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(* Title: 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_equality [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 expand_complex_eq: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y" 
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lemmas complex_Re_Im_cancel_iff = expand_complex_eq 
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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, division_by_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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expand_complex_eq) 

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end 
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subsection {* Exponentiation *} 
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instantiation complex :: recpower 
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begin 

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primrec power_complex where 
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complexpow_0: "z ^ 0 = (1\<Colon>complex)" 

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 complexpow_Suc: "z ^ Suc n = (z\<Colon>complex) * z ^ n" 

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instance by intro_classes simp_all 

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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: expand_complex_eq) 
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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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221 

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lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x" 
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224 

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lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x" 
25712  226 
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25712  228 
instance 
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proof 
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230 
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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232 
by (simp add: expand_complex_eq right_distrib) 
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show "scaleR (a + b) x = scaleR a x + scaleR b x" 
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234 
by (simp add: expand_complex_eq left_distrib) 
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show "scaleR a (scaleR b x) = scaleR (a * b) x" 
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by (simp add: expand_complex_eq mult_assoc) 
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show "scaleR 1 x = x" 
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by (simp add: expand_complex_eq) 
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show "scaleR a x * y = scaleR a (x * y)" 
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by (simp add: expand_complex_eq ring_simps) 
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show "x * scaleR a y = scaleR a (x * y)" 
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by (simp add: expand_complex_eq ring_simps) 
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qed 
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25712  245 
end 
246 

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247 

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subsection{* Properties of Embedding from Reals *} 
14323  249 

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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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253 

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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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256 

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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  263 
lemma Complex_add_complex_of_real [simp]: 
264 
"Complex x y + complex_of_real r = Complex (x+r) y" 

265 
by (simp add: complex_of_real_def) 

266 

267 
lemma complex_of_real_add_Complex [simp]: 

268 
"complex_of_real r + Complex x y = Complex (r+x) y" 

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by (simp add: complex_of_real_def) 
14377  270 

271 
lemma Complex_mult_complex_of_real: 

272 
"Complex x y * complex_of_real r = Complex (x*r) (y*r)" 

273 
by (simp add: complex_of_real_def) 

274 

275 
lemma complex_of_real_mult_Complex: 

276 
"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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14377  279 

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subsection {* Vector Norm *} 
14323  281 

25712  282 
instantiation complex :: real_normed_field 
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begin 
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definition 
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complex_norm_def: "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" 
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abbreviation 
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cmod :: "complex \<Rightarrow> real" where 
25712  290 
"cmod \<equiv> norm" 
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definition 
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complex_sgn_def: "sgn x = x /\<^sub>R cmod x" 
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294 

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lemmas cmod_def = complex_norm_def 
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296 

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lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)" 
25712  298 
by (simp add: complex_norm_def) 
22852  299 

25712  300 
instance 
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proof 
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fix r :: real and x y :: complex 
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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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306 
by (induct x) simp 
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307 
show "norm (x + y) \<le> norm x + norm y" 
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308 
by (induct x, induct y) 
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(simp add: real_sqrt_sum_squares_triangle_ineq) 
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310 
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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313 
show "norm (x * y) = norm x * norm y" 
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314 
by (induct x, induct y) 
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315 
(simp add: real_sqrt_mult [symmetric] power2_eq_square ring_simps) 
24506  316 
show "sgn x = x /\<^sub>R cmod x" by(simp add: complex_sgn_def) 
24520  317 
qed 
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318 

25712  319 
end 
320 

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lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1" 
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322 
by simp 
14323  323 

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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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326 
by (simp add: norm_mult) 
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327 

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328 
lemma complex_Re_le_cmod: "Re x \<le> cmod x" 
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329 
unfolding complex_norm_def 
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330 
by (rule real_sqrt_sum_squares_ge1) 
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331 

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332 
lemma complex_mod_minus_le_complex_mod [simp]: " cmod x \<le> cmod x" 
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333 
by (rule order_trans [OF _ norm_ge_zero], simp) 
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334 

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335 
lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a)  cmod b \<le> cmod a" 
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336 
by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp) 
14323  337 

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lemmas real_sum_squared_expand = power2_sum [where 'a=real] 
14323  339 

26117  340 
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x" 
341 
by (cases x) simp 

342 

343 
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x" 

344 
by (cases x) simp 

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345 

23123  346 
subsection {* Completeness of the Complexes *} 
347 

348 
interpretation Re: bounded_linear ["Re"] 

349 
apply (unfold_locales, simp, simp) 

350 
apply (rule_tac x=1 in exI) 

351 
apply (simp add: complex_norm_def) 

352 
done 

353 

354 
interpretation Im: bounded_linear ["Im"] 

355 
apply (unfold_locales, simp, simp) 

356 
apply (rule_tac x=1 in exI) 

357 
apply (simp add: complex_norm_def) 

358 
done 

359 

360 
lemma LIMSEQ_Complex: 

361 
"\<lbrakk>X > a; Y > b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) > Complex a b" 

362 
apply (rule LIMSEQ_I) 

363 
apply (subgoal_tac "0 < r / sqrt 2") 

364 
apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe) 

365 
apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe) 

366 
apply (rename_tac M N, rule_tac x="max M N" in exI, safe) 

367 
apply (simp add: real_sqrt_sum_squares_less) 

368 
apply (simp add: divide_pos_pos) 

369 
done 

370 

371 
instance complex :: banach 

372 
proof 

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

374 
assume X: "Cauchy X" 

375 
from Re.Cauchy [OF X] have 1: "(\<lambda>n. Re (X n)) > lim (\<lambda>n. Re (X n))" 

376 
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) 

377 
from Im.Cauchy [OF X] have 2: "(\<lambda>n. Im (X n)) > lim (\<lambda>n. Im (X n))" 

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)))" 

380 
using LIMSEQ_Complex [OF 1 2] by simp 

381 
thus "convergent X" 

382 
by (rule convergentI) 

383 
qed 

384 

385 

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386 
subsection {* The Complex Number @{term "\<i>"} *} 
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387 

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388 
definition 
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389 
"ii" :: complex ("\<i>") where 
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390 
i_def: "ii \<equiv> Complex 0 1" 
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391 

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392 
lemma complex_Re_i [simp]: "Re ii = 0" 
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393 
by (simp add: i_def) 
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394 

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395 
lemma complex_Im_i [simp]: "Im ii = 1" 
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396 
by (simp add: i_def) 
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397 

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398 
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)" 
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399 
by (simp add: i_def) 
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400 

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401 
lemma complex_i_not_zero [simp]: "ii \<noteq> 0" 
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402 
by (simp add: expand_complex_eq) 
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403 

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404 
lemma complex_i_not_one [simp]: "ii \<noteq> 1" 
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405 
by (simp add: expand_complex_eq) 
23124  406 

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407 
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w" 
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408 
by (simp add: expand_complex_eq) 
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409 

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410 
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex ( b) a" 
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411 
by (simp add: expand_complex_eq) 
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412 

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413 
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex ( b) a" 
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414 
by (simp add: expand_complex_eq) 
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415 

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416 
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" 
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417 
by (simp add: i_def complex_of_real_def) 
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418 

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419 
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" 
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420 
by (simp add: i_def complex_of_real_def) 
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421 

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422 
lemma i_squared [simp]: "ii * ii = 1" 
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423 
by (simp add: i_def) 
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424 

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425 
lemma power2_i [simp]: "ii\<twosuperior> = 1" 
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426 
by (simp add: power2_eq_square) 
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427 

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428 
lemma inverse_i [simp]: "inverse ii =  ii" 
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429 
by (rule inverse_unique, simp) 
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types complex and hcomplex are now instances of class ringpower:
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430 

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431 

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432 
subsection {* Complex Conjugation *} 
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433 

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434 
definition 
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435 
cnj :: "complex \<Rightarrow> complex" where 
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436 
"cnj z = Complex (Re z) ( Im z)" 
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437 

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438 
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a ( b)" 
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439 
by (simp add: cnj_def) 
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440 

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441 
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x" 
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442 
by (simp add: cnj_def) 
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443 

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444 
lemma complex_Im_cnj [simp]: "Im (cnj x) =  Im x" 
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445 
by (simp add: cnj_def) 
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446 

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447 
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" 
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448 
by (simp add: expand_complex_eq) 
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449 

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450 
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" 
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451 
by (simp add: cnj_def) 
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452 

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453 
lemma complex_cnj_zero [simp]: "cnj 0 = 0" 
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454 
by (simp add: expand_complex_eq) 
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455 

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456 
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" 
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457 
by (simp add: expand_complex_eq) 
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458 

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459 
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y" 
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460 
by (simp add: expand_complex_eq) 
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461 

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462 
lemma complex_cnj_diff: "cnj (x  y) = cnj x  cnj y" 
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463 
by (simp add: expand_complex_eq) 
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464 

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465 
lemma complex_cnj_minus: "cnj ( x) =  cnj x" 
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466 
by (simp add: expand_complex_eq) 
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467 

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468 
lemma complex_cnj_one [simp]: "cnj 1 = 1" 
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469 
by (simp add: expand_complex_eq) 
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470 

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471 
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y" 
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472 
by (simp add: expand_complex_eq) 
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473 

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474 
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)" 
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475 
by (simp add: complex_inverse_def) 
14323  476 

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477 
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y" 
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478 
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) 
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479 

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480 
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n" 
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481 
by (induct n, simp_all add: complex_cnj_mult) 
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482 

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483 
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" 
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484 
by (simp add: expand_complex_eq) 
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485 

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486 
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" 
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487 
by (simp add: expand_complex_eq) 
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488 

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489 
lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w" 
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490 
by (simp add: expand_complex_eq) 
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491 

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492 
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)" 
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493 
by (simp add: expand_complex_eq) 
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494 

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495 
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" 
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496 
by (simp add: complex_norm_def) 
14323  497 

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498 
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" 
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499 
by (simp add: expand_complex_eq) 
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500 

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501 
lemma complex_cnj_i [simp]: "cnj ii =  ii" 
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502 
by (simp add: expand_complex_eq) 
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503 

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504 
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" 
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505 
by (simp add: expand_complex_eq) 
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506 

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507 
lemma complex_diff_cnj: "z  cnj z = complex_of_real (2 * Im z) * ii" 
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508 
by (simp add: expand_complex_eq) 
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types complex and hcomplex are now instances of class ringpower:
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509 

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510 
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" 
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511 
by (simp add: expand_complex_eq power2_eq_square) 
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512 

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513 
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>" 
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514 
by (simp add: norm_mult power2_eq_square) 
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515 

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516 
interpretation cnj: bounded_linear ["cnj"] 
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517 
apply (unfold_locales) 
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518 
apply (rule complex_cnj_add) 
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519 
apply (rule complex_cnj_scaleR) 
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520 
apply (rule_tac x=1 in exI, simp) 
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521 
done 
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522 

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523 

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generalized sgn function to work on any real normed vector space
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524 
subsection{*The Functions @{term sgn} and @{term arg}*} 
14323  525 

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526 
text {* Argand *} 
20557
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complex_of_real abbreviates of_real::real=>complex;
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527 

21404
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more robust syntax for definition/abbreviation/notation;
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528 
definition 
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529 
arg :: "complex => real" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
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530 
"arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & pi < a & a \<le> pi)" 
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531 

14374  532 
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" 
24506  533 
by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute) 
14323  534 

535 
lemma i_mult_eq: "ii * ii = complex_of_real (1)" 

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536 
by (simp add: i_def complex_of_real_def) 
14323  537 

14374  538 
lemma i_mult_eq2 [simp]: "ii * ii = (1::complex)" 
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539 
by (simp add: i_def complex_one_def) 
14323  540 

14374  541 
lemma complex_eq_cancel_iff2 [simp]: 
14377  542 
"(Complex x y = complex_of_real xa) = (x = xa & y = 0)" 
543 
by (simp add: complex_of_real_def) 

14323  544 

14374  545 
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" 
24506  546 
by (simp add: complex_sgn_def divide_inverse) 
14323  547 

14374  548 
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" 
24506  549 
by (simp add: complex_sgn_def divide_inverse) 
14323  550 

551 
lemma complex_inverse_complex_split: 

552 
"inverse(complex_of_real x + ii * complex_of_real y) = 

553 
complex_of_real(x/(x ^ 2 + y ^ 2))  

554 
ii * complex_of_real(y/(x ^ 2 + y ^ 2))" 

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555 
by (simp add: complex_of_real_def i_def diff_minus divide_inverse) 
14323  556 

557 
(**) 

558 
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) 

559 
(* many of the theorems are not used  so should they be kept? *) 

560 
(**) 

561 

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562 
lemma cos_arg_i_mult_zero_pos: 
14377  563 
"0 < y ==> cos (arg(Complex 0 y)) = 0" 
14373  564 
apply (simp add: arg_def abs_if) 
14334  565 
apply (rule_tac a = "pi/2" in someI2, auto) 
566 
apply (rule order_less_trans [of _ 0], auto) 

14323  567 
done 
568 

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types complex and hcomplex are now instances of class ringpower:
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569 
lemma cos_arg_i_mult_zero_neg: 
14377  570 
"y < 0 ==> cos (arg(Complex 0 y)) = 0" 
14373  571 
apply (simp add: arg_def abs_if) 
14334  572 
apply (rule_tac a = " pi/2" in someI2, auto) 
573 
apply (rule order_trans [of _ 0], auto) 

14323  574 
done 
575 

14374  576 
lemma cos_arg_i_mult_zero [simp]: 
14377  577 
"y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0" 
578 
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) 

14323  579 

580 

581 
subsection{*Finally! Polar Form for Complex Numbers*} 

582 

20557
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complex_of_real abbreviates of_real::real=>complex;
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changeset

583 
definition 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
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parents:
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diff
changeset

584 

81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
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changeset

585 
(* abbreviation for (cos a + i sin a) *) 
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changeset

586 
cis :: "real => complex" where 
20557
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complex_of_real abbreviates of_real::real=>complex;
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changeset

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

588 

21404
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changeset

589 
definition 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
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changeset

590 
(* abbreviation for r*(cos a + i sin a) *) 
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wenzelm
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changeset

591 
rcis :: "[real, real] => complex" where 
20557
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complex_of_real abbreviates of_real::real=>complex;
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changeset

592 
"rcis r a = complex_of_real r * cis a" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
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parents:
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diff
changeset

593 

21404
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changeset

594 
definition 
20557
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complex_of_real abbreviates of_real::real=>complex;
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parents:
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diff
changeset

595 
(* e ^ (x + iy) *) 
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596 
expi :: "complex => complex" where 
20557
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complex_of_real abbreviates of_real::real=>complex;
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changeset

597 
"expi z = complex_of_real(exp (Re z)) * cis (Im z)" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
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changeset

598 

14374  599 
lemma complex_split_polar: 
14377  600 
"\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))" 
20725
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instance complex :: real_normed_field; cleaned up
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changeset

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

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605 
lemma rcis_Ex: "\<exists>r a. z = rcis r a" 
20725
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instance complex :: real_normed_field; cleaned up
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diff
changeset

606 
apply (induct z) 
14377  607 
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex) 
14323  608 
done 
609 

14374  610 
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" 
14373  611 
by (simp add: rcis_def cis_def) 
14323  612 

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changeset

613 
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" 
14373  614 
by (simp add: rcis_def cis_def) 
14323  615 

14377  616 
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>" 
617 
proof  

618 
have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)" 

20725
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instance complex :: real_normed_field; cleaned up
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diff
changeset

619 
by (simp only: power_mult_distrib right_distrib) 
14377  620 
thus ?thesis by simp 
621 
qed 

14323  622 

14374  623 
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" 
14377  624 
by (simp add: rcis_def cis_def sin_cos_squared_add2_mult) 
14323  625 

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changeset

626 
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" 
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diff
changeset

627 
by (simp add: cmod_def power2_eq_square) 
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cleaned up proofs; reorganized sections; removed redundant lemmas
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changeset

628 

14374  629 
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" 
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huffman
parents:
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diff
changeset

630 
by simp 
14323  631 

632 

633 
(**) 

634 
(* (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b) *) 

635 
(**) 

636 

637 
lemma cis_rcis_eq: "cis a = rcis 1 a" 

14373  638 
by (simp add: rcis_def) 
14323  639 

14374  640 
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" 
15013  641 
by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib 
642 
complex_of_real_def) 

14323  643 

644 
lemma cis_mult: "cis a * cis b = cis (a + b)" 

14373  645 
by (simp add: cis_rcis_eq rcis_mult) 
14323  646 

14374  647 
lemma cis_zero [simp]: "cis 0 = 1" 
14377  648 
by (simp add: cis_def complex_one_def) 
14323  649 

14374  650 
lemma rcis_zero_mod [simp]: "rcis 0 a = 0" 
14373  651 
by (simp add: rcis_def) 
14323  652 

14374  653 
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" 
14373  654 
by (simp add: rcis_def) 
14323  655 

656 
lemma complex_of_real_minus_one: 

657 
"complex_of_real ((1::real)) = (1::complex)" 

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

658 
by (simp add: complex_of_real_def complex_one_def) 
14323  659 

14374  660 
lemma complex_i_mult_minus [simp]: "ii * (ii * x) =  x" 
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huffman
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diff
changeset

661 
by (simp add: mult_assoc [symmetric]) 
14323  662 

663 

664 
lemma cis_real_of_nat_Suc_mult: 

665 
"cis (real (Suc n) * a) = cis a * cis (real n * a)" 

14377  666 
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib) 
14323  667 

668 
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" 

669 
apply (induct_tac "n") 

670 
apply (auto simp add: cis_real_of_nat_Suc_mult) 

671 
done 

672 

14374  673 
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" 
22890  674 
by (simp add: rcis_def power_mult_distrib DeMoivre) 
14323  675 

14374  676 
lemma cis_inverse [simp]: "inverse(cis a) = cis (a)" 
20725
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instance complex :: real_normed_field; cleaned up
huffman
parents:
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diff
changeset

677 
by (simp add: cis_def complex_inverse_complex_split diff_minus) 
14323  678 

679 
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (a)" 

22884  680 
by (simp add: divide_inverse rcis_def) 
14323  681 

682 
lemma cis_divide: "cis a / cis b = cis (a  b)" 

14373  683 
by (simp add: complex_divide_def cis_mult real_diff_def) 
14323  684 

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

685 
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a  b)" 
14373  686 
apply (simp add: complex_divide_def) 
687 
apply (case_tac "r2=0", simp) 

688 
apply (simp add: rcis_inverse rcis_mult real_diff_def) 

14323  689 
done 
690 

14374  691 
lemma Re_cis [simp]: "Re(cis a) = cos a" 
14373  692 
by (simp add: cis_def) 
14323  693 

14374  694 
lemma Im_cis [simp]: "Im(cis a) = sin a" 
14373  695 
by (simp add: cis_def) 
14323  696 

697 
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" 

14334  698 
by (auto simp add: DeMoivre) 
14323  699 

700 
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" 

14334  701 
by (auto simp add: DeMoivre) 
14323  702 

703 
lemma expi_add: "expi(a + b) = expi(a) * expi(b)" 

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

704 
by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac) 
14323  705 

14374  706 
lemma expi_zero [simp]: "expi (0::complex) = 1" 
14373  707 
by (simp add: expi_def) 
14323  708 

14374  709 
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a" 
14373  710 
apply (insert rcis_Ex [of z]) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

711 
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric]) 
14334  712 
apply (rule_tac x = "ii * complex_of_real a" in exI, auto) 
14323  713 
done 
714 

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

715 
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1" 
23125
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cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

716 
by (simp add: expi_def cis_def) 
14387
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Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

717 

13957  718 
end 