src/HOL/Complex.thy
author wenzelm
Tue Sep 26 20:54:40 2017 +0200 (22 months ago)
changeset 66695 91500c024c7f
parent 65583 8d53b3bebab4
child 66793 deabce3ccf1f
permissions -rw-r--r--
tuned;
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(*  Title:       HOL/Complex.thy
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    Author:      Jacques D. Fleuriot, 2001 University of Edinburgh
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    Author:      Lawrence C Paulson, 2003/4
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*)
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section \<open>Complex Numbers: Rectangular and Polar Representations\<close>
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theory Complex
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imports Transcendental
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begin
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text \<open>
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  We use the \<^theory_text>\<open>codatatype\<close> command to define the type of complex numbers. This
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  allows us to use \<^theory_text>\<open>primcorec\<close> to define complex functions by defining their
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  real and imaginary result separately.
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\<close>
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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?]: "Re x = Re y \<Longrightarrow> Im x = Im y \<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 \<open>Addition and Subtraction\<close>
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instantiation complex :: ab_group_add
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begin
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primcorec zero_complex
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  where
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    "Re 0 = 0"
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  | "Im 0 = 0"
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primcorec plus_complex
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  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
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  where
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    "Re (- x) = - Re x"
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  | "Im (- x) = - Im x"
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primcorec minus_complex
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  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 standard (simp_all add: complex_eq_iff)
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end
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subsection \<open>Multiplication and Division\<close>
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instantiation complex :: field
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begin
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primcorec one_complex
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  where
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    "Re 1 = 1"
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  | "Im 1 = 0"
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primcorec times_complex
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  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
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  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 div y = x * inverse y" for x y :: complex
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instance
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  by standard
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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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  by (simp add: divide_complex_def 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 add: divide_simps)
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lemma Complex_divide:
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    "(x / y) = Complex ((Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2))
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                       ((Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2))"
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  by (metis Im_divide Re_divide complex_surj)
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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 [simp]: "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 [simp]: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0"
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  by (induct n) simp_all
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subsection \<open>Scalar Multiplication\<close>
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instantiation complex :: real_field
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begin
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primcorec scaleR_complex
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  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 \<open>Numerals, Arithmetic, and Embedding from R\<close>
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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 "of_real :: real \<Rightarrow> complex"]]
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declare [[coercion "of_rat :: rat \<Rightarrow> complex"]]
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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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lemma Re_divide_numeral [simp]: "Re (z / numeral w) = Re z / numeral w"
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  by (simp add: Re_divide sqr_conv_mult)
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lemma Im_divide_numeral [simp]: "Im (z / numeral w) = Im z / numeral w"
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  by (simp add: Im_divide sqr_conv_mult)
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lemma Re_divide_of_nat [simp]: "Re (z / of_nat n) = Re z / of_nat n"
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  by (cases n) (simp_all add: Re_divide divide_simps power2_eq_square del: of_nat_Suc)
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lemma Im_divide_of_nat [simp]: "Im (z / of_nat n) = Im z / of_nat n"
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  by (cases n) (simp_all add: Im_divide divide_simps power2_eq_square del: of_nat_Suc)
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lemma of_real_Re [simp]: "z \<in> \<real> \<Longrightarrow> of_real (Re z) = z"
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  by (auto simp: Reals_def)
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lemma complex_Re_fact [simp]: "Re (fact n) = fact n"
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proof -
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  have "(fact n :: complex) = of_real (fact n)"
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    by simp
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  also have "Re \<dots> = fact n"
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    by (subst Re_complex_of_real) simp_all
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  finally show ?thesis .
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qed
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lemma complex_Im_fact [simp]: "Im (fact n) = 0"
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  by (subst of_nat_fact [symmetric]) (simp only: complex_Im_of_nat)
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subsection \<open>The Complex Number $i$\<close>
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primcorec imaginary_unit :: complex  ("\<i>")
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  where
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    "Re \<i> = 0"
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  | "Im \<i> = 1"
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lemma Complex_eq: "Complex a b = a + \<i> * b"
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  by (simp add: complex_eq_iff)
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lemma complex_eq: "a = Re a + \<i> * Im a"
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  by (simp add: complex_eq_iff)
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lemma fun_complex_eq: "f = (\<lambda>x. Re (f x) + \<i> * Im (f x))"
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  by (simp add: fun_eq_iff complex_eq)
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lemma i_squared [simp]: "\<i> * \<i> = -1"
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  by (simp add: complex_eq_iff)
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lemma power2_i [simp]: "\<i>\<^sup>2 = -1"
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  by (simp add: power2_eq_square)
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lemma inverse_i [simp]: "inverse \<i> = - \<i>"
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  by (rule inverse_unique) simp
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lemma divide_i [simp]: "x / \<i> = - \<i> * x"
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  by (simp add: divide_complex_def)
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lemma complex_i_mult_minus [simp]: "\<i> * (\<i> * x) = - x"
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  by (simp add: mult.assoc [symmetric])
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lemma complex_i_not_zero [simp]: "\<i> \<noteq> 0"
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  by (simp add: complex_eq_iff)
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lemma complex_i_not_one [simp]: "\<i> \<noteq> 1"
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  by (simp add: complex_eq_iff)
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lemma complex_i_not_numeral [simp]: "\<i> \<noteq> numeral w"
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  by (simp add: complex_eq_iff)
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lemma complex_i_not_neg_numeral [simp]: "\<i> \<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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lemma i_even_power [simp]: "\<i> ^ (n * 2) = (-1) ^ n"
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  by (metis mult.commute power2_i power_mult)
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lemma Re_i_times [simp]: "Re (\<i> * z) = - Im z"
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  by simp
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lemma Im_i_times [simp]: "Im (\<i> * z) = Re z"
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  by simp
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lemma i_times_eq_iff: "\<i> * w = z \<longleftrightarrow> w = - (\<i> * z)"
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  by auto
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lemma divide_numeral_i [simp]: "z / (numeral n * \<i>) = - (\<i> * z) / numeral n"
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  by (metis divide_divide_eq_left divide_i mult.commute mult_minus_right)
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lemma imaginary_eq_real_iff [simp]:
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  assumes "y \<in> Reals" "x \<in> Reals"
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  shows "\<i> * y = x \<longleftrightarrow> x=0 \<and> y=0"
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    using assms
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    unfolding Reals_def
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    apply clarify
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      apply (rule iffI)
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    apply (metis Im_complex_of_real Im_i_times Re_complex_of_real mult_eq_0_iff of_real_0)
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    by simp
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lemma real_eq_imaginary_iff [simp]:
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  assumes "y \<in> Reals" "x \<in> Reals"
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  shows "x = \<i> * y  \<longleftrightarrow> x=0 \<and> y=0"
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    using assms imaginary_eq_real_iff by fastforce
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subsection \<open>Vector Norm\<close>
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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: "sgn x = x /\<^sub>R cmod x"
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definition dist_complex_def: "dist x y = cmod (x - y)"
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definition uniformity_complex_def [code del]:
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  "(uniformity :: (complex \<times> complex) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})"
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definition open_complex_def [code del]:
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  "open (U :: complex set) \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"
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instance
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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]
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        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]
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        power2_eq_square algebra_simps)
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qed (rule complex_sgn_def dist_complex_def open_complex_def uniformity_complex_def)+
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end
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declare uniformity_Abort[where 'a = complex, code]
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lemma norm_ii [simp]: "norm \<i> = 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>"
hoelzl@56889
   330
  by (simp add: norm_mult cmod_unit_one)
huffman@22861
   331
huffman@22861
   332
lemma complex_Re_le_cmod: "Re x \<le> cmod x"
wenzelm@63569
   333
  unfolding norm_complex_def by (rule real_sqrt_sum_squares_ge1)
huffman@22861
   334
huffman@44761
   335
lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
hoelzl@56889
   336
  by (rule order_trans [OF _ norm_ge_zero]) simp
huffman@22861
   337
hoelzl@56889
   338
lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \<le> cmod a"
hoelzl@56889
   339
  by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp
paulson@14323
   340
chaieb@26117
   341
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
hoelzl@56889
   342
  by (simp add: norm_complex_def)
chaieb@26117
   343
chaieb@26117
   344
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
hoelzl@56889
   345
  by (simp add: norm_complex_def)
hoelzl@56889
   346
hoelzl@57259
   347
lemma cmod_le: "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>"
hoelzl@57259
   348
  apply (subst complex_eq)
hoelzl@57259
   349
  apply (rule order_trans)
wenzelm@63569
   350
   apply (rule norm_triangle_ineq)
hoelzl@57259
   351
  apply (simp add: norm_mult)
hoelzl@57259
   352
  done
hoelzl@57259
   353
hoelzl@56889
   354
lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>"
hoelzl@56889
   355
  by (simp add: norm_complex_def)
hoelzl@56889
   356
hoelzl@56889
   357
lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>"
hoelzl@56889
   358
  by (simp add: norm_complex_def)
huffman@44724
   359
wenzelm@63569
   360
lemma cmod_power2: "(cmod z)\<^sup>2 = (Re z)\<^sup>2 + (Im z)\<^sup>2"
hoelzl@56889
   361
  by (simp add: norm_complex_def)
hoelzl@56889
   362
hoelzl@56889
   363
lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z = - cmod z"
hoelzl@56889
   364
  using abs_Re_le_cmod[of z] by auto
hoelzl@56889
   365
wenzelm@63569
   366
lemma cmod_Re_le_iff: "Im x = Im y \<Longrightarrow> cmod x \<le> cmod y \<longleftrightarrow> \<bar>Re x\<bar> \<le> \<bar>Re y\<bar>"
lp15@62379
   367
  by (metis add.commute add_le_cancel_left norm_complex_def real_sqrt_abs real_sqrt_le_iff)
lp15@62379
   368
wenzelm@63569
   369
lemma cmod_Im_le_iff: "Re x = Re y \<Longrightarrow> cmod x \<le> cmod y \<longleftrightarrow> \<bar>Im x\<bar> \<le> \<bar>Im y\<bar>"
lp15@62379
   370
  by (metis add_le_cancel_left norm_complex_def real_sqrt_abs real_sqrt_le_iff)
lp15@62379
   371
hoelzl@56889
   372
lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0"
wenzelm@63569
   373
  by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2]) (auto simp add: norm_complex_def)
hoelzl@56369
   374
wenzelm@63569
   375
lemma abs_sqrt_wlog: "(\<And>x. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)) \<Longrightarrow> P \<bar>x\<bar> (x\<^sup>2)"
wenzelm@63569
   376
  for x::"'a::linordered_idom"
wenzelm@63569
   377
  by (metis abs_ge_zero power2_abs)
hoelzl@56369
   378
hoelzl@56369
   379
lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z"
hoelzl@56889
   380
  unfolding norm_complex_def
hoelzl@56369
   381
  apply (rule abs_sqrt_wlog [where x="Re z"])
hoelzl@56369
   382
  apply (rule abs_sqrt_wlog [where x="Im z"])
hoelzl@56369
   383
  apply (rule power2_le_imp_le)
wenzelm@63569
   384
   apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])
hoelzl@56369
   385
  done
hoelzl@56369
   386
lp15@59741
   387
lemma complex_unit_circle: "z \<noteq> 0 \<Longrightarrow> (Re z / cmod z)\<^sup>2 + (Im z / cmod z)\<^sup>2 = 1"
lp15@59741
   388
  by (simp add: norm_complex_def divide_simps complex_eq_iff)
lp15@59741
   389
hoelzl@56369
   390
wenzelm@60758
   391
text \<open>Properties of complex signum.\<close>
huffman@44843
   392
huffman@44843
   393
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
haftmann@57512
   394
  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute)
huffman@44843
   395
huffman@44843
   396
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
huffman@44843
   397
  by (simp add: complex_sgn_def divide_inverse)
huffman@44843
   398
huffman@44843
   399
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
huffman@44843
   400
  by (simp add: complex_sgn_def divide_inverse)
huffman@44843
   401
paulson@14354
   402
haftmann@64290
   403
subsection \<open>Absolute value\<close>
haftmann@64290
   404
haftmann@64290
   405
instantiation complex :: field_abs_sgn
haftmann@64290
   406
begin
haftmann@64290
   407
haftmann@64290
   408
definition abs_complex :: "complex \<Rightarrow> complex"
haftmann@64290
   409
  where "abs_complex = of_real \<circ> norm"
haftmann@64290
   410
haftmann@64290
   411
instance
haftmann@64290
   412
  apply standard
haftmann@64290
   413
         apply (auto simp add: abs_complex_def complex_sgn_def norm_mult)
haftmann@64290
   414
  apply (auto simp add: scaleR_conv_of_real field_simps)
haftmann@64290
   415
  done
haftmann@64290
   416
haftmann@64290
   417
end
haftmann@64290
   418
haftmann@64290
   419
wenzelm@60758
   420
subsection \<open>Completeness of the Complexes\<close>
huffman@23123
   421
huffman@44290
   422
lemma bounded_linear_Re: "bounded_linear Re"
wenzelm@63569
   423
  by (rule bounded_linear_intro [where K=1]) (simp_all add: norm_complex_def)
huffman@44290
   424
huffman@44290
   425
lemma bounded_linear_Im: "bounded_linear Im"
wenzelm@63569
   426
  by (rule bounded_linear_intro [where K=1]) (simp_all add: norm_complex_def)
huffman@23123
   427
huffman@44290
   428
lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
huffman@44290
   429
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
hoelzl@56381
   430
lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]
hoelzl@56381
   431
lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]
hoelzl@56381
   432
lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
hoelzl@56381
   433
lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
hoelzl@56381
   434
lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]
hoelzl@56381
   435
lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]
hoelzl@56381
   436
lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]
hoelzl@56381
   437
lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]
hoelzl@56381
   438
lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]
hoelzl@56381
   439
lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]
hoelzl@56381
   440
lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]
hoelzl@56381
   441
lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]
hoelzl@56369
   442
huffman@36825
   443
lemma tendsto_Complex [tendsto_intros]:
wenzelm@61973
   444
  "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) \<longlongrightarrow> Complex a b) F"
lp15@65274
   445
  unfolding Complex_eq by (auto intro!: tendsto_intros)
hoelzl@56369
   446
hoelzl@56369
   447
lemma tendsto_complex_iff:
wenzelm@61973
   448
  "(f \<longlongrightarrow> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F \<and> ((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F)"
hoelzl@56889
   449
proof safe
wenzelm@61973
   450
  assume "((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F" "((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F"
wenzelm@61973
   451
  from tendsto_Complex[OF this] show "(f \<longlongrightarrow> x) F"
hoelzl@56889
   452
    unfolding complex.collapse .
hoelzl@56889
   453
qed (auto intro: tendsto_intros)
hoelzl@56369
   454
wenzelm@63569
   455
lemma continuous_complex_iff:
wenzelm@63569
   456
  "continuous F f \<longleftrightarrow> continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x))"
wenzelm@63569
   457
  by (simp only: continuous_def tendsto_complex_iff)
hoelzl@57259
   458
lp15@64773
   459
lemma continuous_on_of_real_o_iff [simp]:
lp15@64773
   460
     "continuous_on S (\<lambda>x. complex_of_real (g x)) = continuous_on S g"
lp15@64773
   461
  using continuous_on_Re continuous_on_of_real  by fastforce
lp15@64773
   462
lp15@64773
   463
lemma continuous_on_of_real_id [simp]:
lp15@64773
   464
     "continuous_on S (of_real :: real \<Rightarrow> 'a::real_normed_algebra_1)"
lp15@64773
   465
  by (rule continuous_on_of_real [OF continuous_on_id])
lp15@64773
   466
hoelzl@57259
   467
lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \<longleftrightarrow>
hoelzl@57259
   468
    ((\<lambda>x. Re (f x)) has_field_derivative (Re x)) F \<and>
hoelzl@57259
   469
    ((\<lambda>x. Im (f x)) has_field_derivative (Im x)) F"
wenzelm@63569
   470
  by (simp add: has_vector_derivative_def has_field_derivative_def has_derivative_def
wenzelm@63569
   471
      tendsto_complex_iff field_simps bounded_linear_scaleR_left bounded_linear_mult_right)
hoelzl@57259
   472
hoelzl@57259
   473
lemma has_field_derivative_Re[derivative_intros]:
hoelzl@57259
   474
  "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Re (f x)) has_field_derivative (Re D)) F"
hoelzl@57259
   475
  unfolding has_vector_derivative_complex_iff by safe
hoelzl@57259
   476
hoelzl@57259
   477
lemma has_field_derivative_Im[derivative_intros]:
hoelzl@57259
   478
  "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Im (f x)) has_field_derivative (Im D)) F"
hoelzl@57259
   479
  unfolding has_vector_derivative_complex_iff by safe
hoelzl@57259
   480
huffman@23123
   481
instance complex :: banach
huffman@23123
   482
proof
huffman@23123
   483
  fix X :: "nat \<Rightarrow> complex"
huffman@23123
   484
  assume X: "Cauchy X"
wenzelm@63569
   485
  then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) \<longlonglongrightarrow>
wenzelm@63569
   486
    Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
wenzelm@63569
   487
    by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1]
wenzelm@63569
   488
        Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)
hoelzl@56889
   489
  then show "convergent X"
hoelzl@56889
   490
    unfolding complex.collapse by (rule convergentI)
huffman@23123
   491
qed
huffman@23123
   492
wenzelm@63569
   493
declare DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]
wenzelm@63569
   494
lp15@56238
   495
wenzelm@60758
   496
subsection \<open>Complex Conjugation\<close>
huffman@23125
   497
wenzelm@63569
   498
primcorec cnj :: "complex \<Rightarrow> complex"
wenzelm@63569
   499
  where
wenzelm@63569
   500
    "Re (cnj z) = Re z"
wenzelm@63569
   501
  | "Im (cnj z) = - Im z"
huffman@23125
   502
wenzelm@63569
   503
lemma complex_cnj_cancel_iff [simp]: "cnj x = cnj y \<longleftrightarrow> x = y"
huffman@44724
   504
  by (simp add: complex_eq_iff)
huffman@23125
   505
huffman@23125
   506
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
hoelzl@56889
   507
  by (simp add: complex_eq_iff)
huffman@23125
   508
huffman@23125
   509
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
huffman@44724
   510
  by (simp add: complex_eq_iff)
huffman@23125
   511
wenzelm@63569
   512
lemma complex_cnj_zero_iff [iff]: "cnj z = 0 \<longleftrightarrow> z = 0"
huffman@44724
   513
  by (simp add: complex_eq_iff)
huffman@23125
   514
hoelzl@56889
   515
lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y"
huffman@44724
   516
  by (simp add: complex_eq_iff)
huffman@23125
   517
nipkow@64267
   518
lemma cnj_sum [simp]: "cnj (sum f s) = (\<Sum>x\<in>s. cnj (f x))"
hoelzl@56889
   519
  by (induct s rule: infinite_finite_induct) auto
hoelzl@56369
   520
hoelzl@56889
   521
lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y"
huffman@44724
   522
  by (simp add: complex_eq_iff)
huffman@23125
   523
hoelzl@56889
   524
lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x"
huffman@44724
   525
  by (simp add: complex_eq_iff)
huffman@23125
   526
huffman@23125
   527
lemma complex_cnj_one [simp]: "cnj 1 = 1"
huffman@44724
   528
  by (simp add: complex_eq_iff)
huffman@23125
   529
hoelzl@56889
   530
lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y"
huffman@44724
   531
  by (simp add: complex_eq_iff)
huffman@23125
   532
nipkow@64272
   533
lemma cnj_prod [simp]: "cnj (prod f s) = (\<Prod>x\<in>s. cnj (f x))"
hoelzl@56889
   534
  by (induct s rule: infinite_finite_induct) auto
hoelzl@56369
   535
hoelzl@56889
   536
lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)"
hoelzl@56889
   537
  by (simp add: complex_eq_iff)
paulson@14323
   538
hoelzl@56889
   539
lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y"
hoelzl@56889
   540
  by (simp add: divide_complex_def)
huffman@23125
   541
hoelzl@56889
   542
lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n"
hoelzl@56889
   543
  by (induct n) simp_all
huffman@23125
   544
huffman@23125
   545
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
huffman@44724
   546
  by (simp add: complex_eq_iff)
huffman@23125
   547
huffman@23125
   548
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
huffman@44724
   549
  by (simp add: complex_eq_iff)
huffman@23125
   550
huffman@47108
   551
lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
huffman@47108
   552
  by (simp add: complex_eq_iff)
huffman@47108
   553
haftmann@54489
   554
lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
huffman@44724
   555
  by (simp add: complex_eq_iff)
huffman@23125
   556
hoelzl@56889
   557
lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)"
huffman@44724
   558
  by (simp add: complex_eq_iff)
huffman@23125
   559
huffman@23125
   560
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
hoelzl@56889
   561
  by (simp add: norm_complex_def)
paulson@14323
   562
huffman@23125
   563
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
huffman@44724
   564
  by (simp add: complex_eq_iff)
huffman@23125
   565
wenzelm@63569
   566
lemma complex_cnj_i [simp]: "cnj \<i> = - \<i>"
huffman@44724
   567
  by (simp add: complex_eq_iff)
huffman@23125
   568
huffman@23125
   569
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
huffman@44724
   570
  by (simp add: complex_eq_iff)
huffman@23125
   571
wenzelm@63569
   572
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * \<i>"
huffman@44724
   573
  by (simp add: complex_eq_iff)
paulson@14354
   574
wenzelm@53015
   575
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
huffman@44724
   576
  by (simp add: complex_eq_iff power2_eq_square)
huffman@23125
   577
wenzelm@53015
   578
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"
huffman@44724
   579
  by (simp add: norm_mult power2_eq_square)
huffman@23125
   580
huffman@44827
   581
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
hoelzl@56889
   582
  by (simp add: norm_complex_def power2_eq_square)
huffman@44827
   583
huffman@44827
   584
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
huffman@44827
   585
  by simp
huffman@44827
   586
eberlm@61531
   587
lemma complex_cnj_fact [simp]: "cnj (fact n) = fact n"
eberlm@61531
   588
  by (subst of_nat_fact [symmetric], subst complex_cnj_of_nat) simp
eberlm@61531
   589
eberlm@61531
   590
lemma complex_cnj_pochhammer [simp]: "cnj (pochhammer z n) = pochhammer (cnj z) n"
wenzelm@63569
   591
  by (induct n arbitrary: z) (simp_all add: pochhammer_rec)
eberlm@61531
   592
huffman@44290
   593
lemma bounded_linear_cnj: "bounded_linear cnj"
wenzelm@63569
   594
  using complex_cnj_add complex_cnj_scaleR by (rule bounded_linear_intro [where K=1]) simp
paulson@14354
   595
hoelzl@56381
   596
lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]
wenzelm@63569
   597
  and isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]
wenzelm@63569
   598
  and continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]
wenzelm@63569
   599
  and continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]
wenzelm@63569
   600
  and has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]
huffman@44290
   601
wenzelm@61973
   602
lemma lim_cnj: "((\<lambda>x. cnj(f x)) \<longlongrightarrow> cnj l) F \<longleftrightarrow> (f \<longlongrightarrow> l) F"
hoelzl@56889
   603
  by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)
hoelzl@56369
   604
hoelzl@56369
   605
lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
nipkow@64267
   606
  by (simp add: sums_def lim_cnj cnj_sum [symmetric] del: cnj_sum)
hoelzl@56369
   607
paulson@14354
   608
wenzelm@63569
   609
subsection \<open>Basic Lemmas\<close>
lp15@55734
   610
lp15@55734
   611
lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"
hoelzl@56889
   612
  by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)
lp15@55734
   613
lp15@55734
   614
lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"
hoelzl@56889
   615
  by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)
lp15@55734
   616
lp15@55734
   617
lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"
wenzelm@63569
   618
  by (cases z)
wenzelm@63569
   619
    (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric]
wenzelm@63569
   620
      simp del: of_real_power)
lp15@55734
   621
wenzelm@63569
   622
lemma complex_div_cnj: "a / b = (a * cnj b) / (norm b)\<^sup>2"
paulson@61104
   623
  using complex_norm_square by auto
paulson@61104
   624
lp15@59741
   625
lemma Re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0"
hoelzl@56889
   626
  by (auto simp add: Re_divide)
lp15@59613
   627
lp15@59741
   628
lemma Im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0"
hoelzl@56889
   629
  by (auto simp add: Im_divide)
hoelzl@56889
   630
wenzelm@63569
   631
lemma complex_div_gt_0: "(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)"
wenzelm@63569
   632
proof (cases "b = 0")
wenzelm@63569
   633
  case True
wenzelm@63569
   634
  then show ?thesis by auto
lp15@55734
   635
next
wenzelm@63569
   636
  case False
hoelzl@56889
   637
  then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2"
hoelzl@56889
   638
    by (simp add: complex_eq_iff sum_power2_gt_zero_iff)
hoelzl@56889
   639
  then show ?thesis
hoelzl@56889
   640
    by (simp add: Re_divide Im_divide zero_less_divide_iff)
lp15@55734
   641
qed
lp15@55734
   642
lp15@59741
   643
lemma Re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0"
lp15@59741
   644
  and Im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0"
hoelzl@56889
   645
  using complex_div_gt_0 by auto
lp15@55734
   646
wenzelm@63569
   647
lemma Re_complex_div_ge_0: "Re (a / b) \<ge> 0 \<longleftrightarrow> Re (a * cnj b) \<ge> 0"
lp15@59741
   648
  by (metis le_less Re_complex_div_eq_0 Re_complex_div_gt_0)
lp15@55734
   649
wenzelm@63569
   650
lemma Im_complex_div_ge_0: "Im (a / b) \<ge> 0 \<longleftrightarrow> Im (a * cnj b) \<ge> 0"
lp15@59741
   651
  by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 le_less)
lp15@55734
   652
wenzelm@63569
   653
lemma Re_complex_div_lt_0: "Re (a / b) < 0 \<longleftrightarrow> Re (a * cnj b) < 0"
lp15@59741
   654
  by (metis less_asym neq_iff Re_complex_div_eq_0 Re_complex_div_gt_0)
lp15@55734
   655
wenzelm@63569
   656
lemma Im_complex_div_lt_0: "Im (a / b) < 0 \<longleftrightarrow> Im (a * cnj b) < 0"
lp15@59741
   657
  by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 less_asym neq_iff)
lp15@55734
   658
wenzelm@63569
   659
lemma Re_complex_div_le_0: "Re (a / b) \<le> 0 \<longleftrightarrow> Re (a * cnj b) \<le> 0"
lp15@59741
   660
  by (metis not_le Re_complex_div_gt_0)
lp15@55734
   661
wenzelm@63569
   662
lemma Im_complex_div_le_0: "Im (a / b) \<le> 0 \<longleftrightarrow> Im (a * cnj b) \<le> 0"
lp15@59741
   663
  by (metis Im_complex_div_gt_0 not_le)
lp15@55734
   664
paulson@61104
   665
lemma Re_divide_of_real [simp]: "Re (z / of_real r) = Re z / r"
paulson@61104
   666
  by (simp add: Re_divide power2_eq_square)
paulson@61104
   667
paulson@61104
   668
lemma Im_divide_of_real [simp]: "Im (z / of_real r) = Im z / r"
paulson@61104
   669
  by (simp add: Im_divide power2_eq_square)
paulson@61104
   670
lp15@65578
   671
lemma Re_divide_Reals [simp]: "r \<in> \<real> \<Longrightarrow> Re (z / r) = Re z / Re r"
paulson@61104
   672
  by (metis Re_divide_of_real of_real_Re)
paulson@61104
   673
lp15@65578
   674
lemma Im_divide_Reals [simp]: "r \<in> \<real> \<Longrightarrow> Im (z / r) = Im z / Re r"
paulson@61104
   675
  by (metis Im_divide_of_real of_real_Re)
paulson@61104
   676
nipkow@64267
   677
lemma Re_sum[simp]: "Re (sum f s) = (\<Sum>x\<in>s. Re (f x))"
hoelzl@56369
   678
  by (induct s rule: infinite_finite_induct) auto
lp15@55734
   679
nipkow@64267
   680
lemma Im_sum[simp]: "Im (sum f s) = (\<Sum>x\<in>s. Im(f x))"
hoelzl@56369
   681
  by (induct s rule: infinite_finite_induct) auto
hoelzl@56369
   682
hoelzl@56369
   683
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)"
nipkow@64267
   684
  unfolding sums_def tendsto_complex_iff Im_sum Re_sum ..
lp15@59613
   685
hoelzl@56369
   686
lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and>  summable (\<lambda>x. Im (f x))"
hoelzl@56889
   687
  unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel)
hoelzl@56369
   688
hoelzl@56369
   689
lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f"
hoelzl@56369
   690
  unfolding summable_complex_iff by simp
hoelzl@56369
   691
hoelzl@56369
   692
lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))"
hoelzl@56369
   693
  unfolding summable_complex_iff by blast
hoelzl@56369
   694
hoelzl@56369
   695
lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))"
hoelzl@56369
   696
  unfolding summable_complex_iff by blast
lp15@56217
   697
paulson@61104
   698
lemma complex_is_Nat_iff: "z \<in> \<nat> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_nat i)"
paulson@61104
   699
  by (auto simp: Nats_def complex_eq_iff)
paulson@61104
   700
paulson@61104
   701
lemma complex_is_Int_iff: "z \<in> \<int> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_int i)"
paulson@61104
   702
  by (auto simp: Ints_def complex_eq_iff)
paulson@61104
   703
hoelzl@56889
   704
lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
hoelzl@56889
   705
  by (auto simp: Reals_def complex_eq_iff)
lp15@55734
   706
lp15@55734
   707
lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z"
hoelzl@56889
   708
  by (auto simp: complex_is_Real_iff complex_eq_iff)
lp15@55734
   709
wenzelm@61944
   710
lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm z = \<bar>Re z\<bar>"
hoelzl@56889
   711
  by (simp add: complex_is_Real_iff norm_complex_def)
hoelzl@56369
   712
lp15@65578
   713
lemma Re_Reals_divide: "r \<in> \<real> \<Longrightarrow> Re (r / z) = Re r * Re z / (norm z)\<^sup>2"
lp15@65578
   714
  by (simp add: Re_divide complex_is_Real_iff cmod_power2)
lp15@65578
   715
lp15@65578
   716
lemma Im_Reals_divide: "r \<in> \<real> \<Longrightarrow> Im (r / z) = -Re r * Im z / (norm z)\<^sup>2"
lp15@65578
   717
  by (simp add: Im_divide complex_is_Real_iff cmod_power2)
lp15@65578
   718
hoelzl@56369
   719
lemma series_comparison_complex:
hoelzl@56369
   720
  fixes f:: "nat \<Rightarrow> 'a::banach"
hoelzl@56369
   721
  assumes sg: "summable g"
wenzelm@63569
   722
    and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"
wenzelm@63569
   723
    and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"
hoelzl@56369
   724
  shows "summable f"
hoelzl@56369
   725
proof -
wenzelm@63569
   726
  have g: "\<And>n. cmod (g n) = Re (g n)"
wenzelm@63569
   727
    using assms by (metis abs_of_nonneg in_Reals_norm)
hoelzl@56369
   728
  show ?thesis
hoelzl@56369
   729
    apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])
hoelzl@56369
   730
    using sg
wenzelm@63569
   731
     apply (auto simp: summable_def)
wenzelm@63569
   732
     apply (rule_tac x = "Re s" in exI)
wenzelm@63569
   733
     apply (auto simp: g sums_Re)
hoelzl@56369
   734
    apply (metis fg g)
hoelzl@56369
   735
    done
hoelzl@56369
   736
qed
lp15@55734
   737
wenzelm@63569
   738
wenzelm@63569
   739
subsection \<open>Polar Form for Complex Numbers\<close>
lp15@59746
   740
lp15@62620
   741
lemma complex_unimodular_polar:
wenzelm@63569
   742
  assumes "norm z = 1"
wenzelm@63569
   743
  obtains t where "0 \<le> t" "t < 2 * pi" "z = Complex (cos t) (sin t)"
wenzelm@63569
   744
  by (metis cmod_power2 one_power2 complex_surj sincos_total_2pi [of "Re z" "Im z"] assms)
wenzelm@63569
   745
paulson@14323
   746
wenzelm@60758
   747
subsubsection \<open>$\cos \theta + i \sin \theta$\<close>
huffman@20557
   748
wenzelm@63569
   749
primcorec cis :: "real \<Rightarrow> complex"
wenzelm@63569
   750
  where
wenzelm@63569
   751
    "Re (cis a) = cos a"
wenzelm@63569
   752
  | "Im (cis a) = sin a"
huffman@44827
   753
huffman@44827
   754
lemma cis_zero [simp]: "cis 0 = 1"
hoelzl@56889
   755
  by (simp add: complex_eq_iff)
huffman@44827
   756
huffman@44828
   757
lemma norm_cis [simp]: "norm (cis a) = 1"
hoelzl@56889
   758
  by (simp add: norm_complex_def)
huffman@44828
   759
huffman@44828
   760
lemma sgn_cis [simp]: "sgn (cis a) = cis a"
huffman@44828
   761
  by (simp add: sgn_div_norm)
huffman@44828
   762
huffman@44828
   763
lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
huffman@44828
   764
  by (metis norm_cis norm_zero zero_neq_one)
huffman@44828
   765
huffman@44827
   766
lemma cis_mult: "cis a * cis b = cis (a + b)"
hoelzl@56889
   767
  by (simp add: complex_eq_iff cos_add sin_add)
huffman@44827
   768
huffman@44827
   769
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
wenzelm@63569
   770
  by (induct n) (simp_all add: algebra_simps cis_mult)
huffman@44827
   771
wenzelm@63569
   772
lemma cis_inverse [simp]: "inverse (cis a) = cis (- a)"
hoelzl@56889
   773
  by (simp add: complex_eq_iff)
huffman@44827
   774
huffman@44827
   775
lemma cis_divide: "cis a / cis b = cis (a - b)"
hoelzl@56889
   776
  by (simp add: divide_complex_def cis_mult)
huffman@44827
   777
wenzelm@63569
   778
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re (cis a ^ n)"
huffman@44827
   779
  by (auto simp add: DeMoivre)
huffman@44827
   780
wenzelm@63569
   781
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im (cis a ^ n)"
huffman@44827
   782
  by (auto simp add: DeMoivre)
huffman@44827
   783
hoelzl@56889
   784
lemma cis_pi: "cis pi = -1"
hoelzl@56889
   785
  by (simp add: complex_eq_iff)
hoelzl@56889
   786
wenzelm@63569
   787
wenzelm@60758
   788
subsubsection \<open>$r(\cos \theta + i \sin \theta)$\<close>
huffman@44715
   789
wenzelm@63569
   790
definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex"
wenzelm@63569
   791
  where "rcis r a = complex_of_real r * cis a"
huffman@20557
   792
huffman@44827
   793
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
huffman@44828
   794
  by (simp add: rcis_def)
huffman@44827
   795
huffman@44827
   796
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
huffman@44828
   797
  by (simp add: rcis_def)
huffman@44827
   798
huffman@44827
   799
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
huffman@44828
   800
  by (simp add: complex_eq_iff polar_Ex)
huffman@44827
   801
wenzelm@61944
   802
lemma complex_mod_rcis [simp]: "cmod (rcis r a) = \<bar>r\<bar>"
huffman@44828
   803
  by (simp add: rcis_def norm_mult)
huffman@44827
   804
huffman@44827
   805
lemma cis_rcis_eq: "cis a = rcis 1 a"
huffman@44827
   806
  by (simp add: rcis_def)
huffman@44827
   807
wenzelm@63569
   808
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1 * r2) (a + b)"
huffman@44828
   809
  by (simp add: rcis_def cis_mult)
huffman@44827
   810
huffman@44827
   811
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
huffman@44827
   812
  by (simp add: rcis_def)
huffman@44827
   813
huffman@44827
   814
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
huffman@44827
   815
  by (simp add: rcis_def)
huffman@44827
   816
huffman@44828
   817
lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
huffman@44828
   818
  by (simp add: rcis_def)
huffman@44828
   819
huffman@44827
   820
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
huffman@44827
   821
  by (simp add: rcis_def power_mult_distrib DeMoivre)
huffman@44827
   822
wenzelm@63569
   823
lemma rcis_inverse: "inverse(rcis r a) = rcis (1 / r) (- a)"
huffman@44827
   824
  by (simp add: divide_inverse rcis_def)
huffman@44827
   825
wenzelm@63569
   826
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1 / r2) (a - b)"
huffman@44828
   827
  by (simp add: rcis_def cis_divide [symmetric])
huffman@44827
   828
wenzelm@63569
   829
wenzelm@60758
   830
subsubsection \<open>Complex exponential\<close>
huffman@44827
   831
hoelzl@56889
   832
lemma cis_conv_exp: "cis b = exp (\<i> * b)"
hoelzl@56889
   833
proof -
wenzelm@63569
   834
  have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n =
wenzelm@63569
   835
      of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)"
wenzelm@63569
   836
    for n :: nat
wenzelm@63569
   837
  proof -
hoelzl@56889
   838
    have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)"
hoelzl@56889
   839
      by (induct n)
wenzelm@63569
   840
        (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps
wenzelm@63569
   841
          power2_eq_square add_nonneg_eq_0_iff)
wenzelm@63569
   842
    then show ?thesis
wenzelm@63569
   843
      by (simp add: field_simps)
wenzelm@63569
   844
  qed
wenzelm@63569
   845
  then show ?thesis
wenzelm@63569
   846
    using sin_converges [of b] cos_converges [of b]
lp15@65274
   847
    by (auto simp add: Complex_eq cis.ctr exp_def simp del: of_real_mult
wenzelm@63569
   848
        intro!: sums_unique sums_add sums_mult sums_of_real)
huffman@44291
   849
qed
huffman@44291
   850
lp15@61762
   851
lemma exp_eq_polar: "exp z = exp (Re z) * cis (Im z)"
wenzelm@63569
   852
  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp
lp15@65274
   853
  by (cases z) (simp add: Complex_eq)
huffman@20557
   854
huffman@44828
   855
lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
lp15@61762
   856
  unfolding exp_eq_polar by simp
huffman@44828
   857
huffman@44828
   858
lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
lp15@61762
   859
  unfolding exp_eq_polar by simp
huffman@44828
   860
lp15@59746
   861
lemma norm_cos_sin [simp]: "norm (Complex (cos t) (sin t)) = 1"
lp15@59746
   862
  by (simp add: norm_complex_def)
lp15@59746
   863
lp15@59746
   864
lemma norm_exp_eq_Re [simp]: "norm (exp z) = exp (Re z)"
lp15@65274
   865
  by (simp add: cis.code cmod_complex_polar exp_eq_polar Complex_eq)
lp15@59746
   866
lp15@61762
   867
lemma complex_exp_exists: "\<exists>a r. z = complex_of_real r * exp a"
lp15@59746
   868
  apply (insert rcis_Ex [of z])
lp15@61762
   869
  apply (auto simp add: exp_eq_polar rcis_def mult.assoc [symmetric])
wenzelm@63569
   870
  apply (rule_tac x = "\<i> * complex_of_real a" in exI)
wenzelm@63569
   871
  apply auto
lp15@59746
   872
  done
paulson@14323
   873
wenzelm@63569
   874
lemma exp_pi_i [simp]: "exp (of_real pi * \<i>) = -1"
lp15@61848
   875
  by (metis cis_conv_exp cis_pi mult.commute)
lp15@61848
   876
wenzelm@63569
   877
lemma exp_pi_i' [simp]: "exp (\<i> * of_real pi) = -1"
lp15@63114
   878
  using cis_conv_exp cis_pi by auto
lp15@63114
   879
wenzelm@63569
   880
lemma exp_two_pi_i [simp]: "exp (2 * of_real pi * \<i>) = 1"
lp15@61762
   881
  by (simp add: exp_eq_polar complex_eq_iff)
paulson@14387
   882
lp15@63114
   883
lemma exp_two_pi_i' [simp]: "exp (\<i> * (of_real pi * 2)) = 1"
lp15@63114
   884
  by (metis exp_two_pi_i mult.commute)
lp15@63114
   885
wenzelm@63569
   886
wenzelm@60758
   887
subsubsection \<open>Complex argument\<close>
huffman@44844
   888
wenzelm@63569
   889
definition arg :: "complex \<Rightarrow> real"
wenzelm@63569
   890
  where "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> - pi < a \<and> a \<le> pi))"
huffman@44844
   891
huffman@44844
   892
lemma arg_zero: "arg 0 = 0"
huffman@44844
   893
  by (simp add: arg_def)
huffman@44844
   894
huffman@44844
   895
lemma arg_unique:
huffman@44844
   896
  assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
huffman@44844
   897
  shows "arg z = x"
huffman@44844
   898
proof -
huffman@44844
   899
  from assms have "z \<noteq> 0" by auto
huffman@44844
   900
  have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
huffman@44844
   901
  proof
wenzelm@63040
   902
    fix a
wenzelm@63040
   903
    define d where "d = a - x"
huffman@44844
   904
    assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
huffman@44844
   905
    from a assms have "- (2*pi) < d \<and> d < 2*pi"
huffman@44844
   906
      unfolding d_def by simp
wenzelm@63569
   907
    moreover
wenzelm@63569
   908
    from a assms have "cos a = cos x" and "sin a = sin x"
huffman@44844
   909
      by (simp_all add: complex_eq_iff)
wenzelm@63569
   910
    then have cos: "cos d = 1"
wenzelm@63569
   911
      by (simp add: d_def cos_diff)
wenzelm@63569
   912
    moreover from cos have "sin d = 0"
wenzelm@63569
   913
      by (rule cos_one_sin_zero)
huffman@44844
   914
    ultimately have "d = 0"
wenzelm@63569
   915
      by (auto simp: sin_zero_iff elim!: evenE dest!: less_2_cases)
wenzelm@63569
   916
    then show "a = x"
wenzelm@63569
   917
      by (simp add: d_def)
huffman@44844
   918
  qed (simp add: assms del: Re_sgn Im_sgn)
wenzelm@60758
   919
  with \<open>z \<noteq> 0\<close> show "arg z = x"
wenzelm@63569
   920
    by (simp add: arg_def)
huffman@44844
   921
qed
huffman@44844
   922
huffman@44844
   923
lemma arg_correct:
wenzelm@63569
   924
  assumes "z \<noteq> 0"
wenzelm@63569
   925
  shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
huffman@44844
   926
proof (simp add: arg_def assms, rule someI_ex)
wenzelm@63569
   927
  obtain r a where z: "z = rcis r a"
wenzelm@63569
   928
    using rcis_Ex by fast
huffman@44844
   929
  with assms have "r \<noteq> 0" by auto
wenzelm@63040
   930
  define b where "b = (if 0 < r then a else a + pi)"
huffman@44844
   931
  have b: "sgn z = cis b"
wenzelm@63569
   932
    using \<open>r \<noteq> 0\<close> by (simp add: z b_def rcis_def of_real_def sgn_scaleR sgn_if complex_eq_iff)
wenzelm@63569
   933
  have cis_2pi_nat: "cis (2 * pi * real_of_nat n) = 1" for n
wenzelm@63569
   934
    by (induct n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)
wenzelm@63569
   935
  have cis_2pi_int: "cis (2 * pi * real_of_int x) = 1" for x
wenzelm@63569
   936
    by (cases x rule: int_diff_cases)
wenzelm@63569
   937
      (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
wenzelm@63040
   938
  define c where "c = b - 2 * pi * of_int \<lceil>(b - pi) / (2 * pi)\<rceil>"
huffman@44844
   939
  have "sgn z = cis c"
wenzelm@63569
   940
    by (simp add: b c_def cis_divide [symmetric] cis_2pi_int)
huffman@44844
   941
  moreover have "- pi < c \<and> c \<le> pi"
huffman@44844
   942
    using ceiling_correct [of "(b - pi) / (2*pi)"]
lp15@61649
   943
    by (simp add: c_def less_divide_eq divide_le_eq algebra_simps del: le_of_int_ceiling)
wenzelm@63569
   944
  ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi"
wenzelm@63569
   945
    by fast
huffman@44844
   946
qed
huffman@44844
   947
huffman@44844
   948
lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
hoelzl@56889
   949
  by (cases "z = 0") (simp_all add: arg_zero arg_correct)
huffman@44844
   950
huffman@44844
   951
lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
huffman@44844
   952
  by (simp add: arg_correct)
huffman@44844
   953
huffman@44844
   954
lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
hoelzl@56889
   955
  by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
hoelzl@56889
   956
hoelzl@56889
   957
lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0"
hoelzl@56889
   958
  using cis_arg [of y] by (simp add: complex_eq_iff)
hoelzl@56889
   959
wenzelm@63569
   960
wenzelm@60758
   961
subsection \<open>Square root of complex numbers\<close>
hoelzl@56889
   962
wenzelm@63569
   963
primcorec csqrt :: "complex \<Rightarrow> complex"
wenzelm@63569
   964
  where
wenzelm@63569
   965
    "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"
wenzelm@63569
   966
  | "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)"
hoelzl@56889
   967
hoelzl@56889
   968
lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)"
hoelzl@56889
   969
  by (simp add: complex_eq_iff norm_complex_def)
hoelzl@56889
   970
hoelzl@56889
   971
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
   972
  by (simp add: complex_eq_iff norm_complex_def)
hoelzl@56889
   973
lp15@59862
   974
lemma of_real_sqrt: "x \<ge> 0 \<Longrightarrow> of_real (sqrt x) = csqrt (of_real x)"
lp15@59862
   975
  by (simp add: complex_eq_iff norm_complex_def)
lp15@59862
   976
hoelzl@56889
   977
lemma csqrt_0 [simp]: "csqrt 0 = 0"
hoelzl@56889
   978
  by simp
hoelzl@56889
   979
hoelzl@56889
   980
lemma csqrt_1 [simp]: "csqrt 1 = 1"
hoelzl@56889
   981
  by simp
hoelzl@56889
   982
hoelzl@56889
   983
lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2"
hoelzl@56889
   984
  by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt)
huffman@44844
   985
lp15@59741
   986
lemma power2_csqrt[simp,algebra]: "(csqrt z)\<^sup>2 = z"
wenzelm@63569
   987
proof (cases "Im z = 0")
wenzelm@63569
   988
  case True
wenzelm@63569
   989
  then show ?thesis
hoelzl@56889
   990
    using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"]
hoelzl@56889
   991
    by (cases "0::real" "Re z" rule: linorder_cases)
wenzelm@63569
   992
      (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)
hoelzl@56889
   993
next
wenzelm@63569
   994
  case False
wenzelm@63569
   995
  moreover have "cmod z * cmod z - Re z * Re z = Im z * Im z"
hoelzl@56889
   996
    by (simp add: norm_complex_def power2_eq_square)
wenzelm@63569
   997
  moreover have "\<bar>Re z\<bar> \<le> cmod z"
hoelzl@56889
   998
    by (simp add: norm_complex_def)
hoelzl@56889
   999
  ultimately show ?thesis
hoelzl@56889
  1000
    by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq
wenzelm@63569
  1001
        field_simps real_sqrt_mult[symmetric] real_sqrt_divide)
hoelzl@56889
  1002
qed
hoelzl@56889
  1003
hoelzl@56889
  1004
lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
hoelzl@56889
  1005
  by auto (metis power2_csqrt power_eq_0_iff)
hoelzl@56889
  1006
hoelzl@56889
  1007
lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
hoelzl@56889
  1008
  by auto (metis power2_csqrt power2_eq_1_iff)
hoelzl@56889
  1009
hoelzl@56889
  1010
lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)"
hoelzl@56889
  1011
  by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)
hoelzl@56889
  1012
hoelzl@56889
  1013
lemma Re_csqrt: "0 \<le> Re (csqrt z)"
hoelzl@56889
  1014
  by (metis csqrt_principal le_less)
hoelzl@56889
  1015
hoelzl@56889
  1016
lemma csqrt_square:
hoelzl@56889
  1017
  assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)"
hoelzl@56889
  1018
  shows "csqrt (b^2) = b"
hoelzl@56889
  1019
proof -
hoelzl@56889
  1020
  have "csqrt (b^2) = b \<or> csqrt (b^2) = - b"
wenzelm@63569
  1021
    by (simp add: power2_eq_iff[symmetric])
hoelzl@56889
  1022
  moreover have "csqrt (b^2) \<noteq> -b \<or> b = 0"
wenzelm@63569
  1023
    using csqrt_principal[of "b ^ 2"] assms
wenzelm@63569
  1024
    by (intro disjCI notI) (auto simp: complex_eq_iff)
hoelzl@56889
  1025
  ultimately show ?thesis
hoelzl@56889
  1026
    by auto
hoelzl@56889
  1027
qed
hoelzl@56889
  1028
wenzelm@63569
  1029
lemma csqrt_unique: "w\<^sup>2 = z \<Longrightarrow> 0 < Re w \<or> Re w = 0 \<and> 0 \<le> Im w \<Longrightarrow> csqrt z = w"
lp15@59746
  1030
  by (auto simp: csqrt_square)
lp15@59746
  1031
lp15@59613
  1032
lemma csqrt_minus [simp]:
hoelzl@56889
  1033
  assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)"
hoelzl@56889
  1034
  shows "csqrt (- x) = \<i> * csqrt x"
hoelzl@56889
  1035
proof -
hoelzl@56889
  1036
  have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x"
hoelzl@56889
  1037
  proof (rule csqrt_square)
hoelzl@56889
  1038
    have "Im (csqrt x) \<le> 0"
hoelzl@56889
  1039
      using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod)
hoelzl@56889
  1040
    then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)"
hoelzl@56889
  1041
      by (auto simp add: Re_csqrt simp del: csqrt.simps)
hoelzl@56889
  1042
  qed
hoelzl@56889
  1043
  also have "(\<i> * csqrt x)^2 = - x"
lp15@59746
  1044
    by (simp add: power_mult_distrib)
hoelzl@56889
  1045
  finally show ?thesis .
hoelzl@56889
  1046
qed
huffman@44844
  1047
wenzelm@63569
  1048
wenzelm@60758
  1049
text \<open>Legacy theorem names\<close>
huffman@44065
  1050
huffman@44065
  1051
lemmas expand_complex_eq = complex_eq_iff
huffman@44065
  1052
lemmas complex_Re_Im_cancel_iff = complex_eq_iff
huffman@44065
  1053
lemmas complex_equality = complex_eqI
hoelzl@56889
  1054
lemmas cmod_def = norm_complex_def
hoelzl@56889
  1055
lemmas complex_norm_def = norm_complex_def
hoelzl@56889
  1056
lemmas complex_divide_def = divide_complex_def
hoelzl@56889
  1057
hoelzl@56889
  1058
lemma legacy_Complex_simps:
hoelzl@56889
  1059
  shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
hoelzl@56889
  1060
    and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"
hoelzl@56889
  1061
    and complex_minus: "- (Complex a b) = Complex (- a) (- b)"
hoelzl@56889
  1062
    and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"
hoelzl@56889
  1063
    and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"
hoelzl@56889
  1064
    and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"
hoelzl@56889
  1065
    and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
hoelzl@56889
  1066
    and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"
hoelzl@56889
  1067
    and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"
hoelzl@56889
  1068
    and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"
hoelzl@56889
  1069
    and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"
wenzelm@63569
  1070
    and Complex_eq_i: "Complex x y = \<i> \<longleftrightarrow> x = 0 \<and> y = 1"
wenzelm@63569
  1071
    and i_mult_Complex: "\<i> * Complex a b = Complex (- b) a"
wenzelm@63569
  1072
    and Complex_mult_i: "Complex a b * \<i> = Complex (- b) a"
wenzelm@63569
  1073
    and i_complex_of_real: "\<i> * complex_of_real r = Complex 0 r"
wenzelm@63569
  1074
    and complex_of_real_i: "complex_of_real r * \<i> = Complex 0 r"
hoelzl@56889
  1075
    and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"
hoelzl@56889
  1076
    and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"
hoelzl@56889
  1077
    and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
hoelzl@56889
  1078
    and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
wenzelm@63569
  1079
    and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa \<and> y = 0)"
hoelzl@56889
  1080
    and complex_cn: "cnj (Complex a b) = Complex a (- b)"
nipkow@64267
  1081
    and Complex_sum': "sum (\<lambda>x. Complex (f x) 0) s = Complex (sum f s) 0"
nipkow@64267
  1082
    and Complex_sum: "Complex (sum f s) 0 = sum (\<lambda>x. Complex (f x) 0) s"
hoelzl@56889
  1083
    and complex_of_real_def: "complex_of_real r = Complex r 0"
hoelzl@56889
  1084
    and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"
lp15@65274
  1085
  by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide)
hoelzl@56889
  1086
hoelzl@56889
  1087
lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"
hoelzl@56889
  1088
  by (metis Reals_of_real complex_of_real_def)
huffman@44065
  1089
paulson@13957
  1090
end