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(* Title: Complex.thy |
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Author: Jacques D. Fleuriot |
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Copyright: 2001 University of Edinburgh |
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Polymorphic treatment of binary arithmetic using axclasses
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Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 |
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*) |
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header {* Complex Numbers: Rectangular and Polar Representations *} |
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theory Complex |
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imports Transcendental |
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begin |
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datatype complex = Complex real real |
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primrec |
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Re :: "complex \<Rightarrow> real" |
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where |
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Re: "Re (Complex x y) = x" |
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primrec |
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Im :: "complex \<Rightarrow> real" |
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where |
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Im: "Im (Complex x y) = y" |
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lemma complex_surj [simp]: "Complex (Re z) (Im z) = z" |
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by (induct z) simp |
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lemma complex_equality [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y" |
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by (induct x, induct y) simp |
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lemma expand_complex_eq: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y" |
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by (induct x, induct y) simp |
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lemmas complex_Re_Im_cancel_iff = expand_complex_eq |
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subsection {* Addition and Subtraction *} |
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instantiation complex :: ab_group_add |
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begin |
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definition |
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complex_zero_def: "0 = Complex 0 0" |
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definition |
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complex_add_def: "x + y = Complex (Re x + Re y) (Im x + Im y)" |
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definition |
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complex_minus_def: "- x = Complex (- Re x) (- Im x)" |
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definition |
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complex_diff_def: "x - (y\<Colon>complex) = x + - y" |
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lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" |
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by (simp add: complex_zero_def) |
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lemma complex_Re_zero [simp]: "Re 0 = 0" |
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by (simp add: complex_zero_def) |
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lemma complex_Im_zero [simp]: "Im 0 = 0" |
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by (simp add: complex_zero_def) |
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lemma complex_add [simp]: |
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"Complex a b + Complex c d = Complex (a + c) (b + d)" |
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by (simp add: complex_add_def) |
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lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y" |
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by (simp add: complex_add_def) |
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lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y" |
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by (simp add: complex_add_def) |
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lemma complex_minus [simp]: |
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"- (Complex a b) = Complex (- a) (- b)" |
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by (simp add: complex_minus_def) |
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lemma complex_Re_minus [simp]: "Re (- x) = - Re x" |
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by (simp add: complex_minus_def) |
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lemma complex_Im_minus [simp]: "Im (- x) = - Im x" |
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by (simp add: complex_minus_def) |
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lemma complex_diff [simp]: |
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"Complex a b - Complex c d = Complex (a - c) (b - d)" |
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by (simp add: complex_diff_def) |
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lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y" |
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by (simp add: complex_diff_def) |
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lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y" |
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by (simp add: complex_diff_def) |
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instance |
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by intro_classes (simp_all add: complex_add_def complex_diff_def) |
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end |
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subsection {* Multiplication and Division *} |
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instantiation complex :: "{field, division_by_zero}" |
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begin |
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definition |
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complex_one_def: "1 = Complex 1 0" |
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definition |
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complex_mult_def: "x * y = |
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Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)" |
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definition |
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complex_inverse_def: "inverse x = |
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Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))" |
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definition |
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complex_divide_def: "x / (y\<Colon>complex) = x * inverse y" |
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lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)" |
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by (simp add: complex_one_def) |
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lemma complex_Re_one [simp]: "Re 1 = 1" |
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by (simp add: complex_one_def) |
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lemma complex_Im_one [simp]: "Im 1 = 0" |
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by (simp add: complex_one_def) |
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lemma complex_mult [simp]: |
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"Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)" |
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by (simp add: complex_mult_def) |
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lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y" |
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by (simp add: complex_mult_def) |
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lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y" |
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by (simp add: complex_mult_def) |
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lemma complex_inverse [simp]: |
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"inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))" |
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by (simp add: complex_inverse_def) |
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lemma complex_Re_inverse: |
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"Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)" |
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by (simp add: complex_inverse_def) |
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lemma complex_Im_inverse: |
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"Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)" |
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by (simp add: complex_inverse_def) |
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instance |
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by intro_classes (simp_all add: complex_mult_def |
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right_distrib left_distrib right_diff_distrib left_diff_distrib |
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complex_inverse_def complex_divide_def |
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power2_eq_square add_divide_distrib [symmetric] |
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expand_complex_eq) |
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end |
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subsection {* Exponentiation *} |
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instantiation complex :: recpower |
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begin |
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primrec power_complex where |
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complexpow_0: "z ^ 0 = (1\<Colon>complex)" |
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instance by intro_classes simp_all |
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end |
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subsection {* Numerals and Arithmetic *} |
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instantiation complex :: number_ring |
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begin |
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definition number_of_complex where |
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complex_number_of_def: "number_of w = (of_int w \<Colon> complex)" |
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instance |
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by intro_classes (simp only: complex_number_of_def) |
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end |
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lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n" |
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by (induct n) simp_all |
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lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0" |
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by (induct n) simp_all |
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lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z" |
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by (cases z rule: int_diff_cases) simp |
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lemma complex_Im_of_int [simp]: "Im (of_int z) = 0" |
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by (cases z rule: int_diff_cases) simp |
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lemma complex_Re_number_of [simp]: "Re (number_of v) = number_of v" |
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lemma complex_Im_number_of [simp]: "Im (number_of v) = 0" |
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|
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lemma Complex_eq_number_of [simp]: |
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"(Complex a b = number_of w) = (a = number_of w \<and> b = 0)" |
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by (simp add: expand_complex_eq) |
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|
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209 |
|
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subsection {* Scalar Multiplication *} |
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|
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instantiation complex :: real_field |
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begin |
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|
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definition |
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complex_scaleR_def: "scaleR r x = Complex (r * Re x) (r * Im x)" |
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|
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lemma complex_scaleR [simp]: |
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"scaleR r (Complex a b) = Complex (r * a) (r * b)" |
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|
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lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x" |
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|
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lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x" |
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|
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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: expand_complex_eq right_distrib) |
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show "scaleR (a + b) x = scaleR a x + scaleR b x" |
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by (simp add: expand_complex_eq left_distrib) |
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show "scaleR a (scaleR b x) = scaleR (a * b) x" |
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by (simp add: expand_complex_eq mult_assoc) |
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show "scaleR 1 x = x" |
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by (simp add: expand_complex_eq) |
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show "scaleR a x * y = scaleR a (x * y)" |
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by (simp add: expand_complex_eq algebra_simps) |
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show "x * scaleR a y = scaleR a (x * y)" |
29667 | 242 |
by (simp add: expand_complex_eq algebra_simps) |
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qed |
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|
25712 | 245 |
end |
246 |
||
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247 |
|
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subsection{* Properties of Embedding from Reals *} |
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|
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250 |
abbreviation |
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complex_of_real :: "real \<Rightarrow> complex" where |
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"complex_of_real \<equiv> of_real" |
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253 |
|
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254 |
lemma complex_of_real_def: "complex_of_real r = Complex r 0" |
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by (simp add: of_real_def complex_scaleR_def) |
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256 |
|
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lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" |
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258 |
by (simp add: complex_of_real_def) |
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259 |
|
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lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" |
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261 |
by (simp add: complex_of_real_def) |
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262 |
|
14377 | 263 |
lemma Complex_add_complex_of_real [simp]: |
264 |
"Complex x y + complex_of_real r = Complex (x+r) y" |
|
265 |
by (simp add: complex_of_real_def) |
|
266 |
||
267 |
lemma complex_of_real_add_Complex [simp]: |
|
268 |
"complex_of_real r + Complex x y = Complex (r+x) y" |
|
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by (simp add: complex_of_real_def) |
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|
271 |
lemma Complex_mult_complex_of_real: |
|
272 |
"Complex x y * complex_of_real r = Complex (x*r) (y*r)" |
|
273 |
by (simp add: complex_of_real_def) |
|
274 |
||
275 |
lemma complex_of_real_mult_Complex: |
|
276 |
"complex_of_real r * Complex x y = Complex (r*x) (r*y)" |
|
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by (simp add: complex_of_real_def) |
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278 |
|
14377 | 279 |
|
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subsection {* Vector Norm *} |
14323 | 281 |
|
25712 | 282 |
instantiation complex :: real_normed_field |
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begin |
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284 |
|
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definition |
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complex_norm_def: "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" |
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|
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288 |
abbreviation |
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cmod :: "complex \<Rightarrow> real" where |
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"cmod \<equiv> norm" |
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|
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definition |
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complex_sgn_def: "sgn x = x /\<^sub>R cmod x" |
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294 |
|
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295 |
lemmas cmod_def = complex_norm_def |
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296 |
|
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297 |
lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)" |
25712 | 298 |
by (simp add: complex_norm_def) |
22852 | 299 |
|
25712 | 300 |
instance |
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301 |
proof |
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fix r :: real and x y :: complex |
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303 |
show "0 \<le> norm x" |
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304 |
by (induct x) simp |
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show "(norm x = 0) = (x = 0)" |
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by (induct x) simp |
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show "norm (x + y) \<le> norm x + norm y" |
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308 |
by (induct x, induct y) |
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309 |
(simp add: real_sqrt_sum_squares_triangle_ineq) |
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310 |
show "norm (scaleR r x) = \<bar>r\<bar> * norm x" |
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by (induct x) |
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312 |
(simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult) |
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313 |
show "norm (x * y) = norm x * norm y" |
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|
314 |
by (induct x, induct y) |
29667 | 315 |
(simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps) |
24506 | 316 |
show "sgn x = x /\<^sub>R cmod x" by(simp add: complex_sgn_def) |
24520 | 317 |
qed |
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318 |
|
25712 | 319 |
end |
320 |
||
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321 |
lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1" |
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by simp |
14323 | 323 |
|
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324 |
lemma cmod_complex_polar [simp]: |
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325 |
"cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r" |
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326 |
by (simp add: norm_mult) |
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327 |
|
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328 |
lemma complex_Re_le_cmod: "Re x \<le> cmod x" |
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329 |
unfolding complex_norm_def |
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330 |
by (rule real_sqrt_sum_squares_ge1) |
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331 |
|
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332 |
lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x" |
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333 |
by (rule order_trans [OF _ norm_ge_zero], simp) |
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334 |
|
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335 |
lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a" |
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336 |
by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp) |
14323 | 337 |
|
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338 |
lemmas real_sum_squared_expand = power2_sum [where 'a=real] |
14323 | 339 |
|
26117 | 340 |
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x" |
341 |
by (cases x) simp |
|
342 |
||
343 |
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x" |
|
344 |
by (cases x) simp |
|
14354
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|
345 |
|
23123 | 346 |
subsection {* Completeness of the Complexes *} |
347 |
||
29233 | 348 |
interpretation Re!: bounded_linear "Re" |
23123 | 349 |
apply (unfold_locales, simp, simp) |
350 |
apply (rule_tac x=1 in exI) |
|
351 |
apply (simp add: complex_norm_def) |
|
352 |
done |
|
353 |
||
29233 | 354 |
interpretation Im!: bounded_linear "Im" |
23123 | 355 |
apply (unfold_locales, simp, simp) |
356 |
apply (rule_tac x=1 in exI) |
|
357 |
apply (simp add: complex_norm_def) |
|
358 |
done |
|
359 |
||
360 |
lemma LIMSEQ_Complex: |
|
361 |
"\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) ----> Complex a b" |
|
362 |
apply (rule LIMSEQ_I) |
|
363 |
apply (subgoal_tac "0 < r / sqrt 2") |
|
364 |
apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe) |
|
365 |
apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe) |
|
366 |
apply (rename_tac M N, rule_tac x="max M N" in exI, safe) |
|
367 |
apply (simp add: real_sqrt_sum_squares_less) |
|
368 |
apply (simp add: divide_pos_pos) |
|
369 |
done |
|
370 |
||
371 |
instance complex :: banach |
|
372 |
proof |
|
373 |
fix X :: "nat \<Rightarrow> complex" |
|
374 |
assume X: "Cauchy X" |
|
375 |
from Re.Cauchy [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))" |
|
376 |
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) |
|
377 |
from Im.Cauchy [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))" |
|
378 |
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) |
|
379 |
have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))" |
|
380 |
using LIMSEQ_Complex [OF 1 2] by simp |
|
381 |
thus "convergent X" |
|
382 |
by (rule convergentI) |
|
383 |
qed |
|
384 |
||
385 |
||
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386 |
subsection {* The Complex Number @{term "\<i>"} *} |
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|
387 |
|
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|
388 |
definition |
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389 |
"ii" :: complex ("\<i>") where |
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390 |
i_def: "ii \<equiv> Complex 0 1" |
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391 |
|
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392 |
lemma complex_Re_i [simp]: "Re ii = 0" |
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393 |
by (simp add: i_def) |
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|
394 |
|
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395 |
lemma complex_Im_i [simp]: "Im ii = 1" |
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396 |
by (simp add: i_def) |
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|
397 |
|
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398 |
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)" |
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399 |
by (simp add: i_def) |
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400 |
|
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401 |
lemma complex_i_not_zero [simp]: "ii \<noteq> 0" |
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402 |
by (simp add: expand_complex_eq) |
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|
403 |
|
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404 |
lemma complex_i_not_one [simp]: "ii \<noteq> 1" |
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405 |
by (simp add: expand_complex_eq) |
23124 | 406 |
|
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407 |
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w" |
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408 |
by (simp add: expand_complex_eq) |
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|
409 |
|
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410 |
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a" |
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411 |
by (simp add: expand_complex_eq) |
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|
412 |
|
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413 |
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a" |
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414 |
by (simp add: expand_complex_eq) |
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415 |
|
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416 |
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" |
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417 |
by (simp add: i_def complex_of_real_def) |
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|
418 |
|
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419 |
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" |
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420 |
by (simp add: i_def complex_of_real_def) |
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|
421 |
|
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422 |
lemma i_squared [simp]: "ii * ii = -1" |
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423 |
by (simp add: i_def) |
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|
424 |
|
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|
425 |
lemma power2_i [simp]: "ii\<twosuperior> = -1" |
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426 |
by (simp add: power2_eq_square) |
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|
427 |
|
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|
428 |
lemma inverse_i [simp]: "inverse ii = - ii" |
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|
429 |
by (rule inverse_unique, simp) |
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|
430 |
|
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|
431 |
|
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432 |
subsection {* Complex Conjugation *} |
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|
433 |
|
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|
434 |
definition |
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435 |
cnj :: "complex \<Rightarrow> complex" where |
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436 |
"cnj z = Complex (Re z) (- Im z)" |
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|
437 |
|
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438 |
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)" |
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439 |
by (simp add: cnj_def) |
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|
440 |
|
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|
441 |
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x" |
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442 |
by (simp add: cnj_def) |
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|
443 |
|
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444 |
lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x" |
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445 |
by (simp add: cnj_def) |
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|
446 |
|
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|
447 |
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" |
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|
448 |
by (simp add: expand_complex_eq) |
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|
449 |
|
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|
450 |
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" |
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|
451 |
by (simp add: cnj_def) |
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|
452 |
|
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|
453 |
lemma complex_cnj_zero [simp]: "cnj 0 = 0" |
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|
454 |
by (simp add: expand_complex_eq) |
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|
455 |
|
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|
456 |
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" |
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|
457 |
by (simp add: expand_complex_eq) |
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|
458 |
|
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|
459 |
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y" |
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|
460 |
by (simp add: expand_complex_eq) |
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|
461 |
|
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|
462 |
lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y" |
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|
463 |
by (simp add: expand_complex_eq) |
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|
464 |
|
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|
465 |
lemma complex_cnj_minus: "cnj (- x) = - cnj x" |
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|
466 |
by (simp add: expand_complex_eq) |
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|
467 |
|
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|
468 |
lemma complex_cnj_one [simp]: "cnj 1 = 1" |
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469 |
by (simp add: expand_complex_eq) |
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|
470 |
|
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|
471 |
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y" |
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472 |
by (simp add: expand_complex_eq) |
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|
473 |
|
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|
474 |
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)" |
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|
475 |
by (simp add: complex_inverse_def) |
14323 | 476 |
|
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|
477 |
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y" |
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|
478 |
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) |
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|
479 |
|
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|
480 |
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n" |
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|
481 |
by (induct n, simp_all add: complex_cnj_mult) |
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|
482 |
|
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|
483 |
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" |
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484 |
by (simp add: expand_complex_eq) |
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|
485 |
|
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|
486 |
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" |
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|
487 |
by (simp add: expand_complex_eq) |
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|
488 |
|
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|
489 |
lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w" |
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|
490 |
by (simp add: expand_complex_eq) |
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|
491 |
|
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|
492 |
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)" |
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|
493 |
by (simp add: expand_complex_eq) |
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|
494 |
|
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|
495 |
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" |
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|
496 |
by (simp add: complex_norm_def) |
14323 | 497 |
|
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|
498 |
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" |
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|
499 |
by (simp add: expand_complex_eq) |
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|
500 |
|
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|
501 |
lemma complex_cnj_i [simp]: "cnj ii = - ii" |
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|
502 |
by (simp add: expand_complex_eq) |
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|
503 |
|
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|
504 |
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" |
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|
505 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
506 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
507 |
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
508 |
by (simp add: expand_complex_eq) |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
509 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
510 |
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
511 |
by (simp add: expand_complex_eq power2_eq_square) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
512 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
513 |
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
514 |
by (simp add: norm_mult power2_eq_square) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
515 |
|
29233 | 516 |
interpretation cnj!: bounded_linear "cnj" |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
517 |
apply (unfold_locales) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
518 |
apply (rule complex_cnj_add) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
519 |
apply (rule complex_cnj_scaleR) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
520 |
apply (rule_tac x=1 in exI, simp) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
521 |
done |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
522 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
523 |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22968
diff
changeset
|
524 |
subsection{*The Functions @{term sgn} and @{term arg}*} |
14323 | 525 |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22968
diff
changeset
|
526 |
text {*------------ Argand -------------*} |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
527 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
528 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
529 |
arg :: "complex => real" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
530 |
"arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
531 |
|
14374 | 532 |
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" |
24506 | 533 |
by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute) |
14323 | 534 |
|
535 |
lemma i_mult_eq: "ii * ii = complex_of_real (-1)" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
536 |
by (simp add: i_def complex_of_real_def) |
14323 | 537 |
|
14374 | 538 |
lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
539 |
by (simp add: i_def complex_one_def) |
14323 | 540 |
|
14374 | 541 |
lemma complex_eq_cancel_iff2 [simp]: |
14377 | 542 |
"(Complex x y = complex_of_real xa) = (x = xa & y = 0)" |
543 |
by (simp add: complex_of_real_def) |
|
14323 | 544 |
|
14374 | 545 |
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" |
24506 | 546 |
by (simp add: complex_sgn_def divide_inverse) |
14323 | 547 |
|
14374 | 548 |
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" |
24506 | 549 |
by (simp add: complex_sgn_def divide_inverse) |
14323 | 550 |
|
551 |
lemma complex_inverse_complex_split: |
|
552 |
"inverse(complex_of_real x + ii * complex_of_real y) = |
|
553 |
complex_of_real(x/(x ^ 2 + y ^ 2)) - |
|
554 |
ii * complex_of_real(y/(x ^ 2 + y ^ 2))" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
555 |
by (simp add: complex_of_real_def i_def diff_minus divide_inverse) |
14323 | 556 |
|
557 |
(*----------------------------------------------------------------------------*) |
|
558 |
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) |
|
559 |
(* many of the theorems are not used - so should they be kept? *) |
|
560 |
(*----------------------------------------------------------------------------*) |
|
561 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
562 |
lemma cos_arg_i_mult_zero_pos: |
14377 | 563 |
"0 < y ==> cos (arg(Complex 0 y)) = 0" |
14373 | 564 |
apply (simp add: arg_def abs_if) |
14334 | 565 |
apply (rule_tac a = "pi/2" in someI2, auto) |
566 |
apply (rule order_less_trans [of _ 0], auto) |
|
14323 | 567 |
done |
568 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
569 |
lemma cos_arg_i_mult_zero_neg: |
14377 | 570 |
"y < 0 ==> cos (arg(Complex 0 y)) = 0" |
14373 | 571 |
apply (simp add: arg_def abs_if) |
14334 | 572 |
apply (rule_tac a = "- pi/2" in someI2, auto) |
573 |
apply (rule order_trans [of _ 0], auto) |
|
14323 | 574 |
done |
575 |
||
14374 | 576 |
lemma cos_arg_i_mult_zero [simp]: |
14377 | 577 |
"y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0" |
578 |
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) |
|
14323 | 579 |
|
580 |
||
581 |
subsection{*Finally! Polar Form for Complex Numbers*} |
|
582 |
||
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
583 |
definition |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
584 |
|
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
585 |
(* abbreviation for (cos a + i sin a) *) |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
586 |
cis :: "real => complex" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
587 |
"cis a = Complex (cos a) (sin a)" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
588 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
589 |
definition |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
590 |
(* abbreviation for r*(cos a + i sin a) *) |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
591 |
rcis :: "[real, real] => complex" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
592 |
"rcis r a = complex_of_real r * cis a" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
593 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
594 |
definition |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
595 |
(* e ^ (x + iy) *) |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
596 |
expi :: "complex => complex" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
597 |
"expi z = complex_of_real(exp (Re z)) * cis (Im z)" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
598 |
|
14374 | 599 |
lemma complex_split_polar: |
14377 | 600 |
"\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
601 |
apply (induct z) |
14377 | 602 |
apply (auto simp add: polar_Ex complex_of_real_mult_Complex) |
14323 | 603 |
done |
604 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
605 |
lemma rcis_Ex: "\<exists>r a. z = rcis r a" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
606 |
apply (induct z) |
14377 | 607 |
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex) |
14323 | 608 |
done |
609 |
||
14374 | 610 |
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" |
14373 | 611 |
by (simp add: rcis_def cis_def) |
14323 | 612 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
613 |
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" |
14373 | 614 |
by (simp add: rcis_def cis_def) |
14323 | 615 |
|
14377 | 616 |
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>" |
617 |
proof - |
|
618 |
have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
619 |
by (simp only: power_mult_distrib right_distrib) |
14377 | 620 |
thus ?thesis by simp |
621 |
qed |
|
14323 | 622 |
|
14374 | 623 |
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" |
14377 | 624 |
by (simp add: rcis_def cis_def sin_cos_squared_add2_mult) |
14323 | 625 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
626 |
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
627 |
by (simp add: cmod_def power2_eq_square) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
628 |
|
14374 | 629 |
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
630 |
by simp |
14323 | 631 |
|
632 |
||
633 |
(*---------------------------------------------------------------------------*) |
|
634 |
(* (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b) *) |
|
635 |
(*---------------------------------------------------------------------------*) |
|
636 |
||
637 |
lemma cis_rcis_eq: "cis a = rcis 1 a" |
|
14373 | 638 |
by (simp add: rcis_def) |
14323 | 639 |
|
14374 | 640 |
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" |
15013 | 641 |
by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib |
642 |
complex_of_real_def) |
|
14323 | 643 |
|
644 |
lemma cis_mult: "cis a * cis b = cis (a + b)" |
|
14373 | 645 |
by (simp add: cis_rcis_eq rcis_mult) |
14323 | 646 |
|
14374 | 647 |
lemma cis_zero [simp]: "cis 0 = 1" |
14377 | 648 |
by (simp add: cis_def complex_one_def) |
14323 | 649 |
|
14374 | 650 |
lemma rcis_zero_mod [simp]: "rcis 0 a = 0" |
14373 | 651 |
by (simp add: rcis_def) |
14323 | 652 |
|
14374 | 653 |
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" |
14373 | 654 |
by (simp add: rcis_def) |
14323 | 655 |
|
656 |
lemma complex_of_real_minus_one: |
|
657 |
"complex_of_real (-(1::real)) = -(1::complex)" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
658 |
by (simp add: complex_of_real_def complex_one_def) |
14323 | 659 |
|
14374 | 660 |
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x" |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
661 |
by (simp add: mult_assoc [symmetric]) |
14323 | 662 |
|
663 |
||
664 |
lemma cis_real_of_nat_Suc_mult: |
|
665 |
"cis (real (Suc n) * a) = cis a * cis (real n * a)" |
|
14377 | 666 |
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib) |
14323 | 667 |
|
668 |
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" |
|
669 |
apply (induct_tac "n") |
|
670 |
apply (auto simp add: cis_real_of_nat_Suc_mult) |
|
671 |
done |
|
672 |
||
14374 | 673 |
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" |
22890 | 674 |
by (simp add: rcis_def power_mult_distrib DeMoivre) |
14323 | 675 |
|
14374 | 676 |
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
677 |
by (simp add: cis_def complex_inverse_complex_split diff_minus) |
14323 | 678 |
|
679 |
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)" |
|
22884 | 680 |
by (simp add: divide_inverse rcis_def) |
14323 | 681 |
|
682 |
lemma cis_divide: "cis a / cis b = cis (a - b)" |
|
14373 | 683 |
by (simp add: complex_divide_def cis_mult real_diff_def) |
14323 | 684 |
|
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
685 |
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)" |
14373 | 686 |
apply (simp add: complex_divide_def) |
687 |
apply (case_tac "r2=0", simp) |
|
688 |
apply (simp add: rcis_inverse rcis_mult real_diff_def) |
|
14323 | 689 |
done |
690 |
||
14374 | 691 |
lemma Re_cis [simp]: "Re(cis a) = cos a" |
14373 | 692 |
by (simp add: cis_def) |
14323 | 693 |
|
14374 | 694 |
lemma Im_cis [simp]: "Im(cis a) = sin a" |
14373 | 695 |
by (simp add: cis_def) |
14323 | 696 |
|
697 |
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" |
|
14334 | 698 |
by (auto simp add: DeMoivre) |
14323 | 699 |
|
700 |
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" |
|
14334 | 701 |
by (auto simp add: DeMoivre) |
14323 | 702 |
|
703 |
lemma expi_add: "expi(a + b) = expi(a) * expi(b)" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
704 |
by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac) |
14323 | 705 |
|
14374 | 706 |
lemma expi_zero [simp]: "expi (0::complex) = 1" |
14373 | 707 |
by (simp add: expi_def) |
14323 | 708 |
|
14374 | 709 |
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a" |
14373 | 710 |
apply (insert rcis_Ex [of z]) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
711 |
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric]) |
14334 | 712 |
apply (rule_tac x = "ii * complex_of_real a" in exI, auto) |
14323 | 713 |
done |
714 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
715 |
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1" |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
716 |
by (simp add: expi_def cis_def) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
717 |
|
13957 | 718 |
end |