src/HOL/Complex.thy
author huffman
Mon Aug 08 10:26:26 2011 -0700 (2011-08-08)
changeset 44065 eb64ffccfc75
parent 41959 b460124855b8
child 44127 7b57b9295d98
permissions -rw-r--r--
standard theorem naming scheme: complex_eqI, complex_eq_iff
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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
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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_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
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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_inverse_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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  complex_eq_iff)
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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: 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
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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: complex_eq_iff right_distrib)
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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 left_distrib)
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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
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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 complex_norm_def:
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  "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 complex_sgn_def:
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  "sgn x = x /\<^sub>R cmod x"
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definition dist_complex_def:
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  "dist x y = cmod (x - y)"
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definition open_complex_def:
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  "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
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lemmas cmod_def = complex_norm_def
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lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)"
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  by (simp add: complex_norm_def)
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instance proof
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  fix r :: real and x y :: complex and S :: "complex set"
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  show "0 \<le> norm x"
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    by (induct x) simp
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  show "(norm x = 0) = (x = 0)"
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    by (induct x) simp
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  show "norm (x + y) \<le> norm x + norm y"
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    by (induct x, induct y)
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       (simp add: real_sqrt_sum_squares_triangle_ineq)
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  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
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    by (induct x)
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       (simp add: power_mult_distrib 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 algebra_simps)
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  show "sgn x = x /\<^sub>R cmod x"
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    by (rule complex_sgn_def)
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  show "dist x y = cmod (x - y)"
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    by (rule dist_complex_def)
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  show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
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    by (rule open_complex_def)
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qed
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end
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lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1"
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by simp
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lemma cmod_complex_polar [simp]:
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     "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
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by (simp add: norm_mult)
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lemma complex_Re_le_cmod: "Re x \<le> cmod x"
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unfolding complex_norm_def
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by (rule real_sqrt_sum_squares_ge1)
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lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x"
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by (rule order_trans [OF _ norm_ge_zero], simp)
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lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a"
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by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
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lemmas real_sum_squared_expand = power2_sum [where 'a=real]
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lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
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by (cases x) simp
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lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
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by (cases x) simp
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subsection {* Completeness of the Complexes *}
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interpretation Re: bounded_linear "Re"
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apply (unfold_locales, simp, simp)
huffman@23123
   344
apply (rule_tac x=1 in exI)
huffman@23123
   345
apply (simp add: complex_norm_def)
huffman@23123
   346
done
huffman@23123
   347
wenzelm@30729
   348
interpretation Im: bounded_linear "Im"
huffman@23123
   349
apply (unfold_locales, simp, simp)
huffman@23123
   350
apply (rule_tac x=1 in exI)
huffman@23123
   351
apply (simp add: complex_norm_def)
huffman@23123
   352
done
huffman@23123
   353
huffman@36825
   354
lemma tendsto_Complex [tendsto_intros]:
huffman@36825
   355
  assumes "(f ---> a) net" and "(g ---> b) net"
huffman@36825
   356
  shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) net"
huffman@36825
   357
proof (rule tendstoI)
huffman@36825
   358
  fix r :: real assume "0 < r"
huffman@36825
   359
  hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)
huffman@36825
   360
  have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) net"
huffman@36825
   361
    using `(f ---> a) net` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   362
  moreover
huffman@36825
   363
  have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) net"
huffman@36825
   364
    using `(g ---> b) net` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   365
  ultimately
huffman@36825
   366
  show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) net"
huffman@36825
   367
    by (rule eventually_elim2)
huffman@36825
   368
       (simp add: dist_norm real_sqrt_sum_squares_less)
huffman@36825
   369
qed
huffman@36825
   370
huffman@23123
   371
lemma LIMSEQ_Complex:
huffman@23123
   372
  "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) ----> Complex a b"
huffman@36825
   373
by (rule tendsto_Complex)
huffman@23123
   374
huffman@23123
   375
instance complex :: banach
huffman@23123
   376
proof
huffman@23123
   377
  fix X :: "nat \<Rightarrow> complex"
huffman@23123
   378
  assume X: "Cauchy X"
huffman@23123
   379
  from Re.Cauchy [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"
huffman@23123
   380
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@23123
   381
  from Im.Cauchy [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
huffman@23123
   382
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@23123
   383
  have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
huffman@23123
   384
    using LIMSEQ_Complex [OF 1 2] by simp
huffman@23123
   385
  thus "convergent X"
huffman@23123
   386
    by (rule convergentI)
huffman@23123
   387
qed
huffman@23123
   388
huffman@23123
   389
huffman@23125
   390
subsection {* The Complex Number @{term "\<i>"} *}
huffman@23125
   391
huffman@23125
   392
definition
huffman@23125
   393
  "ii" :: complex  ("\<i>") where
huffman@23125
   394
  i_def: "ii \<equiv> Complex 0 1"
huffman@23125
   395
huffman@23125
   396
lemma complex_Re_i [simp]: "Re ii = 0"
huffman@23125
   397
by (simp add: i_def)
paulson@14354
   398
huffman@23125
   399
lemma complex_Im_i [simp]: "Im ii = 1"
huffman@23125
   400
by (simp add: i_def)
huffman@23125
   401
huffman@23125
   402
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
huffman@23125
   403
by (simp add: i_def)
huffman@23125
   404
huffman@23125
   405
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
huffman@44065
   406
by (simp add: complex_eq_iff)
huffman@23125
   407
huffman@23125
   408
lemma complex_i_not_one [simp]: "ii \<noteq> 1"
huffman@44065
   409
by (simp add: complex_eq_iff)
huffman@23124
   410
huffman@23125
   411
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w"
huffman@44065
   412
by (simp add: complex_eq_iff)
huffman@23125
   413
huffman@23125
   414
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
huffman@44065
   415
by (simp add: complex_eq_iff)
huffman@23125
   416
huffman@23125
   417
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
huffman@44065
   418
by (simp add: complex_eq_iff)
huffman@23125
   419
huffman@23125
   420
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
huffman@23125
   421
by (simp add: i_def complex_of_real_def)
huffman@23125
   422
huffman@23125
   423
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
huffman@23125
   424
by (simp add: i_def complex_of_real_def)
huffman@23125
   425
huffman@23125
   426
lemma i_squared [simp]: "ii * ii = -1"
huffman@23125
   427
by (simp add: i_def)
huffman@23125
   428
huffman@23125
   429
lemma power2_i [simp]: "ii\<twosuperior> = -1"
huffman@23125
   430
by (simp add: power2_eq_square)
huffman@23125
   431
huffman@23125
   432
lemma inverse_i [simp]: "inverse ii = - ii"
huffman@23125
   433
by (rule inverse_unique, simp)
paulson@14354
   434
paulson@14354
   435
huffman@23125
   436
subsection {* Complex Conjugation *}
huffman@23125
   437
huffman@23125
   438
definition
huffman@23125
   439
  cnj :: "complex \<Rightarrow> complex" where
huffman@23125
   440
  "cnj z = Complex (Re z) (- Im z)"
huffman@23125
   441
huffman@23125
   442
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
huffman@23125
   443
by (simp add: cnj_def)
huffman@23125
   444
huffman@23125
   445
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
huffman@23125
   446
by (simp add: cnj_def)
huffman@23125
   447
huffman@23125
   448
lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
huffman@23125
   449
by (simp add: cnj_def)
huffman@23125
   450
huffman@23125
   451
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
huffman@44065
   452
by (simp add: complex_eq_iff)
huffman@23125
   453
huffman@23125
   454
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
huffman@23125
   455
by (simp add: cnj_def)
huffman@23125
   456
huffman@23125
   457
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
huffman@44065
   458
by (simp add: complex_eq_iff)
huffman@23125
   459
huffman@23125
   460
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
huffman@44065
   461
by (simp add: complex_eq_iff)
huffman@23125
   462
huffman@23125
   463
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
huffman@44065
   464
by (simp add: complex_eq_iff)
huffman@23125
   465
huffman@23125
   466
lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
huffman@44065
   467
by (simp add: complex_eq_iff)
huffman@23125
   468
huffman@23125
   469
lemma complex_cnj_minus: "cnj (- x) = - cnj x"
huffman@44065
   470
by (simp add: complex_eq_iff)
huffman@23125
   471
huffman@23125
   472
lemma complex_cnj_one [simp]: "cnj 1 = 1"
huffman@44065
   473
by (simp add: complex_eq_iff)
huffman@23125
   474
huffman@23125
   475
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
huffman@44065
   476
by (simp add: complex_eq_iff)
huffman@23125
   477
huffman@23125
   478
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
huffman@23125
   479
by (simp add: complex_inverse_def)
paulson@14323
   480
huffman@23125
   481
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
huffman@23125
   482
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
huffman@23125
   483
huffman@23125
   484
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
huffman@23125
   485
by (induct n, simp_all add: complex_cnj_mult)
huffman@23125
   486
huffman@23125
   487
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
huffman@44065
   488
by (simp add: complex_eq_iff)
huffman@23125
   489
huffman@23125
   490
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
huffman@44065
   491
by (simp add: complex_eq_iff)
huffman@23125
   492
huffman@23125
   493
lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w"
huffman@44065
   494
by (simp add: complex_eq_iff)
huffman@23125
   495
huffman@23125
   496
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
huffman@44065
   497
by (simp add: complex_eq_iff)
huffman@23125
   498
huffman@23125
   499
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
huffman@23125
   500
by (simp add: complex_norm_def)
paulson@14323
   501
huffman@23125
   502
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
huffman@44065
   503
by (simp add: complex_eq_iff)
huffman@23125
   504
huffman@23125
   505
lemma complex_cnj_i [simp]: "cnj ii = - ii"
huffman@44065
   506
by (simp add: complex_eq_iff)
huffman@23125
   507
huffman@23125
   508
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
huffman@44065
   509
by (simp add: complex_eq_iff)
huffman@23125
   510
huffman@23125
   511
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
huffman@44065
   512
by (simp add: complex_eq_iff)
paulson@14354
   513
huffman@23125
   514
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
huffman@44065
   515
by (simp add: complex_eq_iff power2_eq_square)
huffman@23125
   516
huffman@23125
   517
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
huffman@23125
   518
by (simp add: norm_mult power2_eq_square)
huffman@23125
   519
wenzelm@30729
   520
interpretation cnj: bounded_linear "cnj"
huffman@23125
   521
apply (unfold_locales)
huffman@23125
   522
apply (rule complex_cnj_add)
huffman@23125
   523
apply (rule complex_cnj_scaleR)
huffman@23125
   524
apply (rule_tac x=1 in exI, simp)
huffman@23125
   525
done
paulson@14354
   526
paulson@14354
   527
huffman@22972
   528
subsection{*The Functions @{term sgn} and @{term arg}*}
paulson@14323
   529
huffman@22972
   530
text {*------------ Argand -------------*}
huffman@20557
   531
wenzelm@21404
   532
definition
wenzelm@21404
   533
  arg :: "complex => real" where
huffman@20557
   534
  "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)"
huffman@20557
   535
paulson@14374
   536
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
nipkow@24506
   537
by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute)
paulson@14323
   538
paulson@14323
   539
lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
huffman@20725
   540
by (simp add: i_def complex_of_real_def)
paulson@14323
   541
paulson@14374
   542
lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)"
huffman@20725
   543
by (simp add: i_def complex_one_def)
paulson@14323
   544
paulson@14374
   545
lemma complex_eq_cancel_iff2 [simp]:
paulson@14377
   546
     "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
paulson@14377
   547
by (simp add: complex_of_real_def)
paulson@14323
   548
paulson@14374
   549
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
nipkow@24506
   550
by (simp add: complex_sgn_def divide_inverse)
paulson@14323
   551
paulson@14374
   552
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
nipkow@24506
   553
by (simp add: complex_sgn_def divide_inverse)
paulson@14323
   554
paulson@14323
   555
lemma complex_inverse_complex_split:
paulson@14323
   556
     "inverse(complex_of_real x + ii * complex_of_real y) =
paulson@14323
   557
      complex_of_real(x/(x ^ 2 + y ^ 2)) -
paulson@14323
   558
      ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
huffman@20725
   559
by (simp add: complex_of_real_def i_def diff_minus divide_inverse)
paulson@14323
   560
paulson@14323
   561
(*----------------------------------------------------------------------------*)
paulson@14323
   562
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
paulson@14323
   563
(* many of the theorems are not used - so should they be kept?                *)
paulson@14323
   564
(*----------------------------------------------------------------------------*)
paulson@14323
   565
paulson@14354
   566
lemma cos_arg_i_mult_zero_pos:
paulson@14377
   567
   "0 < y ==> cos (arg(Complex 0 y)) = 0"
paulson@14373
   568
apply (simp add: arg_def abs_if)
paulson@14334
   569
apply (rule_tac a = "pi/2" in someI2, auto)
paulson@14334
   570
apply (rule order_less_trans [of _ 0], auto)
paulson@14323
   571
done
paulson@14323
   572
paulson@14354
   573
lemma cos_arg_i_mult_zero_neg:
paulson@14377
   574
   "y < 0 ==> cos (arg(Complex 0 y)) = 0"
paulson@14373
   575
apply (simp add: arg_def abs_if)
paulson@14334
   576
apply (rule_tac a = "- pi/2" in someI2, auto)
paulson@14334
   577
apply (rule order_trans [of _ 0], auto)
paulson@14323
   578
done
paulson@14323
   579
paulson@14374
   580
lemma cos_arg_i_mult_zero [simp]:
paulson@14377
   581
     "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
paulson@14377
   582
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
paulson@14323
   583
paulson@14323
   584
paulson@14323
   585
subsection{*Finally! Polar Form for Complex Numbers*}
paulson@14323
   586
huffman@20557
   587
definition
huffman@20557
   588
huffman@20557
   589
  (* abbreviation for (cos a + i sin a) *)
wenzelm@21404
   590
  cis :: "real => complex" where
huffman@20557
   591
  "cis a = Complex (cos a) (sin a)"
huffman@20557
   592
wenzelm@21404
   593
definition
huffman@20557
   594
  (* abbreviation for r*(cos a + i sin a) *)
wenzelm@21404
   595
  rcis :: "[real, real] => complex" where
huffman@20557
   596
  "rcis r a = complex_of_real r * cis a"
huffman@20557
   597
wenzelm@21404
   598
definition
huffman@20557
   599
  (* e ^ (x + iy) *)
wenzelm@21404
   600
  expi :: "complex => complex" where
huffman@20557
   601
  "expi z = complex_of_real(exp (Re z)) * cis (Im z)"
huffman@20557
   602
paulson@14374
   603
lemma complex_split_polar:
paulson@14377
   604
     "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
huffman@20725
   605
apply (induct z)
paulson@14377
   606
apply (auto simp add: polar_Ex complex_of_real_mult_Complex)
paulson@14323
   607
done
paulson@14323
   608
paulson@14354
   609
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
huffman@20725
   610
apply (induct z)
paulson@14377
   611
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex)
paulson@14323
   612
done
paulson@14323
   613
paulson@14374
   614
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
paulson@14373
   615
by (simp add: rcis_def cis_def)
paulson@14323
   616
paulson@14348
   617
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
paulson@14373
   618
by (simp add: rcis_def cis_def)
paulson@14323
   619
paulson@14377
   620
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>"
paulson@14377
   621
proof -
paulson@14377
   622
  have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)"
huffman@20725
   623
    by (simp only: power_mult_distrib right_distrib)
paulson@14377
   624
  thus ?thesis by simp
paulson@14377
   625
qed
paulson@14323
   626
paulson@14374
   627
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
paulson@14377
   628
by (simp add: rcis_def cis_def sin_cos_squared_add2_mult)
paulson@14323
   629
huffman@23125
   630
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
huffman@23125
   631
by (simp add: cmod_def power2_eq_square)
huffman@23125
   632
paulson@14374
   633
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
huffman@23125
   634
by simp
paulson@14323
   635
paulson@14323
   636
paulson@14323
   637
(*---------------------------------------------------------------------------*)
paulson@14323
   638
(*  (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b)                      *)
paulson@14323
   639
(*---------------------------------------------------------------------------*)
paulson@14323
   640
paulson@14323
   641
lemma cis_rcis_eq: "cis a = rcis 1 a"
paulson@14373
   642
by (simp add: rcis_def)
paulson@14323
   643
paulson@14374
   644
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
paulson@15013
   645
by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib
paulson@15013
   646
              complex_of_real_def)
paulson@14323
   647
paulson@14323
   648
lemma cis_mult: "cis a * cis b = cis (a + b)"
paulson@14373
   649
by (simp add: cis_rcis_eq rcis_mult)
paulson@14323
   650
paulson@14374
   651
lemma cis_zero [simp]: "cis 0 = 1"
paulson@14377
   652
by (simp add: cis_def complex_one_def)
paulson@14323
   653
paulson@14374
   654
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
paulson@14373
   655
by (simp add: rcis_def)
paulson@14323
   656
paulson@14374
   657
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
paulson@14373
   658
by (simp add: rcis_def)
paulson@14323
   659
paulson@14323
   660
lemma complex_of_real_minus_one:
paulson@14323
   661
   "complex_of_real (-(1::real)) = -(1::complex)"
huffman@20725
   662
by (simp add: complex_of_real_def complex_one_def)
paulson@14323
   663
paulson@14374
   664
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
huffman@23125
   665
by (simp add: mult_assoc [symmetric])
paulson@14323
   666
paulson@14323
   667
paulson@14323
   668
lemma cis_real_of_nat_Suc_mult:
paulson@14323
   669
   "cis (real (Suc n) * a) = cis a * cis (real n * a)"
paulson@14377
   670
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib)
paulson@14323
   671
paulson@14323
   672
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
paulson@14323
   673
apply (induct_tac "n")
paulson@14323
   674
apply (auto simp add: cis_real_of_nat_Suc_mult)
paulson@14323
   675
done
paulson@14323
   676
paulson@14374
   677
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
huffman@22890
   678
by (simp add: rcis_def power_mult_distrib DeMoivre)
paulson@14323
   679
paulson@14374
   680
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
huffman@20725
   681
by (simp add: cis_def complex_inverse_complex_split diff_minus)
paulson@14323
   682
paulson@14323
   683
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
huffman@22884
   684
by (simp add: divide_inverse rcis_def)
paulson@14323
   685
paulson@14323
   686
lemma cis_divide: "cis a / cis b = cis (a - b)"
haftmann@37887
   687
by (simp add: complex_divide_def cis_mult diff_minus)
paulson@14323
   688
paulson@14354
   689
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
paulson@14373
   690
apply (simp add: complex_divide_def)
paulson@14373
   691
apply (case_tac "r2=0", simp)
haftmann@37887
   692
apply (simp add: rcis_inverse rcis_mult diff_minus)
paulson@14323
   693
done
paulson@14323
   694
paulson@14374
   695
lemma Re_cis [simp]: "Re(cis a) = cos a"
paulson@14373
   696
by (simp add: cis_def)
paulson@14323
   697
paulson@14374
   698
lemma Im_cis [simp]: "Im(cis a) = sin a"
paulson@14373
   699
by (simp add: cis_def)
paulson@14323
   700
paulson@14323
   701
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
paulson@14334
   702
by (auto simp add: DeMoivre)
paulson@14323
   703
paulson@14323
   704
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
paulson@14334
   705
by (auto simp add: DeMoivre)
paulson@14323
   706
paulson@14323
   707
lemma expi_add: "expi(a + b) = expi(a) * expi(b)"
huffman@20725
   708
by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac)
paulson@14323
   709
paulson@14374
   710
lemma expi_zero [simp]: "expi (0::complex) = 1"
paulson@14373
   711
by (simp add: expi_def)
paulson@14323
   712
paulson@14374
   713
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
paulson@14373
   714
apply (insert rcis_Ex [of z])
huffman@23125
   715
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
paulson@14334
   716
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
paulson@14323
   717
done
paulson@14323
   718
paulson@14387
   719
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
huffman@23125
   720
by (simp add: expi_def cis_def)
paulson@14387
   721
huffman@44065
   722
text {* Legacy theorem names *}
huffman@44065
   723
huffman@44065
   724
lemmas expand_complex_eq = complex_eq_iff
huffman@44065
   725
lemmas complex_Re_Im_cancel_iff = complex_eq_iff
huffman@44065
   726
lemmas complex_equality = complex_eqI
huffman@44065
   727
paulson@13957
   728
end