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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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fix a b :: real and x y :: complex 
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show "scaleR a (x + y) = scaleR a x + scaleR a y" 
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by (simp add: complex_eq_iff right_distrib) 
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show "scaleR (a + b) x = scaleR a x + scaleR b x" 
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by (simp add: complex_eq_iff left_distrib) 
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show "scaleR a (scaleR b x) = scaleR (a * b) x" 
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by (simp add: complex_eq_iff mult_assoc) 
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show "scaleR 1 x = x" 
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by (simp add: complex_eq_iff) 
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show "scaleR a x * y = scaleR a (x * y)" 
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by (simp add: complex_eq_iff algebra_simps) 
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show "x * scaleR a y = scaleR a (x * y)" 
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by (simp add: complex_eq_iff algebra_simps) 
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qed 
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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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lemma complex_of_real_def: "complex_of_real r = Complex r 0" 
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by (simp add: of_real_def complex_scaleR_def) 
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lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" 
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by (simp add: complex_of_real_def) 
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lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" 
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by (simp add: complex_of_real_def) 
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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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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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definition complex_norm_def: 
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"norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" 
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abbreviation 
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cmod :: "complex \<Rightarrow> real" where 
25712  274 
"cmod \<equiv> norm" 
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definition complex_sgn_def: 
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"sgn x = x /\<^sub>R cmod x" 
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definition dist_complex_def: 
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"dist x y = cmod (x  y)" 
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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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lemmas cmod_def = complex_norm_def 
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286 

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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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show "0 \<le> norm x" 
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by (induct x) simp 
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show "(norm x = 0) = (x = 0)" 
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by (induct x) simp 
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show "norm (x + y) \<le> norm x + norm y" 
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by (induct x, induct y) 
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(simp add: real_sqrt_sum_squares_triangle_ineq) 
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show "norm (scaleR r x) = \<bar>r\<bar> * norm x" 
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by (induct x) 
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(simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult) 
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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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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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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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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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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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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 
interpretation Re: bounded_linear "Re" 
44127  343 
by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def) 
23123  344 

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

36825  348 
lemma tendsto_Complex [tendsto_intros]: 
349 
assumes "(f > a) net" and "(g > b) net" 

350 
shows "((\<lambda>x. Complex (f x) (g x)) > Complex a b) net" 

351 
proof (rule tendstoI) 

352 
fix r :: real assume "0 < r" 

353 
hence "0 < r / sqrt 2" by (simp add: divide_pos_pos) 

354 
have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) net" 

355 
using `(f > a) net` and `0 < r / sqrt 2` by (rule tendstoD) 

356 
moreover 

357 
have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) net" 

358 
using `(g > b) net` and `0 < r / sqrt 2` by (rule tendstoD) 

359 
ultimately 

360 
show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) net" 

361 
by (rule eventually_elim2) 

362 
(simp add: dist_norm real_sqrt_sum_squares_less) 

363 
qed 

364 

23123  365 
lemma LIMSEQ_Complex: 
366 
"\<lbrakk>X > a; Y > b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) > Complex a b" 

36825  367 
by (rule tendsto_Complex) 
23123  368 

369 
instance complex :: banach 

370 
proof 

371 
fix X :: "nat \<Rightarrow> complex" 

372 
assume X: "Cauchy X" 

373 
from Re.Cauchy [OF X] have 1: "(\<lambda>n. Re (X n)) > lim (\<lambda>n. Re (X n))" 

374 
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) 

375 
from Im.Cauchy [OF X] have 2: "(\<lambda>n. Im (X n)) > lim (\<lambda>n. Im (X n))" 

376 
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) 

377 
have "X > Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))" 

378 
using LIMSEQ_Complex [OF 1 2] by simp 

379 
thus "convergent X" 

380 
by (rule convergentI) 

381 
qed 

382 

383 

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384 
subsection {* The Complex Number @{term "\<i>"} *} 
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385 

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386 
definition 
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387 
"ii" :: complex ("\<i>") where 
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388 
i_def: "ii \<equiv> Complex 0 1" 
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389 

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390 
lemma complex_Re_i [simp]: "Re ii = 0" 
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391 
by (simp add: i_def) 
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392 

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393 
lemma complex_Im_i [simp]: "Im ii = 1" 
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394 
by (simp add: i_def) 
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395 

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396 
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)" 
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397 
by (simp add: i_def) 
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398 

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lemma complex_i_not_zero [simp]: "ii \<noteq> 0" 
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400 
by (simp add: complex_eq_iff) 
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huffman
parents:
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diff
changeset

401 

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

402 
lemma complex_i_not_one [simp]: "ii \<noteq> 1" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
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diff
changeset

403 
by (simp add: complex_eq_iff) 
23124  404 

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

405 
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
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diff
changeset

406 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

407 

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

408 
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex ( b) a" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

409 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

410 

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

411 
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex ( b) a" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

412 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

413 

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

414 
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

415 
by (simp add: i_def complex_of_real_def) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

416 

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

417 
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

418 
by (simp add: i_def complex_of_real_def) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

419 

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

420 
lemma i_squared [simp]: "ii * ii = 1" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

421 
by (simp add: i_def) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

422 

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

423 
lemma power2_i [simp]: "ii\<twosuperior> = 1" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

424 
by (simp add: power2_eq_square) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

425 

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

426 
lemma inverse_i [simp]: "inverse ii =  ii" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

427 
by (rule inverse_unique, simp) 
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset

428 

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

429 

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

430 
subsection {* Complex Conjugation *} 
6f7b5b96241f
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huffman
parents:
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diff
changeset

431 

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parents:
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432 
definition 
6f7b5b96241f
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huffman
parents:
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diff
changeset

433 
cnj :: "complex \<Rightarrow> complex" where 
6f7b5b96241f
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huffman
parents:
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diff
changeset

434 
"cnj z = Complex (Re z) ( Im z)" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

435 

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parents:
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436 
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a ( b)" 
6f7b5b96241f
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huffman
parents:
23124
diff
changeset

437 
by (simp add: cnj_def) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

438 

6f7b5b96241f
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parents:
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diff
changeset

439 
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

440 
by (simp add: cnj_def) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

441 

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

442 
lemma complex_Im_cnj [simp]: "Im (cnj x) =  Im x" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

443 
by (simp add: cnj_def) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

444 

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

445 
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

446 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

447 

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

448 
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

449 
by (simp add: cnj_def) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

450 

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

451 
lemma complex_cnj_zero [simp]: "cnj 0 = 0" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

452 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

453 

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

454 
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

455 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

456 

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

457 
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

458 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

459 

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

460 
lemma complex_cnj_diff: "cnj (x  y) = cnj x  cnj y" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

461 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

462 

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

463 
lemma complex_cnj_minus: "cnj ( x) =  cnj x" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

464 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

465 

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

466 
lemma complex_cnj_one [simp]: "cnj 1 = 1" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

467 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

468 

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

469 
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

470 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

471 

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

472 
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

473 
by (simp add: complex_inverse_def) 
14323  474 

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

475 
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

476 
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

477 

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

478 
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

479 
by (induct n, simp_all add: complex_cnj_mult) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

480 

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

481 
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

482 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

483 

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

484 
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

485 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

486 

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

487 
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

488 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

489 

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

490 
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

491 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

492 

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

493 
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

494 
by (simp add: complex_norm_def) 
14323  495 

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

496 
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

497 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

498 

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

499 
lemma complex_cnj_i [simp]: "cnj ii =  ii" 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

500 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

501 

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

502 
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

503 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

504 

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

505 
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

506 
by (simp add: complex_eq_iff) 
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset

507 

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

508 
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

509 
by (simp add: complex_eq_iff power2_eq_square) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

510 

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

511 
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

512 
by (simp add: norm_mult power2_eq_square) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

513 

30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30273
diff
changeset

514 
interpretation cnj: bounded_linear "cnj" 
44127  515 
using complex_cnj_add complex_cnj_scaleR 
516 
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

517 

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

518 

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

519 
subsection{*The Functions @{term sgn} and @{term arg}*} 
14323  520 

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

521 
text {* Argand *} 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

522 

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

523 
definition 
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset

524 
arg :: "complex => real" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

525 
"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

526 

14374  527 
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" 
24506  528 
by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute) 
14323  529 

530 
lemma i_mult_eq: "ii * ii = complex_of_real (1)" 

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

531 
by (simp add: i_def complex_of_real_def) 
14323  532 

14374  533 
lemma i_mult_eq2 [simp]: "ii * ii = (1::complex)" 
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

534 
by (simp add: i_def complex_one_def) 
14323  535 

14374  536 
lemma complex_eq_cancel_iff2 [simp]: 
14377  537 
"(Complex x y = complex_of_real xa) = (x = xa & y = 0)" 
538 
by (simp add: complex_of_real_def) 

14323  539 

14374  540 
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" 
24506  541 
by (simp add: complex_sgn_def divide_inverse) 
14323  542 

14374  543 
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" 
24506  544 
by (simp add: complex_sgn_def divide_inverse) 
14323  545 

546 
lemma complex_inverse_complex_split: 

547 
"inverse(complex_of_real x + ii * complex_of_real y) = 

548 
complex_of_real(x/(x ^ 2 + y ^ 2))  

549 
ii * complex_of_real(y/(x ^ 2 + y ^ 2))" 

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

550 
by (simp add: complex_of_real_def i_def diff_minus divide_inverse) 
14323  551 

552 
(**) 

553 
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) 

554 
(* many of the theorems are not used  so should they be kept? *) 

555 
(**) 

556 

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

557 
lemma cos_arg_i_mult_zero_pos: 
14377  558 
"0 < y ==> cos (arg(Complex 0 y)) = 0" 
14373  559 
apply (simp add: arg_def abs_if) 
14334  560 
apply (rule_tac a = "pi/2" in someI2, auto) 
561 
apply (rule order_less_trans [of _ 0], auto) 

14323  562 
done 
563 

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

564 
lemma cos_arg_i_mult_zero_neg: 
14377  565 
"y < 0 ==> cos (arg(Complex 0 y)) = 0" 
14373  566 
apply (simp add: arg_def abs_if) 
14334  567 
apply (rule_tac a = " pi/2" in someI2, auto) 
568 
apply (rule order_trans [of _ 0], auto) 

14323  569 
done 
570 

14374  571 
lemma cos_arg_i_mult_zero [simp]: 
14377  572 
"y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0" 
573 
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) 

14323  574 

575 

576 
subsection{*Finally! Polar Form for Complex Numbers*} 

577 

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

578 
definition 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

579 

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

580 
(* abbreviation for (cos a + i sin a) *) 
21404
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more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset

581 
cis :: "real => complex" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

582 
"cis a = Complex (cos a) (sin a)" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

583 

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

584 
definition 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

585 
(* abbreviation for r*(cos a + i sin a) *) 
21404
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more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset

586 
rcis :: "[real, real] => complex" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
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diff
changeset

587 
"rcis r a = complex_of_real r * cis a" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

588 

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

589 
definition 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

590 
(* e ^ (x + iy) *) 
21404
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more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
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changeset

591 
expi :: "complex => complex" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
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diff
changeset

592 
"expi z = complex_of_real(exp (Re z)) * cis (Im z)" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
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diff
changeset

593 

14374  594 
lemma complex_split_polar: 
14377  595 
"\<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

596 
apply (induct z) 
14377  597 
apply (auto simp add: polar_Ex complex_of_real_mult_Complex) 
14323  598 
done 
599 

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paulson
parents:
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changeset

600 
lemma rcis_Ex: "\<exists>r a. z = rcis r a" 
20725
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instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

601 
apply (induct z) 
14377  602 
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex) 
14323  603 
done 
604 

14374  605 
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" 
14373  606 
by (simp add: rcis_def cis_def) 
14323  607 

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

608 
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" 
14373  609 
by (simp add: rcis_def cis_def) 
14323  610 

14377  611 
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>" 
612 
proof  

613 
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

614 
by (simp only: power_mult_distrib right_distrib) 
14377  615 
thus ?thesis by simp 
616 
qed 

14323  617 

14374  618 
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" 
14377  619 
by (simp add: rcis_def cis_def sin_cos_squared_add2_mult) 
14323  620 

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

621 
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

622 
by (simp add: cmod_def power2_eq_square) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

623 

14374  624 
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

625 
by simp 
14323  626 

627 

628 
(**) 

629 
(* (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b) *) 

630 
(**) 

631 

632 
lemma cis_rcis_eq: "cis a = rcis 1 a" 

14373  633 
by (simp add: rcis_def) 
14323  634 

14374  635 
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" 
15013  636 
by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib 
637 
complex_of_real_def) 

14323  638 

639 
lemma cis_mult: "cis a * cis b = cis (a + b)" 

14373  640 
by (simp add: cis_rcis_eq rcis_mult) 
14323  641 

14374  642 
lemma cis_zero [simp]: "cis 0 = 1" 
14377  643 
by (simp add: cis_def complex_one_def) 
14323  644 

14374  645 
lemma rcis_zero_mod [simp]: "rcis 0 a = 0" 
14373  646 
by (simp add: rcis_def) 
14323  647 

14374  648 
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" 
14373  649 
by (simp add: rcis_def) 
14323  650 

651 
lemma complex_of_real_minus_one: 

652 
"complex_of_real ((1::real)) = (1::complex)" 

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

653 
by (simp add: complex_of_real_def complex_one_def) 
14323  654 

14374  655 
lemma complex_i_mult_minus [simp]: "ii * (ii * x) =  x" 
23125
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huffman
parents:
23124
diff
changeset

656 
by (simp add: mult_assoc [symmetric]) 
14323  657 

658 

659 
lemma cis_real_of_nat_Suc_mult: 

660 
"cis (real (Suc n) * a) = cis a * cis (real n * a)" 

14377  661 
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib) 
14323  662 

663 
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" 

664 
apply (induct_tac "n") 

665 
apply (auto simp add: cis_real_of_nat_Suc_mult) 

666 
done 

667 

14374  668 
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" 
22890  669 
by (simp add: rcis_def power_mult_distrib DeMoivre) 
14323  670 

14374  671 
lemma cis_inverse [simp]: "inverse(cis a) = cis (a)" 
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

672 
by (simp add: cis_def complex_inverse_complex_split diff_minus) 
14323  673 

674 
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (a)" 

22884  675 
by (simp add: divide_inverse rcis_def) 
14323  676 

677 
lemma cis_divide: "cis a / cis b = cis (a  b)" 

37887  678 
by (simp add: complex_divide_def cis_mult diff_minus) 
14323  679 

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

680 
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a  b)" 
14373  681 
apply (simp add: complex_divide_def) 
682 
apply (case_tac "r2=0", simp) 

37887  683 
apply (simp add: rcis_inverse rcis_mult diff_minus) 
14323  684 
done 
685 

14374  686 
lemma Re_cis [simp]: "Re(cis a) = cos a" 
14373  687 
by (simp add: cis_def) 
14323  688 

14374  689 
lemma Im_cis [simp]: "Im(cis a) = sin a" 
14373  690 
by (simp add: cis_def) 
14323  691 

692 
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" 

14334  693 
by (auto simp add: DeMoivre) 
14323  694 

695 
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" 

14334  696 
by (auto simp add: DeMoivre) 
14323  697 

698 
lemma expi_add: "expi(a + b) = expi(a) * expi(b)" 

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

699 
by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac) 
14323  700 

14374  701 
lemma expi_zero [simp]: "expi (0::complex) = 1" 
14373  702 
by (simp add: expi_def) 
14323  703 

14374  704 
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a" 
14373  705 
apply (insert rcis_Ex [of z]) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

706 
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric]) 
14334  707 
apply (rule_tac x = "ii * complex_of_real a" in exI, auto) 
14323  708 
done 
709 

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

710 
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

711 
by (simp add: expi_def cis_def) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

712 

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

713 
text {* Legacy theorem names *} 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

714 

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

715 
lemmas expand_complex_eq = complex_eq_iff 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

716 
lemmas complex_Re_Im_cancel_iff = complex_eq_iff 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

717 
lemmas complex_equality = complex_eqI 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

718 

13957  719 
end 