author  paulson 
Thu, 03 Sep 2015 20:27:53 +0100  
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permissions  rwrr 
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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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section \<open>Complex Numbers: Rectangular and Polar Representations\<close> 
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theory Complex 
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imports Transcendental 
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begin 
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text \<open> 
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We use the @{text codatatype} command to define the type of complex numbers. This allows us to use 
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@{text primcorec} to define complex functions by defining their real and imaginary result 

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separately. 

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\<close> 
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codatatype complex = Complex (Re: real) (Im: real) 
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lemma complex_surj: "Complex (Re z) (Im z) = z" 
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by (rule complex.collapse) 
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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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by (rule complex.expand) simp 
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lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y" 
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by (auto intro: complex.expand) 
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subsection \<open>Addition and Subtraction\<close> 
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instantiation complex :: ab_group_add 
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begin 
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primcorec zero_complex where 
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"Re 0 = 0" 
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 "Im 0 = 0" 
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primcorec plus_complex where 
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"Re (x + y) = Re x + Re y" 
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 "Im (x + y) = Im x + Im y" 
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primcorec uminus_complex where 
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"Re ( x) =  Re x" 
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 "Im ( x) =  Im x" 
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primcorec minus_complex where 
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"Re (x  y) = Re x  Re y" 
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 "Im (x  y) = Im x  Im y" 
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instance 
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by intro_classes (simp_all add: complex_eq_iff) 
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end 

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subsection \<open>Multiplication and Division\<close> 
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instantiation complex :: field 
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begin 
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primcorec one_complex where 
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"Re 1 = 1" 
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 "Im 1 = 0" 
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primcorec times_complex where 
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"Re (x * y) = Re x * Re y  Im x * Im y" 
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 "Im (x * y) = Re x * Im y + Im x * Re y" 
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primcorec inverse_complex where 
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"Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)" 
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 "Im (inverse x) =  Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)" 
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definition "x div (y::complex) = x * inverse y" 
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instance 
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The function frac. Various lemmas about limits, series, the exp function, etc.
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by intro_classes 
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(simp_all add: complex_eq_iff divide_complex_def 
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distrib_left distrib_right right_diff_distrib left_diff_distrib 
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power2_eq_square add_divide_distrib [symmetric]) 
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end 
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lemma Re_divide: "Re (x / y) = (Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)" 
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unfolding divide_complex_def by (simp add: add_divide_distrib) 
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lemma Im_divide: "Im (x / y) = (Im x * Re y  Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)" 
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unfolding divide_complex_def times_complex.sel inverse_complex.sel 
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by (simp_all add: divide_simps) 
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lemma Re_power2: "Re (x ^ 2) = (Re x)^2  (Im x)^2" 
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by (simp add: power2_eq_square) 
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lemma Im_power2: "Im (x ^ 2) = 2 * Re x * Im x" 
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by (simp add: power2_eq_square) 
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lemma Re_power_real [simp]: "Im x = 0 \<Longrightarrow> Re (x ^ n) = Re x ^ n " 
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by (induct n) simp_all 
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lemma Im_power_real [simp]: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0" 
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by (induct n) simp_all 
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subsection \<open>Scalar Multiplication\<close> 
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instantiation complex :: real_field 
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begin 
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primcorec scaleR_complex where 
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"Re (scaleR r x) = r * Re x" 
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 "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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show "scaleR 1 x = x" 
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by (simp add: complex_eq_iff) 
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show "scaleR a x * y = scaleR a (x * y)" 
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by (simp add: complex_eq_iff algebra_simps) 
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show "x * scaleR a y = scaleR a (x * y)" 
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by (simp add: complex_eq_iff algebra_simps) 
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qed 
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end 
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subsection \<open>Numerals, Arithmetic, and Embedding from Reals\<close> 
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abbreviation complex_of_real :: "real \<Rightarrow> complex" 
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where "complex_of_real \<equiv> of_real" 

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declare [[coercion "of_real :: real \<Rightarrow> complex"]] 
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declare [[coercion "of_rat :: rat \<Rightarrow> complex"]] 

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declare [[coercion "of_int :: int \<Rightarrow> complex"]] 
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declare [[coercion "of_nat :: nat \<Rightarrow> complex"]] 
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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_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_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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complex_of_real abbreviates of_real::real=>complex;
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complex_of_real abbreviates of_real::real=>complex;
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158 
lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" 
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159 
by (simp add: of_real_def) 
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160 

81dd3679f92c
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161 
lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" 
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162 
by (simp add: of_real_def) 
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163 

59613
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The function frac. Various lemmas about limits, series, the exp function, etc.
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164 
lemma Re_divide_numeral [simp]: "Re (z / numeral w) = Re z / numeral w" 
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
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165 
by (simp add: Re_divide sqr_conv_mult) 
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
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166 

7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
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167 
lemma Im_divide_numeral [simp]: "Im (z / numeral w) = Im z / numeral w" 
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
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168 
by (simp add: Im_divide sqr_conv_mult) 
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
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169 

60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
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170 
lemma of_real_Re [simp]: 
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
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parents:
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171 
"z \<in> \<real> \<Longrightarrow> of_real (Re z) = z" 
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
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parents:
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172 
by (auto simp: Reals_def) 
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Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
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173 

60758  174 
subsection \<open>The Complex Number $i$\<close> 
56889
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175 

48a745e1bde7
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176 
primcorec "ii" :: complex ("\<i>") where 
48a745e1bde7
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177 
"Re ii = 0" 
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178 
 "Im ii = 1" 
20557
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179 

57259
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180 
lemma Complex_eq[simp]: "Complex a b = a + \<i> * b" 
3a448982a74a
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181 
by (simp add: complex_eq_iff) 
3a448982a74a
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182 

3a448982a74a
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183 
lemma complex_eq: "a = Re a + \<i> * Im a" 
3a448982a74a
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184 
by (simp add: complex_eq_iff) 
3a448982a74a
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hoelzl
parents:
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diff
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185 

3a448982a74a
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diff
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186 
lemma fun_complex_eq: "f = (\<lambda>x. Re (f x) + \<i> * Im (f x))" 
3a448982a74a
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hoelzl
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diff
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187 
by (simp add: fun_eq_iff complex_eq) 
3a448982a74a
add more derivative and continuity rules for complexvalues functions
hoelzl
parents:
56889
diff
changeset

188 

56889
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avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
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189 
lemma i_squared [simp]: "ii * ii = 1" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
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190 
by (simp add: complex_eq_iff) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
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diff
changeset

191 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
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192 
lemma power2_i [simp]: "ii\<^sup>2 = 1" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
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diff
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193 
by (simp add: power2_eq_square) 
14377  194 

56889
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avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
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diff
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195 
lemma inverse_i [simp]: "inverse ii =  ii" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
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196 
by (rule inverse_unique) simp 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
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diff
changeset

197 

48a745e1bde7
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hoelzl
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diff
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198 
lemma divide_i [simp]: "x / ii =  ii * x" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
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diff
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199 
by (simp add: divide_complex_def) 
14377  200 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
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diff
changeset

201 
lemma complex_i_mult_minus [simp]: "ii * (ii * x) =  x" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57259
diff
changeset

202 
by (simp add: mult.assoc [symmetric]) 
14377  203 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

204 
lemma complex_i_not_zero [simp]: "ii \<noteq> 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

205 
by (simp add: complex_eq_iff) 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

206 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

207 
lemma complex_i_not_one [simp]: "ii \<noteq> 1" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
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diff
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208 
by (simp add: complex_eq_iff) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

209 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

210 
lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

211 
by (simp add: complex_eq_iff) 
44841  212 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
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213 
lemma complex_i_not_neg_numeral [simp]: "ii \<noteq>  numeral w" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
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214 
by (simp add: complex_eq_iff) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

215 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

216 
lemma complex_split_polar: "\<exists>r a. z = complex_of_real r * (cos a + \<i> * sin a)" 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

217 
by (simp add: complex_eq_iff polar_Ex) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

218 

59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59000
diff
changeset

219 
lemma i_even_power [simp]: "\<i> ^ (n * 2) = (1) ^ n" 
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59000
diff
changeset

220 
by (metis mult.commute power2_i power_mult) 
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59000
diff
changeset

221 

59741
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

222 
lemma Re_ii_times [simp]: "Re (ii*z) =  Im z" 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

223 
by simp 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

224 

5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

225 
lemma Im_ii_times [simp]: "Im (ii*z) = Re z" 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

226 
by simp 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

227 

5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

228 
lemma ii_times_eq_iff: "ii*w = z \<longleftrightarrow> w = (ii*z)" 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

229 
by auto 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

230 

5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

231 
lemma divide_numeral_i [simp]: "z / (numeral n * ii) = (ii*z) / numeral n" 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

232 
by (metis divide_divide_eq_left divide_i mult.commute mult_minus_right) 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

233 

60758  234 
subsection \<open>Vector Norm\<close> 
14323  235 

25712  236 
instantiation complex :: real_normed_field 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
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237 
begin 
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset

238 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
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239 
definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)" 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset

240 

44724  241 
abbreviation cmod :: "complex \<Rightarrow> real" 
242 
where "cmod \<equiv> norm" 

25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset

243 

31413
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31292
diff
changeset

244 
definition complex_sgn_def: 
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31292
diff
changeset

245 
"sgn x = x /\<^sub>R cmod x" 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset

246 

31413
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31292
diff
changeset

247 
definition dist_complex_def: 
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31292
diff
changeset

248 
"dist x y = cmod (x  y)" 
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31292
diff
changeset

249 

37767  250 
definition open_complex_def: 
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31419
diff
changeset

251 
"open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" 
31292  252 

31413
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
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diff
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253 
instance proof 
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31419
diff
changeset

254 
fix r :: real and x y :: complex and S :: "complex set" 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

255 
show "(norm x = 0) = (x = 0)" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

256 
by (simp add: norm_complex_def complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

257 
show "norm (x + y) \<le> norm x + norm y" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

258 
by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

259 
show "norm (scaleR r x) = \<bar>r\<bar> * norm x" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

260 
by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] real_sqrt_mult) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

261 
show "norm (x * y) = norm x * norm y" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

262 
by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] power2_eq_square algebra_simps) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

263 
qed (rule complex_sgn_def dist_complex_def open_complex_def)+ 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

264 

25712  265 
end 
266 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
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parents:
56541
diff
changeset

267 
lemma norm_ii [simp]: "norm ii = 1" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

268 
by (simp add: norm_complex_def) 
14323  269 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

270 
lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

271 
by (simp add: norm_complex_def) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

272 

48a745e1bde7
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hoelzl
parents:
56541
diff
changeset

273 
lemma cmod_complex_polar: "cmod (r * (cos a + \<i> * sin a)) = \<bar>r\<bar>" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

274 
by (simp add: norm_mult cmod_unit_one) 
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset

275 

8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset

276 
lemma complex_Re_le_cmod: "Re x \<le> cmod x" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

277 
unfolding norm_complex_def 
44724  278 
by (rule real_sqrt_sum_squares_ge1) 
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset

279 

44761  280 
lemma complex_mod_minus_le_complex_mod: " cmod x \<le> cmod x" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

281 
by (rule order_trans [OF _ norm_ge_zero]) simp 
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset

282 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

283 
lemma complex_mod_triangle_ineq2: "cmod (b + a)  cmod b \<le> cmod a" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

284 
by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp 
14323  285 

26117  286 
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

287 
by (simp add: norm_complex_def) 
26117  288 

289 
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x" 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
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290 
by (simp add: norm_complex_def) 
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291 

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292 
lemma cmod_le: "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>" 
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293 
apply (subst complex_eq) 
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294 
apply (rule order_trans) 
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295 
apply (rule norm_triangle_ineq) 
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296 
apply (simp add: norm_mult) 
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297 
done 
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298 

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299 
lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>" 
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300 
by (simp add: norm_complex_def) 
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301 

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302 
lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>" 
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303 
by (simp add: norm_complex_def) 
44724  304 

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305 
lemma cmod_power2: "cmod z ^ 2 = (Re z)^2 + (Im z)^2" 
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306 
by (simp add: norm_complex_def) 
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307 

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308 
lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z =  cmod z" 
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309 
using abs_Re_le_cmod[of z] by auto 
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310 

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311 
lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0" 
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312 
by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2]) 
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313 
(auto simp add: norm_complex_def) 
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314 

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315 
lemma abs_sqrt_wlog: 
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316 
fixes x::"'a::linordered_idom" 
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317 
assumes "\<And>x::'a. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)" 
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318 
by (metis abs_ge_zero assms power2_abs) 
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319 

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320 
lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z" 
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321 
unfolding norm_complex_def 
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322 
apply (rule abs_sqrt_wlog [where x="Re z"]) 
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323 
apply (rule abs_sqrt_wlog [where x="Im z"]) 
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324 
apply (rule power2_le_imp_le) 
57512
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325 
apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric]) 
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326 
done 
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327 

59741
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328 
lemma complex_unit_circle: "z \<noteq> 0 \<Longrightarrow> (Re z / cmod z)\<^sup>2 + (Im z / cmod z)\<^sup>2 = 1" 
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329 
by (simp add: norm_complex_def divide_simps complex_eq_iff) 
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330 

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331 

60758  332 
text \<open>Properties of complex signum.\<close> 
44843  333 

334 
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" 

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335 
by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute) 
44843  336 

337 
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" 

338 
by (simp add: complex_sgn_def divide_inverse) 

339 

340 
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" 

341 
by (simp add: complex_sgn_def divide_inverse) 

342 

14354
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343 

60758  344 
subsection \<open>Completeness of the Complexes\<close> 
23123  345 

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346 
lemma bounded_linear_Re: "bounded_linear Re" 
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347 
by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def) 
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348 

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349 
lemma bounded_linear_Im: "bounded_linear Im" 
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350 
by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def) 
23123  351 

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352 
lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re] 
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353 
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im] 
56381
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354 
lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re] 
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355 
lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im] 
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356 
lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re] 
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357 
lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im] 
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358 
lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re] 
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359 
lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im] 
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360 
lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re] 
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361 
lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im] 
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362 
lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re] 
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363 
lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im] 
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364 
lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re] 
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365 
lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im] 
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366 

36825  367 
lemma tendsto_Complex [tendsto_intros]: 
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368 
"(f > a) F \<Longrightarrow> (g > b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) > Complex a b) F" 
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369 
by (auto intro!: tendsto_intros) 
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370 

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371 
lemma tendsto_complex_iff: 
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372 
"(f > x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) > Re x) F \<and> ((\<lambda>x. Im (f x)) > Im x) F)" 
56889
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373 
proof safe 
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374 
assume "((\<lambda>x. Re (f x)) > Re x) F" "((\<lambda>x. Im (f x)) > Im x) F" 
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375 
from tendsto_Complex[OF this] show "(f > x) F" 
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376 
unfolding complex.collapse . 
48a745e1bde7
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377 
qed (auto intro: tendsto_intros) 
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378 

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379 
lemma continuous_complex_iff: "continuous F f \<longleftrightarrow> 
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380 
continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x))" 
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381 
unfolding continuous_def tendsto_complex_iff .. 
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382 

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383 
lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \<longleftrightarrow> 
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384 
((\<lambda>x. Re (f x)) has_field_derivative (Re x)) F \<and> 
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385 
((\<lambda>x. Im (f x)) has_field_derivative (Im x)) F" 
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changeset

386 
unfolding has_vector_derivative_def has_field_derivative_def has_derivative_def tendsto_complex_iff 
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387 
by (simp add: field_simps bounded_linear_scaleR_left bounded_linear_mult_right) 
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388 

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389 
lemma has_field_derivative_Re[derivative_intros]: 
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390 
"(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Re (f x)) has_field_derivative (Re D)) F" 
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changeset

391 
unfolding has_vector_derivative_complex_iff by safe 
3a448982a74a
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diff
changeset

392 

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393 
lemma has_field_derivative_Im[derivative_intros]: 
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394 
"(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Im (f x)) has_field_derivative (Im D)) F" 
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changeset

395 
unfolding has_vector_derivative_complex_iff by safe 
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396 

23123  397 
instance complex :: banach 
398 
proof 

399 
fix X :: "nat \<Rightarrow> complex" 

400 
assume X: "Cauchy X" 

56889
48a745e1bde7
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hoelzl
parents:
56541
diff
changeset

401 
then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) > Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))" 
48a745e1bde7
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hoelzl
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diff
changeset

402 
by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
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diff
changeset

403 
then show "convergent X" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

404 
unfolding complex.collapse by (rule convergentI) 
23123  405 
qed 
406 

56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56217
diff
changeset

407 
declare 
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56369
diff
changeset

408 
DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros] 
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56217
diff
changeset

409 

60758  410 
subsection \<open>Complex Conjugation\<close> 
23125
6f7b5b96241f
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huffman
parents:
23124
diff
changeset

411 

56889
48a745e1bde7
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hoelzl
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56541
diff
changeset

412 
primcorec cnj :: "complex \<Rightarrow> complex" where 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

413 
"Re (cnj z) = Re z" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

414 
 "Im (cnj z) =  Im z" 
23125
6f7b5b96241f
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huffman
parents:
23124
diff
changeset

415 

6f7b5b96241f
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huffman
parents:
23124
diff
changeset

416 
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" 
44724  417 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
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huffman
parents:
23124
diff
changeset

418 

6f7b5b96241f
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huffman
parents:
23124
diff
changeset

419 
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

420 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
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huffman
parents:
23124
diff
changeset

421 

6f7b5b96241f
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huffman
parents:
23124
diff
changeset

422 
lemma complex_cnj_zero [simp]: "cnj 0 = 0" 
44724  423 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
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huffman
parents:
23124
diff
changeset

424 

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

425 
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" 
44724  426 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

427 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

428 
lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y" 
44724  429 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

430 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

431 
lemma cnj_setsum [simp]: "cnj (setsum f s) = (\<Sum>x\<in>s. cnj (f x))" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

432 
by (induct s rule: infinite_finite_induct) auto 
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

433 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

434 
lemma complex_cnj_diff [simp]: "cnj (x  y) = cnj x  cnj y" 
44724  435 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

436 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

437 
lemma complex_cnj_minus [simp]: "cnj ( x) =  cnj x" 
44724  438 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

439 

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

440 
lemma complex_cnj_one [simp]: "cnj 1 = 1" 
44724  441 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

442 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

443 
lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y" 
44724  444 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

445 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

446 
lemma cnj_setprod [simp]: "cnj (setprod f s) = (\<Prod>x\<in>s. cnj (f x))" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

447 
by (induct s rule: infinite_finite_induct) auto 
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

448 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

449 
lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

450 
by (simp add: complex_eq_iff) 
14323  451 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

452 
lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

453 
by (simp add: divide_complex_def) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

454 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

455 
lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

456 
by (induct n) simp_all 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

457 

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

458 
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" 
44724  459 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

460 

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

461 
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" 
44724  462 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

463 

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

464 
lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
44902
diff
changeset

465 
by (simp add: complex_eq_iff) 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
44902
diff
changeset

466 

54489
03ff4d1e6784
eliminiated neg_numeral in favour of  (numeral _)
haftmann
parents:
54230
diff
changeset

467 
lemma complex_cnj_neg_numeral [simp]: "cnj ( numeral w) =  numeral w" 
44724  468 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

469 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

470 
lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)" 
44724  471 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

472 

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

473 
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

474 
by (simp add: norm_complex_def) 
14323  475 

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

476 
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" 
44724  477 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

478 

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

479 
lemma complex_cnj_i [simp]: "cnj ii =  ii" 
44724  480 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

481 

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

482 
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" 
44724  483 
by (simp add: complex_eq_iff) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

484 

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

485 
lemma complex_diff_cnj: "z  cnj z = complex_of_real (2 * Im z) * ii" 
44724  486 
by (simp add: complex_eq_iff) 
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset

487 

53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51002
diff
changeset

488 
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)" 
44724  489 
by (simp add: complex_eq_iff power2_eq_square) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

490 

53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51002
diff
changeset

491 
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2" 
44724  492 
by (simp add: norm_mult power2_eq_square) 
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset

493 

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

494 
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

495 
by (simp add: norm_complex_def power2_eq_square) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

496 

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

497 
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

498 
by simp 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

499 

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

500 
lemma bounded_linear_cnj: "bounded_linear cnj" 
44127  501 
using complex_cnj_add complex_cnj_scaleR 
502 
by (rule bounded_linear_intro [where K=1], simp) 

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

503 

56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56369
diff
changeset

504 
lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj] 
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56369
diff
changeset

505 
lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj] 
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56369
diff
changeset

506 
lemmas continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj] 
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56369
diff
changeset

507 
lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj] 
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56369
diff
changeset

508 
lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj] 
44290
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset

509 

56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

510 
lemma lim_cnj: "((\<lambda>x. cnj(f x)) > cnj l) F \<longleftrightarrow> (f > l) F" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

511 
by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff) 
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

512 

2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

513 
lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

514 
by (simp add: sums_def lim_cnj cnj_setsum [symmetric] del: cnj_setsum) 
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

515 

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

516 

60758  517 
subsection\<open>Basic Lemmas\<close> 
55734  518 

519 
lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0" 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

520 
by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff) 
55734  521 

522 
lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0" 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

523 
by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff) 
55734  524 

525 
lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z" 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

526 
by (cases z) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

527 
(auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric] 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

528 
simp del: of_real_power) 
55734  529 

61104
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

530 
lemma complex_div_cnj: "a / b = (a * cnj b) / (norm b)^2" 
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

531 
using complex_norm_square by auto 
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

532 

59741
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

533 
lemma Re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

534 
by (auto simp add: Re_divide) 
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59000
diff
changeset

535 

59741
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

536 
lemma Im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

537 
by (auto simp add: Im_divide) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

538 

59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59000
diff
changeset

539 
lemma complex_div_gt_0: 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

540 
"(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

541 
proof cases 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

542 
assume "b = 0" then show ?thesis by auto 
55734  543 
next 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

544 
assume "b \<noteq> 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

545 
then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

546 
by (simp add: complex_eq_iff sum_power2_gt_zero_iff) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

547 
then show ?thesis 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

548 
by (simp add: Re_divide Im_divide zero_less_divide_iff) 
55734  549 
qed 
550 

59741
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

551 
lemma Re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0" 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

552 
and Im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

553 
using complex_div_gt_0 by auto 
55734  554 

59741
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

555 
lemma Re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0" 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

556 
by (metis le_less Re_complex_div_eq_0 Re_complex_div_gt_0) 
55734  557 

59741
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

558 
lemma Im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0" 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

559 
by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 le_less) 
55734  560 

59741
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

561 
lemma Re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0" 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

562 
by (metis less_asym neq_iff Re_complex_div_eq_0 Re_complex_div_gt_0) 
55734  563 

59741
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

564 
lemma Im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0" 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

565 
by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 less_asym neq_iff) 
55734  566 

59741
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

567 
lemma Re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0" 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

568 
by (metis not_le Re_complex_div_gt_0) 
55734  569 

59741
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

570 
lemma Im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0" 
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

571 
by (metis Im_complex_div_gt_0 not_le) 
55734  572 

61104
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

573 
lemma Re_divide_of_real [simp]: "Re (z / of_real r) = Re z / r" 
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

574 
by (simp add: Re_divide power2_eq_square) 
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

575 

3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

576 
lemma Im_divide_of_real [simp]: "Im (z / of_real r) = Im z / r" 
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

577 
by (simp add: Im_divide power2_eq_square) 
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

578 

3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

579 
lemma Re_divide_Reals: "r \<in> Reals \<Longrightarrow> Re (z / r) = Re z / Re r" 
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

580 
by (metis Re_divide_of_real of_real_Re) 
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

581 

3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

582 
lemma Im_divide_Reals: "r \<in> Reals \<Longrightarrow> Im (z / r) = Im z / Re r" 
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

583 
by (metis Im_divide_of_real of_real_Re) 
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

584 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

585 
lemma Re_setsum[simp]: "Re (setsum f s) = (\<Sum>x\<in>s. Re (f x))" 
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

586 
by (induct s rule: infinite_finite_induct) auto 
55734  587 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

588 
lemma Im_setsum[simp]: "Im (setsum f s) = (\<Sum>x\<in>s. Im(f x))" 
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

589 
by (induct s rule: infinite_finite_induct) auto 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

590 

2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

591 
lemma sums_complex_iff: "f sums x \<longleftrightarrow> ((\<lambda>x. Re (f x)) sums Re x) \<and> ((\<lambda>x. Im (f x)) sums Im x)" 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

592 
unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum .. 
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59000
diff
changeset

593 

56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

594 
lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and> summable (\<lambda>x. Im (f x))" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

595 
unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel) 
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

596 

2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

597 
lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f" 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

598 
unfolding summable_complex_iff by simp 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

599 

2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

600 
lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))" 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

601 
unfolding summable_complex_iff by blast 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

602 

2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

603 
lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))" 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

604 
unfolding summable_complex_iff by blast 
56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
55759
diff
changeset

605 

61104
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

606 
lemma complex_is_Nat_iff: "z \<in> \<nat> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_nat i)" 
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

607 
by (auto simp: Nats_def complex_eq_iff) 
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

608 

3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

609 
lemma complex_is_Int_iff: "z \<in> \<int> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_int i)" 
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

610 
by (auto simp: Ints_def complex_eq_iff) 
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
61076
diff
changeset

611 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

612 
lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

613 
by (auto simp: Reals_def complex_eq_iff) 
55734  614 

615 
lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z" 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

616 
by (auto simp: complex_is_Real_iff complex_eq_iff) 
55734  617 

618 
lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm(z) = abs(Re z)" 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

619 
by (simp add: complex_is_Real_iff norm_complex_def) 
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

620 

2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

621 
lemma series_comparison_complex: 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

622 
fixes f:: "nat \<Rightarrow> 'a::banach" 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

623 
assumes sg: "summable g" 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

624 
and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0" 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

625 
and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)" 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

626 
shows "summable f" 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

627 
proof  
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

628 
have g: "\<And>n. cmod (g n) = Re (g n)" using assms 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

629 
by (metis abs_of_nonneg in_Reals_norm) 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

630 
show ?thesis 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

631 
apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N]) 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

632 
using sg 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

633 
apply (auto simp: summable_def) 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

634 
apply (rule_tac x="Re s" in exI) 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

635 
apply (auto simp: g sums_Re) 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

636 
apply (metis fg g) 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

637 
done 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56331
diff
changeset

638 
qed 
55734  639 

60758  640 
subsection\<open>Polar Form for Complex Numbers\<close> 
59746
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

641 

ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

642 
lemma complex_unimodular_polar: "(norm z = 1) \<Longrightarrow> \<exists>x. z = Complex (cos x) (sin x)" 
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

643 
using sincos_total_2pi [of "Re z" "Im z"] 
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

644 
by auto (metis cmod_power2 complex_eq power_one) 
14323  645 

60758  646 
subsubsection \<open>$\cos \theta + i \sin \theta$\<close> 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

647 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

648 
primcorec cis :: "real \<Rightarrow> complex" where 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

649 
"Re (cis a) = cos a" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

650 
 "Im (cis a) = sin a" 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

651 

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

652 
lemma cis_zero [simp]: "cis 0 = 1" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

653 
by (simp add: complex_eq_iff) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

654 

44828  655 
lemma norm_cis [simp]: "norm (cis a) = 1" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

656 
by (simp add: norm_complex_def) 
44828  657 

658 
lemma sgn_cis [simp]: "sgn (cis a) = cis a" 

659 
by (simp add: sgn_div_norm) 

660 

661 
lemma cis_neq_zero [simp]: "cis a \<noteq> 0" 

662 
by (metis norm_cis norm_zero zero_neq_one) 

663 

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

664 
lemma cis_mult: "cis a * cis b = cis (a + b)" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

665 
by (simp add: complex_eq_iff cos_add sin_add) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

666 

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

667 
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

668 
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

669 

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

670 
lemma cis_inverse [simp]: "inverse(cis a) = cis (a)" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

671 
by (simp add: complex_eq_iff) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

672 

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

673 
lemma cis_divide: "cis a / cis b = cis (a  b)" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

674 
by (simp add: divide_complex_def cis_mult) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

675 

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

676 
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

677 
by (auto simp add: DeMoivre) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

678 

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

679 
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

680 
by (auto simp add: DeMoivre) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

681 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

682 
lemma cis_pi: "cis pi = 1" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

683 
by (simp add: complex_eq_iff) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

684 

60758  685 
subsubsection \<open>$r(\cos \theta + i \sin \theta)$\<close> 
44715  686 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

687 
definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex" where 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

688 
"rcis r a = complex_of_real r * cis a" 
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

689 

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

690 
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" 
44828  691 
by (simp add: rcis_def) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

692 

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

693 
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" 
44828  694 
by (simp add: rcis_def) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

695 

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

696 
lemma rcis_Ex: "\<exists>r a. z = rcis r a" 
44828  697 
by (simp add: complex_eq_iff polar_Ex) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

698 

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

699 
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" 
44828  700 
by (simp add: rcis_def norm_mult) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

701 

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

702 
lemma cis_rcis_eq: "cis a = rcis 1 a" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

703 
by (simp add: rcis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

704 

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

705 
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" 
44828  706 
by (simp add: rcis_def cis_mult) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

707 

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

708 
lemma rcis_zero_mod [simp]: "rcis 0 a = 0" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

709 
by (simp add: rcis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

710 

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

711 
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

712 
by (simp add: rcis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

713 

44828  714 
lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0" 
715 
by (simp add: rcis_def) 

716 

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

717 
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

718 
by (simp add: rcis_def power_mult_distrib DeMoivre) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

719 

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

720 
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (a)" 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

721 
by (simp add: divide_inverse rcis_def) 
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

722 

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

723 
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a  b)" 
44828  724 
by (simp add: rcis_def cis_divide [symmetric]) 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

725 

60758  726 
subsubsection \<open>Complex exponential\<close> 
44827
4d1384a1fc82
Complex.thy: move theorems into appropriate subsections
huffman
parents:
44825
diff
changeset

727 

59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset

728 
abbreviation Exp :: "complex \<Rightarrow> complex" 
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset

729 
where "Exp \<equiv> exp" 
44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

730 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

731 
lemma cis_conv_exp: "cis b = exp (\<i> * b)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

732 
proof  
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

733 
{ fix n :: nat 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

734 
have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

735 
by (induct n) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

736 
(simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

737 
power2_eq_square real_of_nat_Suc add_nonneg_eq_0_iff 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

738 
real_of_nat_def[symmetric]) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

739 
then have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n = 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

740 
of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

741 
by (simp add: field_simps) } 
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset

742 
then show ?thesis using sin_converges [of b] cos_converges [of b] 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

743 
by (auto simp add: cis.ctr exp_def simp del: of_real_mult 
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset

744 
intro!: sums_unique sums_add sums_mult sums_of_real) 
44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

745 
qed 
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset

746 

59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset

747 
lemma Exp_eq_polar: "Exp z = exp (Re z) * cis (Im z)" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

748 
unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by (cases z) simp 
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset

749 

44828  750 
lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)" 
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset

751 
unfolding Exp_eq_polar by simp 
44828  752 

753 
lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)" 

59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset

754 
unfolding Exp_eq_polar by simp 
44828  755 

59746
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

756 
lemma norm_cos_sin [simp]: "norm (Complex (cos t) (sin t)) = 1" 
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

757 
by (simp add: norm_complex_def) 
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

758 

ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

759 
lemma norm_exp_eq_Re [simp]: "norm (exp z) = exp (Re z)" 
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

760 
by (simp add: cis.code cmod_complex_polar Exp_eq_polar) 
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

761 

59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset

762 
lemma complex_Exp_Ex: "\<exists>a r. z = complex_of_real r * Exp a" 
59746
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

763 
apply (insert rcis_Ex [of z]) 
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

764 
apply (auto simp add: Exp_eq_polar rcis_def mult.assoc [symmetric]) 
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

765 
apply (rule_tac x = "ii * complex_of_real a" in exI, auto) 
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

766 
done 
14323  767 

59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset

768 
lemma Exp_two_pi_i [simp]: "Exp((2::complex) * complex_of_real pi * ii) = 1" 
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset

769 
by (simp add: Exp_eq_polar complex_eq_iff) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

770 

60758  771 
subsubsection \<open>Complex argument\<close> 
44844
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

772 

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

773 
definition arg :: "complex \<Rightarrow> real" where 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

774 
"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

775 

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

776 
lemma arg_zero: "arg 0 = 0" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

777 
by (simp add: arg_def) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

778 

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

779 
lemma arg_unique: 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

780 
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

781 
shows "arg z = x" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

782 
proof  
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

783 
from assms have "z \<noteq> 0" by auto 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

784 
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

785 
proof 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

786 
fix a def d \<equiv> "a  x" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

787 
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

788 
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

789 
unfolding d_def by simp 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

790 
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

791 
by (simp_all add: complex_eq_iff) 
53374
a14d2a854c02
tuned proofs  clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset

792 
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

793 
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

794 
ultimately have "d = 0" 
58709
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58146
diff
changeset

795 
unfolding sin_zero_iff 
58740  796 
by (auto elim!: evenE dest!: less_2_cases) 
44844
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

797 
thus "a = x" unfolding d_def by simp 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

798 
qed (simp add: assms del: Re_sgn Im_sgn) 
60758  799 
with \<open>z \<noteq> 0\<close> show "arg z = x" 
44844
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

800 
unfolding arg_def by simp 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

801 
qed 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

802 

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

803 
lemma arg_correct: 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

804 
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

805 
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

806 
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

807 
with assms have "r \<noteq> 0" by auto 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

808 
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

809 
have b: "sgn z = cis b" 
60758  810 
unfolding z b_def rcis_def using \<open>r \<noteq> 0\<close> 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

811 
by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff) 
44844
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

812 
have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

813 
by (induct_tac n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff) 
44844
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

814 
have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

815 
by (case_tac x rule: int_diff_cases) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

816 
(simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat) 
44844
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

817 
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

818 
have "sgn z = cis c" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

819 
unfolding b c_def 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

820 
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

821 
moreover have " pi < c \<and> c \<le> pi" 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

822 
using ceiling_correct [of "(b  pi) / (2*pi)"] 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

823 
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

824 
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

825 
qed 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

826 

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

827 
lemma arg_bounded: " pi < arg z \<and> arg z \<le> pi" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

828 
by (cases "z = 0") (simp_all add: arg_zero arg_correct) 
44844
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

829 

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

830 
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

831 
by (simp add: arg_correct) 
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

832 

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

833 
lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

834 
by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

835 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

836 
lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

837 
using cis_arg [of y] by (simp add: complex_eq_iff) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

838 

60758  839 
subsection \<open>Square root of complex numbers\<close> 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

840 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

841 
primcorec csqrt :: "complex \<Rightarrow> complex" where 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

842 
"Re (csqrt z) = sqrt ((cmod z + Re z) / 2)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

843 
 "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z  Re z) / 2)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

844 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

845 
lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

846 
by (simp add: complex_eq_iff norm_complex_def) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

847 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

848 
lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

849 
by (simp add: complex_eq_iff norm_complex_def) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

850 

59862  851 
lemma of_real_sqrt: "x \<ge> 0 \<Longrightarrow> of_real (sqrt x) = csqrt (of_real x)" 
852 
by (simp add: complex_eq_iff norm_complex_def) 

853 

56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

854 
lemma csqrt_0 [simp]: "csqrt 0 = 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

855 
by simp 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

856 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

857 
lemma csqrt_1 [simp]: "csqrt 1 = 1" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

858 
by simp 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

859 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

860 
lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

861 
by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt) 
44844
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

862 

59741
5b762cd73a8e
Lots of new material on complexvalued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset

863 
lemma power2_csqrt[simp,algebra]: "(csqrt z)\<^sup>2 = z" 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

864 
proof cases 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

865 
assume "Im z = 0" then show ?thesis 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

866 
using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of " Re z"] 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

867 
by (cases "0::real" "Re z" rule: linorder_cases) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

868 
(simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

869 
next 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

870 
assume "Im z \<noteq> 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

871 
moreover 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

872 
have "cmod z * cmod z  Re z * Re z = Im z * Im z" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

873 
by (simp add: norm_complex_def power2_eq_square) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

874 
moreover 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

875 
have "\<bar>Re z\<bar> \<le> cmod z" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

876 
by (simp add: norm_complex_def) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

877 
ultimately show ?thesis 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

878 
by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

879 
field_simps real_sqrt_mult[symmetric] real_sqrt_divide) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

880 
qed 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

881 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

882 
lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

883 
by auto (metis power2_csqrt power_eq_0_iff) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

884 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

885 
lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

886 
by auto (metis power2_csqrt power2_eq_1_iff) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

887 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

888 
lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

889 
by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

890 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

891 
lemma Re_csqrt: "0 \<le> Re (csqrt z)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

892 
by (metis csqrt_principal le_less) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

893 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

894 
lemma csqrt_square: 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

895 
assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

896 
shows "csqrt (b^2) = b" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

897 
proof  
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

898 
have "csqrt (b^2) = b \<or> csqrt (b^2) =  b" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

899 
unfolding power2_eq_iff[symmetric] by (simp add: power2_csqrt) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

900 
moreover have "csqrt (b^2) \<noteq> b \<or> b = 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

901 
using csqrt_principal[of "b ^ 2"] assms by (intro disjCI notI) (auto simp: complex_eq_iff) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

902 
ultimately show ?thesis 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

903 
by auto 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

904 
qed 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

905 

59746
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

906 
lemma csqrt_unique: 
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

907 
"w^2 = z \<Longrightarrow> (0 < Re w \<or> Re w = 0 \<and> 0 \<le> Im w) \<Longrightarrow> csqrt z = w" 
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

908 
by (auto simp: csqrt_square) 
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

909 

59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59000
diff
changeset

910 
lemma csqrt_minus [simp]: 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

911 
assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

912 
shows "csqrt ( x) = \<i> * csqrt x" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

913 
proof  
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

914 
have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

915 
proof (rule csqrt_square) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

916 
have "Im (csqrt x) \<le> 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

917 
using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

918 
then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

919 
by (auto simp add: Re_csqrt simp del: csqrt.simps) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

920 
qed 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

921 
also have "(\<i> * csqrt x)^2 =  x" 
59746
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset

922 
by (simp add: power_mult_distrib) 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

923 
finally show ?thesis . 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

924 
qed 
44844
f74a4175a3a8
prove existence, uniqueness, and other properties of complex arg function
huffman
parents:
44843
diff
changeset

925 

60758  926 
text \<open>Legacy theorem names\<close> 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

927 

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

928 
lemmas expand_complex_eq = complex_eq_iff 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

929 
lemmas complex_Re_Im_cancel_iff = complex_eq_iff 
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

930 
lemmas complex_equality = complex_eqI 
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

931 
lemmas cmod_def = norm_complex_def 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

932 
lemmas complex_norm_def = norm_complex_def 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

933 
lemmas complex_divide_def = divide_complex_def 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

934 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

935 
lemma legacy_Complex_simps: 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

936 
shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

937 
and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

938 
and complex_minus: " (Complex a b) = Complex ( a) ( b)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

939 
and complex_diff: "Complex a b  Complex c d = Complex (a  c) (b  d)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

940 
and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

941 
and Complex_eq_neg_1: "Complex a b =  1 \<longleftrightarrow> a =  1 \<and> b = 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

942 
and complex_mult: "Complex a b * Complex c d = Complex (a * c  b * d) (a * d + b * c)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

943 
and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) ( b / (a\<^sup>2 + b\<^sup>2))" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

944 
and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

945 
and Complex_eq_neg_numeral: "Complex a b =  numeral w \<longleftrightarrow> a =  numeral w \<and> b = 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

946 
and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

947 
and Complex_eq_i: "(Complex x y = ii) = (x = 0 \<and> y = 1)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

948 
and i_mult_Complex: "ii * Complex a b = Complex ( b) a" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

949 
and Complex_mult_i: "Complex a b * ii = Complex ( b) a" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

950 
and i_complex_of_real: "ii * complex_of_real r = Complex 0 r" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

951 
and complex_of_real_i: "complex_of_real r * ii = Complex 0 r" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

952 
and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

953 
and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

954 
and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

955 
and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

956 
and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

957 
and complex_cn: "cnj (Complex a b) = Complex a ( b)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

958 
and Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

959 
and Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

960 
and complex_of_real_def: "complex_of_real r = Complex r 0" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

961 
and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

962 
by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide del: Complex_eq) 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

963 

48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

964 
lemma Complex_in_Reals: "Complex x 0 \<in> \<real>" 
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56541
diff
changeset

965 
by (metis Reals_of_real complex_of_real_def) 
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset

966 

13957  967 
end 