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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 "../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) = (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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instance complex :: zero 
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complex_zero_def: 
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"0 \<equiv> Complex 0 0" .. 
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instance complex :: plus 

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complex_add_def: 

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"x + y \<equiv> Complex (Re x + Re y) (Im x + Im y)" .. 
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instance complex :: minus 
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complex_minus_def: 
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" x \<equiv> Complex ( Re x) ( Im x)" 
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complex_diff_def: 
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"x  y \<equiv> x +  y" .. 
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lemma Complex_eq_0 [simp]: "(Complex a b = 0) = (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]: " (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 complex :: ab_group_add 
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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 add_assoc) 
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show "x + y = y + x" 
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by (simp add: expand_complex_eq add_commute) 
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show "0 + x = x" 
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by (simp add: expand_complex_eq) 
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show " x + x = 0" 
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by (simp add: expand_complex_eq) 
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show "x  y = x +  y" 
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qed 
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subsection {* Multiplication and Division *} 
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instance complex :: one 
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complex_one_def: 
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"1 \<equiv> Complex 1 0" .. 
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instance complex :: times 
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complex_mult_def: 
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"x * y \<equiv> Complex (Re x * Re y  Im x * Im y) (Re x * Im y + Im x * Re y)" .. 
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instance complex :: inverse 
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complex_inverse_def: 
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"inverse x \<equiv> 
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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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complex_divide_def: 
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"x / y \<equiv> 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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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_eq_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_eq_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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176 

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177 

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subsection {* Exponentiation *} 
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179 

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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  191 
qed 
14335  192 

14323  193 

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subsection {* Numerals and Arithmetic *} 
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instance complex :: number 
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complex_number_of_def: 
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"number_of w \<equiv> of_int w" .. 
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instance complex :: number_ring 
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by (intro_classes, simp only: complex_number_of_def) 
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202 

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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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unfolding number_ring_class.axioms by (rule complex_Re_of_int) 
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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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224 

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225 

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subsection {* Scalar Multiplication *} 
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instance complex :: scaleR 
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complex_scaleR_def: 
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"scaleR r x \<equiv> 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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235 

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lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x" 
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238 

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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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251 
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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254 
by (simp add: expand_complex_eq ring_eq_simps) 
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255 
show "x * scaleR a y = scaleR a (x * y)" 
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256 
by (simp add: expand_complex_eq ring_eq_simps) 
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257 
qed 
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258 

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259 

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subsection{* Properties of Embedding from Reals *} 
14323  261 

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abbreviation 
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complex_of_real :: "real \<Rightarrow> complex" where 
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264 
"complex_of_real \<equiv> of_real" 
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265 

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lemma complex_of_real_def: "complex_of_real r = Complex r 0" 
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267 
by (simp add: of_real_def complex_scaleR_def) 
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268 

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lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" 
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270 
by (simp add: complex_of_real_def) 
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271 

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lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" 
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273 
by (simp add: complex_of_real_def) 
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274 

14377  275 
lemma Complex_add_complex_of_real [simp]: 
276 
"Complex x y + complex_of_real r = Complex (x+r) y" 

277 
by (simp add: complex_of_real_def) 

278 

279 
lemma complex_of_real_add_Complex [simp]: 

280 
"complex_of_real r + Complex x y = Complex (r+x) y" 

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281 
by (simp add: complex_of_real_def) 
14377  282 

283 
lemma Complex_mult_complex_of_real: 

284 
"Complex x y * complex_of_real r = Complex (x*r) (y*r)" 

285 
by (simp add: complex_of_real_def) 

286 

287 
lemma complex_of_real_mult_Complex: 

288 
"complex_of_real r * Complex x y = Complex (r*x) (r*y)" 

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289 
by (simp add: complex_of_real_def) 
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290 

14377  291 

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292 
subsection {* Vector Norm *} 
14323  293 

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294 
instance complex :: norm 
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complex_norm_def: 
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"norm z \<equiv> sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" .. 
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297 

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298 
abbreviation 
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cmod :: "complex \<Rightarrow> real" where 
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300 
"cmod \<equiv> norm" 
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301 

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lemmas cmod_def = complex_norm_def 
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303 

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304 
lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)" 
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305 
by (simp add: complex_norm_def) 
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307 
instance complex :: real_normed_field 
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308 
proof 
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309 
fix r :: real and x y :: complex 
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310 
show "0 \<le> norm x" 
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311 
by (induct x) simp 
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312 
show "(norm x = 0) = (x = 0)" 
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313 
by (induct x) simp 
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314 
show "norm (x + y) \<le> norm x + norm y" 
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315 
by (induct x, induct y) 
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316 
(simp add: real_sqrt_sum_squares_triangle_ineq) 
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317 
show "norm (scaleR r x) = \<bar>r\<bar> * norm x" 
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318 
by (induct x) 
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319 
(simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult) 
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320 
show "norm (x * y) = norm x * norm y" 
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321 
by (induct x, induct y) 
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322 
(simp add: real_sqrt_mult [symmetric] power2_eq_square ring_eq_simps) 
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323 
qed 
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324 

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325 
lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1" 
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326 
by simp 
14323  327 

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328 
lemma cmod_complex_polar [simp]: 
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329 
"cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r" 
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330 
by (simp add: norm_mult) 
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331 

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332 
lemma complex_Re_le_cmod: "Re x \<le> cmod x" 
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333 
unfolding complex_norm_def 
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334 
by (rule real_sqrt_sum_squares_ge1) 
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335 

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336 
lemma complex_mod_minus_le_complex_mod [simp]: " cmod x \<le> cmod x" 
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337 
by (rule order_trans [OF _ norm_ge_zero], simp) 
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338 

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339 
lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a)  cmod b \<le> cmod a" 
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340 
by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp) 
14323  341 

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342 
lemmas real_sum_squared_expand = power2_sum [where 'a=real] 
14323  343 

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344 

23123  345 
subsection {* Completeness of the Complexes *} 
346 

347 
interpretation Re: bounded_linear ["Re"] 

348 
apply (unfold_locales, simp, simp) 

349 
apply (rule_tac x=1 in exI) 

350 
apply (simp add: complex_norm_def) 

351 
done 

352 

353 
interpretation Im: bounded_linear ["Im"] 

354 
apply (unfold_locales, simp, simp) 

355 
apply (rule_tac x=1 in exI) 

356 
apply (simp add: complex_norm_def) 

357 
done 

358 

359 
lemma LIMSEQ_Complex: 

360 
"\<lbrakk>X > a; Y > b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) > Complex a b" 

361 
apply (rule LIMSEQ_I) 

362 
apply (subgoal_tac "0 < r / sqrt 2") 

363 
apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe) 

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

365 
apply (rename_tac M N, rule_tac x="max M N" in exI, safe) 

366 
apply (simp add: real_sqrt_sum_squares_less) 

367 
apply (simp add: divide_pos_pos) 

368 
done 

369 

370 
instance complex :: banach 

371 
proof 

372 
fix X :: "nat \<Rightarrow> complex" 

373 
assume X: "Cauchy X" 

374 
from Re.Cauchy [OF X] have 1: "(\<lambda>n. Re (X n)) > lim (\<lambda>n. Re (X n))" 

375 
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) 

376 
from Im.Cauchy [OF X] have 2: "(\<lambda>n. Im (X n)) > lim (\<lambda>n. Im (X n))" 

377 
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) 

378 
have "X > Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))" 

379 
using LIMSEQ_Complex [OF 1 2] by simp 

380 
thus "convergent X" 

381 
by (rule convergentI) 

382 
qed 

383 

384 

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385 
subsection {* The Complex Number @{term "\<i>"} *} 
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386 

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387 
definition 
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388 
"ii" :: complex ("\<i>") where 
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389 
i_def: "ii \<equiv> Complex 0 1" 
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390 

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

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394 
lemma complex_Im_i [simp]: "Im ii = 1" 
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395 
by (simp add: i_def) 
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396 

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

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400 
lemma complex_i_not_zero [simp]: "ii \<noteq> 0" 
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401 
by (simp add: expand_complex_eq) 
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402 

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

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406 
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w" 
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407 
by (simp add: expand_complex_eq) 
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408 

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

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

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

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

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421 
lemma i_squared [simp]: "ii * ii = 1" 
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422 
by (simp add: i_def) 
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423 

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

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427 
lemma inverse_i [simp]: "inverse ii =  ii" 
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428 
by (rule inverse_unique, simp) 
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429 

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430 

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431 
subsection {* Complex Conjugation *} 
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432 

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433 
definition 
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434 
cnj :: "complex \<Rightarrow> complex" where 
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435 
"cnj z = Complex (Re z) ( Im z)" 
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436 

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437 
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a ( b)" 
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438 
by (simp add: cnj_def) 
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439 

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440 
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x" 
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441 
by (simp add: cnj_def) 
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442 

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443 
lemma complex_Im_cnj [simp]: "Im (cnj x) =  Im x" 
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444 
by (simp add: cnj_def) 
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445 

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446 
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" 
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447 
by (simp add: expand_complex_eq) 
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448 

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449 
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" 
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450 
by (simp add: cnj_def) 
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451 

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452 
lemma complex_cnj_zero [simp]: "cnj 0 = 0" 
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453 
by (simp add: expand_complex_eq) 
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454 

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455 
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" 
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456 
by (simp add: expand_complex_eq) 
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457 

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458 
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y" 
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459 
by (simp add: expand_complex_eq) 
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460 

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461 
lemma complex_cnj_diff: "cnj (x  y) = cnj x  cnj y" 
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462 
by (simp add: expand_complex_eq) 
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463 

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464 
lemma complex_cnj_minus: "cnj ( x) =  cnj x" 
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465 
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466 

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467 
lemma complex_cnj_one [simp]: "cnj 1 = 1" 
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468 
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469 

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470 
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y" 
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471 
by (simp add: expand_complex_eq) 
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472 

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473 
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)" 
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474 
by (simp add: complex_inverse_def) 
14323  475 

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476 
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y" 
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477 
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) 
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478 

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479 
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n" 
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480 
by (induct n, simp_all add: complex_cnj_mult) 
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481 

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482 
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" 
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483 
by (simp add: expand_complex_eq) 
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484 

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485 
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" 
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486 
by (simp add: expand_complex_eq) 
6f7b5b96241f
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huffman
parents:
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diff
changeset

487 

6f7b5b96241f
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huffman
parents:
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diff
changeset

488 
lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

489 
by (simp add: expand_complex_eq) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

490 

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

491 
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

492 
by (simp add: expand_complex_eq) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

493 

6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
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parents:
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494 
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" 
6f7b5b96241f
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changeset

495 
by (simp add: complex_norm_def) 
14323  496 

23125
6f7b5b96241f
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parents:
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diff
changeset

497 
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

498 
by (simp add: expand_complex_eq) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

499 

6f7b5b96241f
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diff
changeset

500 
lemma complex_cnj_i [simp]: "cnj ii =  ii" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

501 
by (simp add: expand_complex_eq) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

502 

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

503 
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" 
6f7b5b96241f
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huffman
parents:
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diff
changeset

504 
by (simp add: expand_complex_eq) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

505 

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

506 
lemma complex_diff_cnj: "z  cnj z = complex_of_real (2 * Im z) * ii" 
6f7b5b96241f
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changeset

507 
by (simp add: expand_complex_eq) 
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
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parents:
14353
diff
changeset

508 

23125
6f7b5b96241f
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huffman
parents:
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changeset

509 
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" 
6f7b5b96241f
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huffman
parents:
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changeset

510 
by (simp add: expand_complex_eq power2_eq_square) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

511 

6f7b5b96241f
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512 
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>" 
6f7b5b96241f
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513 
by (simp add: norm_mult power2_eq_square) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
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diff
changeset

514 

6f7b5b96241f
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changeset

515 
interpretation cnj: bounded_linear ["cnj"] 
6f7b5b96241f
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changeset

516 
apply (unfold_locales) 
6f7b5b96241f
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parents:
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changeset

517 
apply (rule complex_cnj_add) 
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changeset

518 
apply (rule complex_cnj_scaleR) 
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519 
apply (rule_tac x=1 in exI, simp) 
6f7b5b96241f
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520 
done 
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
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521 

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types complex and hcomplex are now instances of class ringpower:
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522 

22972
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generalized sgn function to work on any real normed vector space
huffman
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523 
subsection{*The Functions @{term sgn} and @{term arg}*} 
14323  524 

22972
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generalized sgn function to work on any real normed vector space
huffman
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525 
text {* Argand *} 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

526 

21404
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changeset

527 
definition 
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
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528 
arg :: "complex => real" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

529 
"arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & pi < a & a \<le> pi)" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

530 

14374  531 
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" 
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22968
diff
changeset

532 
by (simp add: sgn_def divide_inverse scaleR_conv_of_real mult_commute) 
14323  533 

534 
lemma i_mult_eq: "ii * ii = complex_of_real (1)" 

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

535 
by (simp add: i_def complex_of_real_def) 
14323  536 

14374  537 
lemma i_mult_eq2 [simp]: "ii * ii = (1::complex)" 
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

538 
by (simp add: i_def complex_one_def) 
14323  539 

14374  540 
lemma complex_eq_cancel_iff2 [simp]: 
14377  541 
"(Complex x y = complex_of_real xa) = (x = xa & y = 0)" 
542 
by (simp add: complex_of_real_def) 

14323  543 

14374  544 
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" 
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22968
diff
changeset

545 
unfolding sgn_def by (simp add: divide_inverse) 
14323  546 

14374  547 
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" 
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22968
diff
changeset

548 
unfolding sgn_def by (simp add: divide_inverse) 
14323  549 

550 
lemma complex_inverse_complex_split: 

551 
"inverse(complex_of_real x + ii * complex_of_real y) = 

552 
complex_of_real(x/(x ^ 2 + y ^ 2))  

553 
ii * complex_of_real(y/(x ^ 2 + y ^ 2))" 

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

554 
by (simp add: complex_of_real_def i_def diff_minus divide_inverse) 
14323  555 

556 
(**) 

557 
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) 

558 
(* many of the theorems are not used  so should they be kept? *) 

559 
(**) 

560 

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

561 
lemma cos_arg_i_mult_zero_pos: 
14377  562 
"0 < y ==> cos (arg(Complex 0 y)) = 0" 
14373  563 
apply (simp add: arg_def abs_if) 
14334  564 
apply (rule_tac a = "pi/2" in someI2, auto) 
565 
apply (rule order_less_trans [of _ 0], auto) 

14323  566 
done 
567 

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

568 
lemma cos_arg_i_mult_zero_neg: 
14377  569 
"y < 0 ==> cos (arg(Complex 0 y)) = 0" 
14373  570 
apply (simp add: arg_def abs_if) 
14334  571 
apply (rule_tac a = " pi/2" in someI2, auto) 
572 
apply (rule order_trans [of _ 0], auto) 

14323  573 
done 
574 

14374  575 
lemma cos_arg_i_mult_zero [simp]: 
14377  576 
"y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0" 
577 
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) 

14323  578 

579 

580 
subsection{*Finally! Polar Form for Complex Numbers*} 

581 

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

582 
definition 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

583 

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

584 
(* abbreviation for (cos a + i sin a) *) 
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset

585 
cis :: "real => complex" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

586 
"cis a = Complex (cos a) (sin a)" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

587 

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

588 
definition 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

589 
(* abbreviation for r*(cos a + i sin a) *) 
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset

590 
rcis :: "[real, real] => complex" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

591 
"rcis r a = complex_of_real r * cis a" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

592 

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

593 
definition 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

594 
(* e ^ (x + iy) *) 
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset

595 
expi :: "complex => complex" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

596 
"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

597 

14374  598 
lemma complex_split_polar: 
14377  599 
"\<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

600 
apply (induct z) 
14377  601 
apply (auto simp add: polar_Ex complex_of_real_mult_Complex) 
14323  602 
done 
603 

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

604 
lemma rcis_Ex: "\<exists>r a. z = rcis r a" 
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

605 
apply (induct z) 
14377  606 
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex) 
14323  607 
done 
608 

14374  609 
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" 
14373  610 
by (simp add: rcis_def cis_def) 
14323  611 

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

612 
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" 
14373  613 
by (simp add: rcis_def cis_def) 
14323  614 

14377  615 
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>" 
616 
proof  

617 
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

618 
by (simp only: power_mult_distrib right_distrib) 
14377  619 
thus ?thesis by simp 
620 
qed 

14323  621 

14374  622 
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" 
14377  623 
by (simp add: rcis_def cis_def sin_cos_squared_add2_mult) 
14323  624 

14374  625 
lemma complex_Re_cnj [simp]: "Re(cnj z) = Re z" 
14373  626 
by (induct z, simp add: complex_cnj) 
14323  627 

14374  628 
lemma complex_Im_cnj [simp]: "Im(cnj z) =  Im z" 
629 
by (induct z, simp add: complex_cnj) 

630 

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

631 
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

632 
by (simp add: cmod_def power2_eq_square) 
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

633 

14374  634 
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

635 
by simp 
14323  636 

637 

638 
(**) 

639 
(* (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b) *) 

640 
(**) 

641 

642 
lemma cis_rcis_eq: "cis a = rcis 1 a" 

14373  643 
by (simp add: rcis_def) 
14323  644 

14374  645 
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" 
15013  646 
by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib 
647 
complex_of_real_def) 

14323  648 

649 
lemma cis_mult: "cis a * cis b = cis (a + b)" 

14373  650 
by (simp add: cis_rcis_eq rcis_mult) 
14323  651 

14374  652 
lemma cis_zero [simp]: "cis 0 = 1" 
14377  653 
by (simp add: cis_def complex_one_def) 
14323  654 

14374  655 
lemma rcis_zero_mod [simp]: "rcis 0 a = 0" 
14373  656 
by (simp add: rcis_def) 
14323  657 

14374  658 
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" 
14373  659 
by (simp add: rcis_def) 
14323  660 

661 
lemma complex_of_real_minus_one: 

662 
"complex_of_real ((1::real)) = (1::complex)" 

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

663 
by (simp add: complex_of_real_def complex_one_def) 
14323  664 

14374  665 
lemma complex_i_mult_minus [simp]: "ii * (ii * x) =  x" 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

666 
by (simp add: mult_assoc [symmetric]) 
14323  667 

668 

669 
lemma cis_real_of_nat_Suc_mult: 

670 
"cis (real (Suc n) * a) = cis a * cis (real n * a)" 

14377  671 
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib) 
14323  672 

673 
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" 

674 
apply (induct_tac "n") 

675 
apply (auto simp add: cis_real_of_nat_Suc_mult) 

676 
done 

677 

14374  678 
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" 
22890  679 
by (simp add: rcis_def power_mult_distrib DeMoivre) 
14323  680 

14374  681 
lemma cis_inverse [simp]: "inverse(cis a) = cis (a)" 
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset

682 
by (simp add: cis_def complex_inverse_complex_split diff_minus) 
14323  683 

684 
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (a)" 

22884  685 
by (simp add: divide_inverse rcis_def) 
14323  686 

687 
lemma cis_divide: "cis a / cis b = cis (a  b)" 

14373  688 
by (simp add: complex_divide_def cis_mult real_diff_def) 
14323  689 

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

690 
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a  b)" 
14373  691 
apply (simp add: complex_divide_def) 
692 
apply (case_tac "r2=0", simp) 

693 
apply (simp add: rcis_inverse rcis_mult real_diff_def) 

14323  694 
done 
695 

14374  696 
lemma Re_cis [simp]: "Re(cis a) = cos a" 
14373  697 
by (simp add: cis_def) 
14323  698 

14374  699 
lemma Im_cis [simp]: "Im(cis a) = sin a" 
14373  700 
by (simp add: cis_def) 
14323  701 

702 
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" 

14334  703 
by (auto simp add: DeMoivre) 
14323  704 

705 
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" 

14334  706 
by (auto simp add: DeMoivre) 
14323  707 

708 
lemma expi_add: "expi(a + b) = expi(a) * expi(b)" 

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

709 
by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac) 
14323  710 

14374  711 
lemma expi_zero [simp]: "expi (0::complex) = 1" 
14373  712 
by (simp add: expi_def) 
14323  713 

14374  714 
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a" 
14373  715 
apply (insert rcis_Ex [of z]) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

716 
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric]) 
14334  717 
apply (rule_tac x = "ii * complex_of_real a" in exI, auto) 
14323  718 
done 
719 

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

720 
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

721 
by (simp add: expi_def cis_def) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

722 

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

723 
(*examples: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

724 
print_depth 22 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

725 
set timing; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

726 
set trace_simp; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

727 
fun test s = (Goal s, by (Simp_tac 1)); 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

728 

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

729 
test "23 * ii + 45 * ii= (x::complex)"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

730 

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

731 
test "5 * ii + 12  45 * ii= (x::complex)"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

732 
test "5 * ii + 40  12 * ii + 9 = (x::complex) + 89 * ii"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

733 
test "5 * ii + 40  12 * ii + 9  78 = (x::complex) + 89 * ii"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

734 

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

735 
test "l + 10 * ii + 90 + 3*l + 9 + 45 * ii= (x::complex)"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

736 
test "87 + 10 * ii + 90 + 3*7 + 9 + 45 * ii= (x::complex)"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

737 

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

738 

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

739 
fun test s = (Goal s; by (Asm_simp_tac 1)); 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

740 

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

741 
test "x*k = k*(y::complex)"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

742 
test "k = k*(y::complex)"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

743 
test "a*(b*c) = (b::complex)"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

744 
test "a*(b*c) = d*(b::complex)*(x*a)"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

745 

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

746 

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

747 
test "(x*k) / (k*(y::complex)) = (uu::complex)"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

748 
test "(k) / (k*(y::complex)) = (uu::complex)"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

749 
test "(a*(b*c)) / ((b::complex)) = (uu::complex)"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

750 
test "(a*(b*c)) / (d*(b::complex)*(x*a)) = (uu::complex)"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

751 

15003  752 
FIXME: what do we do about this? 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

753 
test "a*(b*c)/(y*z) = d*(b::complex)*(x*a)/z"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

754 
*) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

755 

13957  756 
end 