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(* Title: HOL/Complex.thy 
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Author: Jacques D. Fleuriot 
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Copyright: 2001 University of Edinburgh 

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Polymorphic treatment of binary arithmetic using axclasses
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Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 
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*) 
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header {* Complex Numbers: Rectangular and Polar Representations *} 
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theory Complex 
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imports Transcendental 
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begin 
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datatype complex = Complex real real 
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primrec Re :: "complex \<Rightarrow> real" 
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where Re: "Re (Complex x y) = x" 

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

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

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by (induct z) simp 

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lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y" 
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lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y" 
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subsection {* Addition and Subtraction *} 
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instantiation complex :: ab_group_add 
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begin 
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definition complex_zero_def: 
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"0 = Complex 0 0" 

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

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

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

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

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

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

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by (simp add: complex_add_def) 

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

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

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by (simp add: complex_add_def) 

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

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

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end 

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

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

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definition complex_inverse_def: 
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"inverse x = 

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Complex (Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)) ( Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2))" 
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definition complex_divide_def: 
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"x / (y\<Colon>complex) = x * inverse y" 

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lemma Complex_eq_1 [simp]: 
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"Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0" 
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by (simp add: complex_one_def) 
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lemma Complex_eq_neg_1 [simp]: 
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"Complex a b =  1 \<longleftrightarrow> 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\<^sup>2 + b\<^sup>2)) ( b / (a\<^sup>2 + b\<^sup>2))" 
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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)\<^sup>2 + (Im x)\<^sup>2)" 
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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)\<^sup>2 + (Im x)\<^sup>2)" 
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by (simp add: complex_inverse_def) 
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instance 
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by intro_classes (simp_all add: complex_mult_def 

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distrib_left distrib_right right_diff_distrib left_diff_distrib 
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complex_inverse_def complex_divide_def 
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power2_eq_square add_divide_distrib [symmetric] 

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complex_eq_iff) 

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end 
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subsection {* Numerals and Arithmetic *} 
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lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n" 
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by (induct n) simp_all 
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lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0" 
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lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z" 
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lemma complex_Im_of_int [simp]: "Im (of_int z) = 0" 
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lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v" 
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using complex_Re_of_int [of "numeral v"] by simp 
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lemma complex_Re_neg_numeral [simp]: "Re ( numeral v) =  numeral v" 
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lemma complex_Im_numeral [simp]: "Im (numeral v) = 0" 
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using complex_Im_of_int [of "numeral v"] by simp 
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lemma complex_Im_neg_numeral [simp]: "Im ( numeral v) = 0" 
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lemma Complex_eq_numeral [simp]: 
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"Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0" 
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by (simp add: complex_eq_iff) 
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lemma Complex_eq_neg_numeral [simp]: 
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"Complex a b =  numeral w \<longleftrightarrow> a =  numeral w \<and> b = 0" 
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subsection {* Scalar Multiplication *} 
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instantiation complex :: real_field 
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definition complex_scaleR_def: 
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"scaleR r x = Complex (r * Re x) (r * Im x)" 

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lemma complex_scaleR [simp]: 
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"scaleR r (Complex a b) = Complex (r * a) (r * b)" 
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lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x" 
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lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x" 
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instance 
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proof 
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fix a b :: real and x y :: complex 
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show "scaleR a (x + y) = scaleR a x + scaleR a y" 
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by (simp add: complex_eq_iff distrib_left) 
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show "scaleR (a + b) x = scaleR a x + scaleR b x" 
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by (simp add: complex_eq_iff distrib_right) 
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show "scaleR a (scaleR b x) = scaleR (a * b) x" 
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by (simp add: complex_eq_iff mult_assoc) 
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219 
show "scaleR 1 x = x" 
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by (simp add: complex_eq_iff) 
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221 
show "scaleR a x * y = scaleR a (x * y)" 
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222 
by (simp add: complex_eq_iff algebra_simps) 
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223 
show "x * scaleR a y = scaleR a (x * y)" 
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224 
by (simp add: complex_eq_iff algebra_simps) 
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225 
qed 
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226 

25712  227 
end 
228 

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229 

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230 
subsection{* Properties of Embedding from Reals *} 
14323  231 

44724  232 
abbreviation complex_of_real :: "real \<Rightarrow> complex" 
233 
where "complex_of_real \<equiv> of_real" 

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234 

56331  235 
declare [[coercion complex_of_real]] 
236 

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lemma complex_of_real_def: "complex_of_real r = Complex r 0" 
44724  238 
by (simp add: of_real_def complex_scaleR_def) 
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239 

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240 
lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" 
44724  241 
by (simp add: complex_of_real_def) 
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242 

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lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" 
44724  244 
by (simp add: complex_of_real_def) 
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14377  246 
lemma Complex_add_complex_of_real [simp]: 
44724  247 
shows "Complex x y + complex_of_real r = Complex (x+r) y" 
248 
by (simp add: complex_of_real_def) 

14377  249 

250 
lemma complex_of_real_add_Complex [simp]: 

44724  251 
shows "complex_of_real r + Complex x y = Complex (r+x) y" 
252 
by (simp add: complex_of_real_def) 

14377  253 

254 
lemma Complex_mult_complex_of_real: 

44724  255 
shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)" 
256 
by (simp add: complex_of_real_def) 

14377  257 

258 
lemma complex_of_real_mult_Complex: 

44724  259 
shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)" 
260 
by (simp add: complex_of_real_def) 

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261 

44841  262 
lemma complex_eq_cancel_iff2 [simp]: 
263 
shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)" 

264 
by (simp add: complex_of_real_def) 

265 

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lemma complex_split_polar: 
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"\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))" 
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268 
by (simp add: complex_eq_iff polar_Ex) 
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269 

14377  270 

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subsection {* Vector Norm *} 
14323  272 

25712  273 
instantiation complex :: real_normed_field 
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begin 
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275 

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definition complex_norm_def: 
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"norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)" 
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44724  279 
abbreviation cmod :: "complex \<Rightarrow> real" 
280 
where "cmod \<equiv> norm" 

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definition complex_sgn_def: 
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"sgn x = x /\<^sub>R cmod x" 
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284 

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285 
definition dist_complex_def: 
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286 
"dist x y = cmod (x  y)" 
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287 

37767  288 
definition open_complex_def: 
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"open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" 
31292  290 

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291 
lemmas cmod_def = complex_norm_def 
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292 

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lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)" 
25712  294 
by (simp add: complex_norm_def) 
22852  295 

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instance proof 
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fix r :: real and x y :: complex and S :: "complex set" 
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298 
show "(norm x = 0) = (x = 0)" 
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299 
by (induct x) simp 
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300 
show "norm (x + y) \<le> norm x + norm y" 
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301 
by (induct x, induct y) 
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302 
(simp add: real_sqrt_sum_squares_triangle_ineq) 
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303 
show "norm (scaleR r x) = \<bar>r\<bar> * norm x" 
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304 
by (induct x) 
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305 
(simp add: power_mult_distrib distrib_left [symmetric] real_sqrt_mult) 
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306 
show "norm (x * y) = norm x * norm y" 
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307 
by (induct x, induct y) 
29667  308 
(simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps) 
31292  309 
show "sgn x = x /\<^sub>R cmod x" 
310 
by (rule complex_sgn_def) 

311 
show "dist x y = cmod (x  y)" 

312 
by (rule dist_complex_def) 

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313 
show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" 
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314 
by (rule open_complex_def) 
24520  315 
qed 
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316 

25712  317 
end 
318 

44761  319 
lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1" 
44724  320 
by simp 
14323  321 

44761  322 
lemma cmod_complex_polar: 
44724  323 
"cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r" 
324 
by (simp add: norm_mult) 

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325 

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326 
lemma complex_Re_le_cmod: "Re x \<le> cmod x" 
44724  327 
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328 
by (rule real_sqrt_sum_squares_ge1) 

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329 

44761  330 
lemma complex_mod_minus_le_complex_mod: " cmod x \<le> cmod x" 
44724  331 
by (rule order_trans [OF _ norm_ge_zero], simp) 
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332 

44761  333 
lemma complex_mod_triangle_ineq2: "cmod(b + a)  cmod b \<le> cmod a" 
44724  334 
by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp) 
14323  335 

26117  336 
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x" 
44724  337 
by (cases x) simp 
26117  338 

339 
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x" 

44724  340 
by (cases x) simp 
341 

44843  342 
text {* Properties of complex signum. *} 
343 

344 
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" 

345 
by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute) 

346 

347 
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" 

348 
by (simp add: complex_sgn_def divide_inverse) 

349 

350 
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" 

351 
by (simp add: complex_sgn_def divide_inverse) 

352 

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353 

23123  354 
subsection {* Completeness of the Complexes *} 
355 

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356 
lemma bounded_linear_Re: "bounded_linear Re" 
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357 
by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def) 
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358 

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359 
lemma bounded_linear_Im: "bounded_linear Im" 
44127  360 
by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def) 
23123  361 

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362 
lemmas tendsto_Re [tendsto_intros] = 
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363 
bounded_linear.tendsto [OF bounded_linear_Re] 
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364 

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365 
lemmas tendsto_Im [tendsto_intros] = 
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366 
bounded_linear.tendsto [OF bounded_linear_Im] 
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367 

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368 
lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re] 
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369 
lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im] 
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370 
lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re] 
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371 
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im] 
23123  372 

36825  373 
lemma tendsto_Complex [tendsto_intros]: 
44724  374 
assumes "(f > a) F" and "(g > b) F" 
375 
shows "((\<lambda>x. Complex (f x) (g x)) > Complex a b) F" 

36825  376 
proof (rule tendstoI) 
377 
fix r :: real assume "0 < r" 

378 
hence "0 < r / sqrt 2" by (simp add: divide_pos_pos) 

44724  379 
have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F" 
380 
using `(f > a) F` and `0 < r / sqrt 2` by (rule tendstoD) 

36825  381 
moreover 
44724  382 
have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F" 
383 
using `(g > b) F` and `0 < r / sqrt 2` by (rule tendstoD) 

36825  384 
ultimately 
44724  385 
show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F" 
36825  386 
by (rule eventually_elim2) 
387 
(simp add: dist_norm real_sqrt_sum_squares_less) 

388 
qed 

389 

23123  390 
instance complex :: banach 
391 
proof 

392 
fix X :: "nat \<Rightarrow> complex" 

393 
assume X: "Cauchy X" 

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394 
from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) > lim (\<lambda>n. Re (X n))" 
23123  395 
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) 
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396 
from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) > lim (\<lambda>n. Im (X n))" 
23123  397 
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) 
398 
have "X > Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))" 

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399 
using tendsto_Complex [OF 1 2] by simp 
23123  400 
thus "convergent X" 
401 
by (rule convergentI) 

402 
qed 

403 

56238
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404 
declare 
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405 
DERIV_power[where 'a=complex, THEN DERIV_cong, 
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406 
unfolded of_nat_def[symmetric], DERIV_intros] 
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407 

23123  408 

44827
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409 
subsection {* The Complex Number $i$ *} 
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410 

44724  411 
definition "ii" :: complex ("\<i>") 
412 
where i_def: "ii \<equiv> Complex 0 1" 

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413 

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414 
lemma complex_Re_i [simp]: "Re ii = 0" 
44724  415 
by (simp add: i_def) 
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416 

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417 
lemma complex_Im_i [simp]: "Im ii = 1" 
44724  418 
by (simp add: i_def) 
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419 

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420 
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)" 
44724  421 
by (simp add: i_def) 
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422 

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423 
lemma norm_ii [simp]: "norm ii = 1" 
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424 
by (simp add: i_def) 
9ba11d41cd1f
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425 

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426 
lemma complex_i_not_zero [simp]: "ii \<noteq> 0" 
44724  427 
by (simp add: complex_eq_iff) 
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428 

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429 
lemma complex_i_not_one [simp]: "ii \<noteq> 1" 
44724  430 
by (simp add: complex_eq_iff) 
23124  431 

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432 
lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w" 
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433 
by (simp add: complex_eq_iff) 
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434 

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435 
lemma complex_i_not_neg_numeral [simp]: "ii \<noteq>  numeral w" 
44724  436 
by (simp add: complex_eq_iff) 
23125
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437 

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438 
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex ( b) a" 
44724  439 
by (simp add: complex_eq_iff) 
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440 

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441 
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex ( b) a" 
44724  442 
by (simp add: complex_eq_iff) 
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443 

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444 
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" 
44724  445 
by (simp add: i_def complex_of_real_def) 
23125
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446 

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447 
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" 
44724  448 
by (simp add: i_def complex_of_real_def) 
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449 

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450 
lemma i_squared [simp]: "ii * ii = 1" 
44724  451 
by (simp add: i_def) 
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452 

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453 
lemma power2_i [simp]: "ii\<^sup>2 = 1" 
44724  454 
by (simp add: power2_eq_square) 
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455 

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456 
lemma inverse_i [simp]: "inverse ii =  ii" 
44724  457 
by (rule inverse_unique, simp) 
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458 

44827
4d1384a1fc82
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459 
lemma complex_i_mult_minus [simp]: "ii * (ii * x) =  x" 
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Complex.thy: move theorems into appropriate subsections
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460 
by (simp add: mult_assoc [symmetric]) 
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461 

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462 

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463 
subsection {* Complex Conjugation *} 
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464 

44724  465 
definition cnj :: "complex \<Rightarrow> complex" where 
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466 
"cnj z = Complex (Re z) ( Im z)" 
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467 

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468 
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a ( b)" 
44724  469 
by (simp add: cnj_def) 
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470 

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471 
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x" 
44724  472 
by (simp add: cnj_def) 
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473 

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474 
lemma complex_Im_cnj [simp]: "Im (cnj x) =  Im x" 
44724  475 
by (simp add: cnj_def) 
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476 

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477 
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" 
44724  478 
by (simp add: complex_eq_iff) 
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479 

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480 
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" 
44724  481 
by (simp add: cnj_def) 
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482 

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483 
lemma complex_cnj_zero [simp]: "cnj 0 = 0" 
44724  484 
by (simp add: complex_eq_iff) 
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485 

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486 
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" 
44724  487 
by (simp add: complex_eq_iff) 
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488 

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489 
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y" 
44724  490 
by (simp add: complex_eq_iff) 
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491 

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492 
lemma complex_cnj_diff: "cnj (x  y) = cnj x  cnj y" 
44724  493 
by (simp add: complex_eq_iff) 
23125
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494 

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495 
lemma complex_cnj_minus: "cnj ( x) =  cnj x" 
44724  496 
by (simp add: complex_eq_iff) 
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497 

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498 
lemma complex_cnj_one [simp]: "cnj 1 = 1" 
44724  499 
by (simp add: complex_eq_iff) 
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500 

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501 
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y" 
44724  502 
by (simp add: complex_eq_iff) 
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503 

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504 
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)" 
44724  505 
by (simp add: complex_inverse_def) 
14323  506 

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507 
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y" 
44724  508 
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) 
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509 

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510 
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n" 
44724  511 
by (induct n, simp_all add: complex_cnj_mult) 
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512 

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513 
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" 
44724  514 
by (simp add: complex_eq_iff) 
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515 

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516 
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" 
44724  517 
by (simp add: complex_eq_iff) 
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518 

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519 
lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w" 
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520 
by (simp add: complex_eq_iff) 
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521 

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522 
lemma complex_cnj_neg_numeral [simp]: "cnj ( numeral w) =  numeral w" 
44724  523 
by (simp add: complex_eq_iff) 
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524 

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525 
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)" 
44724  526 
by (simp add: complex_eq_iff) 
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527 

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528 
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" 
44724  529 
by (simp add: complex_norm_def) 
14323  530 

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531 
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" 
44724  532 
by (simp add: complex_eq_iff) 
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533 

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534 
lemma complex_cnj_i [simp]: "cnj ii =  ii" 
44724  535 
by (simp add: complex_eq_iff) 
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536 

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537 
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" 
44724  538 
by (simp add: complex_eq_iff) 
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539 

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540 
lemma complex_diff_cnj: "z  cnj z = complex_of_real (2 * Im z) * ii" 
44724  541 
by (simp add: complex_eq_iff) 
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542 

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543 
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)" 
44724  544 
by (simp add: complex_eq_iff power2_eq_square) 
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545 

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546 
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2" 
44724  547 
by (simp add: norm_mult power2_eq_square) 
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548 

44827
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549 
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" 
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Complex.thy: move theorems into appropriate subsections
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550 
by (simp add: cmod_def power2_eq_square) 
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551 

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552 
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" 
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553 
by simp 
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554 

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555 
lemma bounded_linear_cnj: "bounded_linear cnj" 
44127  556 
using complex_cnj_add complex_cnj_scaleR 
557 
by (rule bounded_linear_intro [where K=1], simp) 

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558 

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559 
lemmas tendsto_cnj [tendsto_intros] = 
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560 
bounded_linear.tendsto [OF bounded_linear_cnj] 
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561 

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562 
lemmas isCont_cnj [simp] = 
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563 
bounded_linear.isCont [OF bounded_linear_cnj] 
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564 

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565 

55734  566 
subsection{*Basic Lemmas*} 
567 

568 
lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0" 

569 
by (metis complex_Im_zero complex_Re_zero complex_eqI sum_power2_eq_zero_iff) 

570 

571 
lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0" 

572 
by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff) 

573 

574 
lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z" 

575 
apply (cases z, auto) 

576 
by (metis complex_of_real_def of_real_add of_real_power power2_eq_square) 

577 

578 
lemma complex_div_eq_0: 

579 
"(Re(a / b) = 0 \<longleftrightarrow> Re(a * cnj b) = 0) & (Im(a / b) = 0 \<longleftrightarrow> Im(a * cnj b) = 0)" 

580 
proof (cases "b=0") 

581 
case True then show ?thesis by auto 

582 
next 

583 
case False 

584 
show ?thesis 

585 
proof (cases b) 

586 
case (Complex x y) 

587 
then have "x\<^sup>2 + y\<^sup>2 > 0" 

588 
by (metis Complex_eq_0 False sum_power2_gt_zero_iff) 

589 
then have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)" 

590 
by (metis add_divide_distrib) 

591 
with Complex False show ?thesis 

592 
by (auto simp: complex_divide_def) 

593 
qed 

594 
qed 

595 

596 
lemma re_complex_div_eq_0: "Re(a / b) = 0 \<longleftrightarrow> Re(a * cnj b) = 0" 

597 
and im_complex_div_eq_0: "Im(a / b) = 0 \<longleftrightarrow> Im(a * cnj b) = 0" 

598 
using complex_div_eq_0 by auto 

599 

600 

601 
lemma complex_div_gt_0: 

602 
"(Re(a / b) > 0 \<longleftrightarrow> Re(a * cnj b) > 0) & (Im(a / b) > 0 \<longleftrightarrow> Im(a * cnj b) > 0)" 

603 
proof (cases "b=0") 

604 
case True then show ?thesis by auto 

605 
next 

606 
case False 

607 
show ?thesis 

608 
proof (cases b) 

609 
case (Complex x y) 

610 
then have "x\<^sup>2 + y\<^sup>2 > 0" 

611 
by (metis Complex_eq_0 False sum_power2_gt_zero_iff) 

612 
moreover have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)" 

613 
by (metis add_divide_distrib) 

614 
ultimately show ?thesis using Complex False `0 < x\<^sup>2 + y\<^sup>2` 

615 
apply (simp add: complex_divide_def zero_less_divide_iff less_divide_eq) 

616 
apply (metis less_divide_eq mult_zero_left diff_conv_add_uminus minus_divide_left) 

617 
done 

618 
qed 

619 
qed 

620 

621 
lemma re_complex_div_gt_0: "Re(a / b) > 0 \<longleftrightarrow> Re(a * cnj b) > 0" 

622 
and im_complex_div_gt_0: "Im(a / b) > 0 \<longleftrightarrow> Im(a * cnj b) > 0" 

623 
using complex_div_gt_0 by auto 

624 

625 
lemma re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0" 

626 
by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0) 

627 

628 
lemma im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0" 

629 
by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less) 

630 

631 
lemma re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0" 

55759
fe3d8f585c20
replaced smtbased proof with metis proof that requires no external tool
boehmes
parents:
55734
diff
changeset

632 
by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0) 
55734  633 

634 
lemma im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0" 

635 
by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff) 

636 

637 
lemma re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0" 

638 
by (metis not_le re_complex_div_gt_0) 

639 

640 
lemma im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0" 

641 
by (metis im_complex_div_gt_0 not_le) 

642 

56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

643 
lemma Re_setsum: "Re(setsum f s) = setsum (%x. Re(f x)) s" 
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

644 
apply (cases "finite s") 
55734  645 
by (induct s rule: finite_induct) auto 
646 

56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

647 
lemma Im_setsum: "Im(setsum f s) = setsum (%x. Im(f x)) s" 
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

648 
apply (cases "finite s") 
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

649 
by (induct s rule: finite_induct) auto 
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

650 

dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

651 
lemma Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0" 
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

652 
apply (cases "finite s") 
55734  653 
by (induct s rule: finite_induct) auto 
654 

56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

655 
lemma Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s" 
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

656 
by (metis Complex_setsum') 
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

657 

dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

658 
lemma cnj_setsum: "cnj (setsum f s) = setsum (%x. cnj (f x)) s" 
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

659 
apply (cases "finite s") 
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

660 
by (induct s rule: finite_induct) (auto simp: complex_cnj_add) 
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

661 

dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

662 
lemma of_real_setsum: "of_real (setsum f s) = setsum (%x. of_real(f x)) s" 
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

663 
apply (cases "finite s") 
55734  664 
by (induct s rule: finite_induct) auto 
665 

56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

666 
lemma of_real_setprod: "of_real (setprod f s) = setprod (%x. of_real(f x)) s" 
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

667 
apply (cases "finite s") 
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

668 
by (induct s rule: finite_induct) auto 
55734  669 

670 
lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z" 

671 
by (metis Reals_cases Reals_of_real complex.exhaust complex.inject complex_cnj 

672 
complex_of_real_def equal_neg_zero) 

673 

674 
lemma Complex_in_Reals: "Complex x 0 \<in> \<real>" 

675 
by (metis Reals_of_real complex_of_real_def) 

676 

677 
lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm(z) = abs(Re z)" 

678 
by (metis Re_complex_of_real Reals_cases norm_of_real) 

679 

680 

14323  681 
subsection{*Finally! Polar Form for Complex Numbers*} 
682 

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

683 
subsubsection {* $\cos \theta + i \sin \theta$ *} 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

684 

44715  685 
definition cis :: "real \<Rightarrow> complex" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

686 
"cis a = Complex (cos a) (sin a)" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

687 

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

688 
lemma Re_cis [simp]: "Re (cis a) = cos a" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

689 
by (simp add: cis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

690 

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

691 
lemma Im_cis [simp]: "Im (cis a) = sin a" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

692 
by (simp add: cis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

693 

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

694 
lemma cis_zero [simp]: "cis 0 = 1" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

695 
by (simp add: cis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

696 

44828  697 
lemma norm_cis [simp]: "norm (cis a) = 1" 
698 
by (simp add: cis_def) 

699 

700 
lemma sgn_cis [simp]: "sgn (cis a) = cis a" 

701 
by (simp add: sgn_div_norm) 

702 

703 
lemma cis_neq_zero [simp]: "cis a \<noteq> 0" 

704 
by (metis norm_cis norm_zero zero_neq_one) 

705 

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

706 
lemma cis_mult: "cis a * cis b = cis (a + b)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

707 
by (simp add: cis_def cos_add sin_add) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

708 

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

709 
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

710 
by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

711 

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

712 
lemma cis_inverse [simp]: "inverse(cis a) = cis (a)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

713 
by (simp add: cis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

714 

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

715 
lemma cis_divide: "cis a / cis b = cis (a  b)" 
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53374
diff
changeset

716 
by (simp add: complex_divide_def cis_mult) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

717 

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

718 
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

719 
by (auto simp add: DeMoivre) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

720 

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

721 
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

722 
by (auto simp add: DeMoivre) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

723 

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

724 
subsubsection {* $r(\cos \theta + i \sin \theta)$ *} 
44715  725 

726 
definition rcis :: "[real, real] \<Rightarrow> complex" where 

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

727 
"rcis r a = complex_of_real r * cis a" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

728 

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

729 
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" 
44828  730 
by (simp add: rcis_def) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

731 

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

732 
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" 
44828  733 
by (simp add: rcis_def) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

734 

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

735 
lemma rcis_Ex: "\<exists>r a. z = rcis r a" 
44828  736 
by (simp add: complex_eq_iff polar_Ex) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

737 

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

738 
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" 
44828  739 
by (simp add: rcis_def norm_mult) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

740 

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

741 
lemma cis_rcis_eq: "cis a = rcis 1 a" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

742 
by (simp add: rcis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

743 

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

744 
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" 
44828  745 
by (simp add: rcis_def cis_mult) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

746 

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

747 
lemma rcis_zero_mod [simp]: "rcis 0 a = 0" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

748 
by (simp add: rcis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

749 

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

750 
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

751 
by (simp add: rcis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

752 

44828  753 
lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0" 
754 
by (simp add: rcis_def) 

755 

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

756 
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

757 
by (simp add: rcis_def power_mult_distrib DeMoivre) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

758 

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

759 
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (a)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

760 
by (simp add: divide_inverse rcis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

761 

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

762 
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a  b)" 
44828  763 
by (simp add: rcis_def cis_divide [symmetric]) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

764 

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

765 
subsubsection {* Complex exponential *} 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

766 

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

767 
abbreviation expi :: "complex \<Rightarrow> complex" 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

768 
where "expi \<equiv> exp" 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

769 

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

770 
lemma cis_conv_exp: "cis b = exp (Complex 0 b)" 
44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

771 
proof (rule complex_eqI) 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

772 
{ fix n have "Complex 0 b ^ n = 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

773 
real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)" 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

774 
apply (induct n) 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

775 
apply (simp add: cos_coeff_def sin_coeff_def) 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

776 
apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc) 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

777 
done } note * = this 
44712
1e490e891c88
replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents:
44711
diff
changeset

778 
show "Re (cis b) = Re (exp (Complex 0 b))" 
44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

779 
unfolding exp_def cis_def cos_def 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

780 
by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic], 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

781 
simp add: * mult_assoc [symmetric]) 
44712
1e490e891c88
replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents:
44711
diff
changeset

782 
show "Im (cis b) = Im (exp (Complex 0 b))" 
44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

783 
unfolding exp_def cis_def sin_def 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

784 
by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic], 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

785 
simp add: * mult_assoc [symmetric]) 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

786 
qed 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

787 

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

788 
lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)" 
44712
1e490e891c88
replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents:
44711
diff
changeset

789 
unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

790 

44828  791 
lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)" 
792 
unfolding expi_def by simp 

793 

794 
lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)" 

795 
unfolding expi_def by simp 

796 

14374  797 
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a" 
14373  798 
apply (insert rcis_Ex [of z]) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

799 
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric]) 
14334  800 
apply (rule_tac x = "ii * complex_of_real a" in exI, auto) 
14323  801 
done 
802 

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

803 
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1" 
44724  804 
by (simp add: expi_def cis_def) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

805 

44844
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

806 
subsubsection {* Complex argument *} 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

807 

f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

808 
definition arg :: "complex \<Rightarrow> real" where 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

809 
"arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> pi < a \<and> a \<le> pi))" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

810 

f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

811 
lemma arg_zero: "arg 0 = 0" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

812 
by (simp add: arg_def) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

813 

f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

814 
lemma of_nat_less_of_int_iff: (* TODO: move *) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

815 
"(of_nat n :: 'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

816 
by (metis of_int_of_nat_eq of_int_less_iff) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

817 

47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
44902
diff
changeset

818 
lemma real_of_nat_less_numeral_iff [simp]: (* TODO: move *) 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
44902
diff
changeset

819 
"real (n::nat) < numeral w \<longleftrightarrow> n < numeral w" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
44902
diff
changeset

820 
using of_nat_less_of_int_iff [of n "numeral w", where 'a=real] 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
44902
diff
changeset

821 
by (simp add: real_of_nat_def zless_nat_eq_int_zless [symmetric]) 
44844
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

822 

f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

823 
lemma arg_unique: 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

824 
assumes "sgn z = cis x" and "pi < x" and "x \<le> pi" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

825 
shows "arg z = x" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

826 
proof  
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

827 
from assms have "z \<noteq> 0" by auto 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

828 
have "(SOME a. sgn z = cis a \<and> pi < a \<and> a \<le> pi) = x" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

829 
proof 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

830 
fix a def d \<equiv> "a  x" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

831 
assume a: "sgn z = cis a \<and>  pi < a \<and> a \<le> pi" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

832 
from a assms have " (2*pi) < d \<and> d < 2*pi" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

833 
unfolding d_def by simp 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

834 
moreover from a assms have "cos a = cos x" and "sin a = sin x" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

835 
by (simp_all add: complex_eq_iff) 
53374
a14d2a854c02
tuned proofs  clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset

836 
hence cos: "cos d = 1" unfolding d_def cos_diff by simp 
a14d2a854c02
tuned proofs  clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset

837 
moreover from cos have "sin d = 0" by (rule cos_one_sin_zero) 
44844
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

838 
ultimately have "d = 0" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

839 
unfolding sin_zero_iff even_mult_two_ex 
53374
a14d2a854c02
tuned proofs  clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset

840 
by (auto simp add: numeral_2_eq_2 less_Suc_eq) 
44844
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

841 
thus "a = x" unfolding d_def by simp 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

842 
qed (simp add: assms del: Re_sgn Im_sgn) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

843 
with `z \<noteq> 0` show "arg z = x" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

844 
unfolding arg_def by simp 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

845 
qed 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

846 

f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

847 
lemma arg_correct: 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

848 
assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> pi < arg z \<and> arg z \<le> pi" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

849 
proof (simp add: arg_def assms, rule someI_ex) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

850 
obtain r a where z: "z = rcis r a" using rcis_Ex by fast 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

851 
with assms have "r \<noteq> 0" by auto 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

852 
def b \<equiv> "if 0 < r then a else a + pi" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

853 
have b: "sgn z = cis b" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

854 
unfolding z b_def rcis_def using `r \<noteq> 0` 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

855 
by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

856 
have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1" 
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
47108
diff
changeset

857 
by (induct_tac n, simp_all add: distrib_left cis_mult [symmetric], 
44844
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

858 
simp add: cis_def) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

859 
have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

860 
by (case_tac x rule: int_diff_cases, 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

861 
simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

862 
def c \<equiv> "b  2*pi * of_int \<lceil>(b  pi) / (2*pi)\<rceil>" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

863 
have "sgn z = cis c" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

864 
unfolding b c_def 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

865 
by (simp add: cis_divide [symmetric] cis_2pi_int) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

866 
moreover have " pi < c \<and> c \<le> pi" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

867 
using ceiling_correct [of "(b  pi) / (2*pi)"] 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

868 
by (simp add: c_def less_divide_eq divide_le_eq algebra_simps) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

869 
ultimately show "\<exists>a. sgn z = cis a \<and> pi < a \<and> a \<le> pi" by fast 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

870 
qed 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

871 

f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

872 
lemma arg_bounded: " pi < arg z \<and> arg z \<le> pi" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

873 
by (cases "z = 0", simp_all add: arg_zero arg_correct) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

874 

f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

875 
lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

876 
by (simp add: arg_correct) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

877 

f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

878 
lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

879 
by (cases "z = 0", simp_all add: rcis_def cis_arg sgn_div_norm of_real_def) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

880 

f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

881 
lemma cos_arg_i_mult_zero [simp]: 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

882 
"y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

883 
using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

884 

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

885 
text {* Legacy theorem names *} 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

886 

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

887 
lemmas expand_complex_eq = complex_eq_iff 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

888 
lemmas complex_Re_Im_cancel_iff = complex_eq_iff 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

889 
lemmas complex_equality = complex_eqI 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

890 

13957  891 
end 