src/HOL/Complex.thy
author huffman
Mon Sep 05 08:38:50 2011 -0700 (2011-09-05)
changeset 44724 0b900a9d8023
parent 44715 1a17d8913976
child 44748 7f6838b3474a
permissions -rw-r--r--
tuned indentation
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(*  Title:       HOL/Complex.thy
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    Author:      Jacques D. Fleuriot
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    Copyright:   2001 University of Edinburgh
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
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*)
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header {* Complex Numbers: Rectangular and Polar Representations *}
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theory Complex
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imports Transcendental
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begin
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datatype complex = Complex real real
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primrec Re :: "complex \<Rightarrow> real"
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  where Re: "Re (Complex x y) = x"
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primrec Im :: "complex \<Rightarrow> real"
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  where Im: "Im (Complex x y) = y"
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lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"
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  by (induct z) simp
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lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"
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  by (induct x, induct y) simp
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lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"
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  by (induct x, induct y) simp
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subsection {* Addition and Subtraction *}
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instantiation complex :: ab_group_add
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begin
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definition complex_zero_def:
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  "0 = Complex 0 0"
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definition complex_add_def:
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  "x + y = Complex (Re x + Re y) (Im x + Im y)"
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definition complex_minus_def:
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  "- x = Complex (- Re x) (- Im x)"
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definition complex_diff_def:
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  "x - (y\<Colon>complex) = x + - y"
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lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
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  by (simp add: complex_zero_def)
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lemma complex_Re_zero [simp]: "Re 0 = 0"
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  by (simp add: complex_zero_def)
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lemma complex_Im_zero [simp]: "Im 0 = 0"
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  by (simp add: complex_zero_def)
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lemma complex_add [simp]:
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  "Complex a b + Complex c d = Complex (a + c) (b + d)"
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  by (simp add: complex_add_def)
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lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y"
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  by (simp add: complex_add_def)
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lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y"
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  by (simp add: complex_add_def)
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lemma complex_minus [simp]:
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  "- (Complex a b) = Complex (- a) (- b)"
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  by (simp add: complex_minus_def)
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lemma complex_Re_minus [simp]: "Re (- x) = - Re x"
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  by (simp add: complex_minus_def)
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lemma complex_Im_minus [simp]: "Im (- x) = - Im x"
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  by (simp add: complex_minus_def)
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lemma complex_diff [simp]:
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  "Complex a b - Complex c d = Complex (a - c) (b - d)"
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  by (simp add: complex_diff_def)
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lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y"
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  by (simp add: complex_diff_def)
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lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y"
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  by (simp add: complex_diff_def)
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instance
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  by intro_classes (simp_all add: complex_add_def complex_diff_def)
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end
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subsection {* Multiplication and Division *}
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instantiation complex :: field_inverse_zero
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begin
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definition complex_one_def:
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  "1 = Complex 1 0"
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definition complex_mult_def:
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  "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
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definition complex_inverse_def:
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  "inverse x =
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    Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))"
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definition complex_divide_def:
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  "x / (y\<Colon>complex) = x * inverse y"
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lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"
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  by (simp add: complex_one_def)
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lemma complex_Re_one [simp]: "Re 1 = 1"
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  by (simp add: complex_one_def)
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lemma complex_Im_one [simp]: "Im 1 = 0"
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  by (simp add: complex_one_def)
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lemma complex_mult [simp]:
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  "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
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  by (simp add: complex_mult_def)
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lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"
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  by (simp add: complex_mult_def)
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lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"
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  by (simp add: complex_mult_def)
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lemma complex_inverse [simp]:
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  "inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))"
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  by (simp add: complex_inverse_def)
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lemma complex_Re_inverse:
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  "Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
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  by (simp add: complex_inverse_def)
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lemma complex_Im_inverse:
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  "Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
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  by (simp add: complex_inverse_def)
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instance
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  by intro_classes (simp_all add: complex_mult_def
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    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 complex_number_of_def:
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  "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 complex_scaleR_def:
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  "scaleR r x = Complex (r * Re x) (r * Im x)"
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lemma complex_scaleR [simp]:
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  "scaleR r (Complex a b) = Complex (r * a) (r * b)"
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  unfolding complex_scaleR_def by simp
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lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"
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  unfolding complex_scaleR_def by simp
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lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"
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  unfolding complex_scaleR_def by simp
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instance
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proof
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  fix a b :: real and x y :: complex
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  show "scaleR a (x + y) = scaleR a x + scaleR a y"
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    by (simp add: complex_eq_iff 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 complex_of_real :: "real \<Rightarrow> complex"
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  where "complex_of_real \<equiv> of_real"
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lemma complex_of_real_def: "complex_of_real r = Complex r 0"
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  by (simp add: of_real_def complex_scaleR_def)
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lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
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  by (simp add: complex_of_real_def)
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lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
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  by (simp add: complex_of_real_def)
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lemma Complex_add_complex_of_real [simp]:
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  shows "Complex x y + complex_of_real r = Complex (x+r) y"
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  by (simp add: complex_of_real_def)
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lemma complex_of_real_add_Complex [simp]:
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  shows "complex_of_real r + Complex x y = Complex (r+x) y"
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  by (simp add: complex_of_real_def)
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lemma Complex_mult_complex_of_real:
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  shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
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  by (simp add: complex_of_real_def)
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lemma complex_of_real_mult_Complex:
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  shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
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  by (simp add: complex_of_real_def)
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subsection {* Vector Norm *}
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instantiation complex :: real_normed_field
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begin
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definition complex_norm_def:
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  "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
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abbreviation cmod :: "complex \<Rightarrow> real"
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  where "cmod \<equiv> norm"
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definition complex_sgn_def:
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  "sgn x = x /\<^sub>R cmod x"
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definition dist_complex_def:
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  "dist x y = cmod (x - y)"
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definition open_complex_def:
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  "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
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lemmas cmod_def = complex_norm_def
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lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)"
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  by (simp add: complex_norm_def)
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instance proof
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  fix r :: real and x y :: complex and S :: "complex set"
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  show "0 \<le> norm x"
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    by (induct x) simp
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  show "(norm x = 0) = (x = 0)"
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    by (induct x) simp
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  show "norm (x + y) \<le> norm x + norm y"
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    by (induct x, induct y)
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       (simp add: real_sqrt_sum_squares_triangle_ineq)
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  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
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    by (induct x)
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       (simp add: power_mult_distrib 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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lemma bounded_linear_Re: "bounded_linear Re"
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  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
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lemma bounded_linear_Im: "bounded_linear Im"
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  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@23123
   340
huffman@44290
   341
lemmas tendsto_Re [tendsto_intros] =
huffman@44290
   342
  bounded_linear.tendsto [OF bounded_linear_Re]
huffman@44290
   343
huffman@44290
   344
lemmas tendsto_Im [tendsto_intros] =
huffman@44290
   345
  bounded_linear.tendsto [OF bounded_linear_Im]
huffman@44290
   346
huffman@44290
   347
lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
huffman@44290
   348
lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
huffman@44290
   349
lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
huffman@44290
   350
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
huffman@23123
   351
huffman@36825
   352
lemma tendsto_Complex [tendsto_intros]:
huffman@44724
   353
  assumes "(f ---> a) F" and "(g ---> b) F"
huffman@44724
   354
  shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
huffman@36825
   355
proof (rule tendstoI)
huffman@36825
   356
  fix r :: real assume "0 < r"
huffman@36825
   357
  hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)
huffman@44724
   358
  have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F"
huffman@44724
   359
    using `(f ---> a) F` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   360
  moreover
huffman@44724
   361
  have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F"
huffman@44724
   362
    using `(g ---> b) F` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   363
  ultimately
huffman@44724
   364
  show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F"
huffman@36825
   365
    by (rule eventually_elim2)
huffman@36825
   366
       (simp add: dist_norm real_sqrt_sum_squares_less)
huffman@36825
   367
qed
huffman@36825
   368
huffman@23123
   369
lemma LIMSEQ_Complex:
huffman@23123
   370
  "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) ----> Complex a b"
huffman@44724
   371
  by (rule tendsto_Complex)
huffman@23123
   372
huffman@23123
   373
instance complex :: banach
huffman@23123
   374
proof
huffman@23123
   375
  fix X :: "nat \<Rightarrow> complex"
huffman@23123
   376
  assume X: "Cauchy X"
huffman@44290
   377
  from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"
huffman@23123
   378
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@44290
   379
  from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
huffman@23123
   380
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@23123
   381
  have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
huffman@23123
   382
    using LIMSEQ_Complex [OF 1 2] by simp
huffman@23123
   383
  thus "convergent X"
huffman@23123
   384
    by (rule convergentI)
huffman@23123
   385
qed
huffman@23123
   386
huffman@23123
   387
huffman@23125
   388
subsection {* The Complex Number @{term "\<i>"} *}
huffman@23125
   389
huffman@44724
   390
definition "ii" :: complex  ("\<i>")
huffman@44724
   391
  where i_def: "ii \<equiv> Complex 0 1"
huffman@23125
   392
huffman@23125
   393
lemma complex_Re_i [simp]: "Re ii = 0"
huffman@44724
   394
  by (simp add: i_def)
paulson@14354
   395
huffman@23125
   396
lemma complex_Im_i [simp]: "Im ii = 1"
huffman@44724
   397
  by (simp add: i_def)
huffman@23125
   398
huffman@23125
   399
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
huffman@44724
   400
  by (simp add: i_def)
huffman@23125
   401
huffman@23125
   402
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
huffman@44724
   403
  by (simp add: complex_eq_iff)
huffman@23125
   404
huffman@23125
   405
lemma complex_i_not_one [simp]: "ii \<noteq> 1"
huffman@44724
   406
  by (simp add: complex_eq_iff)
huffman@23124
   407
huffman@23125
   408
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w"
huffman@44724
   409
  by (simp add: complex_eq_iff)
huffman@23125
   410
huffman@23125
   411
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
huffman@44724
   412
  by (simp add: complex_eq_iff)
huffman@23125
   413
huffman@23125
   414
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
huffman@44724
   415
  by (simp add: complex_eq_iff)
huffman@23125
   416
huffman@23125
   417
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
huffman@44724
   418
  by (simp add: i_def complex_of_real_def)
huffman@23125
   419
huffman@23125
   420
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
huffman@44724
   421
  by (simp add: i_def complex_of_real_def)
huffman@23125
   422
huffman@23125
   423
lemma i_squared [simp]: "ii * ii = -1"
huffman@44724
   424
  by (simp add: i_def)
huffman@23125
   425
huffman@23125
   426
lemma power2_i [simp]: "ii\<twosuperior> = -1"
huffman@44724
   427
  by (simp add: power2_eq_square)
huffman@23125
   428
huffman@23125
   429
lemma inverse_i [simp]: "inverse ii = - ii"
huffman@44724
   430
  by (rule inverse_unique, simp)
paulson@14354
   431
paulson@14354
   432
huffman@23125
   433
subsection {* Complex Conjugation *}
huffman@23125
   434
huffman@44724
   435
definition cnj :: "complex \<Rightarrow> complex" where
huffman@23125
   436
  "cnj z = Complex (Re z) (- Im z)"
huffman@23125
   437
huffman@23125
   438
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
huffman@44724
   439
  by (simp add: cnj_def)
huffman@23125
   440
huffman@23125
   441
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
huffman@44724
   442
  by (simp add: cnj_def)
huffman@23125
   443
huffman@23125
   444
lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
huffman@44724
   445
  by (simp add: cnj_def)
huffman@23125
   446
huffman@23125
   447
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
huffman@44724
   448
  by (simp add: complex_eq_iff)
huffman@23125
   449
huffman@23125
   450
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
huffman@44724
   451
  by (simp add: cnj_def)
huffman@23125
   452
huffman@23125
   453
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
huffman@44724
   454
  by (simp add: complex_eq_iff)
huffman@23125
   455
huffman@23125
   456
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
huffman@44724
   457
  by (simp add: complex_eq_iff)
huffman@23125
   458
huffman@23125
   459
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
huffman@44724
   460
  by (simp add: complex_eq_iff)
huffman@23125
   461
huffman@23125
   462
lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
huffman@44724
   463
  by (simp add: complex_eq_iff)
huffman@23125
   464
huffman@23125
   465
lemma complex_cnj_minus: "cnj (- x) = - cnj x"
huffman@44724
   466
  by (simp add: complex_eq_iff)
huffman@23125
   467
huffman@23125
   468
lemma complex_cnj_one [simp]: "cnj 1 = 1"
huffman@44724
   469
  by (simp add: complex_eq_iff)
huffman@23125
   470
huffman@23125
   471
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
huffman@44724
   472
  by (simp add: complex_eq_iff)
huffman@23125
   473
huffman@23125
   474
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
huffman@44724
   475
  by (simp add: complex_inverse_def)
paulson@14323
   476
huffman@23125
   477
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
huffman@44724
   478
  by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
huffman@23125
   479
huffman@23125
   480
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
huffman@44724
   481
  by (induct n, simp_all add: complex_cnj_mult)
huffman@23125
   482
huffman@23125
   483
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
huffman@44724
   484
  by (simp add: complex_eq_iff)
huffman@23125
   485
huffman@23125
   486
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
huffman@44724
   487
  by (simp add: complex_eq_iff)
huffman@23125
   488
huffman@23125
   489
lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w"
huffman@44724
   490
  by (simp add: complex_eq_iff)
huffman@23125
   491
huffman@23125
   492
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
huffman@44724
   493
  by (simp add: complex_eq_iff)
huffman@23125
   494
huffman@23125
   495
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
huffman@44724
   496
  by (simp add: complex_norm_def)
paulson@14323
   497
huffman@23125
   498
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
huffman@44724
   499
  by (simp add: complex_eq_iff)
huffman@23125
   500
huffman@23125
   501
lemma complex_cnj_i [simp]: "cnj ii = - ii"
huffman@44724
   502
  by (simp add: complex_eq_iff)
huffman@23125
   503
huffman@23125
   504
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
huffman@44724
   505
  by (simp add: complex_eq_iff)
huffman@23125
   506
huffman@23125
   507
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
huffman@44724
   508
  by (simp add: complex_eq_iff)
paulson@14354
   509
huffman@23125
   510
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
huffman@44724
   511
  by (simp add: complex_eq_iff power2_eq_square)
huffman@23125
   512
huffman@23125
   513
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
huffman@44724
   514
  by (simp add: norm_mult power2_eq_square)
huffman@23125
   515
huffman@44290
   516
lemma bounded_linear_cnj: "bounded_linear cnj"
huffman@44127
   517
  using complex_cnj_add complex_cnj_scaleR
huffman@44127
   518
  by (rule bounded_linear_intro [where K=1], simp)
paulson@14354
   519
huffman@44290
   520
lemmas tendsto_cnj [tendsto_intros] =
huffman@44290
   521
  bounded_linear.tendsto [OF bounded_linear_cnj]
huffman@44290
   522
huffman@44290
   523
lemmas isCont_cnj [simp] =
huffman@44290
   524
  bounded_linear.isCont [OF bounded_linear_cnj]
huffman@44290
   525
paulson@14354
   526
huffman@22972
   527
subsection{*The Functions @{term sgn} and @{term arg}*}
paulson@14323
   528
huffman@22972
   529
text {*------------ Argand -------------*}
huffman@20557
   530
huffman@44724
   531
definition arg :: "complex => real" where
huffman@20557
   532
  "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)"
huffman@20557
   533
paulson@14374
   534
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
huffman@44724
   535
  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
paulson@14323
   536
paulson@14323
   537
lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
huffman@44724
   538
  by (simp add: i_def complex_of_real_def)
paulson@14323
   539
paulson@14374
   540
lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)"
huffman@44724
   541
  by (simp add: i_def complex_one_def)
paulson@14323
   542
paulson@14374
   543
lemma complex_eq_cancel_iff2 [simp]:
huffman@44724
   544
  shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
huffman@44724
   545
  by (simp add: complex_of_real_def)
paulson@14323
   546
paulson@14374
   547
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
huffman@44724
   548
  by (simp add: complex_sgn_def divide_inverse)
paulson@14323
   549
paulson@14374
   550
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
huffman@44724
   551
  by (simp add: complex_sgn_def divide_inverse)
paulson@14323
   552
paulson@14323
   553
lemma complex_inverse_complex_split:
paulson@14323
   554
     "inverse(complex_of_real x + ii * complex_of_real y) =
paulson@14323
   555
      complex_of_real(x/(x ^ 2 + y ^ 2)) -
paulson@14323
   556
      ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
huffman@44724
   557
  by (simp add: complex_of_real_def i_def diff_minus divide_inverse)
paulson@14323
   558
paulson@14323
   559
(*----------------------------------------------------------------------------*)
paulson@14323
   560
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
paulson@14323
   561
(* many of the theorems are not used - so should they be kept?                *)
paulson@14323
   562
(*----------------------------------------------------------------------------*)
paulson@14323
   563
paulson@14354
   564
lemma cos_arg_i_mult_zero_pos:
paulson@14377
   565
   "0 < y ==> cos (arg(Complex 0 y)) = 0"
paulson@14373
   566
apply (simp add: arg_def abs_if)
paulson@14334
   567
apply (rule_tac a = "pi/2" in someI2, auto)
paulson@14334
   568
apply (rule order_less_trans [of _ 0], auto)
paulson@14323
   569
done
paulson@14323
   570
paulson@14354
   571
lemma cos_arg_i_mult_zero_neg:
paulson@14377
   572
   "y < 0 ==> cos (arg(Complex 0 y)) = 0"
paulson@14373
   573
apply (simp add: arg_def abs_if)
paulson@14334
   574
apply (rule_tac a = "- pi/2" in someI2, auto)
paulson@14334
   575
apply (rule order_trans [of _ 0], auto)
paulson@14323
   576
done
paulson@14323
   577
paulson@14374
   578
lemma cos_arg_i_mult_zero [simp]:
paulson@14377
   579
     "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
paulson@14377
   580
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
paulson@14323
   581
paulson@14323
   582
paulson@14323
   583
subsection{*Finally! Polar Form for Complex Numbers*}
paulson@14323
   584
huffman@44715
   585
text {* An abbreviation for @{text "cos a + i sin a"}. *}
huffman@20557
   586
huffman@44715
   587
definition cis :: "real \<Rightarrow> complex" where
huffman@20557
   588
  "cis a = Complex (cos a) (sin a)"
huffman@20557
   589
huffman@44715
   590
text {* An abbreviation for @{text "r(cos a + i sin a)"}. *}
huffman@44715
   591
huffman@44715
   592
definition rcis :: "[real, real] \<Rightarrow> complex" where
huffman@20557
   593
  "rcis r a = complex_of_real r * cis a"
huffman@20557
   594
huffman@44291
   595
abbreviation expi :: "complex \<Rightarrow> complex"
huffman@44291
   596
  where "expi \<equiv> exp"
huffman@44291
   597
huffman@44712
   598
lemma cis_conv_exp: "cis b = exp (Complex 0 b)"
huffman@44291
   599
proof (rule complex_eqI)
huffman@44291
   600
  { fix n have "Complex 0 b ^ n =
huffman@44291
   601
    real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"
huffman@44291
   602
      apply (induct n)
huffman@44291
   603
      apply (simp add: cos_coeff_def sin_coeff_def)
huffman@44291
   604
      apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)
huffman@44291
   605
      done } note * = this
huffman@44712
   606
  show "Re (cis b) = Re (exp (Complex 0 b))"
huffman@44291
   607
    unfolding exp_def cis_def cos_def
huffman@44291
   608
    by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],
huffman@44291
   609
      simp add: * mult_assoc [symmetric])
huffman@44712
   610
  show "Im (cis b) = Im (exp (Complex 0 b))"
huffman@44291
   611
    unfolding exp_def cis_def sin_def
huffman@44291
   612
    by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],
huffman@44291
   613
      simp add: * mult_assoc [symmetric])
huffman@44291
   614
qed
huffman@44291
   615
huffman@44291
   616
lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"
huffman@44712
   617
  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp
huffman@20557
   618
paulson@14374
   619
lemma complex_split_polar:
paulson@14377
   620
     "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
huffman@20725
   621
apply (induct z)
paulson@14377
   622
apply (auto simp add: polar_Ex complex_of_real_mult_Complex)
paulson@14323
   623
done
paulson@14323
   624
paulson@14354
   625
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
huffman@20725
   626
apply (induct z)
paulson@14377
   627
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex)
paulson@14323
   628
done
paulson@14323
   629
paulson@14374
   630
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
huffman@44724
   631
  by (simp add: rcis_def cis_def)
paulson@14323
   632
paulson@14348
   633
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
huffman@44724
   634
  by (simp add: rcis_def cis_def)
paulson@14323
   635
paulson@14377
   636
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>"
paulson@14377
   637
proof -
paulson@14377
   638
  have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)"
huffman@20725
   639
    by (simp only: power_mult_distrib right_distrib)
paulson@14377
   640
  thus ?thesis by simp
paulson@14377
   641
qed
paulson@14323
   642
paulson@14374
   643
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
huffman@44724
   644
  by (simp add: rcis_def cis_def sin_cos_squared_add2_mult)
paulson@14323
   645
huffman@23125
   646
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
huffman@44724
   647
  by (simp add: cmod_def power2_eq_square)
huffman@23125
   648
paulson@14374
   649
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
huffman@44724
   650
  by simp
paulson@14323
   651
paulson@14323
   652
lemma cis_rcis_eq: "cis a = rcis 1 a"
huffman@44724
   653
  by (simp add: rcis_def)
paulson@14323
   654
paulson@14374
   655
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
huffman@44724
   656
  by (simp add: rcis_def cis_def cos_add sin_add right_distrib
huffman@44724
   657
    right_diff_distrib complex_of_real_def)
paulson@14323
   658
paulson@14323
   659
lemma cis_mult: "cis a * cis b = cis (a + b)"
huffman@44724
   660
  by (simp add: cis_rcis_eq rcis_mult)
paulson@14323
   661
paulson@14374
   662
lemma cis_zero [simp]: "cis 0 = 1"
huffman@44724
   663
  by (simp add: cis_def complex_one_def)
paulson@14323
   664
paulson@14374
   665
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
huffman@44724
   666
  by (simp add: rcis_def)
paulson@14323
   667
paulson@14374
   668
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
huffman@44724
   669
  by (simp add: rcis_def)
paulson@14323
   670
paulson@14323
   671
lemma complex_of_real_minus_one:
paulson@14323
   672
   "complex_of_real (-(1::real)) = -(1::complex)"
huffman@44724
   673
  by (simp add: complex_of_real_def complex_one_def)
paulson@14323
   674
paulson@14374
   675
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
huffman@44724
   676
  by (simp add: mult_assoc [symmetric])
paulson@14323
   677
paulson@14323
   678
paulson@14323
   679
lemma cis_real_of_nat_Suc_mult:
paulson@14323
   680
   "cis (real (Suc n) * a) = cis a * cis (real n * a)"
huffman@44724
   681
  by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib)
paulson@14323
   682
paulson@14323
   683
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
paulson@14323
   684
apply (induct_tac "n")
paulson@14323
   685
apply (auto simp add: cis_real_of_nat_Suc_mult)
paulson@14323
   686
done
paulson@14323
   687
paulson@14374
   688
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
huffman@44724
   689
  by (simp add: rcis_def power_mult_distrib DeMoivre)
paulson@14323
   690
paulson@14374
   691
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
huffman@44724
   692
  by (simp add: cis_def complex_inverse_complex_split diff_minus)
paulson@14323
   693
paulson@14323
   694
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
huffman@44724
   695
  by (simp add: divide_inverse rcis_def)
paulson@14323
   696
paulson@14323
   697
lemma cis_divide: "cis a / cis b = cis (a - b)"
huffman@44724
   698
  by (simp add: complex_divide_def cis_mult diff_minus)
paulson@14323
   699
paulson@14354
   700
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
paulson@14373
   701
apply (simp add: complex_divide_def)
paulson@14373
   702
apply (case_tac "r2=0", simp)
haftmann@37887
   703
apply (simp add: rcis_inverse rcis_mult diff_minus)
paulson@14323
   704
done
paulson@14323
   705
paulson@14374
   706
lemma Re_cis [simp]: "Re(cis a) = cos a"
huffman@44724
   707
  by (simp add: cis_def)
paulson@14323
   708
paulson@14374
   709
lemma Im_cis [simp]: "Im(cis a) = sin a"
huffman@44724
   710
  by (simp add: cis_def)
paulson@14323
   711
paulson@14323
   712
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
huffman@44724
   713
  by (auto simp add: DeMoivre)
paulson@14323
   714
paulson@14323
   715
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
huffman@44724
   716
  by (auto simp add: DeMoivre)
paulson@14323
   717
paulson@14374
   718
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
paulson@14373
   719
apply (insert rcis_Ex [of z])
huffman@23125
   720
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
paulson@14334
   721
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
paulson@14323
   722
done
paulson@14323
   723
paulson@14387
   724
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
huffman@44724
   725
  by (simp add: expi_def cis_def)
paulson@14387
   726
huffman@44065
   727
text {* Legacy theorem names *}
huffman@44065
   728
huffman@44065
   729
lemmas expand_complex_eq = complex_eq_iff
huffman@44065
   730
lemmas complex_Re_Im_cancel_iff = complex_eq_iff
huffman@44065
   731
lemmas complex_equality = complex_eqI
huffman@44065
   732
paulson@13957
   733
end