author | paulson |
Thu, 05 Feb 2004 10:45:28 +0100 | |
changeset 14377 | f454b3004f8f |
parent 14374 | 61de62096768 |
child 14387 | e96d5c42c4b0 |
permissions | -rw-r--r-- |
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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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*) |
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header {* Complex Numbers: Rectangular and Polar Representations *} |
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theory Complex = HLog: |
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datatype complex = Complex real real |
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instance complex :: zero .. |
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instance complex :: one .. |
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instance complex :: plus .. |
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instance complex :: times .. |
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instance complex :: minus .. |
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instance complex :: inverse .. |
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instance complex :: 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 (Complex x y) = x" |
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consts Im :: "complex => real" |
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primrec "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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constdefs |
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(*----------- modulus ------------*) |
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cmod :: "complex => real" |
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"cmod z == sqrt(Re(z) ^ 2 + Im(z) ^ 2)" |
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(*----- injection from reals -----*) |
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complex_of_real :: "real => complex" |
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"complex_of_real r == Complex r 0" |
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(*------- complex conjugate ------*) |
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cnj :: "complex => complex" |
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"cnj z == Complex (Re z) (-Im z)" |
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(*------------ Argand -------------*) |
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sgn :: "complex => complex" |
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"sgn z == z / complex_of_real(cmod z)" |
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arg :: "complex => real" |
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"arg z == @a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi" |
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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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constdefs |
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(* abbreviation for (cos a + i sin a) *) |
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cis :: "real => complex" |
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"cis a == Complex (cos a) (sin a)" |
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(* abbreviation for r*(cos a + i sin a) *) |
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rcis :: "[real, real] => complex" |
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"rcis r a == complex_of_real r * cis a" |
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(* e ^ (x + iy) *) |
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expi :: "complex => complex" |
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"expi z == complex_of_real(exp (Re z)) * cis (Im 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 Re [simp]: "Re(Complex x y) = x" |
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by simp |
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lemma Im [simp]: "Im(Complex x y) = y" |
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by 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_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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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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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: complex_mult complex_inverse complex_one_def |
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complex_zero_def add_divide_distrib [symmetric] power2_eq_square mult_ac) |
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apply (drule_tac y = y in real_sum_squares_not_zero) |
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apply (drule_tac [2] x = x in real_sum_squares_not_zero2, auto) |
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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 real_diff_def add_ac) |
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show "z+u = z+v ==> u=v" |
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proof - |
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assume eq: "z+u = z+v" |
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hence "(-z + z) + u = (-z + z) + v" by (simp only: eq complex_add_assoc) |
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thus "u = v" by (simp add: complex_add_minus_left complex_add_zero_left) |
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qed |
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assume neq: "w \<noteq> 0" |
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thus "z / w = z * inverse w" |
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by (simp add: complex_divide_def) |
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show "inverse w * w = 1" |
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by (simp add: neq 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 inv: "inverse 0 = (0::complex)" |
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by (simp add: complex_inverse_def complex_zero_def) |
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fix x :: complex |
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show "x/0 = 0" |
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by (simp add: complex_divide_def inv) |
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qed |
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subsection{*Embedding Properties for @{term complex_of_real} Map*} |
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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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lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" |
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by (simp add: i_def complex_of_real_def) |
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lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" |
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by (simp add: i_def complex_of_real_def) |
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lemma complex_of_real_one [simp]: "complex_of_real 1 = 1" |
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by (simp add: complex_one_def complex_of_real_def) |
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lemma complex_of_real_zero [simp]: "complex_of_real 0 = 0" |
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by (simp add: complex_zero_def complex_of_real_def) |
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lemma complex_of_real_eq_iff [iff]: |
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"(complex_of_real x = complex_of_real y) = (x = y)" |
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by (simp add: complex_of_real_def) |
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lemma complex_of_real_minus: "complex_of_real(-x) = - complex_of_real x" |
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by (simp add: complex_of_real_def complex_minus) |
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lemma complex_of_real_inverse: |
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"complex_of_real(inverse x) = inverse(complex_of_real x)" |
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apply (case_tac "x=0", simp) |
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apply (simp add: complex_inverse complex_of_real_def real_divide_def |
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inverse_mult_distrib power2_eq_square) |
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done |
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lemma complex_of_real_add: |
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"complex_of_real x + complex_of_real y = complex_of_real (x + y)" |
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by (simp add: complex_add complex_of_real_def) |
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lemma complex_of_real_diff: |
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"complex_of_real x - complex_of_real y = complex_of_real (x - y)" |
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by (simp add: complex_of_real_minus [symmetric] complex_diff_def |
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complex_of_real_add) |
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lemma complex_of_real_mult: |
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"complex_of_real x * complex_of_real y = complex_of_real (x * y)" |
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by (simp add: complex_mult complex_of_real_def) |
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lemma complex_of_real_divide: |
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"complex_of_real x / complex_of_real y = complex_of_real(x/y)" |
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apply (simp add: complex_divide_def) |
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apply (case_tac "y=0", simp) |
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apply (simp add: complex_of_real_mult [symmetric] complex_of_real_inverse |
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real_divide_def) |
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done |
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lemma complex_mod [simp]: "cmod (Complex x y) = sqrt(x ^ 2 + y ^ 2)" |
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by (simp add: cmod_def) |
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lemma complex_mod_zero [simp]: "cmod(0) = 0" |
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by (simp add: cmod_def) |
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lemma complex_mod_one [simp]: "cmod(1) = 1" |
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by (simp add: cmod_def power2_eq_square) |
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lemma complex_mod_complex_of_real [simp]: "cmod(complex_of_real x) = abs x" |
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by (simp add: complex_of_real_def power2_eq_square complex_mod) |
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lemma complex_of_real_abs: |
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"complex_of_real (abs x) = complex_of_real(cmod(complex_of_real x))" |
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by simp |
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subsection{*The Functions @{term Re} and @{term Im}*} |
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lemma complex_Re_mult_eq: "Re (w * z) = Re w * Re z - Im w * Im z" |
|
357 |
by (induct z, induct w, simp add: complex_mult) |
|
358 |
||
359 |
lemma complex_Im_mult_eq: "Im (w * z) = Re w * Im z + Im w * Re z" |
|
360 |
by (induct z, induct w, simp add: complex_mult) |
|
361 |
||
362 |
lemma Re_i_times [simp]: "Re(ii * z) = - Im z" |
|
363 |
by (simp add: complex_Re_mult_eq) |
|
364 |
||
365 |
lemma Re_times_i [simp]: "Re(z * ii) = - Im z" |
|
366 |
by (simp add: complex_Re_mult_eq) |
|
367 |
||
368 |
lemma Im_i_times [simp]: "Im(ii * z) = Re z" |
|
369 |
by (simp add: complex_Im_mult_eq) |
|
370 |
||
371 |
lemma Im_times_i [simp]: "Im(z * ii) = Re z" |
|
372 |
by (simp add: complex_Im_mult_eq) |
|
373 |
||
374 |
lemma complex_Re_mult: "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)" |
|
375 |
by (simp add: complex_Re_mult_eq) |
|
376 |
||
377 |
lemma complex_Re_mult_complex_of_real [simp]: |
|
378 |
"Re (z * complex_of_real c) = Re(z) * c" |
|
379 |
by (simp add: complex_Re_mult_eq) |
|
380 |
||
381 |
lemma complex_Im_mult_complex_of_real [simp]: |
|
382 |
"Im (z * complex_of_real c) = Im(z) * c" |
|
383 |
by (simp add: complex_Im_mult_eq) |
|
384 |
||
385 |
lemma complex_Re_mult_complex_of_real2 [simp]: |
|
386 |
"Re (complex_of_real c * z) = c * Re(z)" |
|
387 |
by (simp add: complex_Re_mult_eq) |
|
388 |
||
389 |
lemma complex_Im_mult_complex_of_real2 [simp]: |
|
390 |
"Im (complex_of_real c * z) = c * Im(z)" |
|
391 |
by (simp add: complex_Im_mult_eq) |
|
392 |
||
393 |
||
14323 | 394 |
subsection{*Conjugation is an Automorphism*} |
395 |
||
14373 | 396 |
lemma complex_cnj: "cnj (Complex x y) = Complex x (-y)" |
397 |
by (simp add: cnj_def) |
|
14323 | 398 |
|
14374 | 399 |
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" |
14373 | 400 |
by (simp add: cnj_def complex_Re_Im_cancel_iff) |
14323 | 401 |
|
14374 | 402 |
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" |
14373 | 403 |
by (simp add: cnj_def) |
14323 | 404 |
|
14374 | 405 |
lemma complex_cnj_complex_of_real [simp]: |
14373 | 406 |
"cnj (complex_of_real x) = complex_of_real x" |
407 |
by (simp add: complex_of_real_def complex_cnj) |
|
14323 | 408 |
|
14374 | 409 |
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" |
14373 | 410 |
by (induct z, simp add: complex_cnj complex_mod power2_eq_square) |
14323 | 411 |
|
412 |
lemma complex_cnj_minus: "cnj (-z) = - cnj z" |
|
14373 | 413 |
by (simp add: cnj_def complex_minus complex_Re_minus complex_Im_minus) |
14323 | 414 |
|
415 |
lemma complex_cnj_inverse: "cnj(inverse z) = inverse(cnj z)" |
|
14373 | 416 |
by (induct z, simp add: complex_cnj complex_inverse power2_eq_square) |
14323 | 417 |
|
418 |
lemma complex_cnj_add: "cnj(w + z) = cnj(w) + cnj(z)" |
|
14373 | 419 |
by (induct w, induct z, simp add: complex_cnj complex_add) |
14323 | 420 |
|
421 |
lemma complex_cnj_diff: "cnj(w - z) = cnj(w) - cnj(z)" |
|
14373 | 422 |
by (simp add: complex_diff_def complex_cnj_add complex_cnj_minus) |
14323 | 423 |
|
424 |
lemma complex_cnj_mult: "cnj(w * z) = cnj(w) * cnj(z)" |
|
14373 | 425 |
by (induct w, induct z, simp add: complex_cnj complex_mult) |
14323 | 426 |
|
427 |
lemma complex_cnj_divide: "cnj(w / z) = (cnj w)/(cnj z)" |
|
14373 | 428 |
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) |
14323 | 429 |
|
14374 | 430 |
lemma complex_cnj_one [simp]: "cnj 1 = 1" |
14373 | 431 |
by (simp add: cnj_def complex_one_def) |
14323 | 432 |
|
433 |
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re(z))" |
|
14373 | 434 |
by (induct z, simp add: complex_add complex_cnj complex_of_real_def) |
14323 | 435 |
|
436 |
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im(z)) * ii" |
|
14373 | 437 |
apply (induct z) |
14374 | 438 |
apply (simp add: complex_add complex_cnj complex_of_real_def complex_diff_def |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
439 |
complex_minus i_def complex_mult) |
14323 | 440 |
done |
441 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
442 |
lemma complex_cnj_zero [simp]: "cnj 0 = 0" |
14334 | 443 |
by (simp add: cnj_def complex_zero_def) |
14323 | 444 |
|
14374 | 445 |
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" |
14373 | 446 |
by (induct z, simp add: complex_zero_def complex_cnj) |
14323 | 447 |
|
448 |
lemma complex_mult_cnj: "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)" |
|
14374 | 449 |
by (induct z, |
450 |
simp add: complex_cnj complex_mult complex_of_real_def power2_eq_square) |
|
14323 | 451 |
|
452 |
||
453 |
subsection{*Modulus*} |
|
454 |
||
14374 | 455 |
lemma complex_mod_eq_zero_cancel [simp]: "(cmod x = 0) = (x = 0)" |
14373 | 456 |
apply (induct x) |
14374 | 457 |
apply (auto intro: real_sum_squares_cancel real_sum_squares_cancel2 |
14373 | 458 |
simp add: complex_mod complex_zero_def power2_eq_square) |
14323 | 459 |
done |
460 |
||
14374 | 461 |
lemma complex_mod_complex_of_real_of_nat [simp]: |
14373 | 462 |
"cmod (complex_of_real(real (n::nat))) = real n" |
463 |
by simp |
|
14323 | 464 |
|
14374 | 465 |
lemma complex_mod_minus [simp]: "cmod (-x) = cmod(x)" |
14373 | 466 |
by (induct x, simp add: complex_mod complex_minus power2_eq_square) |
14323 | 467 |
|
468 |
lemma complex_mod_mult_cnj: "cmod(z * cnj(z)) = cmod(z) ^ 2" |
|
14373 | 469 |
apply (induct z, simp add: complex_mod complex_cnj complex_mult) |
470 |
apply (simp add: power2_eq_square real_abs_def) |
|
14323 | 471 |
done |
472 |
||
14373 | 473 |
lemma complex_mod_squared: "cmod(Complex x y) ^ 2 = x ^ 2 + y ^ 2" |
474 |
by (simp add: cmod_def) |
|
14323 | 475 |
|
14374 | 476 |
lemma complex_mod_ge_zero [simp]: "0 \<le> cmod x" |
14373 | 477 |
by (simp add: cmod_def) |
14323 | 478 |
|
14374 | 479 |
lemma abs_cmod_cancel [simp]: "abs(cmod x) = cmod x" |
480 |
by (simp add: abs_if linorder_not_less) |
|
14323 | 481 |
|
482 |
lemma complex_mod_mult: "cmod(x*y) = cmod(x) * cmod(y)" |
|
14373 | 483 |
apply (induct x, induct y) |
14377 | 484 |
apply (auto simp add: complex_mult complex_mod real_sqrt_mult_distrib2[symmetric]) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
485 |
apply (rule_tac n = 1 in power_inject_base) |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
486 |
apply (auto simp add: power2_eq_square [symmetric] simp del: realpow_Suc) |
14374 | 487 |
apply (auto simp add: real_diff_def power2_eq_square right_distrib left_distrib |
488 |
add_ac mult_ac) |
|
14323 | 489 |
done |
490 |
||
14377 | 491 |
lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1" |
492 |
by (simp add: cmod_def) |
|
493 |
||
494 |
lemma cmod_complex_polar [simp]: |
|
495 |
"cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r" |
|
496 |
by (simp only: cmod_unit_one complex_mod_mult, simp) |
|
497 |
||
14374 | 498 |
lemma complex_mod_add_squared_eq: |
499 |
"cmod(x + y) ^ 2 = cmod(x) ^ 2 + cmod(y) ^ 2 + 2 * Re(x * cnj y)" |
|
14373 | 500 |
apply (induct x, induct y) |
14323 | 501 |
apply (auto simp add: complex_add complex_mod_squared complex_mult complex_cnj real_diff_def simp del: realpow_Suc) |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
502 |
apply (auto simp add: right_distrib left_distrib power2_eq_square mult_ac add_ac) |
14323 | 503 |
done |
504 |
||
14374 | 505 |
lemma complex_Re_mult_cnj_le_cmod [simp]: "Re(x * cnj y) \<le> cmod(x * cnj y)" |
14373 | 506 |
apply (induct x, induct y) |
14323 | 507 |
apply (auto simp add: complex_mod complex_mult complex_cnj real_diff_def simp del: realpow_Suc) |
508 |
done |
|
509 |
||
14374 | 510 |
lemma complex_Re_mult_cnj_le_cmod2 [simp]: "Re(x * cnj y) \<le> cmod(x * y)" |
14373 | 511 |
by (insert complex_Re_mult_cnj_le_cmod [of x y], simp add: complex_mod_mult) |
14323 | 512 |
|
14374 | 513 |
lemma real_sum_squared_expand: |
514 |
"((x::real) + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y" |
|
14373 | 515 |
by (simp add: left_distrib right_distrib power2_eq_square) |
14323 | 516 |
|
14374 | 517 |
lemma complex_mod_triangle_squared [simp]: |
518 |
"cmod (x + y) ^ 2 \<le> (cmod(x) + cmod(y)) ^ 2" |
|
14373 | 519 |
by (simp add: real_sum_squared_expand complex_mod_add_squared_eq real_mult_assoc complex_mod_mult [symmetric]) |
14323 | 520 |
|
14374 | 521 |
lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x" |
14373 | 522 |
by (rule order_trans [OF _ complex_mod_ge_zero], simp) |
14323 | 523 |
|
14374 | 524 |
lemma complex_mod_triangle_ineq [simp]: "cmod (x + y) \<le> cmod(x) + cmod(y)" |
14334 | 525 |
apply (rule_tac n = 1 in realpow_increasing) |
14323 | 526 |
apply (auto intro: order_trans [OF _ complex_mod_ge_zero] |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
527 |
simp add: power2_eq_square [symmetric]) |
14323 | 528 |
done |
529 |
||
14374 | 530 |
lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a" |
14373 | 531 |
by (insert complex_mod_triangle_ineq [THEN add_right_mono, of b a"-cmod b"], simp) |
14323 | 532 |
|
533 |
lemma complex_mod_diff_commute: "cmod (x - y) = cmod (y - x)" |
|
14373 | 534 |
apply (induct x, induct y) |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset
|
535 |
apply (auto simp add: complex_diff complex_mod right_diff_distrib power2_eq_square left_diff_distrib add_ac mult_ac) |
14323 | 536 |
done |
537 |
||
14374 | 538 |
lemma complex_mod_add_less: |
539 |
"[| cmod x < r; cmod y < s |] ==> cmod (x + y) < r + s" |
|
14334 | 540 |
by (auto intro: order_le_less_trans complex_mod_triangle_ineq) |
14323 | 541 |
|
14374 | 542 |
lemma complex_mod_mult_less: |
543 |
"[| cmod x < r; cmod y < s |] ==> cmod (x * y) < r * s" |
|
14334 | 544 |
by (auto intro: real_mult_less_mono' simp add: complex_mod_mult) |
14323 | 545 |
|
14374 | 546 |
lemma complex_mod_diff_ineq [simp]: "cmod(a) - cmod(b) \<le> cmod(a + b)" |
14323 | 547 |
apply (rule linorder_cases [of "cmod(a)" "cmod (b)"]) |
548 |
apply auto |
|
14334 | 549 |
apply (rule order_trans [of _ 0], rule order_less_imp_le) |
14374 | 550 |
apply (simp add: compare_rls, simp) |
14323 | 551 |
apply (simp add: compare_rls) |
552 |
apply (rule complex_mod_minus [THEN subst]) |
|
553 |
apply (rule order_trans) |
|
554 |
apply (rule_tac [2] complex_mod_triangle_ineq) |
|
14373 | 555 |
apply (auto simp add: add_ac) |
14323 | 556 |
done |
557 |
||
14374 | 558 |
lemma complex_Re_le_cmod [simp]: "Re z \<le> cmod z" |
14373 | 559 |
by (induct z, simp add: complex_mod del: realpow_Suc) |
14323 | 560 |
|
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
561 |
lemma complex_mod_gt_zero: "z \<noteq> 0 ==> 0 < cmod z" |
14373 | 562 |
apply (insert complex_mod_ge_zero [of z]) |
14334 | 563 |
apply (drule order_le_imp_less_or_eq, auto) |
14323 | 564 |
done |
565 |
||
566 |
||
567 |
subsection{*A Few More Theorems*} |
|
568 |
||
569 |
lemma complex_mod_inverse: "cmod(inverse x) = inverse(cmod x)" |
|
14373 | 570 |
apply (case_tac "x=0", simp) |
14323 | 571 |
apply (rule_tac c1 = "cmod x" in real_mult_left_cancel [THEN iffD1]) |
572 |
apply (auto simp add: complex_mod_mult [symmetric]) |
|
573 |
done |
|
574 |
||
14373 | 575 |
lemma complex_mod_divide: "cmod(x/y) = cmod(x)/(cmod y)" |
14377 | 576 |
by (simp add: complex_divide_def real_divide_def complex_mod_mult complex_mod_inverse) |
14323 | 577 |
|
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
578 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
579 |
subsection{*Exponentiation*} |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
580 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
581 |
primrec |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
582 |
complexpow_0: "z ^ 0 = 1" |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
583 |
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
|
584 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
585 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
586 |
instance complex :: ringpower |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
587 |
proof |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
588 |
fix z :: complex |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
589 |
fix n :: nat |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
590 |
show "z^0 = 1" by simp |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
591 |
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
|
592 |
qed |
14323 | 593 |
|
594 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
595 |
lemma complex_of_real_pow: "complex_of_real (x ^ n) = (complex_of_real x) ^ n" |
14323 | 596 |
apply (induct_tac "n") |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
597 |
apply (auto simp add: complex_of_real_mult [symmetric]) |
14323 | 598 |
done |
599 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
600 |
lemma complex_cnj_pow: "cnj(z ^ n) = cnj(z) ^ n" |
14323 | 601 |
apply (induct_tac "n") |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
602 |
apply (auto simp add: complex_cnj_mult) |
14323 | 603 |
done |
604 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
605 |
lemma complex_mod_complexpow: "cmod(x ^ n) = cmod(x) ^ n" |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
606 |
apply (induct_tac "n") |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
607 |
apply (auto simp add: complex_mod_mult) |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
608 |
done |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
609 |
|
14374 | 610 |
lemma complexpow_minus: |
611 |
"(-x::complex) ^ n = (if even n then (x ^ n) else -(x ^ n))" |
|
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
612 |
by (induct_tac "n", auto) |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
613 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
614 |
lemma complexpow_i_squared [simp]: "ii ^ 2 = -(1::complex)" |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
615 |
by (simp add: i_def complex_mult complex_one_def complex_minus numeral_2_eq_2) |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
616 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
617 |
lemma complex_i_not_zero [simp]: "ii \<noteq> 0" |
14373 | 618 |
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
|
619 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
620 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
621 |
subsection{*The Function @{term sgn}*} |
14323 | 622 |
|
14374 | 623 |
lemma sgn_zero [simp]: "sgn 0 = 0" |
14373 | 624 |
by (simp add: sgn_def) |
14323 | 625 |
|
14374 | 626 |
lemma sgn_one [simp]: "sgn 1 = 1" |
14373 | 627 |
by (simp add: sgn_def) |
14323 | 628 |
|
629 |
lemma sgn_minus: "sgn (-z) = - sgn(z)" |
|
14373 | 630 |
by (simp add: sgn_def) |
14323 | 631 |
|
14374 | 632 |
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" |
14377 | 633 |
by (simp add: sgn_def) |
14323 | 634 |
|
635 |
lemma i_mult_eq: "ii * ii = complex_of_real (-1)" |
|
14373 | 636 |
by (simp add: i_def complex_of_real_def complex_mult complex_add) |
14323 | 637 |
|
14374 | 638 |
lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)" |
14373 | 639 |
by (simp add: i_def complex_one_def complex_mult complex_minus) |
14323 | 640 |
|
14374 | 641 |
lemma complex_eq_cancel_iff2 [simp]: |
14377 | 642 |
"(Complex x y = complex_of_real xa) = (x = xa & y = 0)" |
643 |
by (simp add: complex_of_real_def) |
|
14323 | 644 |
|
14374 | 645 |
lemma complex_eq_cancel_iff2a [simp]: |
14377 | 646 |
"(Complex x y = complex_of_real xa) = (x = xa & y = 0)" |
647 |
by (simp add: complex_of_real_def) |
|
14323 | 648 |
|
14377 | 649 |
lemma Complex_eq_0 [simp]: "(Complex x y = 0) = (x = 0 & y = 0)" |
650 |
by (simp add: complex_zero_def) |
|
14323 | 651 |
|
14377 | 652 |
lemma Complex_eq_1 [simp]: "(Complex x y = 1) = (x = 1 & y = 0)" |
653 |
by (simp add: complex_one_def) |
|
14323 | 654 |
|
14377 | 655 |
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 & y = 1)" |
656 |
by (simp add: i_def) |
|
14323 | 657 |
|
14374 | 658 |
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" |
14373 | 659 |
apply (induct z) |
660 |
apply (simp add: sgn_def complex_divide_def complex_of_real_inverse [symmetric]) |
|
661 |
apply (simp add: complex_of_real_def complex_mult real_divide_def) |
|
14323 | 662 |
done |
663 |
||
14374 | 664 |
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" |
14373 | 665 |
apply (induct z) |
666 |
apply (simp add: sgn_def complex_divide_def complex_of_real_inverse [symmetric]) |
|
667 |
apply (simp add: complex_of_real_def complex_mult real_divide_def) |
|
14323 | 668 |
done |
669 |
||
670 |
lemma complex_inverse_complex_split: |
|
671 |
"inverse(complex_of_real x + ii * complex_of_real y) = |
|
672 |
complex_of_real(x/(x ^ 2 + y ^ 2)) - |
|
673 |
ii * complex_of_real(y/(x ^ 2 + y ^ 2))" |
|
14374 | 674 |
by (simp add: complex_of_real_def i_def complex_mult complex_add |
14373 | 675 |
complex_diff_def complex_minus complex_inverse real_divide_def) |
14323 | 676 |
|
677 |
(*----------------------------------------------------------------------------*) |
|
678 |
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) |
|
679 |
(* many of the theorems are not used - so should they be kept? *) |
|
680 |
(*----------------------------------------------------------------------------*) |
|
681 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
682 |
lemma complex_of_real_zero_iff [simp]: "(complex_of_real y = 0) = (y = 0)" |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
683 |
by (auto simp add: complex_zero_def complex_of_real_def) |
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
684 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
685 |
lemma cos_arg_i_mult_zero_pos: |
14377 | 686 |
"0 < y ==> cos (arg(Complex 0 y)) = 0" |
14373 | 687 |
apply (simp add: arg_def abs_if) |
14334 | 688 |
apply (rule_tac a = "pi/2" in someI2, auto) |
689 |
apply (rule order_less_trans [of _ 0], auto) |
|
14323 | 690 |
done |
691 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
692 |
lemma cos_arg_i_mult_zero_neg: |
14377 | 693 |
"y < 0 ==> cos (arg(Complex 0 y)) = 0" |
14373 | 694 |
apply (simp add: arg_def abs_if) |
14334 | 695 |
apply (rule_tac a = "- pi/2" in someI2, auto) |
696 |
apply (rule order_trans [of _ 0], auto) |
|
14323 | 697 |
done |
698 |
||
14374 | 699 |
lemma cos_arg_i_mult_zero [simp]: |
14377 | 700 |
"y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0" |
701 |
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) |
|
14323 | 702 |
|
703 |
||
704 |
subsection{*Finally! Polar Form for Complex Numbers*} |
|
705 |
||
14374 | 706 |
lemma complex_split_polar: |
14377 | 707 |
"\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))" |
708 |
apply (induct z) |
|
709 |
apply (auto simp add: polar_Ex complex_of_real_mult_Complex) |
|
14323 | 710 |
done |
711 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
712 |
lemma rcis_Ex: "\<exists>r a. z = rcis r a" |
14377 | 713 |
apply (induct z) |
714 |
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex) |
|
14323 | 715 |
done |
716 |
||
14374 | 717 |
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" |
14373 | 718 |
by (simp add: rcis_def cis_def) |
14323 | 719 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
720 |
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" |
14373 | 721 |
by (simp add: rcis_def cis_def) |
14323 | 722 |
|
14377 | 723 |
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>" |
724 |
proof - |
|
725 |
have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)" |
|
726 |
by (simp only: power_mult_distrib right_distrib) |
|
727 |
thus ?thesis by simp |
|
728 |
qed |
|
14323 | 729 |
|
14374 | 730 |
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" |
14377 | 731 |
by (simp add: rcis_def cis_def sin_cos_squared_add2_mult) |
14323 | 732 |
|
733 |
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" |
|
14373 | 734 |
apply (simp add: cmod_def) |
14323 | 735 |
apply (rule real_sqrt_eq_iff [THEN iffD2]) |
736 |
apply (auto simp add: complex_mult_cnj) |
|
737 |
done |
|
738 |
||
14374 | 739 |
lemma complex_Re_cnj [simp]: "Re(cnj z) = Re z" |
14373 | 740 |
by (induct z, simp add: complex_cnj) |
14323 | 741 |
|
14374 | 742 |
lemma complex_Im_cnj [simp]: "Im(cnj z) = - Im z" |
743 |
by (induct z, simp add: complex_cnj) |
|
744 |
||
745 |
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" |
|
14373 | 746 |
by (induct z, simp add: complex_cnj complex_mult) |
14323 | 747 |
|
748 |
||
749 |
(*---------------------------------------------------------------------------*) |
|
750 |
(* (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b) *) |
|
751 |
(*---------------------------------------------------------------------------*) |
|
752 |
||
753 |
lemma cis_rcis_eq: "cis a = rcis 1 a" |
|
14373 | 754 |
by (simp add: rcis_def) |
14323 | 755 |
|
14374 | 756 |
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" |
14377 | 757 |
by (simp add: rcis_def cis_def complex_of_real_mult_Complex cos_add sin_add right_distrib right_diff_distrib) |
14323 | 758 |
|
759 |
lemma cis_mult: "cis a * cis b = cis (a + b)" |
|
14373 | 760 |
by (simp add: cis_rcis_eq rcis_mult) |
14323 | 761 |
|
14374 | 762 |
lemma cis_zero [simp]: "cis 0 = 1" |
14377 | 763 |
by (simp add: cis_def complex_one_def) |
14323 | 764 |
|
14374 | 765 |
lemma rcis_zero_mod [simp]: "rcis 0 a = 0" |
14373 | 766 |
by (simp add: rcis_def) |
14323 | 767 |
|
14374 | 768 |
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" |
14373 | 769 |
by (simp add: rcis_def) |
14323 | 770 |
|
771 |
lemma complex_of_real_minus_one: |
|
772 |
"complex_of_real (-(1::real)) = -(1::complex)" |
|
14377 | 773 |
by (simp add: complex_of_real_def complex_one_def complex_minus) |
14323 | 774 |
|
14374 | 775 |
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x" |
14373 | 776 |
by (simp add: complex_mult_assoc [symmetric]) |
14323 | 777 |
|
778 |
||
779 |
lemma cis_real_of_nat_Suc_mult: |
|
780 |
"cis (real (Suc n) * a) = cis a * cis (real n * a)" |
|
14377 | 781 |
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib) |
14323 | 782 |
|
783 |
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" |
|
784 |
apply (induct_tac "n") |
|
785 |
apply (auto simp add: cis_real_of_nat_Suc_mult) |
|
786 |
done |
|
787 |
||
14374 | 788 |
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" |
789 |
by (simp add: rcis_def power_mult_distrib DeMoivre complex_of_real_pow) |
|
14323 | 790 |
|
14374 | 791 |
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)" |
792 |
by (simp add: cis_def complex_inverse_complex_split complex_of_real_minus |
|
793 |
complex_diff_def) |
|
14323 | 794 |
|
795 |
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)" |
|
14377 | 796 |
by (simp add: divide_inverse_zero rcis_def complex_of_real_inverse) |
14323 | 797 |
|
798 |
lemma cis_divide: "cis a / cis b = cis (a - b)" |
|
14373 | 799 |
by (simp add: complex_divide_def cis_mult real_diff_def) |
14323 | 800 |
|
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
801 |
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)" |
14373 | 802 |
apply (simp add: complex_divide_def) |
803 |
apply (case_tac "r2=0", simp) |
|
804 |
apply (simp add: rcis_inverse rcis_mult real_diff_def) |
|
14323 | 805 |
done |
806 |
||
14374 | 807 |
lemma Re_cis [simp]: "Re(cis a) = cos a" |
14373 | 808 |
by (simp add: cis_def) |
14323 | 809 |
|
14374 | 810 |
lemma Im_cis [simp]: "Im(cis a) = sin a" |
14373 | 811 |
by (simp add: cis_def) |
14323 | 812 |
|
813 |
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" |
|
14334 | 814 |
by (auto simp add: DeMoivre) |
14323 | 815 |
|
816 |
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" |
|
14334 | 817 |
by (auto simp add: DeMoivre) |
14323 | 818 |
|
819 |
lemma expi_add: "expi(a + b) = expi(a) * expi(b)" |
|
14374 | 820 |
by (simp add: expi_def complex_Re_add exp_add complex_Im_add |
821 |
cis_mult [symmetric] complex_of_real_mult mult_ac) |
|
14323 | 822 |
|
14374 | 823 |
lemma expi_zero [simp]: "expi (0::complex) = 1" |
14373 | 824 |
by (simp add: expi_def) |
14323 | 825 |
|
14374 | 826 |
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a" |
14373 | 827 |
apply (insert rcis_Ex [of z]) |
14323 | 828 |
apply (auto simp add: expi_def rcis_def complex_mult_assoc [symmetric] complex_of_real_mult) |
14334 | 829 |
apply (rule_tac x = "ii * complex_of_real a" in exI, auto) |
14323 | 830 |
done |
831 |
||
832 |
||
833 |
||
834 |
ML |
|
835 |
{* |
|
836 |
val complex_zero_def = thm"complex_zero_def"; |
|
837 |
val complex_one_def = thm"complex_one_def"; |
|
838 |
val complex_minus_def = thm"complex_minus_def"; |
|
839 |
val complex_diff_def = thm"complex_diff_def"; |
|
840 |
val complex_divide_def = thm"complex_divide_def"; |
|
841 |
val complex_mult_def = thm"complex_mult_def"; |
|
842 |
val complex_add_def = thm"complex_add_def"; |
|
843 |
val complex_of_real_def = thm"complex_of_real_def"; |
|
844 |
val i_def = thm"i_def"; |
|
845 |
val expi_def = thm"expi_def"; |
|
846 |
val cis_def = thm"cis_def"; |
|
847 |
val rcis_def = thm"rcis_def"; |
|
848 |
val cmod_def = thm"cmod_def"; |
|
849 |
val cnj_def = thm"cnj_def"; |
|
850 |
val sgn_def = thm"sgn_def"; |
|
851 |
val arg_def = thm"arg_def"; |
|
852 |
val complexpow_0 = thm"complexpow_0"; |
|
853 |
val complexpow_Suc = thm"complexpow_Suc"; |
|
854 |
||
855 |
val Re = thm"Re"; |
|
856 |
val Im = thm"Im"; |
|
857 |
val complex_Re_Im_cancel_iff = thm"complex_Re_Im_cancel_iff"; |
|
858 |
val complex_Re_zero = thm"complex_Re_zero"; |
|
859 |
val complex_Im_zero = thm"complex_Im_zero"; |
|
860 |
val complex_Re_one = thm"complex_Re_one"; |
|
861 |
val complex_Im_one = thm"complex_Im_one"; |
|
862 |
val complex_Re_i = thm"complex_Re_i"; |
|
863 |
val complex_Im_i = thm"complex_Im_i"; |
|
864 |
val Re_complex_of_real = thm"Re_complex_of_real"; |
|
865 |
val Im_complex_of_real = thm"Im_complex_of_real"; |
|
866 |
val complex_minus = thm"complex_minus"; |
|
867 |
val complex_Re_minus = thm"complex_Re_minus"; |
|
868 |
val complex_Im_minus = thm"complex_Im_minus"; |
|
869 |
val complex_add = thm"complex_add"; |
|
870 |
val complex_Re_add = thm"complex_Re_add"; |
|
871 |
val complex_Im_add = thm"complex_Im_add"; |
|
872 |
val complex_add_commute = thm"complex_add_commute"; |
|
873 |
val complex_add_assoc = thm"complex_add_assoc"; |
|
874 |
val complex_add_zero_left = thm"complex_add_zero_left"; |
|
875 |
val complex_add_zero_right = thm"complex_add_zero_right"; |
|
876 |
val complex_diff = thm"complex_diff"; |
|
877 |
val complex_mult = thm"complex_mult"; |
|
878 |
val complex_mult_one_left = thm"complex_mult_one_left"; |
|
879 |
val complex_mult_one_right = thm"complex_mult_one_right"; |
|
880 |
val complex_inverse = thm"complex_inverse"; |
|
881 |
val complex_of_real_one = thm"complex_of_real_one"; |
|
882 |
val complex_of_real_zero = thm"complex_of_real_zero"; |
|
883 |
val complex_of_real_eq_iff = thm"complex_of_real_eq_iff"; |
|
884 |
val complex_of_real_minus = thm"complex_of_real_minus"; |
|
885 |
val complex_of_real_inverse = thm"complex_of_real_inverse"; |
|
886 |
val complex_of_real_add = thm"complex_of_real_add"; |
|
887 |
val complex_of_real_diff = thm"complex_of_real_diff"; |
|
888 |
val complex_of_real_mult = thm"complex_of_real_mult"; |
|
889 |
val complex_of_real_divide = thm"complex_of_real_divide"; |
|
890 |
val complex_of_real_pow = thm"complex_of_real_pow"; |
|
891 |
val complex_mod = thm"complex_mod"; |
|
892 |
val complex_mod_zero = thm"complex_mod_zero"; |
|
893 |
val complex_mod_one = thm"complex_mod_one"; |
|
894 |
val complex_mod_complex_of_real = thm"complex_mod_complex_of_real"; |
|
895 |
val complex_of_real_abs = thm"complex_of_real_abs"; |
|
896 |
val complex_cnj = thm"complex_cnj"; |
|
897 |
val complex_cnj_cancel_iff = thm"complex_cnj_cancel_iff"; |
|
898 |
val complex_cnj_cnj = thm"complex_cnj_cnj"; |
|
899 |
val complex_cnj_complex_of_real = thm"complex_cnj_complex_of_real"; |
|
900 |
val complex_mod_cnj = thm"complex_mod_cnj"; |
|
901 |
val complex_cnj_minus = thm"complex_cnj_minus"; |
|
902 |
val complex_cnj_inverse = thm"complex_cnj_inverse"; |
|
903 |
val complex_cnj_add = thm"complex_cnj_add"; |
|
904 |
val complex_cnj_diff = thm"complex_cnj_diff"; |
|
905 |
val complex_cnj_mult = thm"complex_cnj_mult"; |
|
906 |
val complex_cnj_divide = thm"complex_cnj_divide"; |
|
907 |
val complex_cnj_one = thm"complex_cnj_one"; |
|
908 |
val complex_cnj_pow = thm"complex_cnj_pow"; |
|
909 |
val complex_add_cnj = thm"complex_add_cnj"; |
|
910 |
val complex_diff_cnj = thm"complex_diff_cnj"; |
|
911 |
val complex_cnj_zero = thm"complex_cnj_zero"; |
|
912 |
val complex_cnj_zero_iff = thm"complex_cnj_zero_iff"; |
|
913 |
val complex_mult_cnj = thm"complex_mult_cnj"; |
|
914 |
val complex_mod_eq_zero_cancel = thm"complex_mod_eq_zero_cancel"; |
|
915 |
val complex_mod_complex_of_real_of_nat = thm"complex_mod_complex_of_real_of_nat"; |
|
916 |
val complex_mod_minus = thm"complex_mod_minus"; |
|
917 |
val complex_mod_mult_cnj = thm"complex_mod_mult_cnj"; |
|
918 |
val complex_mod_squared = thm"complex_mod_squared"; |
|
919 |
val complex_mod_ge_zero = thm"complex_mod_ge_zero"; |
|
920 |
val abs_cmod_cancel = thm"abs_cmod_cancel"; |
|
921 |
val complex_mod_mult = thm"complex_mod_mult"; |
|
922 |
val complex_mod_add_squared_eq = thm"complex_mod_add_squared_eq"; |
|
923 |
val complex_Re_mult_cnj_le_cmod = thm"complex_Re_mult_cnj_le_cmod"; |
|
924 |
val complex_Re_mult_cnj_le_cmod2 = thm"complex_Re_mult_cnj_le_cmod2"; |
|
925 |
val real_sum_squared_expand = thm"real_sum_squared_expand"; |
|
926 |
val complex_mod_triangle_squared = thm"complex_mod_triangle_squared"; |
|
927 |
val complex_mod_minus_le_complex_mod = thm"complex_mod_minus_le_complex_mod"; |
|
928 |
val complex_mod_triangle_ineq = thm"complex_mod_triangle_ineq"; |
|
929 |
val complex_mod_triangle_ineq2 = thm"complex_mod_triangle_ineq2"; |
|
930 |
val complex_mod_diff_commute = thm"complex_mod_diff_commute"; |
|
931 |
val complex_mod_add_less = thm"complex_mod_add_less"; |
|
932 |
val complex_mod_mult_less = thm"complex_mod_mult_less"; |
|
933 |
val complex_mod_diff_ineq = thm"complex_mod_diff_ineq"; |
|
934 |
val complex_Re_le_cmod = thm"complex_Re_le_cmod"; |
|
935 |
val complex_mod_gt_zero = thm"complex_mod_gt_zero"; |
|
936 |
val complex_mod_complexpow = thm"complex_mod_complexpow"; |
|
937 |
val complexpow_minus = thm"complexpow_minus"; |
|
938 |
val complex_mod_inverse = thm"complex_mod_inverse"; |
|
939 |
val complex_mod_divide = thm"complex_mod_divide"; |
|
940 |
val complexpow_i_squared = thm"complexpow_i_squared"; |
|
941 |
val complex_i_not_zero = thm"complex_i_not_zero"; |
|
942 |
val sgn_zero = thm"sgn_zero"; |
|
943 |
val sgn_one = thm"sgn_one"; |
|
944 |
val sgn_minus = thm"sgn_minus"; |
|
945 |
val sgn_eq = thm"sgn_eq"; |
|
946 |
val i_mult_eq = thm"i_mult_eq"; |
|
947 |
val i_mult_eq2 = thm"i_mult_eq2"; |
|
948 |
val Re_sgn = thm"Re_sgn"; |
|
949 |
val Im_sgn = thm"Im_sgn"; |
|
950 |
val complex_inverse_complex_split = thm"complex_inverse_complex_split"; |
|
951 |
val cos_arg_i_mult_zero = thm"cos_arg_i_mult_zero"; |
|
952 |
val complex_of_real_zero_iff = thm"complex_of_real_zero_iff"; |
|
953 |
val rcis_Ex = thm"rcis_Ex"; |
|
954 |
val Re_rcis = thm"Re_rcis"; |
|
955 |
val Im_rcis = thm"Im_rcis"; |
|
956 |
val complex_mod_rcis = thm"complex_mod_rcis"; |
|
957 |
val complex_mod_sqrt_Re_mult_cnj = thm"complex_mod_sqrt_Re_mult_cnj"; |
|
958 |
val complex_Re_cnj = thm"complex_Re_cnj"; |
|
959 |
val complex_Im_cnj = thm"complex_Im_cnj"; |
|
960 |
val complex_In_mult_cnj_zero = thm"complex_In_mult_cnj_zero"; |
|
961 |
val complex_Re_mult = thm"complex_Re_mult"; |
|
962 |
val complex_Re_mult_complex_of_real = thm"complex_Re_mult_complex_of_real"; |
|
963 |
val complex_Im_mult_complex_of_real = thm"complex_Im_mult_complex_of_real"; |
|
964 |
val complex_Re_mult_complex_of_real2 = thm"complex_Re_mult_complex_of_real2"; |
|
965 |
val complex_Im_mult_complex_of_real2 = thm"complex_Im_mult_complex_of_real2"; |
|
966 |
val cis_rcis_eq = thm"cis_rcis_eq"; |
|
967 |
val rcis_mult = thm"rcis_mult"; |
|
968 |
val cis_mult = thm"cis_mult"; |
|
969 |
val cis_zero = thm"cis_zero"; |
|
970 |
val rcis_zero_mod = thm"rcis_zero_mod"; |
|
971 |
val rcis_zero_arg = thm"rcis_zero_arg"; |
|
972 |
val complex_of_real_minus_one = thm"complex_of_real_minus_one"; |
|
973 |
val complex_i_mult_minus = thm"complex_i_mult_minus"; |
|
974 |
val cis_real_of_nat_Suc_mult = thm"cis_real_of_nat_Suc_mult"; |
|
975 |
val DeMoivre = thm"DeMoivre"; |
|
976 |
val DeMoivre2 = thm"DeMoivre2"; |
|
977 |
val cis_inverse = thm"cis_inverse"; |
|
978 |
val rcis_inverse = thm"rcis_inverse"; |
|
979 |
val cis_divide = thm"cis_divide"; |
|
980 |
val rcis_divide = thm"rcis_divide"; |
|
981 |
val Re_cis = thm"Re_cis"; |
|
982 |
val Im_cis = thm"Im_cis"; |
|
983 |
val cos_n_Re_cis_pow_n = thm"cos_n_Re_cis_pow_n"; |
|
984 |
val sin_n_Im_cis_pow_n = thm"sin_n_Im_cis_pow_n"; |
|
985 |
val expi_add = thm"expi_add"; |
|
986 |
val expi_zero = thm"expi_zero"; |
|
987 |
val complex_Re_mult_eq = thm"complex_Re_mult_eq"; |
|
988 |
val complex_Im_mult_eq = thm"complex_Im_mult_eq"; |
|
989 |
val complex_expi_Ex = thm"complex_expi_Ex"; |
|
990 |
*} |
|
991 |
||
13957 | 992 |
end |
993 |
||
994 |