norm proofs, remove redundant lemmas
huffman
parents: 22852
diff changeset
   339
lemmas real_sum_squared_expand = power2_sum [where 'a=real]
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   340
26117
ca578d3b9f8c Added trivial theorems aboud cmod
chaieb
parents: 25712
diff changeset
   341
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
ca578d3b9f8c Added trivial theorems aboud cmod
chaieb
parents: 25712
diff changeset
   342
by (cases x) simp
ca578d3b9f8c Added trivial theorems aboud cmod
chaieb
parents: 25712
diff changeset
   343
ca578d3b9f8c Added trivial theorems aboud cmod
chaieb
parents: 25712
diff changeset
   344
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
ca578d3b9f8c Added trivial theorems aboud cmod
chaieb
parents: 25712
diff changeset
   345
by (cases x) simp
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   346
23123
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
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   347
subsection {* Completeness of the Complexes *}
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   348
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   349
interpretation Re: bounded_linear ["Re"]
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   350
apply (unfold_locales, simp, simp)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   351
apply (rule_tac x=1 in exI)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   352
apply (simp add: complex_norm_def)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   353
done
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   354
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   355
interpretation Im: bounded_linear ["Im"]
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   356
apply (unfold_locales, simp, simp)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   357
apply (rule_tac x=1 in exI)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   358
apply (simp add: complex_norm_def)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   359
done
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   360
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   361
lemma LIMSEQ_Complex:
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   362
  "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) ----> Complex a b"
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   363
apply (rule LIMSEQ_I)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   364
apply (subgoal_tac "0 < r / sqrt 2")
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   365
apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   366
apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   367
apply (rename_tac M N, rule_tac x="max M N" in exI, safe)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   368
apply (simp add: real_sqrt_sum_squares_less)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   369
apply (simp add: divide_pos_pos)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   370
done
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   371
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   372
instance complex :: banach
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   373
proof
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   374
  fix X :: "nat \<Rightarrow> complex"
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   375
  assume X: "Cauchy X"
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   376
  from Re.Cauchy [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   377
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   378
  from Im.Cauchy [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   379
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   380
  have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   381
    using LIMSEQ_Complex [OF 1 2] by simp
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   382
  thus "convergent X"
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   383
    by (rule convergentI)
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   384
qed
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   385
e2e370bccde1 instance complex :: banach
huffman
parents: 22972
diff changeset
   386
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   387
subsection {* The Complex Number @{term "\<i>"} *}
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   388
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   389
definition
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   390
  "ii" :: complex  ("\<i>") where
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   391
  i_def: "ii \<equiv> Complex 0 1"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   392
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   393
lemma complex_Re_i [simp]: "Re ii = 0"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   394
by (simp add: i_def)
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   395
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   396
lemma complex_Im_i [simp]: "Im ii = 1"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   397
by (simp add: i_def)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   398
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   399
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   400
by (simp add: i_def)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   401
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   402
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   403
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   404
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   405
lemma complex_i_not_one [simp]: "ii \<noteq> 1"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   406
by (simp add: expand_complex_eq)
23124
892e0a4551da use new-style instance declarations
huffman
parents: 23123
diff changeset
   407
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   408
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   409
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   410
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   411
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   412
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   413
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   414
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   415
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   416
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   417
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   418
by (simp add: i_def complex_of_real_def)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   419
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   420
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   421
by (simp add: i_def complex_of_real_def)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   422
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   423
lemma i_squared [simp]: "ii * ii = -1"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   424
by (simp add: i_def)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   425
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   426
lemma power2_i [simp]: "ii\<twosuperior> = -1"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   427
by (simp add: power2_eq_square)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   428
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   429
lemma inverse_i [simp]: "inverse ii = - ii"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   430
by (rule inverse_unique, simp)
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   431
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   432
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   433
subsection {* Complex Conjugation *}
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   434
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   435
definition
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   436
  cnj :: "complex \<Rightarrow> complex" where
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   437
  "cnj z = Complex (Re z) (- Im z)"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   438
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   439
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   440
by (simp add: cnj_def)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   441
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   442
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   443
by (simp add: cnj_def)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   444
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   445
lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   446
by (simp add: cnj_def)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   447
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   448
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   449
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   450
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   451
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   452
by (simp add: cnj_def)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   453
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   454
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   455
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   456
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   457
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   458
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   459
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   460
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   461
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   462
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   463
lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   464
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   465
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   466
lemma complex_cnj_minus: "cnj (- x) = - cnj x"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   467
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   468
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   469
lemma complex_cnj_one [simp]: "cnj 1 = 1"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   470
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   471
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   472
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   473
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   474
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   475
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   476
by (simp add: complex_inverse_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   477
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   478
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   479
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   480
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   481
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   482
by (induct n, simp_all add: complex_cnj_mult)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   483
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   484
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   485
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   486
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   487
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   488
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   489
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   490
lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   491
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   492
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   493
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   494
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   495
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   496
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   497
by (simp add: complex_norm_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   498
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   499
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   500
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   501
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   502
lemma complex_cnj_i [simp]: "cnj ii = - ii"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   503
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   504
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   505
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   506
by (simp add: expand_complex_eq)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   507
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   508
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   509
by (simp add: expand_complex_eq)
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   510
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   511
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   512
by (simp add: expand_complex_eq power2_eq_square)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   513
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   514
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   515
by (simp add: norm_mult power2_eq_square)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   516
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   517
interpretation cnj: bounded_linear ["cnj"]
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   518
apply (unfold_locales)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   519
apply (rule complex_cnj_add)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   520
apply (rule complex_cnj_scaleR)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   521
apply (rule_tac x=1 in exI, simp)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   522
done
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   523
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   524
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22968
diff changeset
   525
subsection{*The Functions @{term sgn} and @{term arg}*}
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   526
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22968
diff changeset
   527
text {*------------ Argand -------------*}
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   528
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   529
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   530
  arg :: "complex => real" where
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   531
  "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   532
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   533
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23477
diff changeset
   534
by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   535
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   536
lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   537
by (simp add: i_def complex_of_real_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   538
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   539
lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   540
by (simp add: i_def complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   541
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   542
lemma complex_eq_cancel_iff2 [simp]:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   543
     "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   544
by (simp add: complex_of_real_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   545
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   546
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23477
diff changeset
   547
by (simp add: complex_sgn_def divide_inverse)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   548
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   549
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23477
diff changeset
   550
by (simp add: complex_sgn_def divide_inverse)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   551
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   552
lemma complex_inverse_complex_split:
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   553
     "inverse(complex_of_real x + ii * complex_of_real y) =
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   554
      complex_of_real(x/(x ^ 2 + y ^ 2)) -
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   555
      ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   556
by (simp add: complex_of_real_def i_def diff_minus divide_inverse)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   557
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   558
(*----------------------------------------------------------------------------*)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   559
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   560
(* many of the theorems are not used - so should they be kept?                *)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   561
(*----------------------------------------------------------------------------*)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   562
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   563
lemma cos_arg_i_mult_zero_pos:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   564
   "0 < y ==> cos (arg(Complex 0 y)) = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   565
apply (simp add: arg_def abs_if)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   566
apply (rule_tac a = "pi/2" in someI2, auto)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   567
apply (rule order_less_trans [of _ 0], auto)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   568
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   569
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   570
lemma cos_arg_i_mult_zero_neg:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   571
   "y < 0 ==> cos (arg(Complex 0 y)) = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   572
apply (simp add: arg_def abs_if)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   573
apply (rule_tac a = "- pi/2" in someI2, auto)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   574
apply (rule order_trans [of _ 0], auto)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   575
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   576
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   577
lemma cos_arg_i_mult_zero [simp]:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   578
     "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   579
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   580
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   581
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   582
subsection{*Finally! Polar Form for Complex Numbers*}
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   583
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   584
definition
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   585
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   586
  (* abbreviation for (cos a + i sin a) *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   587
  cis :: "real => complex" where
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   588
  "cis a = Complex (cos a) (sin a)"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   589
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   590
definition
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   591
  (* abbreviation for r*(cos a + i sin a) *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   592
  rcis :: "[real, real] => complex" where
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   593
  "rcis r a = complex_of_real r * cis a"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   594
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   595
definition
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   596
  (* e ^ (x + iy) *)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 20763
diff changeset
   597
  expi :: "complex => complex" where
20557
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   598
  "expi z = complex_of_real(exp (Re z)) * cis (Im z)"
81dd3679f92c complex_of_real abbreviates of_real::real=>complex;
huffman
parents: 20556
diff changeset
   599
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   600
lemma complex_split_polar:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   601
     "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   602
apply (induct z)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   603
apply (auto simp add: polar_Ex complex_of_real_mult_Complex)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   604
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   605
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   606
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   607
apply (induct z)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   608
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   609
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   610
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   611
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   612
by (simp add: rcis_def cis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   613
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   614
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   615
by (simp add: rcis_def cis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   616
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   617
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   618
proof -
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   619
  have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   620
    by (simp only: power_mult_distrib right_distrib)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   621
  thus ?thesis by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   622
qed
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   623
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   624
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   625
by (simp add: rcis_def cis_def sin_cos_squared_add2_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   626
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   627
lemma complex_Re_cnj [simp]: "Re(cnj z) = Re z"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   628
by (induct z, simp add: complex_cnj)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   629
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   630
lemma complex_Im_cnj [simp]: "Im(cnj z) = - Im z"
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   631
by (induct z, simp add: complex_cnj)
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   632
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   633
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   634
by (simp add: cmod_def power2_eq_square)
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   635
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   636
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   637
by simp
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   638
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   639
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   640
(*---------------------------------------------------------------------------*)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   641
(*  (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b)                      *)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   642
(*---------------------------------------------------------------------------*)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   643
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   644
lemma cis_rcis_eq: "cis a = rcis 1 a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   645
by (simp add: rcis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   646
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   647
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   648
by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   649
              complex_of_real_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   650
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   651
lemma cis_mult: "cis a * cis b = cis (a + b)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   652
by (simp add: cis_rcis_eq rcis_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   653
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   654
lemma cis_zero [simp]: "cis 0 = 1"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   655
by (simp add: cis_def complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   656
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   657
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   658
by (simp add: rcis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   659
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   660
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   661
by (simp add: rcis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   662
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   663
lemma complex_of_real_minus_one:
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   664
   "complex_of_real (-(1::real)) = -(1::complex)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   665
by (simp add: complex_of_real_def complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   666
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   667
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   668
by (simp add: mult_assoc [symmetric])
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   669
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   670
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   671
lemma cis_real_of_nat_Suc_mult:
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   672
   "cis (real (Suc n) * a) = cis a * cis (real n * a)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   673
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   674
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   675
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   676
apply (induct_tac "n")
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   677
apply (auto simp add: cis_real_of_nat_Suc_mult)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   678
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   679
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   680
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
22890
9449c991cc33 remove redundant lemmas
huffman
parents: 22884
diff changeset
   681
by (simp add: rcis_def power_mult_distrib DeMoivre)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   682
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   683
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   684
by (simp add: cis_def complex_inverse_complex_split diff_minus)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   685
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   686
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
22884
5804926e0379 remove redundant lemmas
huffman
parents: 22883
diff changeset
   687
by (simp add: divide_inverse rcis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   688
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   689
lemma cis_divide: "cis a / cis b = cis (a - b)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   690
by (simp add: complex_divide_def cis_mult real_diff_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   691
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   692
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   693
apply (simp add: complex_divide_def)
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   694
apply (case_tac "r2=0", simp)
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   695
apply (simp add: rcis_inverse rcis_mult real_diff_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   696
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   697
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   698
lemma Re_cis [simp]: "Re(cis a) = cos a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   699
by (simp add: cis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   700
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   701
lemma Im_cis [simp]: "Im(cis a) = sin a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   702
by (simp add: cis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   703
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   704
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   705
by (auto simp add: DeMoivre)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   706
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   707
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   708
by (auto simp add: DeMoivre)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   709
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   710
lemma expi_add: "expi(a + b) = expi(a) * expi(b)"
20725
72e20198f834 instance complex :: real_normed_field; cleaned up
huffman
parents: 20560
diff changeset
   711
by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   712
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   713
lemma expi_zero [simp]: "expi (0::complex) = 1"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   714
by (simp add: expi_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   715
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   716
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
   717
apply (insert rcis_Ex [of z])
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   718
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   719
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
   720
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   721
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   722
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
23125
6f7b5b96241f cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents: 23124
diff changeset
   723
by (simp add: expi_def cis_def)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   724
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
   725
end