author | huffman |
Wed, 09 May 2007 01:56:59 +0200 | |
changeset 22884 | 5804926e0379 |
parent 22883 | 005be8dafce0 |
child 22890 | 9449c991cc33 |
permissions | -rw-r--r-- |
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(* Title: Complex.thy |
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ID: $Id$ |
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Author: Jacques D. Fleuriot |
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Copyright: 2001 University of Edinburgh |
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Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 |
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*) |
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header {* Complex Numbers: Rectangular and Polar Representations *} |
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theory Complex |
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imports "../Hyperreal/Transcendental" |
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begin |
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datatype complex = Complex real real |
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instance complex :: "{zero, one, plus, times, minus, inverse, power}" .. |
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consts |
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"ii" :: complex ("\<i>") |
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consts Re :: "complex => real" |
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primrec Re: "Re (Complex x y) = x" |
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consts Im :: "complex => real" |
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primrec Im: "Im (Complex x y) = y" |
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lemma complex_surj [simp]: "Complex (Re z) (Im z) = z" |
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by (induct z) simp |
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defs (overloaded) |
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complex_zero_def: |
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"0 == Complex 0 0" |
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complex_one_def: |
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"1 == Complex 1 0" |
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i_def: "ii == Complex 0 1" |
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complex_minus_def: "- z == Complex (- Re z) (- Im z)" |
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complex_inverse_def: |
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"inverse z == |
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Complex (Re z / ((Re z)\<twosuperior> + (Im z)\<twosuperior>)) (- Im z / ((Re z)\<twosuperior> + (Im z)\<twosuperior>))" |
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complex_add_def: |
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"z + w == Complex (Re z + Re w) (Im z + Im w)" |
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complex_diff_def: |
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"z - w == z + - (w::complex)" |
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complex_mult_def: |
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"z * w == Complex (Re z * Re w - Im z * Im w) (Re z * Im w + Im z * Re w)" |
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complex_divide_def: "w / (z::complex) == w * inverse z" |
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lemma complex_equality [intro?]: "Re z = Re w ==> Im z = Im w ==> z = w" |
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by (induct z, induct w) simp |
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lemma complex_Re_Im_cancel_iff: "(w=z) = (Re(w) = Re(z) & Im(w) = Im(z))" |
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by (induct w, induct z, simp) |
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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_zero_iff [simp]: "(Complex x y = 0) = (x = 0 \<and> y = 0)" |
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unfolding complex_zero_def by simp |
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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_Re_i [simp]: "Re(ii) = 0" |
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by (simp add: i_def) |
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lemma complex_Im_i [simp]: "Im(ii) = 1" |
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by (simp add: i_def) |
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subsection{*Unary Minus*} |
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lemma complex_minus [simp]: "- (Complex x y) = Complex (-x) (-y)" |
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by (simp add: complex_minus_def) |
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lemma complex_Re_minus [simp]: "Re (-z) = - Re z" |
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by (simp add: complex_minus_def) |
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lemma complex_Im_minus [simp]: "Im (-z) = - Im z" |
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by (simp add: complex_minus_def) |
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subsection{*Addition*} |
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lemma complex_add [simp]: |
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"Complex x1 y1 + Complex x2 y2 = Complex (x1+x2) (y1+y2)" |
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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_add_commute: "(u::complex) + v = v + u" |
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by (simp add: complex_add_def add_commute) |
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lemma complex_add_assoc: "((u::complex) + v) + w = u + (v + w)" |
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by (simp add: complex_add_def add_assoc) |
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lemma complex_add_zero_left: "(0::complex) + z = z" |
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by (simp add: complex_add_def complex_zero_def) |
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lemma complex_add_zero_right: "z + (0::complex) = z" |
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by (simp add: complex_add_def complex_zero_def) |
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lemma complex_add_minus_left: "-z + z = (0::complex)" |
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by (simp add: complex_add_def complex_minus_def complex_zero_def) |
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lemma complex_diff: |
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"Complex x1 y1 - Complex x2 y2 = Complex (x1-x2) (y1-y2)" |
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by (simp add: complex_add_def complex_minus_def complex_diff_def) |
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lemma complex_Re_diff [simp]: "Re(x - y) = Re(x) - Re(y)" |
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by (simp add: complex_diff_def) |
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lemma complex_Im_diff [simp]: "Im(x - y) = Im(x) - Im(y)" |
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by (simp add: complex_diff_def) |
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subsection{*Multiplication*} |
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lemma complex_mult [simp]: |
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"Complex x1 y1 * Complex x2 y2 = Complex (x1*x2 - y1*y2) (x1*y2 + y1*x2)" |
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by (simp add: complex_mult_def) |
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lemma complex_mult_commute: "(w::complex) * z = z * w" |
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by (simp add: complex_mult_def mult_commute add_commute) |
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lemma complex_mult_assoc: "((u::complex) * v) * w = u * (v * w)" |
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by (simp add: complex_mult_def mult_ac add_ac |
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right_diff_distrib right_distrib left_diff_distrib left_distrib) |
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lemma complex_mult_one_left: "(1::complex) * z = z" |
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by (simp add: complex_mult_def complex_one_def) |
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lemma complex_mult_one_right: "z * (1::complex) = z" |
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by (simp add: complex_mult_def complex_one_def) |
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subsection{*Inverse*} |
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lemma complex_inverse [simp]: |
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"inverse (Complex x y) = Complex (x/(x ^ 2 + y ^ 2)) (-y/(x ^ 2 + y ^ 2))" |
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by (simp add: complex_inverse_def) |
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lemma complex_mult_inv_left: "z \<noteq> (0::complex) ==> inverse(z) * z = 1" |
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apply (induct z) |
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apply (rename_tac x y) |
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apply (auto simp add: |
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complex_one_def complex_zero_def add_divide_distrib [symmetric] |
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power2_eq_square mult_ac) |
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apply (simp_all add: real_sum_squares_not_zero real_sum_squares_not_zero2) |
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done |
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subsection {* The field of complex numbers *} |
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instance complex :: field |
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proof |
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fix z u v w :: complex |
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show "(u + v) + w = u + (v + w)" |
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by (rule complex_add_assoc) |
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show "z + w = w + z" |
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by (rule complex_add_commute) |
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show "0 + z = z" |
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by (rule complex_add_zero_left) |
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show "-z + z = 0" |
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by (rule complex_add_minus_left) |
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show "z - w = z + -w" |
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by (simp add: complex_diff_def) |
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show "(u * v) * w = u * (v * w)" |
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by (rule complex_mult_assoc) |
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show "z * w = w * z" |
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by (rule complex_mult_commute) |
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show "1 * z = z" |
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by (rule complex_mult_one_left) |
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show "0 \<noteq> (1::complex)" |
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by (simp add: complex_zero_def complex_one_def) |
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show "(u + v) * w = u * w + v * w" |
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by (simp add: complex_mult_def complex_add_def left_distrib |
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diff_minus add_ac) |
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show "z / w = z * inverse w" |
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by (simp add: complex_divide_def) |
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assume "w \<noteq> 0" |
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thus "inverse w * w = 1" |
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by (simp add: complex_mult_inv_left) |
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qed |
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instance complex :: division_by_zero |
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proof |
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show "inverse 0 = (0::complex)" |
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by (simp add: complex_inverse_def complex_zero_def) |
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qed |
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subsection{*The real algebra of complex numbers*} |
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instance complex :: scaleR .. |
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defs (overloaded) |
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complex_scaleR_def: "r *# x == Complex r 0 * x" |
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instance complex :: real_field |
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proof |
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fix a b :: real |
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fix x y :: complex |
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show "a *# (x + y) = a *# x + a *# y" |
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by (simp add: complex_scaleR_def right_distrib) |
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show "(a + b) *# x = a *# x + b *# x" |
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by (simp add: complex_scaleR_def left_distrib [symmetric]) |
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show "a *# b *# x = (a * b) *# x" |
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by (simp add: complex_scaleR_def mult_assoc [symmetric]) |
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show "1 *# x = x" |
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by (simp add: complex_scaleR_def complex_one_def [symmetric]) |
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show "a *# x * y = a *# (x * y)" |
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by (simp add: complex_scaleR_def mult_assoc) |
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show "x * a *# y = a *# (x * y)" |
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by (simp add: complex_scaleR_def mult_left_commute) |
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qed |
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subsection{*Embedding Properties for @{term complex_of_real} Map*} |
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abbreviation |
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complex_of_real :: "real => complex" where |
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"complex_of_real == of_real" |
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lemma complex_of_real_def: "complex_of_real r = Complex r 0" |
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by (simp add: of_real_def complex_scaleR_def) |
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lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" |
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by (simp add: complex_of_real_def) |
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lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" |
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by (simp add: complex_of_real_def) |
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lemma Complex_add_complex_of_real [simp]: |
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"Complex x y + complex_of_real r = Complex (x+r) y" |
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by (simp add: complex_of_real_def) |
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lemma complex_of_real_add_Complex [simp]: |
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"complex_of_real r + Complex x y = Complex (r+x) y" |
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by (simp add: i_def complex_of_real_def) |
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lemma Complex_mult_complex_of_real: |
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"Complex x y * complex_of_real r = Complex (x*r) (y*r)" |
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by (simp add: complex_of_real_def) |
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lemma complex_of_real_mult_Complex: |
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"complex_of_real r * Complex x y = Complex (r*x) (r*y)" |
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by (simp add: i_def complex_of_real_def) |
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||
269 |
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" |
|
270 |
by (simp add: i_def complex_of_real_def) |
|
271 |
||
272 |
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" |
|
273 |
by (simp add: i_def complex_of_real_def) |
|
274 |
||
14323 | 275 |
|
14377 | 276 |
subsection{*The Functions @{term Re} and @{term Im}*} |
277 |
||
278 |
lemma complex_Re_mult_eq: "Re (w * z) = Re w * Re z - Im w * Im z" |
|
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|
279 |
by (induct z, induct w, simp) |
14377 | 280 |
|
281 |
lemma complex_Im_mult_eq: "Im (w * z) = Re w * Im z + Im w * Re z" |
|
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|
282 |
by (induct z, induct w, simp) |
14377 | 283 |
|
284 |
lemma Re_i_times [simp]: "Re(ii * z) = - Im z" |
|
20725
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|
285 |
by (simp add: complex_Re_mult_eq) |
14377 | 286 |
|
287 |
lemma Re_times_i [simp]: "Re(z * ii) = - Im z" |
|
20725
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|
288 |
by (simp add: complex_Re_mult_eq) |
14377 | 289 |
|
290 |
lemma Im_i_times [simp]: "Im(ii * z) = Re z" |
|
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|
291 |
by (simp add: complex_Im_mult_eq) |
14377 | 292 |
|
293 |
lemma Im_times_i [simp]: "Im(z * ii) = Re z" |
|
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|
294 |
by (simp add: complex_Im_mult_eq) |
14377 | 295 |
|
296 |
lemma complex_Re_mult: "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)" |
|
297 |
by (simp add: complex_Re_mult_eq) |
|
298 |
||
299 |
lemma complex_Re_mult_complex_of_real [simp]: |
|
300 |
"Re (z * complex_of_real c) = Re(z) * c" |
|
301 |
by (simp add: complex_Re_mult_eq) |
|
302 |
||
303 |
lemma complex_Im_mult_complex_of_real [simp]: |
|
304 |
"Im (z * complex_of_real c) = Im(z) * c" |
|
305 |
by (simp add: complex_Im_mult_eq) |
|
306 |
||
307 |
lemma complex_Re_mult_complex_of_real2 [simp]: |
|
308 |
"Re (complex_of_real c * z) = c * Re(z)" |
|
309 |
by (simp add: complex_Re_mult_eq) |
|
310 |
||
311 |
lemma complex_Im_mult_complex_of_real2 [simp]: |
|
312 |
"Im (complex_of_real c * z) = c * Im(z)" |
|
313 |
by (simp add: complex_Im_mult_eq) |
|
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huffman
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changeset
|
314 |
|
14377 | 315 |
|
14323 | 316 |
subsection{*Conjugation is an Automorphism*} |
317 |
||
20557
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|
318 |
definition |
21404
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more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
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|
319 |
cnj :: "complex => complex" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
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|
320 |
"cnj z = Complex (Re z) (-Im z)" |
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|
321 |
|
14373 | 322 |
lemma complex_cnj: "cnj (Complex x y) = Complex x (-y)" |
323 |
by (simp add: cnj_def) |
|
14323 | 324 |
|
14374 | 325 |
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" |
14373 | 326 |
by (simp add: cnj_def complex_Re_Im_cancel_iff) |
14323 | 327 |
|
14374 | 328 |
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" |
14373 | 329 |
by (simp add: cnj_def) |
14323 | 330 |
|
14374 | 331 |
lemma complex_cnj_complex_of_real [simp]: |
14373 | 332 |
"cnj (complex_of_real x) = complex_of_real x" |
333 |
by (simp add: complex_of_real_def complex_cnj) |
|
14323 | 334 |
|
335 |
lemma complex_cnj_minus: "cnj (-z) = - cnj z" |
|
20725
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changeset
|
336 |
by (simp add: cnj_def) |
14323 | 337 |
|
338 |
lemma complex_cnj_inverse: "cnj(inverse z) = inverse(cnj z)" |
|
20725
72e20198f834
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huffman
parents:
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diff
changeset
|
339 |
by (induct z, simp add: complex_cnj power2_eq_square) |
14323 | 340 |
|
341 |
lemma complex_cnj_add: "cnj(w + z) = cnj(w) + cnj(z)" |
|
20725
72e20198f834
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huffman
parents:
20560
diff
changeset
|
342 |
by (induct w, induct z, simp add: complex_cnj) |
14323 | 343 |
|
344 |
lemma complex_cnj_diff: "cnj(w - z) = cnj(w) - cnj(z)" |
|
15013 | 345 |
by (simp add: diff_minus complex_cnj_add complex_cnj_minus) |
14323 | 346 |
|
347 |
lemma complex_cnj_mult: "cnj(w * z) = cnj(w) * cnj(z)" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
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diff
changeset
|
348 |
by (induct w, induct z, simp add: complex_cnj) |
14323 | 349 |
|
350 |
lemma complex_cnj_divide: "cnj(w / z) = (cnj w)/(cnj z)" |
|
14373 | 351 |
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) |
14323 | 352 |
|
14374 | 353 |
lemma complex_cnj_one [simp]: "cnj 1 = 1" |
14373 | 354 |
by (simp add: cnj_def complex_one_def) |
14323 | 355 |
|
356 |
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re(z))" |
|
20725
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instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
357 |
by (induct z, simp add: complex_cnj complex_of_real_def) |
14323 | 358 |
|
359 |
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im(z)) * ii" |
|
14373 | 360 |
apply (induct z) |
15013 | 361 |
apply (simp add: complex_add complex_cnj complex_of_real_def diff_minus |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
362 |
complex_minus i_def complex_mult) |
14323 | 363 |
done |
364 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
365 |
lemma complex_cnj_zero [simp]: "cnj 0 = 0" |
14334 | 366 |
by (simp add: cnj_def complex_zero_def) |
14323 | 367 |
|
14374 | 368 |
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" |
14373 | 369 |
by (induct z, simp add: complex_zero_def complex_cnj) |
14323 | 370 |
|
371 |
lemma complex_mult_cnj: "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)" |
|
20725
72e20198f834
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huffman
parents:
20560
diff
changeset
|
372 |
by (induct z, simp add: complex_cnj complex_of_real_def power2_eq_square) |
14323 | 373 |
|
374 |
||
375 |
subsection{*Modulus*} |
|
376 |
||
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changeset
|
377 |
instance complex :: norm |
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changeset
|
378 |
complex_norm_def: "norm z \<equiv> sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" .. |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
379 |
|
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
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diff
changeset
|
380 |
abbreviation |
22861
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changeset
|
381 |
cmod :: "complex \<Rightarrow> real" where |
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huffman
parents:
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diff
changeset
|
382 |
"cmod \<equiv> norm" |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
383 |
|
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
384 |
lemmas cmod_def = complex_norm_def |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
385 |
|
22861
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huffman
parents:
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diff
changeset
|
386 |
lemma complex_mod [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)" |
14373 | 387 |
by (simp add: cmod_def) |
14323 | 388 |
|
22861
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clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
389 |
lemma complex_mod_triangle_ineq [simp]: "cmod (x + y) \<le> cmod x + cmod y" |
8ec47039614e
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huffman
parents:
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diff
changeset
|
390 |
apply (simp add: cmod_def) |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
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diff
changeset
|
391 |
apply (rule real_sqrt_sum_squares_triangle_ineq) |
14323 | 392 |
done |
393 |
||
22861
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clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
394 |
lemma complex_mod_mult: "cmod (x * y) = cmod x * cmod y" |
14373 | 395 |
apply (induct x, induct y) |
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
396 |
apply (simp add: real_sqrt_mult_distrib [symmetric]) |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
397 |
apply (rule_tac f=sqrt in arg_cong) |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
398 |
apply (simp add: power2_sum power2_diff power_mult_distrib ring_distrib) |
14323 | 399 |
done |
400 |
||
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
401 |
lemma complex_mod_complex_of_real: "cmod (complex_of_real x) = \<bar>x\<bar>" |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
402 |
by (simp add: complex_of_real_def) |
14323 | 403 |
|
22852 | 404 |
lemma complex_norm_scaleR: |
405 |
"norm (scaleR a x) = \<bar>a\<bar> * norm (x::complex)" |
|
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
406 |
unfolding scaleR_conv_of_real |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
407 |
by (simp only: complex_mod_mult complex_mod_complex_of_real) |
22852 | 408 |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
409 |
instance complex :: real_normed_field |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
410 |
proof |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
411 |
fix r :: real |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
412 |
fix x y :: complex |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
413 |
show "0 \<le> cmod x" |
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
414 |
by (induct x) simp |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
415 |
show "(cmod x = 0) = (x = 0)" |
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
416 |
by (induct x) simp |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
417 |
show "cmod (x + y) \<le> cmod x + cmod y" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
418 |
by (rule complex_mod_triangle_ineq) |
22852 | 419 |
show "cmod (scaleR r x) = \<bar>r\<bar> * cmod x" |
420 |
by (rule complex_norm_scaleR) |
|
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
421 |
show "cmod (x * y) = cmod x * cmod y" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
422 |
by (rule complex_mod_mult) |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
423 |
qed |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
424 |
|
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
425 |
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
426 |
by (induct z, simp add: complex_cnj) |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
427 |
|
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
428 |
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>" |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
429 |
by (simp add: complex_mod_mult power2_eq_square) |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
430 |
|
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
431 |
lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1" |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
432 |
by simp |
14323 | 433 |
|
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
434 |
lemma cmod_complex_polar [simp]: |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
435 |
"cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r" |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
436 |
apply (simp only: cmod_unit_one complex_mod_mult) |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
437 |
apply (simp add: complex_mod_complex_of_real) |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
438 |
done |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
439 |
|
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
440 |
lemma complex_Re_le_cmod: "Re x \<le> cmod x" |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
441 |
unfolding complex_norm_def |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
442 |
by (rule real_sqrt_sum_squares_ge1) |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
443 |
|
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
444 |
lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x" |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
445 |
by (rule order_trans [OF _ norm_ge_zero], simp) |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
446 |
|
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
447 |
lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a" |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
448 |
by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp) |
14323 | 449 |
|
14374 | 450 |
lemma complex_mod_diff_ineq [simp]: "cmod(a) - cmod(b) \<le> cmod(a + b)" |
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
451 |
(* TODO: generalize *) |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
452 |
proof - |
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
453 |
have "cmod a - cmod b = cmod a - cmod (- b)" by simp |
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
454 |
also have "\<dots> \<le> cmod (a - - b)" by (rule norm_triangle_ineq2) |
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
455 |
also have "\<dots> = cmod (a + b)" by simp |
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
456 |
finally show ?thesis . |
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
457 |
qed |
14323 | 458 |
|
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
459 |
lemmas real_sum_squared_expand = power2_sum [where 'a=real] |
14323 | 460 |
|
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
461 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
462 |
subsection{*Exponentiation*} |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
463 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
464 |
primrec |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
465 |
complexpow_0: "z ^ 0 = 1" |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
466 |
complexpow_Suc: "z ^ (Suc n) = (z::complex) * (z ^ n)" |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
467 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
468 |
|
15003 | 469 |
instance complex :: recpower |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
470 |
proof |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
471 |
fix z :: complex |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
472 |
fix n :: nat |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
473 |
show "z^0 = 1" by simp |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
474 |
show "z^(Suc n) = z * (z^n)" by simp |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
475 |
qed |
14323 | 476 |
|
477 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
478 |
lemma complex_of_real_pow: "complex_of_real (x ^ n) = (complex_of_real x) ^ n" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
479 |
by (rule of_real_power) |
14323 | 480 |
|
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
481 |
lemma complex_cnj_pow: "cnj(z ^ n) = cnj(z) ^ n" |
14323 | 482 |
apply (induct_tac "n") |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
483 |
apply (auto simp add: complex_cnj_mult) |
14323 | 484 |
done |
485 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
486 |
lemma complex_mod_complexpow: "cmod(x ^ n) = cmod(x) ^ n" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
487 |
by (rule norm_power) |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
488 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
489 |
lemma complexpow_i_squared [simp]: "ii ^ 2 = -(1::complex)" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
490 |
by (simp add: i_def complex_one_def numeral_2_eq_2) |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
491 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
492 |
lemma complex_i_not_zero [simp]: "ii \<noteq> 0" |
14373 | 493 |
by (simp add: i_def complex_zero_def) |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
494 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
495 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
496 |
subsection{*The Function @{term sgn}*} |
14323 | 497 |
|
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
498 |
definition |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
499 |
(*------------ Argand -------------*) |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
500 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
501 |
sgn :: "complex => complex" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
502 |
"sgn z = z / complex_of_real(cmod z)" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
503 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
504 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
505 |
arg :: "complex => real" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
506 |
"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
|
507 |
|
14374 | 508 |
lemma sgn_zero [simp]: "sgn 0 = 0" |
14373 | 509 |
by (simp add: sgn_def) |
14323 | 510 |
|
14374 | 511 |
lemma sgn_one [simp]: "sgn 1 = 1" |
14373 | 512 |
by (simp add: sgn_def) |
14323 | 513 |
|
514 |
lemma sgn_minus: "sgn (-z) = - sgn(z)" |
|
14373 | 515 |
by (simp add: sgn_def) |
14323 | 516 |
|
14374 | 517 |
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" |
14377 | 518 |
by (simp add: sgn_def) |
14323 | 519 |
|
520 |
lemma i_mult_eq: "ii * ii = complex_of_real (-1)" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
521 |
by (simp add: i_def complex_of_real_def) |
14323 | 522 |
|
14374 | 523 |
lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
524 |
by (simp add: i_def complex_one_def) |
14323 | 525 |
|
14374 | 526 |
lemma complex_eq_cancel_iff2 [simp]: |
14377 | 527 |
"(Complex x y = complex_of_real xa) = (x = xa & y = 0)" |
528 |
by (simp add: complex_of_real_def) |
|
14323 | 529 |
|
14377 | 530 |
lemma Complex_eq_0 [simp]: "(Complex x y = 0) = (x = 0 & y = 0)" |
531 |
by (simp add: complex_zero_def) |
|
14323 | 532 |
|
14377 | 533 |
lemma Complex_eq_1 [simp]: "(Complex x y = 1) = (x = 1 & y = 0)" |
534 |
by (simp add: complex_one_def) |
|
14323 | 535 |
|
14377 | 536 |
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 & y = 1)" |
537 |
by (simp add: i_def) |
|
14323 | 538 |
|
15013 | 539 |
|
540 |
||
14374 | 541 |
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" |
15013 | 542 |
proof (induct z) |
543 |
case (Complex x y) |
|
544 |
have "sqrt (x\<twosuperior> + y\<twosuperior>) * inverse (x\<twosuperior> + y\<twosuperior>) = inverse (sqrt (x\<twosuperior> + y\<twosuperior>))" |
|
545 |
by (simp add: divide_inverse [symmetric] sqrt_divide_self_eq) |
|
546 |
thus "Re (sgn (Complex x y)) = Re (Complex x y) /cmod (Complex x y)" |
|
547 |
by (simp add: sgn_def complex_of_real_def divide_inverse) |
|
548 |
qed |
|
549 |
||
14323 | 550 |
|
14374 | 551 |
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" |
15013 | 552 |
proof (induct z) |
553 |
case (Complex x y) |
|
554 |
have "sqrt (x\<twosuperior> + y\<twosuperior>) * inverse (x\<twosuperior> + y\<twosuperior>) = inverse (sqrt (x\<twosuperior> + y\<twosuperior>))" |
|
555 |
by (simp add: divide_inverse [symmetric] sqrt_divide_self_eq) |
|
556 |
thus "Im (sgn (Complex x y)) = Im (Complex x y) /cmod (Complex x y)" |
|
557 |
by (simp add: sgn_def complex_of_real_def divide_inverse) |
|
558 |
qed |
|
14323 | 559 |
|
560 |
lemma complex_inverse_complex_split: |
|
561 |
"inverse(complex_of_real x + ii * complex_of_real y) = |
|
562 |
complex_of_real(x/(x ^ 2 + y ^ 2)) - |
|
563 |
ii * complex_of_real(y/(x ^ 2 + y ^ 2))" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
564 |
by (simp add: complex_of_real_def i_def diff_minus divide_inverse) |
14323 | 565 |
|
566 |
(*----------------------------------------------------------------------------*) |
|
567 |
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) |
|
568 |
(* many of the theorems are not used - so should they be kept? *) |
|
569 |
(*----------------------------------------------------------------------------*) |
|
570 |
||
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
571 |
lemma complex_of_real_zero_iff: "(complex_of_real y = 0) = (y = 0)" |
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
572 |
by (rule of_real_eq_0_iff) |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
573 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
574 |
lemma cos_arg_i_mult_zero_pos: |
14377 | 575 |
"0 < y ==> cos (arg(Complex 0 y)) = 0" |
14373 | 576 |
apply (simp add: arg_def abs_if) |
14334 | 577 |
apply (rule_tac a = "pi/2" in someI2, auto) |
578 |
apply (rule order_less_trans [of _ 0], auto) |
|
14323 | 579 |
done |
580 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
581 |
lemma cos_arg_i_mult_zero_neg: |
14377 | 582 |
"y < 0 ==> cos (arg(Complex 0 y)) = 0" |
14373 | 583 |
apply (simp add: arg_def abs_if) |
14334 | 584 |
apply (rule_tac a = "- pi/2" in someI2, auto) |
585 |
apply (rule order_trans [of _ 0], auto) |
|
14323 | 586 |
done |
587 |
||
14374 | 588 |
lemma cos_arg_i_mult_zero [simp]: |
14377 | 589 |
"y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0" |
590 |
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) |
|
14323 | 591 |
|
592 |
||
593 |
subsection{*Finally! Polar Form for Complex Numbers*} |
|
594 |
||
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
595 |
definition |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
596 |
|
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
597 |
(* abbreviation for (cos a + i sin a) *) |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
598 |
cis :: "real => complex" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
599 |
"cis a = Complex (cos a) (sin a)" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
600 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
601 |
definition |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
602 |
(* abbreviation for r*(cos a + i sin a) *) |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
603 |
rcis :: "[real, real] => complex" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
604 |
"rcis r a = complex_of_real r * cis a" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
605 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
606 |
definition |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
607 |
(* e ^ (x + iy) *) |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
608 |
expi :: "complex => complex" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
609 |
"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
|
610 |
|
14374 | 611 |
lemma complex_split_polar: |
14377 | 612 |
"\<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
|
613 |
apply (induct z) |
14377 | 614 |
apply (auto simp add: polar_Ex complex_of_real_mult_Complex) |
14323 | 615 |
done |
616 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
617 |
lemma rcis_Ex: "\<exists>r a. z = rcis r a" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
618 |
apply (induct z) |
14377 | 619 |
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex) |
14323 | 620 |
done |
621 |
||
14374 | 622 |
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" |
14373 | 623 |
by (simp add: rcis_def cis_def) |
14323 | 624 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
625 |
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" |
14373 | 626 |
by (simp add: rcis_def cis_def) |
14323 | 627 |
|
14377 | 628 |
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>" |
629 |
proof - |
|
630 |
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
|
631 |
by (simp only: power_mult_distrib right_distrib) |
14377 | 632 |
thus ?thesis by simp |
633 |
qed |
|
14323 | 634 |
|
14374 | 635 |
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" |
14377 | 636 |
by (simp add: rcis_def cis_def sin_cos_squared_add2_mult) |
14323 | 637 |
|
638 |
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" |
|
14373 | 639 |
apply (simp add: cmod_def) |
14323 | 640 |
apply (rule real_sqrt_eq_iff [THEN iffD2]) |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
641 |
apply (auto simp add: complex_mult_cnj |
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
642 |
simp del: of_real_add) |
14323 | 643 |
done |
644 |
||
14374 | 645 |
lemma complex_Re_cnj [simp]: "Re(cnj z) = Re z" |
14373 | 646 |
by (induct z, simp add: complex_cnj) |
14323 | 647 |
|
14374 | 648 |
lemma complex_Im_cnj [simp]: "Im(cnj z) = - Im z" |
649 |
by (induct z, simp add: complex_cnj) |
|
650 |
||
651 |
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" |
|
14373 | 652 |
by (induct z, simp add: complex_cnj complex_mult) |
14323 | 653 |
|
654 |
||
655 |
(*---------------------------------------------------------------------------*) |
|
656 |
(* (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b) *) |
|
657 |
(*---------------------------------------------------------------------------*) |
|
658 |
||
659 |
lemma cis_rcis_eq: "cis a = rcis 1 a" |
|
14373 | 660 |
by (simp add: rcis_def) |
14323 | 661 |
|
14374 | 662 |
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" |
15013 | 663 |
by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib |
664 |
complex_of_real_def) |
|
14323 | 665 |
|
666 |
lemma cis_mult: "cis a * cis b = cis (a + b)" |
|
14373 | 667 |
by (simp add: cis_rcis_eq rcis_mult) |
14323 | 668 |
|
14374 | 669 |
lemma cis_zero [simp]: "cis 0 = 1" |
14377 | 670 |
by (simp add: cis_def complex_one_def) |
14323 | 671 |
|
14374 | 672 |
lemma rcis_zero_mod [simp]: "rcis 0 a = 0" |
14373 | 673 |
by (simp add: rcis_def) |
14323 | 674 |
|
14374 | 675 |
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" |
14373 | 676 |
by (simp add: rcis_def) |
14323 | 677 |
|
678 |
lemma complex_of_real_minus_one: |
|
679 |
"complex_of_real (-(1::real)) = -(1::complex)" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
680 |
by (simp add: complex_of_real_def complex_one_def) |
14323 | 681 |
|
14374 | 682 |
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x" |
14373 | 683 |
by (simp add: complex_mult_assoc [symmetric]) |
14323 | 684 |
|
685 |
||
686 |
lemma cis_real_of_nat_Suc_mult: |
|
687 |
"cis (real (Suc n) * a) = cis a * cis (real n * a)" |
|
14377 | 688 |
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib) |
14323 | 689 |
|
690 |
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" |
|
691 |
apply (induct_tac "n") |
|
692 |
apply (auto simp add: cis_real_of_nat_Suc_mult) |
|
693 |
done |
|
694 |
||
14374 | 695 |
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" |
696 |
by (simp add: rcis_def power_mult_distrib DeMoivre complex_of_real_pow) |
|
14323 | 697 |
|
14374 | 698 |
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
699 |
by (simp add: cis_def complex_inverse_complex_split diff_minus) |
14323 | 700 |
|
701 |
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)" |
|
22884 | 702 |
by (simp add: divide_inverse rcis_def) |
14323 | 703 |
|
704 |
lemma cis_divide: "cis a / cis b = cis (a - b)" |
|
14373 | 705 |
by (simp add: complex_divide_def cis_mult real_diff_def) |
14323 | 706 |
|
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
707 |
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)" |
14373 | 708 |
apply (simp add: complex_divide_def) |
709 |
apply (case_tac "r2=0", simp) |
|
710 |
apply (simp add: rcis_inverse rcis_mult real_diff_def) |
|
14323 | 711 |
done |
712 |
||
14374 | 713 |
lemma Re_cis [simp]: "Re(cis a) = cos a" |
14373 | 714 |
by (simp add: cis_def) |
14323 | 715 |
|
14374 | 716 |
lemma Im_cis [simp]: "Im(cis a) = sin a" |
14373 | 717 |
by (simp add: cis_def) |
14323 | 718 |
|
719 |
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" |
|
14334 | 720 |
by (auto simp add: DeMoivre) |
14323 | 721 |
|
722 |
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" |
|
14334 | 723 |
by (auto simp add: DeMoivre) |
14323 | 724 |
|
725 |
lemma expi_add: "expi(a + b) = expi(a) * expi(b)" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
726 |
by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac) |
14323 | 727 |
|
14374 | 728 |
lemma expi_zero [simp]: "expi (0::complex) = 1" |
14373 | 729 |
by (simp add: expi_def) |
14323 | 730 |
|
14374 | 731 |
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a" |
14373 | 732 |
apply (insert rcis_Ex [of z]) |
22884 | 733 |
apply (auto simp add: expi_def rcis_def complex_mult_assoc [symmetric]) |
14334 | 734 |
apply (rule_tac x = "ii * complex_of_real a" in exI, auto) |
14323 | 735 |
done |
736 |
||
737 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
738 |
subsection{*Numerals and Arithmetic*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
739 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
740 |
instance complex :: number .. |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
741 |
|
15013 | 742 |
defs (overloaded) |
20485 | 743 |
complex_number_of_def: "(number_of w :: complex) == of_int w" |
15013 | 744 |
--{*the type constraint is essential!*} |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
745 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
746 |
instance complex :: number_ring |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
747 |
by (intro_classes, simp add: complex_number_of_def) |
15013 | 748 |
|
749 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
750 |
text{*Collapse applications of @{term complex_of_real} to @{term number_of}*} |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
751 |
lemma complex_number_of [simp]: "complex_of_real (number_of w) = number_of w" |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
752 |
by (rule of_real_number_of_eq) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
753 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
754 |
text{*This theorem is necessary because theorems such as |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
755 |
@{text iszero_number_of_0} only hold for ordered rings. They cannot |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
756 |
be generalized to fields in general because they fail for finite fields. |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
757 |
They work for type complex because the reals can be embedded in them.*} |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
758 |
(* TODO: generalize and move to Real/RealVector.thy *) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
759 |
lemma iszero_complex_number_of [simp]: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
760 |
"iszero (number_of w :: complex) = iszero (number_of w :: real)" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
761 |
by (simp only: complex_of_real_zero_iff complex_number_of [symmetric] |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
762 |
iszero_def) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
763 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
764 |
lemma complex_number_of_cnj [simp]: "cnj(number_of v :: complex) = number_of v" |
15481 | 765 |
by (simp only: complex_number_of [symmetric] complex_cnj_complex_of_real) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
766 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
767 |
lemma complex_number_of_cmod: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
768 |
"cmod(number_of v :: complex) = abs (number_of v :: real)" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
769 |
by (simp only: complex_number_of [symmetric] complex_mod_complex_of_real) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
770 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
771 |
lemma complex_number_of_Re [simp]: "Re(number_of v :: complex) = number_of v" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
772 |
by (simp only: complex_number_of [symmetric] Re_complex_of_real) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
773 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
774 |
lemma complex_number_of_Im [simp]: "Im(number_of v :: complex) = 0" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
775 |
by (simp only: complex_number_of [symmetric] Im_complex_of_real) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
776 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
777 |
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1" |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
778 |
by (simp add: expi_def complex_Re_mult_eq complex_Im_mult_eq cis_def) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
779 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
780 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
781 |
(*examples: |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
782 |
print_depth 22 |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
783 |
set timing; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
784 |
set trace_simp; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
785 |
fun test s = (Goal s, by (Simp_tac 1)); |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
786 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
787 |
test "23 * ii + 45 * ii= (x::complex)"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
788 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
789 |
test "5 * ii + 12 - 45 * ii= (x::complex)"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
790 |
test "5 * ii + 40 - 12 * ii + 9 = (x::complex) + 89 * ii"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
791 |
test "5 * ii + 40 - 12 * ii + 9 - 78 = (x::complex) + 89 * ii"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
792 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
793 |
test "l + 10 * ii + 90 + 3*l + 9 + 45 * ii= (x::complex)"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
794 |
test "87 + 10 * ii + 90 + 3*7 + 9 + 45 * ii= (x::complex)"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
795 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
796 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
797 |
fun test s = (Goal s; by (Asm_simp_tac 1)); |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
798 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
799 |
test "x*k = k*(y::complex)"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
800 |
test "k = k*(y::complex)"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
801 |
test "a*(b*c) = (b::complex)"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
802 |
test "a*(b*c) = d*(b::complex)*(x*a)"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
803 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
804 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
805 |
test "(x*k) / (k*(y::complex)) = (uu::complex)"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
806 |
test "(k) / (k*(y::complex)) = (uu::complex)"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
807 |
test "(a*(b*c)) / ((b::complex)) = (uu::complex)"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
808 |
test "(a*(b*c)) / (d*(b::complex)*(x*a)) = (uu::complex)"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
809 |
|
15003 | 810 |
FIXME: what do we do about this? |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
811 |
test "a*(b*c)/(y*z) = d*(b::complex)*(x*a)/z"; |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
812 |
*) |
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
813 |
|
13957 | 814 |
end |