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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 Re :: "complex \<Rightarrow> real" 
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where Re: "Re (Complex x y) = x" 

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primrec Im :: "complex \<Rightarrow> real" 
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where Im: "Im (Complex x y) = y" 

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

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by (induct z) simp 

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lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y" 
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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 complex_zero_def: 
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"0 = Complex 0 0" 

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definition complex_add_def: 
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"x + y = Complex (Re x + Re y) (Im x + Im y)" 

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definition complex_minus_def: 
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" x = Complex ( Re x) ( Im x)" 

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definition complex_diff_def: 
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"x  (y\<Colon>complex) = x +  y" 

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lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" 
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by (simp add: complex_zero_def) 

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

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

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by (simp add: complex_add_def) 

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

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

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by (simp add: complex_add_def) 

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

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

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end 

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subsection {* Multiplication and Division *} 
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instantiation complex :: field_inverse_zero 
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begin 
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definition complex_one_def: 
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"1 = Complex 1 0" 

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definition complex_mult_def: 
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"x * y = Complex (Re x * Re y  Im x * Im y) (Re x * Im y + Im x * Re y)" 

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definition complex_inverse_def: 
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"inverse x = 

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Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) ( Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))" 
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definition complex_divide_def: 
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"x / (y\<Colon>complex) = x * inverse y" 

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lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)" 
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by (simp add: complex_one_def) 
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lemma complex_Re_one [simp]: "Re 1 = 1" 
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by (simp add: complex_one_def) 
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lemma complex_Im_one [simp]: "Im 1 = 0" 
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by (simp add: complex_one_def) 
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lemma complex_mult [simp]: 
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"Complex a b * Complex c d = Complex (a * c  b * d) (a * d + b * c)" 
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by (simp add: complex_mult_def) 
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lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y  Im x * Im y" 
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by (simp add: complex_mult_def) 
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lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y" 
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by (simp add: complex_mult_def) 
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lemma complex_inverse [simp]: 
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"inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) ( b / (a\<twosuperior> + b\<twosuperior>))" 
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by (simp add: complex_inverse_def) 
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lemma complex_Re_inverse: 
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"Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)" 
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by (simp add: complex_inverse_def) 
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lemma complex_Im_inverse: 
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"Im (inverse x) =  Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)" 
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by (simp add: complex_inverse_def) 
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instance 
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by intro_classes (simp_all add: complex_mult_def 

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right_distrib left_distrib right_diff_distrib left_diff_distrib 
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complex_inverse_def complex_divide_def 

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power2_eq_square add_divide_distrib [symmetric] 

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complex_eq_iff) 

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end 
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subsection {* Numerals and Arithmetic *} 
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instantiation complex :: number_ring 
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begin 
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definition complex_number_of_def: 
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"number_of w = (of_int w \<Colon> complex)" 

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instance 
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by intro_classes (simp only: complex_number_of_def) 
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end 
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lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n" 
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by (induct n) simp_all 
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lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0" 
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lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z" 
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lemma complex_Im_of_int [simp]: "Im (of_int z) = 0" 
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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 complex_scaleR_def: 
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"scaleR r x = Complex (r * Re x) (r * Im x)" 

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lemma complex_scaleR [simp]: 
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"scaleR r (Complex a b) = Complex (r * a) (r * b)" 
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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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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  223 
end 
224 

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225 

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226 
subsection{* Properties of Embedding from Reals *} 
14323  227 

44724  228 
abbreviation complex_of_real :: "real \<Rightarrow> complex" 
229 
where "complex_of_real \<equiv> of_real" 

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230 

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lemma complex_of_real_def: "complex_of_real r = Complex r 0" 
44724  232 
by (simp add: of_real_def complex_scaleR_def) 
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233 

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lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" 
44724  235 
by (simp add: complex_of_real_def) 
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236 

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lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" 
44724  238 
by (simp add: complex_of_real_def) 
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239 

14377  240 
lemma Complex_add_complex_of_real [simp]: 
44724  241 
shows "Complex x y + complex_of_real r = Complex (x+r) y" 
242 
by (simp add: complex_of_real_def) 

14377  243 

244 
lemma complex_of_real_add_Complex [simp]: 

44724  245 
shows "complex_of_real r + Complex x y = Complex (r+x) y" 
246 
by (simp add: complex_of_real_def) 

14377  247 

248 
lemma Complex_mult_complex_of_real: 

44724  249 
shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)" 
250 
by (simp add: complex_of_real_def) 

14377  251 

252 
lemma complex_of_real_mult_Complex: 

44724  253 
shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)" 
254 
by (simp add: complex_of_real_def) 

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255 

44827
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256 
lemma complex_split_polar: 
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"\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))" 
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258 
by (simp add: complex_eq_iff polar_Ex) 
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259 

14377  260 

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subsection {* Vector Norm *} 
14323  262 

25712  263 
instantiation complex :: real_normed_field 
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begin 
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265 

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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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44724  269 
abbreviation cmod :: "complex \<Rightarrow> real" 
270 
where "cmod \<equiv> norm" 

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271 

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definition complex_sgn_def: 
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"sgn x = x /\<^sub>R cmod x" 
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274 

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definition dist_complex_def: 
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276 
"dist x y = cmod (x  y)" 
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277 

37767  278 
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  280 

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281 
lemmas cmod_def = complex_norm_def 
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282 

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283 
lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)" 
25712  284 
by (simp add: complex_norm_def) 
22852  285 

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instance proof 
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287 
fix r :: real and x y :: complex and S :: "complex set" 
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288 
show "0 \<le> norm x" 
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289 
by (induct x) simp 
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290 
show "(norm x = 0) = (x = 0)" 
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291 
by (induct x) simp 
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292 
show "norm (x + y) \<le> norm x + norm y" 
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293 
by (induct x, induct y) 
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294 
(simp add: real_sqrt_sum_squares_triangle_ineq) 
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295 
show "norm (scaleR r x) = \<bar>r\<bar> * norm x" 
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296 
by (induct x) 
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297 
(simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult) 
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298 
show "norm (x * y) = norm x * norm y" 
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299 
by (induct x, induct y) 
29667  300 
(simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps) 
31292  301 
show "sgn x = x /\<^sub>R cmod x" 
302 
by (rule complex_sgn_def) 

303 
show "dist x y = cmod (x  y)" 

304 
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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306 
by (rule open_complex_def) 
24520  307 
qed 
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308 

25712  309 
end 
310 

44761  311 
lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1" 
44724  312 
by simp 
14323  313 

44761  314 
lemma cmod_complex_polar: 
44724  315 
"cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r" 
316 
by (simp add: norm_mult) 

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317 

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318 
lemma complex_Re_le_cmod: "Re x \<le> cmod x" 
44724  319 
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320 
by (rule real_sqrt_sum_squares_ge1) 

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321 

44761  322 
lemma complex_mod_minus_le_complex_mod: " cmod x \<le> cmod x" 
44724  323 
by (rule order_trans [OF _ norm_ge_zero], simp) 
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324 

44761  325 
lemma complex_mod_triangle_ineq2: "cmod(b + a)  cmod b \<le> cmod a" 
44724  326 
by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp) 
14323  327 

26117  328 
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x" 
44724  329 
by (cases x) simp 
26117  330 

331 
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x" 

44724  332 
by (cases x) simp 
333 

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334 

23123  335 
subsection {* Completeness of the Complexes *} 
336 

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337 
lemma bounded_linear_Re: "bounded_linear Re" 
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338 
by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def) 
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339 

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340 
lemma bounded_linear_Im: "bounded_linear Im" 
44127  341 
by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def) 
23123  342 

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343 
lemmas tendsto_Re [tendsto_intros] = 
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344 
bounded_linear.tendsto [OF bounded_linear_Re] 
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345 

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lemmas tendsto_Im [tendsto_intros] = 
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347 
bounded_linear.tendsto [OF bounded_linear_Im] 
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348 

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349 
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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352 
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im] 
23123  353 

36825  354 
lemma tendsto_Complex [tendsto_intros]: 
44724  355 
assumes "(f > a) F" and "(g > b) F" 
356 
shows "((\<lambda>x. Complex (f x) (g x)) > Complex a b) F" 

36825  357 
proof (rule tendstoI) 
358 
fix r :: real assume "0 < r" 

359 
hence "0 < r / sqrt 2" by (simp add: divide_pos_pos) 

44724  360 
have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F" 
361 
using `(f > a) F` and `0 < r / sqrt 2` by (rule tendstoD) 

36825  362 
moreover 
44724  363 
have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F" 
364 
using `(g > b) F` and `0 < r / sqrt 2` by (rule tendstoD) 

36825  365 
ultimately 
44724  366 
show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F" 
36825  367 
by (rule eventually_elim2) 
368 
(simp add: dist_norm real_sqrt_sum_squares_less) 

369 
qed 

370 

23123  371 
instance complex :: banach 
372 
proof 

373 
fix X :: "nat \<Rightarrow> complex" 

374 
assume X: "Cauchy X" 

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375 
from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) > lim (\<lambda>n. Re (X n))" 
23123  376 
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) 
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377 
from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) > lim (\<lambda>n. Im (X n))" 
23123  378 
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) 
379 
have "X > Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))" 

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380 
using tendsto_Complex [OF 1 2] by simp 
23123  381 
thus "convergent X" 
382 
by (rule convergentI) 

383 
qed 

384 

385 

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386 
subsection {* The Complex Number $i$ *} 
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387 

44724  388 
definition "ii" :: complex ("\<i>") 
389 
where i_def: "ii \<equiv> Complex 0 1" 

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390 

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391 
lemma complex_Re_i [simp]: "Re ii = 0" 
44724  392 
by (simp add: i_def) 
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393 

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394 
lemma complex_Im_i [simp]: "Im ii = 1" 
44724  395 
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396 

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397 
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)" 
44724  398 
by (simp add: i_def) 
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399 

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400 
lemma complex_i_not_zero [simp]: "ii \<noteq> 0" 
44724  401 
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402 

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403 
lemma complex_i_not_one [simp]: "ii \<noteq> 1" 
44724  404 
by (simp add: complex_eq_iff) 
23124  405 

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406 
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w" 
44724  407 
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408 

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409 
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex ( b) a" 
44724  410 
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411 

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412 
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex ( b) a" 
44724  413 
by (simp add: complex_eq_iff) 
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414 

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415 
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" 
44724  416 
by (simp add: i_def complex_of_real_def) 
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417 

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418 
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" 
44724  419 
by (simp add: i_def complex_of_real_def) 
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420 

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421 
lemma i_squared [simp]: "ii * ii = 1" 
44724  422 
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423 

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424 
lemma power2_i [simp]: "ii\<twosuperior> = 1" 
44724  425 
by (simp add: power2_eq_square) 
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426 

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427 
lemma inverse_i [simp]: "inverse ii =  ii" 
44724  428 
by (rule inverse_unique, simp) 
14354
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types complex and hcomplex are now instances of class ringpower:
paulson
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14353
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changeset

429 

44827
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Complex.thy: move theorems into appropriate subsections
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430 
lemma complex_i_mult_minus [simp]: "ii * (ii * x) =  x" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
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44825
diff
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431 
by (simp add: mult_assoc [symmetric]) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
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432 

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types complex and hcomplex are now instances of class ringpower:
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14353
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433 

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changeset

434 
subsection {* Complex Conjugation *} 
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435 

44724  436 
definition cnj :: "complex \<Rightarrow> complex" where 
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437 
"cnj z = Complex (Re z) ( Im z)" 
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438 

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parents:
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439 
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a ( b)" 
44724  440 
by (simp add: cnj_def) 
23125
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changeset

441 

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parents:
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changeset

442 
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x" 
44724  443 
by (simp add: cnj_def) 
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444 

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parents:
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445 
lemma complex_Im_cnj [simp]: "Im (cnj x) =  Im x" 
44724  446 
by (simp add: cnj_def) 
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447 

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parents:
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448 
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" 
44724  449 
by (simp add: complex_eq_iff) 
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huffman
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450 

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parents:
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451 
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" 
44724  452 
by (simp add: cnj_def) 
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huffman
parents:
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453 

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454 
lemma complex_cnj_zero [simp]: "cnj 0 = 0" 
44724  455 
by (simp add: complex_eq_iff) 
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huffman
parents:
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456 

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457 
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" 
44724  458 
by (simp add: complex_eq_iff) 
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459 

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parents:
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460 
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y" 
44724  461 
by (simp add: complex_eq_iff) 
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462 

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parents:
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463 
lemma complex_cnj_diff: "cnj (x  y) = cnj x  cnj y" 
44724  464 
by (simp add: complex_eq_iff) 
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465 

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466 
lemma complex_cnj_minus: "cnj ( x) =  cnj x" 
44724  467 
by (simp add: complex_eq_iff) 
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468 

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469 
lemma complex_cnj_one [simp]: "cnj 1 = 1" 
44724  470 
by (simp add: complex_eq_iff) 
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471 

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472 
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y" 
44724  473 
by (simp add: complex_eq_iff) 
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474 

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parents:
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475 
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)" 
44724  476 
by (simp add: complex_inverse_def) 
14323  477 

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478 
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y" 
44724  479 
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) 
23125
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parents:
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480 

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parents:
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481 
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n" 
44724  482 
by (induct n, simp_all add: complex_cnj_mult) 
23125
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huffman
parents:
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483 

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parents:
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484 
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" 
44724  485 
by (simp add: complex_eq_iff) 
23125
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huffman
parents:
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changeset

486 

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parents:
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487 
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" 
44724  488 
by (simp add: complex_eq_iff) 
23125
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huffman
parents:
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changeset

489 

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parents:
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changeset

490 
lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w" 
44724  491 
by (simp add: complex_eq_iff) 
23125
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huffman
parents:
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changeset

492 

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parents:
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493 
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)" 
44724  494 
by (simp add: complex_eq_iff) 
23125
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huffman
parents:
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changeset

495 

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parents:
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496 
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" 
44724  497 
by (simp add: complex_norm_def) 
14323  498 

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parents:
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499 
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" 
44724  500 
by (simp add: complex_eq_iff) 
23125
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huffman
parents:
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diff
changeset

501 

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huffman
parents:
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diff
changeset

502 
lemma complex_cnj_i [simp]: "cnj ii =  ii" 
44724  503 
by (simp add: complex_eq_iff) 
23125
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cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

504 

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huffman
parents:
23124
diff
changeset

505 
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" 
44724  506 
by (simp add: complex_eq_iff) 
23125
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huffman
parents:
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diff
changeset

507 

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parents:
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diff
changeset

508 
lemma complex_diff_cnj: "z  cnj z = complex_of_real (2 * Im z) * ii" 
44724  509 
by (simp add: complex_eq_iff) 
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset

510 

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

511 
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" 
44724  512 
by (simp add: complex_eq_iff power2_eq_square) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

513 

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huffman
parents:
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diff
changeset

514 
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>" 
44724  515 
by (simp add: norm_mult power2_eq_square) 
23125
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huffman
parents:
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516 

44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
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parents:
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diff
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517 
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

518 
by (simp add: cmod_def power2_eq_square) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

519 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

520 
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

521 
by simp 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

522 

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

523 
lemma bounded_linear_cnj: "bounded_linear cnj" 
44127  524 
using complex_cnj_add complex_cnj_scaleR 
525 
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

526 

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

527 
lemmas tendsto_cnj [tendsto_intros] = 
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset

528 
bounded_linear.tendsto [OF bounded_linear_cnj] 
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset

529 

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

530 
lemmas isCont_cnj [simp] = 
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset

531 
bounded_linear.isCont [OF bounded_linear_cnj] 
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset

532 

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

533 

44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

534 
subsection {* Complex Signum and Argument *} 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

535 

44724  536 
definition arg :: "complex => real" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

537 
"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

538 

14374  539 
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" 
44724  540 
by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute) 
14323  541 

14374  542 
lemma complex_eq_cancel_iff2 [simp]: 
44724  543 
shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)" 
544 
by (simp add: complex_of_real_def) 

14323  545 

14374  546 
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" 
44724  547 
by (simp add: complex_sgn_def divide_inverse) 
14323  548 

14374  549 
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" 
44724  550 
by (simp add: complex_sgn_def divide_inverse) 
14323  551 

552 
lemma complex_inverse_complex_split: 

553 
"inverse(complex_of_real x + ii * complex_of_real y) = 

554 
complex_of_real(x/(x ^ 2 + y ^ 2))  

555 
ii * complex_of_real(y/(x ^ 2 + y ^ 2))" 

44724  556 
by (simp add: complex_of_real_def i_def diff_minus divide_inverse) 
14323  557 

558 
(**) 

559 
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) 

560 
(* many of the theorems are not used  so should they be kept? *) 

561 
(**) 

562 

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

563 
lemma cos_arg_i_mult_zero_pos: 
14377  564 
"0 < y ==> cos (arg(Complex 0 y)) = 0" 
14373  565 
apply (simp add: arg_def abs_if) 
14334  566 
apply (rule_tac a = "pi/2" in someI2, auto) 
567 
apply (rule order_less_trans [of _ 0], auto) 

14323  568 
done 
569 

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

570 
lemma cos_arg_i_mult_zero_neg: 
14377  571 
"y < 0 ==> cos (arg(Complex 0 y)) = 0" 
14373  572 
apply (simp add: arg_def abs_if) 
14334  573 
apply (rule_tac a = " pi/2" in someI2, auto) 
574 
apply (rule order_trans [of _ 0], auto) 

14323  575 
done 
576 

14374  577 
lemma cos_arg_i_mult_zero [simp]: 
14377  578 
"y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0" 
579 
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) 

14323  580 

581 

582 
subsection{*Finally! Polar Form for Complex Numbers*} 

583 

44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

584 
subsubsection {* $\cos \theta + i \sin \theta$ *} 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

585 

44715  586 
definition cis :: "real \<Rightarrow> complex" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

587 
"cis a = Complex (cos a) (sin a)" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

588 

44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

589 
lemma Re_cis [simp]: "Re (cis a) = cos a" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

590 
by (simp add: cis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

591 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

592 
lemma Im_cis [simp]: "Im (cis a) = sin a" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

593 
by (simp add: cis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

594 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

595 
lemma cis_zero [simp]: "cis 0 = 1" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

596 
by (simp add: cis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

597 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

598 
lemma cis_mult: "cis a * cis b = cis (a + b)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

599 
by (simp add: cis_def cos_add sin_add) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

600 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

601 
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

602 
by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

603 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

604 
lemma cis_inverse [simp]: "inverse(cis a) = cis (a)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

605 
by (simp add: cis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

606 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

607 
lemma cis_divide: "cis a / cis b = cis (a  b)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

608 
by (simp add: complex_divide_def cis_mult diff_minus) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

609 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

610 
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

611 
by (auto simp add: DeMoivre) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

612 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

613 
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

614 
by (auto simp add: DeMoivre) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

615 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

616 
subsubsection {* $r(\cos \theta + i \sin \theta)$ *} 
44715  617 

618 
definition rcis :: "[real, real] \<Rightarrow> complex" where 

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

619 
"rcis r a = complex_of_real r * cis a" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

620 

44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

621 
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

622 
by (simp add: rcis_def cis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

623 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

624 
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

625 
by (simp add: rcis_def cis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

626 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

627 
lemma rcis_Ex: "\<exists>r a. z = rcis r a" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

628 
apply (induct z) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

629 
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

630 
done 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

631 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

632 
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

633 
by (simp add: rcis_def cis_def norm_mult) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

634 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

635 
lemma cis_rcis_eq: "cis a = rcis 1 a" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

636 
by (simp add: rcis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

637 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

638 
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

639 
by (simp add: rcis_def cis_def cos_add sin_add right_distrib 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

640 
right_diff_distrib complex_of_real_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

641 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

642 
lemma rcis_zero_mod [simp]: "rcis 0 a = 0" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

643 
by (simp add: rcis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

644 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

645 
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

646 
by (simp add: rcis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

647 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

648 
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

649 
by (simp add: rcis_def power_mult_distrib DeMoivre) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

650 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

651 
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (a)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

652 
by (simp add: divide_inverse rcis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

653 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

654 
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a  b)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

655 
apply (simp add: complex_divide_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

656 
apply (case_tac "r2=0", simp) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

657 
apply (simp add: rcis_inverse rcis_mult diff_minus) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

658 
done 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

659 

4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

660 
subsubsection {* Complex exponential *} 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

661 

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

662 
abbreviation expi :: "complex \<Rightarrow> complex" 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

663 
where "expi \<equiv> exp" 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

664 

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

665 
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

666 
proof (rule complex_eqI) 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

667 
{ fix n have "Complex 0 b ^ n = 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

668 
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

669 
apply (induct n) 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

670 
apply (simp add: cos_coeff_def sin_coeff_def) 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

671 
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

672 
done } note * = this 
44712
1e490e891c88
replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents:
44711
diff
changeset

673 
show "Re (cis b) = Re (exp (Complex 0 b))" 
44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

674 
unfolding exp_def cis_def cos_def 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

675 
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

676 
simp add: * mult_assoc [symmetric]) 
44712
1e490e891c88
replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents:
44711
diff
changeset

677 
show "Im (cis b) = Im (exp (Complex 0 b))" 
44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

678 
unfolding exp_def cis_def sin_def 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

679 
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

680 
simp add: * mult_assoc [symmetric]) 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

681 
qed 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

682 

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

683 
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

684 
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

685 

14374  686 
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a" 
14373  687 
apply (insert rcis_Ex [of z]) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

688 
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric]) 
14334  689 
apply (rule_tac x = "ii * complex_of_real a" in exI, auto) 
14323  690 
done 
691 

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

692 
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1" 
44724  693 
by (simp add: expi_def cis_def) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

694 

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

695 
text {* Legacy theorem names *} 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

696 

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

697 
lemmas expand_complex_eq = complex_eq_iff 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

698 
lemmas complex_Re_Im_cancel_iff = complex_eq_iff 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

699 
lemmas complex_equality = complex_eqI 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

700 

13957  701 
end 