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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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Polymorphic treatment of binary arithmetic using axclasses
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Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 
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*) 
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header {* Complex Numbers: Rectangular and Polar Representations *} 
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theory Complex 
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imports Transcendental 
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begin 
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datatype complex = Complex real real 
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primrec 
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Re :: "complex \<Rightarrow> real" 

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where 

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Re: "Re (Complex x y) = x" 

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primrec 
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Im :: "complex \<Rightarrow> real" 

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where 

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Im: "Im (Complex x y) = y" 

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lemma complex_surj [simp]: "Complex (Re z) (Im z) = z" 

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by (induct z) simp 

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lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y" 
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by (induct x, induct y) simp 
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lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y" 
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subsection {* Addition and Subtraction *} 
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instantiation complex :: ab_group_add 
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begin 
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definition 
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complex_zero_def: "0 = Complex 0 0" 
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definition 
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complex_add_def: "x + y = Complex (Re x + Re y) (Im x + Im y)" 
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definition 
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complex_minus_def: " x = Complex ( Re x) ( Im x)" 
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definition 
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complex_diff_def: "x  (y\<Colon>complex) = x +  y" 
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lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" 
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by (simp add: complex_zero_def) 

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lemma complex_Re_zero [simp]: "Re 0 = 0" 
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by (simp add: complex_zero_def) 
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lemma complex_Im_zero [simp]: "Im 0 = 0" 

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by (simp add: complex_zero_def) 
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lemma complex_add [simp]: 
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"Complex a b + Complex c d = Complex (a + c) (b + d)" 

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by (simp add: complex_add_def) 

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lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y" 
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by (simp add: complex_add_def) 

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lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y" 

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by (simp add: complex_add_def) 

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lemma complex_minus [simp]: 
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" (Complex a b) = Complex ( a) ( b)" 

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by (simp add: complex_minus_def) 
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lemma complex_Re_minus [simp]: "Re ( x) =  Re x" 
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by (simp add: complex_minus_def) 
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lemma complex_Im_minus [simp]: "Im ( x) =  Im x" 
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by (simp add: complex_minus_def) 
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lemma complex_diff [simp]: 
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"Complex a b  Complex c d = Complex (a  c) (b  d)" 
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by (simp add: complex_diff_def) 
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lemma complex_Re_diff [simp]: "Re (x  y) = Re x  Re y" 
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by (simp add: complex_diff_def) 
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lemma complex_Im_diff [simp]: "Im (x  y) = Im x  Im y" 
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by (simp add: complex_diff_def) 
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instance 
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by intro_classes (simp_all add: complex_add_def complex_diff_def) 

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end 

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subsection {* Multiplication and Division *} 
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instantiation complex :: field_inverse_zero 
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begin 
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definition 
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complex_one_def: "1 = Complex 1 0" 
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definition 
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complex_mult_def: "x * y = 
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Complex (Re x * Re y  Im x * Im y) (Re x * Im y + Im x * Re y)" 
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definition 
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complex_inverse_def: "inverse x = 
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Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) ( Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))" 
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definition 
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complex_divide_def: "x / (y\<Colon>complex) = x * inverse y" 
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lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)" 
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by (simp add: complex_one_def) 
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lemma complex_Re_one [simp]: "Re 1 = 1" 
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by (simp add: complex_one_def) 
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lemma complex_Im_one [simp]: "Im 1 = 0" 
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by (simp add: complex_one_def) 
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lemma complex_mult [simp]: 
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"Complex a b * Complex c d = Complex (a * c  b * d) (a * d + b * c)" 
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by (simp add: complex_mult_def) 
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lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y  Im x * Im y" 
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by (simp add: complex_mult_def) 
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lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y" 
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by (simp add: complex_mult_def) 
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lemma complex_inverse [simp]: 
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"inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) ( b / (a\<twosuperior> + b\<twosuperior>))" 
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by (simp add: complex_inverse_def) 
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lemma complex_Re_inverse: 
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"Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)" 
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by (simp add: complex_inverse_def) 
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lemma complex_Im_inverse: 
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"Im (inverse x) =  Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)" 
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by (simp add: complex_inverse_def) 
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instance 
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by intro_classes (simp_all add: complex_mult_def 

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right_distrib left_distrib right_diff_distrib left_diff_distrib 

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complex_inverse_def complex_divide_def 

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power2_eq_square add_divide_distrib [symmetric] 

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complex_eq_iff) 
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end 
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subsection {* Numerals and Arithmetic *} 
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instantiation complex :: number_ring 
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begin 
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definition number_of_complex where 
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complex_number_of_def: "number_of w = (of_int w \<Colon> complex)" 
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instance 
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by intro_classes (simp only: complex_number_of_def) 
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end 
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lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n" 
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by (induct n) simp_all 
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lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0" 
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by (induct n) simp_all 
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lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z" 
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by (cases z rule: int_diff_cases) simp 
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lemma complex_Im_of_int [simp]: "Im (of_int z) = 0" 
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by (cases z rule: int_diff_cases) simp 
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lemma complex_Re_number_of [simp]: "Re (number_of v) = number_of v" 
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lemma complex_Im_number_of [simp]: "Im (number_of v) = 0" 
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lemma Complex_eq_number_of [simp]: 
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"(Complex a b = number_of w) = (a = number_of w \<and> b = 0)" 
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by (simp add: complex_eq_iff) 
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subsection {* Scalar Multiplication *} 
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instantiation complex :: real_field 
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begin 
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definition 
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complex_scaleR_def: "scaleR r x = Complex (r * Re x) (r * Im x)" 
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lemma complex_scaleR [simp]: 
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"scaleR r (Complex a b) = Complex (r * a) (r * b)" 
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lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x" 
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lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x" 
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25712  212 
instance 
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proof 
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show "scaleR a (x + y) = scaleR a x + scaleR a y" 
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by (simp add: complex_eq_iff right_distrib) 
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show "scaleR (a + b) x = scaleR a x + scaleR b x" 
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by (simp add: complex_eq_iff left_distrib) 
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show "scaleR a (scaleR b x) = scaleR (a * b) x" 
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by (simp add: complex_eq_iff mult_assoc) 
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show "scaleR 1 x = x" 
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by (simp add: complex_eq_iff) 
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show "scaleR a x * y = scaleR a (x * y)" 
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by (simp add: complex_eq_iff algebra_simps) 
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show "x * scaleR a y = scaleR a (x * y)" 
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by (simp add: complex_eq_iff algebra_simps) 
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qed 
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25712  229 
end 
230 

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subsection{* Properties of Embedding from Reals *} 
14323  233 

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abbreviation 
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complex_of_real :: "real \<Rightarrow> complex" where 
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"complex_of_real \<equiv> of_real" 
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237 

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lemma complex_of_real_def: "complex_of_real r = Complex r 0" 
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by (simp add: of_real_def complex_scaleR_def) 
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240 

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lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" 
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by (simp add: complex_of_real_def) 
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243 

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lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" 
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by (simp add: complex_of_real_def) 
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246 

14377  247 
lemma Complex_add_complex_of_real [simp]: 
248 
"Complex x y + complex_of_real r = Complex (x+r) y" 

249 
by (simp add: complex_of_real_def) 

250 

251 
lemma complex_of_real_add_Complex [simp]: 

252 
"complex_of_real r + Complex x y = Complex (r+x) y" 

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253 
by (simp add: complex_of_real_def) 
14377  254 

255 
lemma Complex_mult_complex_of_real: 

256 
"Complex x y * complex_of_real r = Complex (x*r) (y*r)" 

257 
by (simp add: complex_of_real_def) 

258 

259 
lemma complex_of_real_mult_Complex: 

260 
"complex_of_real r * Complex x y = Complex (r*x) (r*y)" 

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by (simp add: complex_of_real_def) 
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262 

14377  263 

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subsection {* Vector Norm *} 
14323  265 

25712  266 
instantiation complex :: real_normed_field 
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begin 
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268 

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definition complex_norm_def: 
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"norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" 
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271 

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abbreviation 
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cmod :: "complex \<Rightarrow> real" where 
25712  274 
"cmod \<equiv> norm" 
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275 

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definition complex_sgn_def: 
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"sgn x = x /\<^sub>R cmod x" 
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278 

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definition dist_complex_def: 
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280 
"dist x y = cmod (x  y)" 
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281 

37767  282 
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)" 
31292  284 

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285 
lemmas cmod_def = complex_norm_def 
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286 

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287 
lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)" 
25712  288 
by (simp add: complex_norm_def) 
22852  289 

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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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292 
show "0 \<le> norm x" 
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293 
by (induct x) simp 
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294 
show "(norm x = 0) = (x = 0)" 
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295 
by (induct x) simp 
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296 
show "norm (x + y) \<le> norm x + norm y" 
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297 
by (induct x, induct y) 
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298 
(simp add: real_sqrt_sum_squares_triangle_ineq) 
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299 
show "norm (scaleR r x) = \<bar>r\<bar> * norm x" 
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300 
by (induct x) 
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301 
(simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult) 
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302 
show "norm (x * y) = norm x * norm y" 
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303 
by (induct x, induct y) 
29667  304 
(simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps) 
31292  305 
show "sgn x = x /\<^sub>R cmod x" 
306 
by (rule complex_sgn_def) 

307 
show "dist x y = cmod (x  y)" 

308 
by (rule dist_complex_def) 

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show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" 
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310 
by (rule open_complex_def) 
24520  311 
qed 
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312 

25712  313 
end 
314 

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lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1" 
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316 
by simp 
14323  317 

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lemma cmod_complex_polar [simp]: 
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"cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r" 
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320 
by (simp add: norm_mult) 
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321 

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322 
lemma complex_Re_le_cmod: "Re x \<le> cmod x" 
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323 
unfolding complex_norm_def 
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324 
by (rule real_sqrt_sum_squares_ge1) 
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325 

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326 
lemma complex_mod_minus_le_complex_mod [simp]: " cmod x \<le> cmod x" 
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327 
by (rule order_trans [OF _ norm_ge_zero], simp) 
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328 

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329 
lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a)  cmod b \<le> cmod a" 
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330 
by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp) 
14323  331 

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332 
lemmas real_sum_squared_expand = power2_sum [where 'a=real] 
14323  333 

26117  334 
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x" 
335 
by (cases x) simp 

336 

337 
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x" 

338 
by (cases x) simp 

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339 

23123  340 
subsection {* Completeness of the Complexes *} 
341 

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342 
lemma bounded_linear_Re: "bounded_linear Re" 
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343 
by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def) 
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344 

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345 
lemma bounded_linear_Im: "bounded_linear Im" 
44127  346 
by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def) 
23123  347 

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348 
lemmas tendsto_Re [tendsto_intros] = 
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349 
bounded_linear.tendsto [OF bounded_linear_Re] 
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350 

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351 
lemmas tendsto_Im [tendsto_intros] = 
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352 
bounded_linear.tendsto [OF bounded_linear_Im] 
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353 

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354 
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 Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re] 
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357 
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im] 
23123  358 

36825  359 
lemma tendsto_Complex [tendsto_intros]: 
360 
assumes "(f > a) net" and "(g > b) net" 

361 
shows "((\<lambda>x. Complex (f x) (g x)) > Complex a b) net" 

362 
proof (rule tendstoI) 

363 
fix r :: real assume "0 < r" 

364 
hence "0 < r / sqrt 2" by (simp add: divide_pos_pos) 

365 
have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) net" 

366 
using `(f > a) net` and `0 < r / sqrt 2` by (rule tendstoD) 

367 
moreover 

368 
have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) net" 

369 
using `(g > b) net` and `0 < r / sqrt 2` by (rule tendstoD) 

370 
ultimately 

371 
show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) net" 

372 
by (rule eventually_elim2) 

373 
(simp add: dist_norm real_sqrt_sum_squares_less) 

374 
qed 

375 

23123  376 
lemma LIMSEQ_Complex: 
377 
"\<lbrakk>X > a; Y > b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) > Complex a b" 

36825  378 
by (rule tendsto_Complex) 
23123  379 

380 
instance complex :: banach 

381 
proof 

382 
fix X :: "nat \<Rightarrow> complex" 

383 
assume X: "Cauchy X" 

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384 
from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) > lim (\<lambda>n. Re (X n))" 
23123  385 
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) 
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386 
from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) > lim (\<lambda>n. Im (X n))" 
23123  387 
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) 
388 
have "X > Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))" 

389 
using LIMSEQ_Complex [OF 1 2] by simp 

390 
thus "convergent X" 

391 
by (rule convergentI) 

392 
qed 

393 

394 

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395 
subsection {* The Complex Number @{term "\<i>"} *} 
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396 

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parents:
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397 
definition 
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parents:
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398 
"ii" :: complex ("\<i>") where 
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parents:
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399 
i_def: "ii \<equiv> Complex 0 1" 
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parents:
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400 

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parents:
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401 
lemma complex_Re_i [simp]: "Re ii = 0" 
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402 
by (simp add: i_def) 
14354
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types complex and hcomplex are now instances of class ringpower:
paulson
parents:
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diff
changeset

403 

23125
6f7b5b96241f
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parents:
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diff
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404 
lemma complex_Im_i [simp]: "Im ii = 1" 
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parents:
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changeset

405 
by (simp add: i_def) 
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cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

406 

6f7b5b96241f
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parents:
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diff
changeset

407 
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)" 
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parents:
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408 
by (simp add: i_def) 
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cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

409 

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cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

410 
lemma complex_i_not_zero [simp]: "ii \<noteq> 0" 
44065
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standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
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diff
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411 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

412 

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huffman
parents:
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diff
changeset

413 
lemma complex_i_not_one [simp]: "ii \<noteq> 1" 
44065
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standard theorem naming scheme: complex_eqI, complex_eq_iff
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parents:
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diff
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414 
by (simp add: complex_eq_iff) 
23124  415 

23125
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parents:
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diff
changeset

416 
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w" 
44065
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standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
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diff
changeset

417 
by (simp add: complex_eq_iff) 
23125
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cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

418 

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parents:
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diff
changeset

419 
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex ( b) a" 
44065
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standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
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420 
by (simp add: complex_eq_iff) 
23125
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cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

421 

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huffman
parents:
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diff
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422 
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex ( b) a" 
44065
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standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
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changeset

423 
by (simp add: complex_eq_iff) 
23125
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cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

424 

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parents:
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425 
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" 
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426 
by (simp add: i_def complex_of_real_def) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

427 

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parents:
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changeset

428 
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" 
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parents:
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diff
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429 
by (simp add: i_def complex_of_real_def) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

430 

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parents:
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changeset

431 
lemma i_squared [simp]: "ii * ii = 1" 
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huffman
parents:
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changeset

432 
by (simp add: i_def) 
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cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

433 

6f7b5b96241f
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huffman
parents:
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diff
changeset

434 
lemma power2_i [simp]: "ii\<twosuperior> = 1" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

435 
by (simp add: power2_eq_square) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

436 

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parents:
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437 
lemma inverse_i [simp]: "inverse ii =  ii" 
6f7b5b96241f
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huffman
parents:
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diff
changeset

438 
by (rule inverse_unique, simp) 
14354
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types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset

439 

988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset

440 

23125
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parents:
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diff
changeset

441 
subsection {* Complex Conjugation *} 
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huffman
parents:
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diff
changeset

442 

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parents:
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443 
definition 
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parents:
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444 
cnj :: "complex \<Rightarrow> complex" where 
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parents:
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445 
"cnj z = Complex (Re z) ( Im z)" 
6f7b5b96241f
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huffman
parents:
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diff
changeset

446 

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parents:
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447 
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a ( b)" 
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huffman
parents:
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diff
changeset

448 
by (simp add: cnj_def) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

449 

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parents:
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450 
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x" 
6f7b5b96241f
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huffman
parents:
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diff
changeset

451 
by (simp add: cnj_def) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

452 

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huffman
parents:
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diff
changeset

453 
lemma complex_Im_cnj [simp]: "Im (cnj x) =  Im x" 
6f7b5b96241f
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huffman
parents:
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diff
changeset

454 
by (simp add: cnj_def) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

455 

6f7b5b96241f
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huffman
parents:
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diff
changeset

456 
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

457 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

458 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

459 
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

460 
by (simp add: cnj_def) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

461 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

462 
lemma complex_cnj_zero [simp]: "cnj 0 = 0" 
44065
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standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

463 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

464 

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huffman
parents:
23124
diff
changeset

465 
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" 
44065
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standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

466 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

467 

6f7b5b96241f
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huffman
parents:
23124
diff
changeset

468 
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

469 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

470 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

471 
lemma complex_cnj_diff: "cnj (x  y) = cnj x  cnj y" 
44065
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standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

472 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

473 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

474 
lemma complex_cnj_minus: "cnj ( x) =  cnj x" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

475 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

476 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

477 
lemma complex_cnj_one [simp]: "cnj 1 = 1" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

478 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

479 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

480 
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

481 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

482 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

483 
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

484 
by (simp add: complex_inverse_def) 
14323  485 

23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

486 
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

487 
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

488 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

489 
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

490 
by (induct n, simp_all add: complex_cnj_mult) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

491 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

492 
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

493 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

494 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

495 
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

496 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

497 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

498 
lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

499 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

500 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

501 
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

502 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

503 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

504 
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

505 
by (simp add: complex_norm_def) 
14323  506 

23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

507 
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

508 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

509 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

510 
lemma complex_cnj_i [simp]: "cnj ii =  ii" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

511 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

512 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

513 
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

514 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

515 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

516 
lemma complex_diff_cnj: "z  cnj z = complex_of_real (2 * Im z) * ii" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

517 
by (simp add: complex_eq_iff) 
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset

518 

23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

519 
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

520 
by (simp add: complex_eq_iff power2_eq_square) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

521 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

522 
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

523 
by (simp add: norm_mult power2_eq_square) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

524 

44290
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset

525 
lemma bounded_linear_cnj: "bounded_linear cnj" 
44127  526 
using complex_cnj_add complex_cnj_scaleR 
527 
by (rule bounded_linear_intro [where K=1], simp) 

14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset

528 

44290
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset

529 
lemmas tendsto_cnj [tendsto_intros] = 
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset

530 
bounded_linear.tendsto [OF bounded_linear_cnj] 
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset

531 

23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset

532 
lemmas isCont_cnj [simp] = 
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset

533 
bounded_linear.isCont [OF bounded_linear_cnj] 
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset

534 

14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset

535 

22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22968
diff
changeset

536 
subsection{*The Functions @{term sgn} and @{term arg}*} 
14323  537 

22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22968
diff
changeset

538 
text {* Argand *} 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

539 

21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset

540 
definition 
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset

541 
arg :: "complex => real" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

542 
"arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & pi < a & a \<le> pi)" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

543 

14374  544 
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" 
24506  545 
by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute) 
14323  546 

547 
lemma i_mult_eq: "ii * ii = complex_of_real (1)" 

20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

548 
by (simp add: i_def complex_of_real_def) 
14323  549 

14374  550 
lemma i_mult_eq2 [simp]: "ii * ii = (1::complex)" 
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

551 
by (simp add: i_def complex_one_def) 
14323  552 

14374  553 
lemma complex_eq_cancel_iff2 [simp]: 
14377  554 
"(Complex x y = complex_of_real xa) = (x = xa & y = 0)" 
555 
by (simp add: complex_of_real_def) 

14323  556 

14374  557 
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" 
24506  558 
by (simp add: complex_sgn_def divide_inverse) 
14323  559 

14374  560 
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" 
24506  561 
by (simp add: complex_sgn_def divide_inverse) 
14323  562 

563 
lemma complex_inverse_complex_split: 

564 
"inverse(complex_of_real x + ii * complex_of_real y) = 

565 
complex_of_real(x/(x ^ 2 + y ^ 2))  

566 
ii * complex_of_real(y/(x ^ 2 + y ^ 2))" 

20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

567 
by (simp add: complex_of_real_def i_def diff_minus divide_inverse) 
14323  568 

569 
(**) 

570 
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) 

571 
(* many of the theorems are not used  so should they be kept? *) 

572 
(**) 

573 

14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset

574 
lemma cos_arg_i_mult_zero_pos: 
14377  575 
"0 < y ==> cos (arg(Complex 0 y)) = 0" 
14373  576 
apply (simp add: arg_def abs_if) 
14334  577 
apply (rule_tac a = "pi/2" in someI2, auto) 
578 
apply (rule order_less_trans [of _ 0], auto) 

14323  579 
done 
580 

14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset

581 
lemma cos_arg_i_mult_zero_neg: 
14377  582 
"y < 0 ==> cos (arg(Complex 0 y)) = 0" 
14373  583 
apply (simp add: arg_def abs_if) 
14334  584 
apply (rule_tac a = " pi/2" in someI2, auto) 
585 
apply (rule order_trans [of _ 0], auto) 

14323  586 
done 
587 

14374  588 
lemma cos_arg_i_mult_zero [simp]: 
14377  589 
"y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0" 
590 
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) 

14323  591 

592 

593 
subsection{*Finally! Polar Form for Complex Numbers*} 

594 

20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

595 
definition 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

596 

81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

597 
(* abbreviation for (cos a + i sin a) *) 
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset

598 
cis :: "real => complex" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

599 
"cis a = Complex (cos a) (sin a)" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

600 

21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset

601 
definition 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

602 
(* abbreviation for r*(cos a + i sin a) *) 
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset

603 
rcis :: "[real, real] => complex" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

604 
"rcis r a = complex_of_real r * cis a" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

605 

44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

606 
abbreviation expi :: "complex \<Rightarrow> complex" 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

607 
where "expi \<equiv> exp" 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

608 

44712
1e490e891c88
replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents:
44711
diff
changeset

609 
lemma cis_conv_exp: "cis b = exp (Complex 0 b)" 
44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

610 
proof (rule complex_eqI) 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

611 
{ fix n have "Complex 0 b ^ n = 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

612 
real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)" 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

613 
apply (induct n) 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

614 
apply (simp add: cos_coeff_def sin_coeff_def) 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

615 
apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc) 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

616 
done } note * = this 
44712
1e490e891c88
replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents:
44711
diff
changeset

617 
show "Re (cis b) = Re (exp (Complex 0 b))" 
44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

618 
unfolding exp_def cis_def cos_def 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

619 
by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic], 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

620 
simp add: * mult_assoc [symmetric]) 
44712
1e490e891c88
replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents:
44711
diff
changeset

621 
show "Im (cis b) = Im (exp (Complex 0 b))" 
44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

622 
unfolding exp_def cis_def sin_def 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

623 
by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic], 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

624 
simp add: * mult_assoc [symmetric]) 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

625 
qed 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

626 

dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

627 
lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)" 
44712
1e490e891c88
replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents:
44711
diff
changeset

628 
unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

629 

14374  630 
lemma complex_split_polar: 
14377  631 
"\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))" 
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

632 
apply (induct z) 
14377  633 
apply (auto simp add: polar_Ex complex_of_real_mult_Complex) 
14323  634 
done 
635 

14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset

636 
lemma rcis_Ex: "\<exists>r a. z = rcis r a" 
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

637 
apply (induct z) 
14377  638 
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex) 
14323  639 
done 
640 

14374  641 
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" 
14373  642 
by (simp add: rcis_def cis_def) 
14323  643 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

644 
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" 
14373  645 
by (simp add: rcis_def cis_def) 
14323  646 

14377  647 
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>" 
648 
proof  

649 
have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)" 

20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

650 
by (simp only: power_mult_distrib right_distrib) 
14377  651 
thus ?thesis by simp 
652 
qed 

14323  653 

14374  654 
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" 
14377  655 
by (simp add: rcis_def cis_def sin_cos_squared_add2_mult) 
14323  656 

23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

657 
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

658 
by (simp add: cmod_def power2_eq_square) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

659 

14374  660 
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

661 
by simp 
14323  662 

663 

664 
(**) 

665 
(* (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b) *) 

666 
(**) 

667 

668 
lemma cis_rcis_eq: "cis a = rcis 1 a" 

14373  669 
by (simp add: rcis_def) 
14323  670 

14374  671 
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" 
15013  672 
by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib 
673 
complex_of_real_def) 

14323  674 

675 
lemma cis_mult: "cis a * cis b = cis (a + b)" 

14373  676 
by (simp add: cis_rcis_eq rcis_mult) 
14323  677 

14374  678 
lemma cis_zero [simp]: "cis 0 = 1" 
14377  679 
by (simp add: cis_def complex_one_def) 
14323  680 

14374  681 
lemma rcis_zero_mod [simp]: "rcis 0 a = 0" 
14373  682 
by (simp add: rcis_def) 
14323  683 

14374  684 
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" 
14373  685 
by (simp add: rcis_def) 
14323  686 

687 
lemma complex_of_real_minus_one: 

688 
"complex_of_real ((1::real)) = (1::complex)" 

20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

689 
by (simp add: complex_of_real_def complex_one_def) 
14323  690 

14374  691 
lemma complex_i_mult_minus [simp]: "ii * (ii * x) =  x" 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

692 
by (simp add: mult_assoc [symmetric]) 
14323  693 

694 

695 
lemma cis_real_of_nat_Suc_mult: 

696 
"cis (real (Suc n) * a) = cis a * cis (real n * a)" 

14377  697 
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib) 
14323  698 

699 
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" 

700 
apply (induct_tac "n") 

701 
apply (auto simp add: cis_real_of_nat_Suc_mult) 

702 
done 

703 

14374  704 
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" 
22890  705 
by (simp add: rcis_def power_mult_distrib DeMoivre) 
14323  706 

14374  707 
lemma cis_inverse [simp]: "inverse(cis a) = cis (a)" 
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

708 
by (simp add: cis_def complex_inverse_complex_split diff_minus) 
14323  709 

710 
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (a)" 

22884  711 
by (simp add: divide_inverse rcis_def) 
14323  712 

713 
lemma cis_divide: "cis a / cis b = cis (a  b)" 

37887  714 
by (simp add: complex_divide_def cis_mult diff_minus) 
14323  715 

14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset

716 
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a  b)" 
14373  717 
apply (simp add: complex_divide_def) 
718 
apply (case_tac "r2=0", simp) 

37887  719 
apply (simp add: rcis_inverse rcis_mult diff_minus) 
14323  720 
done 
721 

14374  722 
lemma Re_cis [simp]: "Re(cis a) = cos a" 
14373  723 
by (simp add: cis_def) 
14323  724 

14374  725 
lemma Im_cis [simp]: "Im(cis a) = sin a" 
14373  726 
by (simp add: cis_def) 
14323  727 

728 
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" 

14334  729 
by (auto simp add: DeMoivre) 
14323  730 

731 
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" 

14334  732 
by (auto simp add: DeMoivre) 
14323  733 

14374  734 
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a" 
14373  735 
apply (insert rcis_Ex [of z]) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

736 
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric]) 
14334  737 
apply (rule_tac x = "ii * complex_of_real a" in exI, auto) 
14323  738 
done 
739 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

740 
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1" 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

741 
by (simp add: expi_def cis_def) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

742 

44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

743 
text {* Legacy theorem names *} 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

744 

eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

745 
lemmas expand_complex_eq = complex_eq_iff 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

746 
lemmas complex_Re_Im_cancel_iff = complex_eq_iff 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

747 
lemmas complex_equality = complex_eqI 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

748 

13957  749 
end 