author  haftmann 
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(* Title: Complex.thy 
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ID: $Id$ 
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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 "../Real/Real" "../Hyperreal/Transcendental" 
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begin 
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datatype complex = Complex real real 
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consts Re :: "complex \<Rightarrow> real" 
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primrec Re: "Re (Complex x y) = x" 
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consts Im :: "complex \<Rightarrow> real" 
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primrec 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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by (induct x, induct y) simp 
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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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instance proof 
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fix x y z :: complex 

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show "(x + y) + z = x + (y + z)" 

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by (simp add: expand_complex_eq complex_add_def add_assoc) 

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show "x + y = y + x" 

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by (simp add: expand_complex_eq complex_add_def add_commute) 

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show "0 + x = x" 

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by (simp add: expand_complex_eq complex_add_def complex_zero_def) 

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show " x + x = 0" 

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by (simp add: expand_complex_eq complex_add_def complex_zero_def complex_minus_def) 

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show "x  y = x +  y" 

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by (simp add: expand_complex_eq complex_add_def complex_zero_def complex_diff_def) 

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qed 

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end 
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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_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_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_minus [simp]: " (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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subsection {* Multiplication and Division *} 
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instantiation complex :: "{one, times, inverse}" 
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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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instance .. 
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end 
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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 complex :: field 

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proof 

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fix x y z :: complex 
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show "(x * y) * z = x * (y * z)" 
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by (simp add: expand_complex_eq ring_simps) 
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show "x * y = y * x" 
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by (simp add: expand_complex_eq mult_commute add_commute) 
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show "1 * x = x" 
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by (simp add: expand_complex_eq) 
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show "0 \<noteq> (1::complex)" 
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by (simp add: expand_complex_eq) 
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show "(x + y) * z = x * z + y * z" 
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by (simp add: expand_complex_eq ring_simps) 
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show "x / y = x * inverse y" 
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by (simp only: complex_divide_def) 
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show "x \<noteq> 0 \<Longrightarrow> inverse x * x = 1" 
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by (induct x, simp add: power2_eq_square add_divide_distrib [symmetric]) 
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qed 
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instance complex :: division_by_zero 
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proof 

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show "inverse 0 = (0::complex)" 
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by (simp add: complex_inverse_def) 
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qed 
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subsection {* Exponentiation *} 
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instance complex :: power .. 
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primrec 
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complexpow_0: "z ^ 0 = 1" 
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complexpow_Suc: "z ^ (Suc n) = (z::complex) * (z ^ n)" 
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instance complex :: recpower 
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proof 
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fix x :: complex and n :: nat 
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show "x ^ 0 = 1" by simp 
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show "x ^ Suc n = x * x ^ n" by simp 
14373  199 
qed 
14335  200 

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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 
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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 :: scaleR 
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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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instance .. 
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end 
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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 complex :: real_field 
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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: expand_complex_eq right_distrib) 
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show "scaleR (a + b) x = scaleR a x + scaleR b x" 
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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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subsection{* Properties of Embedding from Reals *} 
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abbreviation 
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complex_of_real :: "real \<Rightarrow> complex" where 
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"complex_of_real \<equiv> of_real" 
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lemma complex_of_real_def: "complex_of_real r = Complex r 0" 
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by (simp add: of_real_def complex_scaleR_def) 
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lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" 
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by (simp add: complex_of_real_def) 
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lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" 
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by (simp add: complex_of_real_def) 
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14377  293 
lemma Complex_add_complex_of_real [simp]: 
294 
"Complex x y + complex_of_real r = Complex (x+r) y" 

295 
by (simp add: complex_of_real_def) 

296 

297 
lemma complex_of_real_add_Complex [simp]: 

298 
"complex_of_real r + Complex x y = Complex (r+x) y" 

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by (simp add: complex_of_real_def) 
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301 
lemma Complex_mult_complex_of_real: 

302 
"Complex x y * complex_of_real r = Complex (x*r) (y*r)" 

303 
by (simp add: complex_of_real_def) 

304 

305 
lemma complex_of_real_mult_Complex: 

306 
"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  309 

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subsection {* Vector Norm *} 
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instantiation complex :: norm 
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begin 
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314 

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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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instance .. 
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end 
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abbreviation 
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cmod :: "complex \<Rightarrow> real" where 
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"cmod \<equiv> norm" 
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instantiation complex :: sgn 
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begin 
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definition 
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complex_sgn_def: "sgn x = x /\<^sub>R cmod x" 
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instance .. 
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end 
24506  335 

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lemmas cmod_def = complex_norm_def 
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337 

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lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)" 
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by (simp add: complex_norm_def) 
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instance complex :: real_normed_field 
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proof 
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fix r :: real and x y :: complex 
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344 
show "0 \<le> norm x" 
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345 
by (induct x) simp 
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show "(norm x = 0) = (x = 0)" 
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347 
by (induct x) simp 
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show "norm (x + y) \<le> norm x + norm y" 
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349 
by (induct x, induct y) 
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350 
(simp add: real_sqrt_sum_squares_triangle_ineq) 
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351 
show "norm (scaleR r x) = \<bar>r\<bar> * norm x" 
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352 
by (induct x) 
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353 
(simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult) 
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354 
show "norm (x * y) = norm x * norm y" 
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355 
by (induct x, induct y) 
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356 
(simp add: real_sqrt_mult [symmetric] power2_eq_square ring_simps) 
24506  357 
show "sgn x = x /\<^sub>R cmod x" by(simp add: complex_sgn_def) 
24520  358 
qed 
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359 

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360 
lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1" 
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361 
by simp 
14323  362 

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363 
lemma cmod_complex_polar [simp]: 
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364 
"cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r" 
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365 
by (simp add: norm_mult) 
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366 

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367 
lemma complex_Re_le_cmod: "Re x \<le> cmod x" 
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368 
unfolding complex_norm_def 
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369 
by (rule real_sqrt_sum_squares_ge1) 
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370 

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371 
lemma complex_mod_minus_le_complex_mod [simp]: " cmod x \<le> cmod x" 
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372 
by (rule order_trans [OF _ norm_ge_zero], simp) 
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373 

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374 
lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a)  cmod b \<le> cmod a" 
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375 
by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp) 
14323  376 

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377 
lemmas real_sum_squared_expand = power2_sum [where 'a=real] 
14323  378 

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379 

23123  380 
subsection {* Completeness of the Complexes *} 
381 

382 
interpretation Re: bounded_linear ["Re"] 

383 
apply (unfold_locales, simp, simp) 

384 
apply (rule_tac x=1 in exI) 

385 
apply (simp add: complex_norm_def) 

386 
done 

387 

388 
interpretation Im: bounded_linear ["Im"] 

389 
apply (unfold_locales, simp, simp) 

390 
apply (rule_tac x=1 in exI) 

391 
apply (simp add: complex_norm_def) 

392 
done 

393 

394 
lemma LIMSEQ_Complex: 

395 
"\<lbrakk>X > a; Y > b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) > Complex a b" 

396 
apply (rule LIMSEQ_I) 

397 
apply (subgoal_tac "0 < r / sqrt 2") 

398 
apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe) 

399 
apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe) 

400 
apply (rename_tac M N, rule_tac x="max M N" in exI, safe) 

401 
apply (simp add: real_sqrt_sum_squares_less) 

402 
apply (simp add: divide_pos_pos) 

403 
done 

404 

405 
instance complex :: banach 

406 
proof 

407 
fix X :: "nat \<Rightarrow> complex" 

408 
assume X: "Cauchy X" 

409 
from Re.Cauchy [OF X] have 1: "(\<lambda>n. Re (X n)) > lim (\<lambda>n. Re (X n))" 

410 
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) 

411 
from Im.Cauchy [OF X] have 2: "(\<lambda>n. Im (X n)) > lim (\<lambda>n. Im (X n))" 

412 
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) 

413 
have "X > Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))" 

414 
using LIMSEQ_Complex [OF 1 2] by simp 

415 
thus "convergent X" 

416 
by (rule convergentI) 

417 
qed 

418 

419 

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420 
subsection {* The Complex Number @{term "\<i>"} *} 
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421 

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422 
definition 
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423 
"ii" :: complex ("\<i>") where 
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424 
i_def: "ii \<equiv> Complex 0 1" 
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425 

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426 
lemma complex_Re_i [simp]: "Re ii = 0" 
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427 
by (simp add: i_def) 
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428 

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429 
lemma complex_Im_i [simp]: "Im ii = 1" 
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430 
by (simp add: i_def) 
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431 

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432 
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)" 
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433 
by (simp add: i_def) 
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434 

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435 
lemma complex_i_not_zero [simp]: "ii \<noteq> 0" 
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436 
by (simp add: expand_complex_eq) 
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437 

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438 
lemma complex_i_not_one [simp]: "ii \<noteq> 1" 
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439 
by (simp add: expand_complex_eq) 
23124  440 

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441 
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w" 
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442 
by (simp add: expand_complex_eq) 
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443 

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444 
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex ( b) a" 
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445 
by (simp add: expand_complex_eq) 
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446 

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447 
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex ( b) a" 
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448 
by (simp add: expand_complex_eq) 
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449 

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450 
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" 
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451 
by (simp add: i_def complex_of_real_def) 
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452 

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453 
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" 
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454 
by (simp add: i_def complex_of_real_def) 
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455 

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456 
lemma i_squared [simp]: "ii * ii = 1" 
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457 
by (simp add: i_def) 
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458 

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459 
lemma power2_i [simp]: "ii\<twosuperior> = 1" 
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460 
by (simp add: power2_eq_square) 
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461 

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462 
lemma inverse_i [simp]: "inverse ii =  ii" 
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463 
by (rule inverse_unique, simp) 
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464 

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465 

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466 
subsection {* Complex Conjugation *} 
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467 

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468 
definition 
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469 
cnj :: "complex \<Rightarrow> complex" where 
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470 
"cnj z = Complex (Re z) ( Im z)" 
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471 

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472 
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a ( b)" 
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473 
by (simp add: cnj_def) 
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474 

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475 
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x" 
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476 
by (simp add: cnj_def) 
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477 

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478 
lemma complex_Im_cnj [simp]: "Im (cnj x) =  Im x" 
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479 
by (simp add: cnj_def) 
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480 

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481 
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" 
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482 
by (simp add: expand_complex_eq) 
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483 

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484 
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" 
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485 
by (simp add: cnj_def) 
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486 

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487 
lemma complex_cnj_zero [simp]: "cnj 0 = 0" 
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488 
by (simp add: expand_complex_eq) 
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489 

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490 
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" 
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491 
by (simp add: expand_complex_eq) 
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492 

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493 
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y" 
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494 
by (simp add: expand_complex_eq) 
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495 

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496 
lemma complex_cnj_diff: "cnj (x  y) = cnj x  cnj y" 
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497 
by (simp add: expand_complex_eq) 
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498 

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499 
lemma complex_cnj_minus: "cnj ( x) =  cnj x" 
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500 
by (simp add: expand_complex_eq) 
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501 

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502 
lemma complex_cnj_one [simp]: "cnj 1 = 1" 
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503 
by (simp add: expand_complex_eq) 
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504 

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505 
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y" 
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506 
by (simp add: expand_complex_eq) 
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507 

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508 
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)" 
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509 
by (simp add: complex_inverse_def) 
14323  510 

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511 
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y" 
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512 
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) 
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513 

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514 
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n" 
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515 
by (induct n, simp_all add: complex_cnj_mult) 
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516 

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517 
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" 
6f7b5b96241f
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518 
by (simp add: expand_complex_eq) 
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changeset

519 

6f7b5b96241f
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520 
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" 
6f7b5b96241f
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521 
by (simp add: expand_complex_eq) 
6f7b5b96241f
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changeset

522 

6f7b5b96241f
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changeset

523 
lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w" 
6f7b5b96241f
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524 
by (simp add: expand_complex_eq) 
6f7b5b96241f
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huffman
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changeset

525 

6f7b5b96241f
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526 
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)" 
6f7b5b96241f
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527 
by (simp add: expand_complex_eq) 
6f7b5b96241f
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huffman
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changeset

528 

6f7b5b96241f
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529 
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" 
6f7b5b96241f
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530 
by (simp add: complex_norm_def) 
14323  531 

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532 
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" 
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533 
by (simp add: expand_complex_eq) 
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changeset

534 

6f7b5b96241f
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535 
lemma complex_cnj_i [simp]: "cnj ii =  ii" 
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536 
by (simp add: expand_complex_eq) 
6f7b5b96241f
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huffman
parents:
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diff
changeset

537 

6f7b5b96241f
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changeset

538 
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" 
6f7b5b96241f
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539 
by (simp add: expand_complex_eq) 
6f7b5b96241f
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changeset

540 

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541 
lemma complex_diff_cnj: "z  cnj z = complex_of_real (2 * Im z) * ii" 
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542 
by (simp add: expand_complex_eq) 
14354
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changeset

543 

23125
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544 
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" 
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545 
by (simp add: expand_complex_eq power2_eq_square) 
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546 

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547 
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>" 
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548 
by (simp add: norm_mult power2_eq_square) 
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549 

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550 
interpretation cnj: bounded_linear ["cnj"] 
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551 
apply (unfold_locales) 
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552 
apply (rule complex_cnj_add) 
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553 
apply (rule complex_cnj_scaleR) 
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554 
apply (rule_tac x=1 in exI, simp) 
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555 
done 
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556 

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557 

22972
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558 
subsection{*The Functions @{term sgn} and @{term arg}*} 
14323  559 

22972
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560 
text {* Argand *} 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
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561 

21404
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562 
definition 
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563 
arg :: "complex => real" where 
20557
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complex_of_real abbreviates of_real::real=>complex;
huffman
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diff
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564 
"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;
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565 

14374  566 
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" 
24506  567 
by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute) 
14323  568 

569 
lemma i_mult_eq: "ii * ii = complex_of_real (1)" 

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

570 
by (simp add: i_def complex_of_real_def) 
14323  571 

14374  572 
lemma i_mult_eq2 [simp]: "ii * ii = (1::complex)" 
20725
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parents:
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diff
changeset

573 
by (simp add: i_def complex_one_def) 
14323  574 

14374  575 
lemma complex_eq_cancel_iff2 [simp]: 
14377  576 
"(Complex x y = complex_of_real xa) = (x = xa & y = 0)" 
577 
by (simp add: complex_of_real_def) 

14323  578 

14374  579 
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" 
24506  580 
by (simp add: complex_sgn_def divide_inverse) 
14323  581 

14374  582 
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" 
24506  583 
by (simp add: complex_sgn_def divide_inverse) 
14323  584 

585 
lemma complex_inverse_complex_split: 

586 
"inverse(complex_of_real x + ii * complex_of_real y) = 

587 
complex_of_real(x/(x ^ 2 + y ^ 2))  

588 
ii * complex_of_real(y/(x ^ 2 + y ^ 2))" 

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

589 
by (simp add: complex_of_real_def i_def diff_minus divide_inverse) 
14323  590 

591 
(**) 

592 
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) 

593 
(* many of the theorems are not used  so should they be kept? *) 

594 
(**) 

595 

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diff
changeset

596 
lemma cos_arg_i_mult_zero_pos: 
14377  597 
"0 < y ==> cos (arg(Complex 0 y)) = 0" 
14373  598 
apply (simp add: arg_def abs_if) 
14334  599 
apply (rule_tac a = "pi/2" in someI2, auto) 
600 
apply (rule order_less_trans [of _ 0], auto) 

14323  601 
done 
602 

14354
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parents:
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diff
changeset

603 
lemma cos_arg_i_mult_zero_neg: 
14377  604 
"y < 0 ==> cos (arg(Complex 0 y)) = 0" 
14373  605 
apply (simp add: arg_def abs_if) 
14334  606 
apply (rule_tac a = " pi/2" in someI2, auto) 
607 
apply (rule order_trans [of _ 0], auto) 

14323  608 
done 
609 

14374  610 
lemma cos_arg_i_mult_zero [simp]: 
14377  611 
"y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0" 
612 
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) 

14323  613 

614 

615 
subsection{*Finally! Polar Form for Complex Numbers*} 

616 

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

617 
definition 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
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diff
changeset

618 

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

619 
(* abbreviation for (cos a + i sin a) *) 
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620 
cis :: "real => complex" where 
20557
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complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

621 
"cis a = Complex (cos a) (sin a)" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
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diff
changeset

622 

21404
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623 
definition 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
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parents:
20556
diff
changeset

624 
(* abbreviation for r*(cos a + i sin a) *) 
21404
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changeset

625 
rcis :: "[real, real] => complex" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

626 
"rcis r a = complex_of_real r * cis a" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
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diff
changeset

627 

21404
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changeset

628 
definition 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

629 
(* e ^ (x + iy) *) 
21404
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changeset

630 
expi :: "complex => complex" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

631 
"expi z = complex_of_real(exp (Re z)) * cis (Im z)" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

632 

14374  633 
lemma complex_split_polar: 
14377  634 
"\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))" 
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

635 
apply (induct z) 
14377  636 
apply (auto simp add: polar_Ex complex_of_real_mult_Complex) 
14323  637 
done 
638 

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

639 
lemma rcis_Ex: "\<exists>r a. z = rcis r a" 
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

640 
apply (induct z) 
14377  641 
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex) 
14323  642 
done 
643 

14374  644 
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" 
14373  645 
by (simp add: rcis_def cis_def) 
14323  646 

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

647 
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" 
14373  648 
by (simp add: rcis_def cis_def) 
14323  649 

14377  650 
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>" 
651 
proof  

652 
have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)" 

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

653 
by (simp only: power_mult_distrib right_distrib) 
14377  654 
thus ?thesis by simp 
655 
qed 

14323  656 

14374  657 
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" 
14377  658 
by (simp add: rcis_def cis_def sin_cos_squared_add2_mult) 
14323  659 

14374  660 
lemma complex_Re_cnj [simp]: "Re(cnj z) = Re z" 
14373  661 
by (induct z, simp add: complex_cnj) 
14323  662 

14374  663 
lemma complex_Im_cnj [simp]: "Im(cnj z) =  Im z" 
664 
by (induct z, simp add: complex_cnj) 

665 

23125
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huffman
parents:
23124
diff
changeset

666 
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" 
6f7b5b96241f
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huffman
parents:
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diff
changeset

667 
by (simp add: cmod_def power2_eq_square) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

668 

14374  669 
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" 
23125
6f7b5b96241f
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huffman
parents:
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diff
changeset

670 
by simp 
14323  671 

672 

673 
(**) 

674 
(* (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b) *) 

675 
(**) 

676 

677 
lemma cis_rcis_eq: "cis a = rcis 1 a" 

14373  678 
by (simp add: rcis_def) 
14323  679 

14374  680 
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" 
15013  681 
by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib 
682 
complex_of_real_def) 

14323  683 

684 
lemma cis_mult: "cis a * cis b = cis (a + b)" 

14373  685 
by (simp add: cis_rcis_eq rcis_mult) 
14323  686 

14374  687 
lemma cis_zero [simp]: "cis 0 = 1" 
14377  688 
by (simp add: cis_def complex_one_def) 
14323  689 

14374  690 
lemma rcis_zero_mod [simp]: "rcis 0 a = 0" 
14373  691 
by (simp add: rcis_def) 
14323  692 

14374  693 
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" 
14373  694 
by (simp add: rcis_def) 
14323  695 

696 
lemma complex_of_real_minus_one: 

697 
"complex_of_real ((1::real)) = (1::complex)" 

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

698 
by (simp add: complex_of_real_def complex_one_def) 
14323  699 

14374  700 
lemma complex_i_mult_minus [simp]: "ii * (ii * x) =  x" 
23125
6f7b5b96241f
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huffman
parents:
23124
diff
changeset

701 
by (simp add: mult_assoc [symmetric]) 
14323  702 

703 

704 
lemma cis_real_of_nat_Suc_mult: 

705 
"cis (real (Suc n) * a) = cis a * cis (real n * a)" 

14377  706 
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib) 
14323  707 

708 
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" 

709 
apply (induct_tac "n") 

710 
apply (auto simp add: cis_real_of_nat_Suc_mult) 

711 
done 

712 

14374  713 
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" 
22890  714 
by (simp add: rcis_def power_mult_distrib DeMoivre) 
14323  715 

14374  716 
lemma cis_inverse [simp]: "inverse(cis a) = cis (a)" 
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

717 
by (simp add: cis_def complex_inverse_complex_split diff_minus) 
14323  718 

719 
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (a)" 

22884  720 
by (simp add: divide_inverse rcis_def) 
14323  721 

722 
lemma cis_divide: "cis a / cis b = cis (a  b)" 

14373  723 
by (simp add: complex_divide_def cis_mult real_diff_def) 
14323  724 

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

725 
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a  b)" 
14373  726 
apply (simp add: complex_divide_def) 
727 
apply (case_tac "r2=0", simp) 

728 
apply (simp add: rcis_inverse rcis_mult real_diff_def) 

14323  729 
done 
730 

14374  731 
lemma Re_cis [simp]: "Re(cis a) = cos a" 
14373  732 
by (simp add: cis_def) 
14323  733 

14374  734 
lemma Im_cis [simp]: "Im(cis a) = sin a" 
14373  735 
by (simp add: cis_def) 
14323  736 

737 
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" 

14334  738 
by (auto simp add: DeMoivre) 
14323  739 

740 
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" 

14334  741 
by (auto simp add: DeMoivre) 
14323  742 

743 
lemma expi_add: "expi(a + b) = expi(a) * expi(b)" 

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

744 
by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac) 
14323  745 

14374  746 
lemma expi_zero [simp]: "expi (0::complex) = 1" 
14373  747 
by (simp add: expi_def) 
14323  748 

14374  749 
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a" 
14373  750 
apply (insert rcis_Ex [of z]) 
23125
6f7b5b96241f
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huffman
parents:
23124
diff
changeset

751 
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric]) 
14334  752 
apply (rule_tac x = "ii * complex_of_real a" in exI, auto) 
14323  753 
done 
754 

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

755 
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1" 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

756 
by (simp add: expi_def cis_def) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

757 

13957  758 
end 