src/HOL/Complex.thy
author hoelzl
Wed May 07 12:25:35 2014 +0200 (2014-05-07)
changeset 56889 48a745e1bde7
parent 56541 0e3abadbef39
child 57259 3a448982a74a
permissions -rw-r--r--
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
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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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text {*
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We use the @{text codatatype}-command to define the type of complex numbers. This might look strange
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at first, but allows us to use @{text primcorec} to define complex-functions by defining their
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real and imaginary result separate.
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*}
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codatatype complex = Complex (Re: real) (Im: real)
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lemma complex_surj: "Complex (Re z) (Im z) = z"
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  by (rule complex.collapse)
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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 (rule complex.expand) 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 (auto intro: complex.expand)
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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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primcorec zero_complex where
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  "Re 0 = 0"
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| "Im 0 = 0"
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primcorec plus_complex where
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  "Re (x + y) = Re x + Re y"
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| "Im (x + y) = Im x + Im y"
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primcorec uminus_complex where
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  "Re (- x) = - Re x"
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| "Im (- x) = - Im x"
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primcorec minus_complex where
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  "Re (x - y) = Re x - Re y"
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| "Im (x - y) = Im x - Im y"
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instance
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  by intro_classes (simp_all add: complex_eq_iff)
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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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primcorec one_complex where
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  "Re 1 = 1"
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| "Im 1 = 0"
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primcorec times_complex where
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  "Re (x * y) = Re x * Re y - Im x * Im y"
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| "Im (x * y) = Re x * Im y + Im x * Re y"
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primcorec inverse_complex where
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  "Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"
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| "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"
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definition "x / (y\<Colon>complex) = x * inverse y"
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instance
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  by intro_classes 
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     (simp_all add: complex_eq_iff divide_complex_def
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      distrib_left distrib_right right_diff_distrib left_diff_distrib
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      power2_eq_square add_divide_distrib [symmetric])
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end
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lemma Re_divide: "Re (x / y) = (Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
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  unfolding divide_complex_def by (simp add: add_divide_distrib)
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lemma Im_divide: "Im (x / y) = (Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
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  unfolding divide_complex_def times_complex.sel inverse_complex.sel
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  by (simp_all add: divide_simps)
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lemma Re_power2: "Re (x ^ 2) = (Re x)^2 - (Im x)^2"
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  by (simp add: power2_eq_square)
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lemma Im_power2: "Im (x ^ 2) = 2 * Re x * Im x"
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  by (simp add: power2_eq_square)
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lemma Re_power_real: "Im x = 0 \<Longrightarrow> Re (x ^ n) = Re x ^ n "
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  by (induct n) simp_all
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lemma Im_power_real: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0"
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  by (induct n) simp_all
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subsection {* Scalar Multiplication *}
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instantiation complex :: real_field
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begin
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primcorec scaleR_complex where
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  "Re (scaleR r x) = r * Re x"
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| "Im (scaleR r x) = r * Im x"
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instance
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proof
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  fix a b :: real and x y :: complex
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  show "scaleR a (x + y) = scaleR a x + scaleR a y"
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    by (simp add: complex_eq_iff distrib_left)
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  show "scaleR (a + b) x = scaleR a x + scaleR b x"
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    by (simp add: complex_eq_iff distrib_right)
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  show "scaleR a (scaleR b x) = scaleR (a * b) x"
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    by (simp add: complex_eq_iff mult_assoc)
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  show "scaleR 1 x = x"
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    by (simp add: complex_eq_iff)
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  show "scaleR a x * y = scaleR a (x * y)"
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    by (simp add: complex_eq_iff algebra_simps)
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  show "x * scaleR a y = scaleR a (x * y)"
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    by (simp add: complex_eq_iff algebra_simps)
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qed
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end
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subsection {* Numerals, Arithmetic, and Embedding from Reals *}
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abbreviation complex_of_real :: "real \<Rightarrow> complex"
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  where "complex_of_real \<equiv> of_real"
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declare [[coercion complex_of_real]]
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declare [[coercion "of_int :: int \<Rightarrow> complex"]]
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declare [[coercion "of_nat :: nat \<Rightarrow> complex"]]
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lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
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  by (induct n) simp_all
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lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
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  by (induct n) simp_all
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lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
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  by (cases z rule: int_diff_cases) simp
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lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
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  by (cases z rule: int_diff_cases) simp
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lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"
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  using complex_Re_of_int [of "numeral v"] by simp
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lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"
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  using complex_Im_of_int [of "numeral v"] by simp
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lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
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  by (simp add: 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: of_real_def)
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subsection {* The Complex Number $i$ *}
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primcorec "ii" :: complex  ("\<i>") where
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  "Re ii = 0"
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| "Im ii = 1"
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lemma i_squared [simp]: "ii * ii = -1"
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  by (simp add: complex_eq_iff)
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lemma power2_i [simp]: "ii\<^sup>2 = -1"
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  by (simp add: power2_eq_square)
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lemma inverse_i [simp]: "inverse ii = - ii"
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  by (rule inverse_unique) simp
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lemma divide_i [simp]: "x / ii = - ii * x"
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  by (simp add: divide_complex_def)
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lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
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  by (simp add: mult_assoc [symmetric])
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lemma Complex_eq[simp]: "Complex a b = a + \<i> * b"
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  by (simp add: complex_eq_iff)
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lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
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  by (simp add: complex_eq_iff)
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lemma complex_i_not_one [simp]: "ii \<noteq> 1"
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  by (simp add: complex_eq_iff)
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lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"
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  by (simp add: complex_eq_iff)
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lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> - numeral w"
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  by (simp add: complex_eq_iff)
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lemma complex_split_polar: "\<exists>r a. z = complex_of_real r * (cos a + \<i> * sin a)"
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  by (simp add: complex_eq_iff polar_Ex)
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subsection {* Vector Norm *}
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instantiation complex :: real_normed_field
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begin
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definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
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abbreviation cmod :: "complex \<Rightarrow> real"
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  where "cmod \<equiv> norm"
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definition complex_sgn_def:
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  "sgn x = x /\<^sub>R cmod x"
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definition dist_complex_def:
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  "dist x y = cmod (x - y)"
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definition open_complex_def:
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  "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
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instance proof
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  fix r :: real and x y :: complex and S :: "complex set"
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  show "(norm x = 0) = (x = 0)"
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    by (simp add: norm_complex_def complex_eq_iff)
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  show "norm (x + y) \<le> norm x + norm y"
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    by (simp add: norm_complex_def complex_eq_iff 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 (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] real_sqrt_mult)
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  show "norm (x * y) = norm x * norm y"
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    by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
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qed (rule complex_sgn_def dist_complex_def open_complex_def)+
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end
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lemma norm_ii [simp]: "norm ii = 1"
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  by (simp add: norm_complex_def)
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lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1"
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  by (simp add: norm_complex_def)
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lemma cmod_complex_polar: "cmod (r * (cos a + \<i> * sin a)) = \<bar>r\<bar>"
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  by (simp add: norm_mult cmod_unit_one)
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lemma complex_Re_le_cmod: "Re x \<le> cmod x"
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  unfolding norm_complex_def
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  by (rule real_sqrt_sum_squares_ge1)
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lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
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  by (rule order_trans [OF _ norm_ge_zero]) simp
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lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \<le> cmod a"
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  by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp
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lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
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  by (simp add: norm_complex_def)
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lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
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  by (simp add: norm_complex_def)
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lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>"
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  by (simp add: norm_complex_def)
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lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>"
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  by (simp add: norm_complex_def)
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lemma cmod_power2: "cmod z ^ 2 = (Re z)^2 + (Im z)^2"
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  by (simp add: norm_complex_def)
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lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z = - cmod z"
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  using abs_Re_le_cmod[of z] by auto
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lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0"
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  by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2])
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     (auto simp add: norm_complex_def)
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lemma abs_sqrt_wlog:
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  fixes x::"'a::linordered_idom"
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  assumes "\<And>x::'a. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"
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by (metis abs_ge_zero assms power2_abs)
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lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z"
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  unfolding norm_complex_def
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  apply (rule abs_sqrt_wlog [where x="Re z"])
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  apply (rule abs_sqrt_wlog [where x="Im z"])
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  apply (rule power2_le_imp_le)
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  apply (simp_all add: power2_sum add_commute sum_squares_bound real_sqrt_mult [symmetric])
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  done
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text {* Properties of complex signum. *}
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lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
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  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
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lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
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  by (simp add: complex_sgn_def divide_inverse)
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lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
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  by (simp add: complex_sgn_def divide_inverse)
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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: norm_complex_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: norm_complex_def)
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lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
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lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
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lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]
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lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]
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lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
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lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
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lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]
hoelzl@56381
   319
lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]
hoelzl@56381
   320
lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]
hoelzl@56381
   321
lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]
hoelzl@56381
   322
lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]
hoelzl@56381
   323
lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]
hoelzl@56381
   324
lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]
hoelzl@56381
   325
lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]
hoelzl@56369
   326
huffman@36825
   327
lemma tendsto_Complex [tendsto_intros]:
hoelzl@56889
   328
  "(f ---> a) F \<Longrightarrow> (g ---> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
hoelzl@56889
   329
  by (auto intro!: tendsto_intros)
hoelzl@56369
   330
hoelzl@56369
   331
lemma tendsto_complex_iff:
hoelzl@56369
   332
  "(f ---> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) ---> Re x) F \<and> ((\<lambda>x. Im (f x)) ---> Im x) F)"
hoelzl@56889
   333
proof safe
hoelzl@56889
   334
  assume "((\<lambda>x. Re (f x)) ---> Re x) F" "((\<lambda>x. Im (f x)) ---> Im x) F"
hoelzl@56889
   335
  from tendsto_Complex[OF this] show "(f ---> x) F"
hoelzl@56889
   336
    unfolding complex.collapse .
hoelzl@56889
   337
qed (auto intro: tendsto_intros)
hoelzl@56369
   338
huffman@23123
   339
instance complex :: banach
huffman@23123
   340
proof
huffman@23123
   341
  fix X :: "nat \<Rightarrow> complex"
huffman@23123
   342
  assume X: "Cauchy X"
hoelzl@56889
   343
  then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
hoelzl@56889
   344
    by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)
hoelzl@56889
   345
  then show "convergent X"
hoelzl@56889
   346
    unfolding complex.collapse by (rule convergentI)
huffman@23123
   347
qed
huffman@23123
   348
lp15@56238
   349
declare
hoelzl@56381
   350
  DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]
lp15@56238
   351
huffman@23125
   352
subsection {* Complex Conjugation *}
huffman@23125
   353
hoelzl@56889
   354
primcorec cnj :: "complex \<Rightarrow> complex" where
hoelzl@56889
   355
  "Re (cnj z) = Re z"
hoelzl@56889
   356
| "Im (cnj z) = - Im z"
huffman@23125
   357
huffman@23125
   358
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
huffman@44724
   359
  by (simp add: complex_eq_iff)
huffman@23125
   360
huffman@23125
   361
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
hoelzl@56889
   362
  by (simp add: complex_eq_iff)
huffman@23125
   363
huffman@23125
   364
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
huffman@44724
   365
  by (simp add: complex_eq_iff)
huffman@23125
   366
huffman@23125
   367
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
huffman@44724
   368
  by (simp add: complex_eq_iff)
huffman@23125
   369
hoelzl@56889
   370
lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y"
huffman@44724
   371
  by (simp add: complex_eq_iff)
huffman@23125
   372
hoelzl@56889
   373
lemma cnj_setsum [simp]: "cnj (setsum f s) = (\<Sum>x\<in>s. cnj (f x))"
hoelzl@56889
   374
  by (induct s rule: infinite_finite_induct) auto
hoelzl@56369
   375
hoelzl@56889
   376
lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y"
huffman@44724
   377
  by (simp add: complex_eq_iff)
huffman@23125
   378
hoelzl@56889
   379
lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x"
huffman@44724
   380
  by (simp add: complex_eq_iff)
huffman@23125
   381
huffman@23125
   382
lemma complex_cnj_one [simp]: "cnj 1 = 1"
huffman@44724
   383
  by (simp add: complex_eq_iff)
huffman@23125
   384
hoelzl@56889
   385
lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y"
huffman@44724
   386
  by (simp add: complex_eq_iff)
huffman@23125
   387
hoelzl@56889
   388
lemma cnj_setprod [simp]: "cnj (setprod f s) = (\<Prod>x\<in>s. cnj (f x))"
hoelzl@56889
   389
  by (induct s rule: infinite_finite_induct) auto
hoelzl@56369
   390
hoelzl@56889
   391
lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)"
hoelzl@56889
   392
  by (simp add: complex_eq_iff)
paulson@14323
   393
hoelzl@56889
   394
lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y"
hoelzl@56889
   395
  by (simp add: divide_complex_def)
huffman@23125
   396
hoelzl@56889
   397
lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n"
hoelzl@56889
   398
  by (induct n) simp_all
huffman@23125
   399
huffman@23125
   400
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
huffman@44724
   401
  by (simp add: complex_eq_iff)
huffman@23125
   402
huffman@23125
   403
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
huffman@44724
   404
  by (simp add: complex_eq_iff)
huffman@23125
   405
huffman@47108
   406
lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
huffman@47108
   407
  by (simp add: complex_eq_iff)
huffman@47108
   408
haftmann@54489
   409
lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
huffman@44724
   410
  by (simp add: complex_eq_iff)
huffman@23125
   411
hoelzl@56889
   412
lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)"
huffman@44724
   413
  by (simp add: complex_eq_iff)
huffman@23125
   414
huffman@23125
   415
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
hoelzl@56889
   416
  by (simp add: norm_complex_def)
paulson@14323
   417
huffman@23125
   418
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
huffman@44724
   419
  by (simp add: complex_eq_iff)
huffman@23125
   420
huffman@23125
   421
lemma complex_cnj_i [simp]: "cnj ii = - ii"
huffman@44724
   422
  by (simp add: complex_eq_iff)
huffman@23125
   423
huffman@23125
   424
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
huffman@44724
   425
  by (simp add: complex_eq_iff)
huffman@23125
   426
huffman@23125
   427
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
huffman@44724
   428
  by (simp add: complex_eq_iff)
paulson@14354
   429
wenzelm@53015
   430
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
huffman@44724
   431
  by (simp add: complex_eq_iff power2_eq_square)
huffman@23125
   432
wenzelm@53015
   433
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"
huffman@44724
   434
  by (simp add: norm_mult power2_eq_square)
huffman@23125
   435
huffman@44827
   436
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
hoelzl@56889
   437
  by (simp add: norm_complex_def power2_eq_square)
huffman@44827
   438
huffman@44827
   439
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
huffman@44827
   440
  by simp
huffman@44827
   441
huffman@44290
   442
lemma bounded_linear_cnj: "bounded_linear cnj"
huffman@44127
   443
  using complex_cnj_add complex_cnj_scaleR
huffman@44127
   444
  by (rule bounded_linear_intro [where K=1], simp)
paulson@14354
   445
hoelzl@56381
   446
lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]
hoelzl@56381
   447
lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]
hoelzl@56381
   448
lemmas continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]
hoelzl@56381
   449
lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]
hoelzl@56381
   450
lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]
huffman@44290
   451
hoelzl@56369
   452
lemma lim_cnj: "((\<lambda>x. cnj(f x)) ---> cnj l) F \<longleftrightarrow> (f ---> l) F"
hoelzl@56889
   453
  by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)
hoelzl@56369
   454
hoelzl@56369
   455
lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
hoelzl@56889
   456
  by (simp add: sums_def lim_cnj cnj_setsum [symmetric] del: cnj_setsum)
hoelzl@56369
   457
paulson@14354
   458
lp15@55734
   459
subsection{*Basic Lemmas*}
lp15@55734
   460
lp15@55734
   461
lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"
hoelzl@56889
   462
  by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)
lp15@55734
   463
lp15@55734
   464
lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"
hoelzl@56889
   465
  by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)
lp15@55734
   466
lp15@55734
   467
lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"
hoelzl@56889
   468
by (cases z)
hoelzl@56889
   469
   (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric]
hoelzl@56889
   470
         simp del: of_real_power)
lp15@55734
   471
hoelzl@56889
   472
lemma re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0"
hoelzl@56889
   473
  by (auto simp add: Re_divide)
hoelzl@56889
   474
  
hoelzl@56889
   475
lemma im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0"
hoelzl@56889
   476
  by (auto simp add: Im_divide)
hoelzl@56889
   477
hoelzl@56889
   478
lemma complex_div_gt_0: 
hoelzl@56889
   479
  "(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)"
hoelzl@56889
   480
proof cases
hoelzl@56889
   481
  assume "b = 0" then show ?thesis by auto
lp15@55734
   482
next
hoelzl@56889
   483
  assume "b \<noteq> 0"
hoelzl@56889
   484
  then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2"
hoelzl@56889
   485
    by (simp add: complex_eq_iff sum_power2_gt_zero_iff)
hoelzl@56889
   486
  then show ?thesis
hoelzl@56889
   487
    by (simp add: Re_divide Im_divide zero_less_divide_iff)
lp15@55734
   488
qed
lp15@55734
   489
hoelzl@56889
   490
lemma re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0"
hoelzl@56889
   491
  and im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0"
hoelzl@56889
   492
  using complex_div_gt_0 by auto
lp15@55734
   493
lp15@55734
   494
lemma re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0"
lp15@55734
   495
  by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0)
lp15@55734
   496
lp15@55734
   497
lemma im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0"
lp15@55734
   498
  by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less)
lp15@55734
   499
lp15@55734
   500
lemma re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0"
boehmes@55759
   501
  by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0)
lp15@55734
   502
lp15@55734
   503
lemma im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0"
lp15@55734
   504
  by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff)
lp15@55734
   505
lp15@55734
   506
lemma re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0"
lp15@55734
   507
  by (metis not_le re_complex_div_gt_0)
lp15@55734
   508
lp15@55734
   509
lemma im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0"
lp15@55734
   510
  by (metis im_complex_div_gt_0 not_le)
lp15@55734
   511
hoelzl@56889
   512
lemma Re_setsum[simp]: "Re (setsum f s) = (\<Sum>x\<in>s. Re (f x))"
hoelzl@56369
   513
  by (induct s rule: infinite_finite_induct) auto
lp15@55734
   514
hoelzl@56889
   515
lemma Im_setsum[simp]: "Im (setsum f s) = (\<Sum>x\<in>s. Im(f x))"
hoelzl@56369
   516
  by (induct s rule: infinite_finite_induct) auto
hoelzl@56369
   517
hoelzl@56369
   518
lemma sums_complex_iff: "f sums x \<longleftrightarrow> ((\<lambda>x. Re (f x)) sums Re x) \<and> ((\<lambda>x. Im (f x)) sums Im x)"
hoelzl@56369
   519
  unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..
hoelzl@56369
   520
  
hoelzl@56369
   521
lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and>  summable (\<lambda>x. Im (f x))"
hoelzl@56889
   522
  unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel)
hoelzl@56369
   523
hoelzl@56369
   524
lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f"
hoelzl@56369
   525
  unfolding summable_complex_iff by simp
hoelzl@56369
   526
hoelzl@56369
   527
lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))"
hoelzl@56369
   528
  unfolding summable_complex_iff by blast
hoelzl@56369
   529
hoelzl@56369
   530
lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))"
hoelzl@56369
   531
  unfolding summable_complex_iff by blast
lp15@56217
   532
hoelzl@56889
   533
lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
hoelzl@56889
   534
  by (auto simp: Reals_def complex_eq_iff)
lp15@55734
   535
lp15@55734
   536
lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z"
hoelzl@56889
   537
  by (auto simp: complex_is_Real_iff complex_eq_iff)
lp15@55734
   538
lp15@55734
   539
lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm(z) = abs(Re z)"
hoelzl@56889
   540
  by (simp add: complex_is_Real_iff norm_complex_def)
hoelzl@56369
   541
hoelzl@56369
   542
lemma series_comparison_complex:
hoelzl@56369
   543
  fixes f:: "nat \<Rightarrow> 'a::banach"
hoelzl@56369
   544
  assumes sg: "summable g"
hoelzl@56369
   545
     and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"
hoelzl@56369
   546
     and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"
hoelzl@56369
   547
  shows "summable f"
hoelzl@56369
   548
proof -
hoelzl@56369
   549
  have g: "\<And>n. cmod (g n) = Re (g n)" using assms
hoelzl@56369
   550
    by (metis abs_of_nonneg in_Reals_norm)
hoelzl@56369
   551
  show ?thesis
hoelzl@56369
   552
    apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])
hoelzl@56369
   553
    using sg
hoelzl@56369
   554
    apply (auto simp: summable_def)
hoelzl@56369
   555
    apply (rule_tac x="Re s" in exI)
hoelzl@56369
   556
    apply (auto simp: g sums_Re)
hoelzl@56369
   557
    apply (metis fg g)
hoelzl@56369
   558
    done
hoelzl@56369
   559
qed
lp15@55734
   560
paulson@14323
   561
subsection{*Finally! Polar Form for Complex Numbers*}
paulson@14323
   562
huffman@44827
   563
subsubsection {* $\cos \theta + i \sin \theta$ *}
huffman@20557
   564
hoelzl@56889
   565
primcorec cis :: "real \<Rightarrow> complex" where
hoelzl@56889
   566
  "Re (cis a) = cos a"
hoelzl@56889
   567
| "Im (cis a) = sin a"
huffman@44827
   568
huffman@44827
   569
lemma cis_zero [simp]: "cis 0 = 1"
hoelzl@56889
   570
  by (simp add: complex_eq_iff)
huffman@44827
   571
huffman@44828
   572
lemma norm_cis [simp]: "norm (cis a) = 1"
hoelzl@56889
   573
  by (simp add: norm_complex_def)
huffman@44828
   574
huffman@44828
   575
lemma sgn_cis [simp]: "sgn (cis a) = cis a"
huffman@44828
   576
  by (simp add: sgn_div_norm)
huffman@44828
   577
huffman@44828
   578
lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
huffman@44828
   579
  by (metis norm_cis norm_zero zero_neq_one)
huffman@44828
   580
huffman@44827
   581
lemma cis_mult: "cis a * cis b = cis (a + b)"
hoelzl@56889
   582
  by (simp add: complex_eq_iff cos_add sin_add)
huffman@44827
   583
huffman@44827
   584
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
huffman@44827
   585
  by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
huffman@44827
   586
huffman@44827
   587
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
hoelzl@56889
   588
  by (simp add: complex_eq_iff)
huffman@44827
   589
huffman@44827
   590
lemma cis_divide: "cis a / cis b = cis (a - b)"
hoelzl@56889
   591
  by (simp add: divide_complex_def cis_mult)
huffman@44827
   592
huffman@44827
   593
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
huffman@44827
   594
  by (auto simp add: DeMoivre)
huffman@44827
   595
huffman@44827
   596
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
huffman@44827
   597
  by (auto simp add: DeMoivre)
huffman@44827
   598
hoelzl@56889
   599
lemma cis_pi: "cis pi = -1"
hoelzl@56889
   600
  by (simp add: complex_eq_iff)
hoelzl@56889
   601
huffman@44827
   602
subsubsection {* $r(\cos \theta + i \sin \theta)$ *}
huffman@44715
   603
hoelzl@56889
   604
definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex" where
huffman@20557
   605
  "rcis r a = complex_of_real r * cis a"
huffman@20557
   606
huffman@44827
   607
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
huffman@44828
   608
  by (simp add: rcis_def)
huffman@44827
   609
huffman@44827
   610
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
huffman@44828
   611
  by (simp add: rcis_def)
huffman@44827
   612
huffman@44827
   613
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
huffman@44828
   614
  by (simp add: complex_eq_iff polar_Ex)
huffman@44827
   615
huffman@44827
   616
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
huffman@44828
   617
  by (simp add: rcis_def norm_mult)
huffman@44827
   618
huffman@44827
   619
lemma cis_rcis_eq: "cis a = rcis 1 a"
huffman@44827
   620
  by (simp add: rcis_def)
huffman@44827
   621
huffman@44827
   622
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
huffman@44828
   623
  by (simp add: rcis_def cis_mult)
huffman@44827
   624
huffman@44827
   625
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
huffman@44827
   626
  by (simp add: rcis_def)
huffman@44827
   627
huffman@44827
   628
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
huffman@44827
   629
  by (simp add: rcis_def)
huffman@44827
   630
huffman@44828
   631
lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
huffman@44828
   632
  by (simp add: rcis_def)
huffman@44828
   633
huffman@44827
   634
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
huffman@44827
   635
  by (simp add: rcis_def power_mult_distrib DeMoivre)
huffman@44827
   636
huffman@44827
   637
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
huffman@44827
   638
  by (simp add: divide_inverse rcis_def)
huffman@44827
   639
huffman@44827
   640
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
huffman@44828
   641
  by (simp add: rcis_def cis_divide [symmetric])
huffman@44827
   642
huffman@44827
   643
subsubsection {* Complex exponential *}
huffman@44827
   644
huffman@44291
   645
abbreviation expi :: "complex \<Rightarrow> complex"
huffman@44291
   646
  where "expi \<equiv> exp"
huffman@44291
   647
hoelzl@56889
   648
lemma cis_conv_exp: "cis b = exp (\<i> * b)"
hoelzl@56889
   649
proof -
hoelzl@56889
   650
  { fix n :: nat
hoelzl@56889
   651
    have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)"
hoelzl@56889
   652
      by (induct n)
hoelzl@56889
   653
         (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps
hoelzl@56889
   654
                        power2_eq_square real_of_nat_Suc add_nonneg_eq_0_iff
hoelzl@56889
   655
                        real_of_nat_def[symmetric])
hoelzl@56889
   656
    then have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n =
hoelzl@56889
   657
        of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)"
hoelzl@56889
   658
      by (simp add: field_simps) }
hoelzl@56889
   659
  then show ?thesis
hoelzl@56889
   660
    by (auto simp add: cis.ctr exp_def simp del: of_real_mult
hoelzl@56889
   661
             intro!: sums_unique sums_add sums_mult sums_of_real sin_converges cos_converges)
huffman@44291
   662
qed
huffman@44291
   663
hoelzl@56889
   664
lemma expi_def: "expi z = exp (Re z) * cis (Im z)"
hoelzl@56889
   665
  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by (cases z) simp
huffman@20557
   666
huffman@44828
   667
lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
huffman@44828
   668
  unfolding expi_def by simp
huffman@44828
   669
huffman@44828
   670
lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
huffman@44828
   671
  unfolding expi_def by simp
huffman@44828
   672
paulson@14374
   673
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
paulson@14373
   674
apply (insert rcis_Ex [of z])
huffman@23125
   675
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
paulson@14334
   676
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
paulson@14323
   677
done
paulson@14323
   678
paulson@14387
   679
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
hoelzl@56889
   680
  by (simp add: expi_def complex_eq_iff)
paulson@14387
   681
huffman@44844
   682
subsubsection {* Complex argument *}
huffman@44844
   683
huffman@44844
   684
definition arg :: "complex \<Rightarrow> real" where
huffman@44844
   685
  "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"
huffman@44844
   686
huffman@44844
   687
lemma arg_zero: "arg 0 = 0"
huffman@44844
   688
  by (simp add: arg_def)
huffman@44844
   689
huffman@44844
   690
lemma arg_unique:
huffman@44844
   691
  assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
huffman@44844
   692
  shows "arg z = x"
huffman@44844
   693
proof -
huffman@44844
   694
  from assms have "z \<noteq> 0" by auto
huffman@44844
   695
  have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
huffman@44844
   696
  proof
huffman@44844
   697
    fix a def d \<equiv> "a - x"
huffman@44844
   698
    assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
huffman@44844
   699
    from a assms have "- (2*pi) < d \<and> d < 2*pi"
huffman@44844
   700
      unfolding d_def by simp
huffman@44844
   701
    moreover from a assms have "cos a = cos x" and "sin a = sin x"
huffman@44844
   702
      by (simp_all add: complex_eq_iff)
wenzelm@53374
   703
    hence cos: "cos d = 1" unfolding d_def cos_diff by simp
wenzelm@53374
   704
    moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)
huffman@44844
   705
    ultimately have "d = 0"
huffman@44844
   706
      unfolding sin_zero_iff even_mult_two_ex
wenzelm@53374
   707
      by (auto simp add: numeral_2_eq_2 less_Suc_eq)
huffman@44844
   708
    thus "a = x" unfolding d_def by simp
huffman@44844
   709
  qed (simp add: assms del: Re_sgn Im_sgn)
huffman@44844
   710
  with `z \<noteq> 0` show "arg z = x"
huffman@44844
   711
    unfolding arg_def by simp
huffman@44844
   712
qed
huffman@44844
   713
huffman@44844
   714
lemma arg_correct:
huffman@44844
   715
  assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
huffman@44844
   716
proof (simp add: arg_def assms, rule someI_ex)
huffman@44844
   717
  obtain r a where z: "z = rcis r a" using rcis_Ex by fast
huffman@44844
   718
  with assms have "r \<noteq> 0" by auto
huffman@44844
   719
  def b \<equiv> "if 0 < r then a else a + pi"
huffman@44844
   720
  have b: "sgn z = cis b"
huffman@44844
   721
    unfolding z b_def rcis_def using `r \<noteq> 0`
hoelzl@56889
   722
    by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff)
huffman@44844
   723
  have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
hoelzl@56889
   724
    by (induct_tac n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)
huffman@44844
   725
  have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
hoelzl@56889
   726
    by (case_tac x rule: int_diff_cases)
hoelzl@56889
   727
       (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
huffman@44844
   728
  def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
huffman@44844
   729
  have "sgn z = cis c"
huffman@44844
   730
    unfolding b c_def
huffman@44844
   731
    by (simp add: cis_divide [symmetric] cis_2pi_int)
huffman@44844
   732
  moreover have "- pi < c \<and> c \<le> pi"
huffman@44844
   733
    using ceiling_correct [of "(b - pi) / (2*pi)"]
huffman@44844
   734
    by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)
huffman@44844
   735
  ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
huffman@44844
   736
qed
huffman@44844
   737
huffman@44844
   738
lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
hoelzl@56889
   739
  by (cases "z = 0") (simp_all add: arg_zero arg_correct)
huffman@44844
   740
huffman@44844
   741
lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
huffman@44844
   742
  by (simp add: arg_correct)
huffman@44844
   743
huffman@44844
   744
lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
hoelzl@56889
   745
  by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
hoelzl@56889
   746
hoelzl@56889
   747
lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0"
hoelzl@56889
   748
  using cis_arg [of y] by (simp add: complex_eq_iff)
hoelzl@56889
   749
hoelzl@56889
   750
subsection {* Square root of complex numbers *}
hoelzl@56889
   751
hoelzl@56889
   752
primcorec csqrt :: "complex \<Rightarrow> complex" where
hoelzl@56889
   753
  "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"
hoelzl@56889
   754
| "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)"
hoelzl@56889
   755
hoelzl@56889
   756
lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)"
hoelzl@56889
   757
  by (simp add: complex_eq_iff norm_complex_def)
hoelzl@56889
   758
hoelzl@56889
   759
lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>"
hoelzl@56889
   760
  by (simp add: complex_eq_iff norm_complex_def)
hoelzl@56889
   761
hoelzl@56889
   762
lemma csqrt_0 [simp]: "csqrt 0 = 0"
hoelzl@56889
   763
  by simp
hoelzl@56889
   764
hoelzl@56889
   765
lemma csqrt_1 [simp]: "csqrt 1 = 1"
hoelzl@56889
   766
  by simp
hoelzl@56889
   767
hoelzl@56889
   768
lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2"
hoelzl@56889
   769
  by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt)
huffman@44844
   770
hoelzl@56889
   771
lemma power2_csqrt[algebra]: "(csqrt z)\<^sup>2 = z"
hoelzl@56889
   772
proof cases
hoelzl@56889
   773
  assume "Im z = 0" then show ?thesis
hoelzl@56889
   774
    using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"]
hoelzl@56889
   775
    by (cases "0::real" "Re z" rule: linorder_cases)
hoelzl@56889
   776
       (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)
hoelzl@56889
   777
next
hoelzl@56889
   778
  assume "Im z \<noteq> 0"
hoelzl@56889
   779
  moreover
hoelzl@56889
   780
  have "cmod z * cmod z - Re z * Re z = Im z * Im z"
hoelzl@56889
   781
    by (simp add: norm_complex_def power2_eq_square)
hoelzl@56889
   782
  moreover
hoelzl@56889
   783
  have "\<bar>Re z\<bar> \<le> cmod z"
hoelzl@56889
   784
    by (simp add: norm_complex_def)
hoelzl@56889
   785
  ultimately show ?thesis
hoelzl@56889
   786
    by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq
hoelzl@56889
   787
                  field_simps real_sqrt_mult[symmetric] real_sqrt_divide)
hoelzl@56889
   788
qed
hoelzl@56889
   789
hoelzl@56889
   790
lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
hoelzl@56889
   791
  by auto (metis power2_csqrt power_eq_0_iff)
hoelzl@56889
   792
hoelzl@56889
   793
lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
hoelzl@56889
   794
  by auto (metis power2_csqrt power2_eq_1_iff)
hoelzl@56889
   795
hoelzl@56889
   796
lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)"
hoelzl@56889
   797
  by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)
hoelzl@56889
   798
hoelzl@56889
   799
lemma Re_csqrt: "0 \<le> Re (csqrt z)"
hoelzl@56889
   800
  by (metis csqrt_principal le_less)
hoelzl@56889
   801
hoelzl@56889
   802
lemma csqrt_square:
hoelzl@56889
   803
  assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)"
hoelzl@56889
   804
  shows "csqrt (b^2) = b"
hoelzl@56889
   805
proof -
hoelzl@56889
   806
  have "csqrt (b^2) = b \<or> csqrt (b^2) = - b"
hoelzl@56889
   807
    unfolding power2_eq_iff[symmetric] by (simp add: power2_csqrt)
hoelzl@56889
   808
  moreover have "csqrt (b^2) \<noteq> -b \<or> b = 0"
hoelzl@56889
   809
    using csqrt_principal[of "b ^ 2"] assms by (intro disjCI notI) (auto simp: complex_eq_iff)
hoelzl@56889
   810
  ultimately show ?thesis
hoelzl@56889
   811
    by auto
hoelzl@56889
   812
qed
hoelzl@56889
   813
hoelzl@56889
   814
lemma csqrt_minus [simp]: 
hoelzl@56889
   815
  assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)"
hoelzl@56889
   816
  shows "csqrt (- x) = \<i> * csqrt x"
hoelzl@56889
   817
proof -
hoelzl@56889
   818
  have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x"
hoelzl@56889
   819
  proof (rule csqrt_square)
hoelzl@56889
   820
    have "Im (csqrt x) \<le> 0"
hoelzl@56889
   821
      using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod)
hoelzl@56889
   822
    then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)"
hoelzl@56889
   823
      by (auto simp add: Re_csqrt simp del: csqrt.simps)
hoelzl@56889
   824
  qed
hoelzl@56889
   825
  also have "(\<i> * csqrt x)^2 = - x"
hoelzl@56889
   826
    by (simp add: power2_csqrt power_mult_distrib)
hoelzl@56889
   827
  finally show ?thesis .
hoelzl@56889
   828
qed
huffman@44844
   829
huffman@44065
   830
text {* Legacy theorem names *}
huffman@44065
   831
huffman@44065
   832
lemmas expand_complex_eq = complex_eq_iff
huffman@44065
   833
lemmas complex_Re_Im_cancel_iff = complex_eq_iff
huffman@44065
   834
lemmas complex_equality = complex_eqI
hoelzl@56889
   835
lemmas cmod_def = norm_complex_def
hoelzl@56889
   836
lemmas complex_norm_def = norm_complex_def
hoelzl@56889
   837
lemmas complex_divide_def = divide_complex_def
hoelzl@56889
   838
hoelzl@56889
   839
lemma legacy_Complex_simps:
hoelzl@56889
   840
  shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
hoelzl@56889
   841
    and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"
hoelzl@56889
   842
    and complex_minus: "- (Complex a b) = Complex (- a) (- b)"
hoelzl@56889
   843
    and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"
hoelzl@56889
   844
    and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"
hoelzl@56889
   845
    and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"
hoelzl@56889
   846
    and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
hoelzl@56889
   847
    and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"
hoelzl@56889
   848
    and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"
hoelzl@56889
   849
    and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"
hoelzl@56889
   850
    and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"
hoelzl@56889
   851
    and Complex_eq_i: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
hoelzl@56889
   852
    and i_mult_Complex: "ii * Complex a b = Complex (- b) a"
hoelzl@56889
   853
    and Complex_mult_i: "Complex a b * ii = Complex (- b) a"
hoelzl@56889
   854
    and i_complex_of_real: "ii * complex_of_real r = Complex 0 r"
hoelzl@56889
   855
    and complex_of_real_i: "complex_of_real r * ii = Complex 0 r"
hoelzl@56889
   856
    and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"
hoelzl@56889
   857
    and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"
hoelzl@56889
   858
    and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
hoelzl@56889
   859
    and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
hoelzl@56889
   860
    and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
hoelzl@56889
   861
    and complex_cn: "cnj (Complex a b) = Complex a (- b)"
hoelzl@56889
   862
    and Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"
hoelzl@56889
   863
    and Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"
hoelzl@56889
   864
    and complex_of_real_def: "complex_of_real r = Complex r 0"
hoelzl@56889
   865
    and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"
hoelzl@56889
   866
  by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide del: Complex_eq)
hoelzl@56889
   867
hoelzl@56889
   868
lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"
hoelzl@56889
   869
  by (metis Reals_of_real complex_of_real_def)
huffman@44065
   870
paulson@13957
   871
end