src/HOL/Complex/Complex.thy
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add instance for real_algebra_1 and real_normed_div_algebra
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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/HLog"
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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 (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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definition
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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 = (SOME 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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definition
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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: times_divide_eq complex_mult complex_inverse 
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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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20556
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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_algebra_1
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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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   286
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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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   293
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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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   297
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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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   304
by (simp add: i_def complex_of_real_def)
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   305
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   306
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
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   307
by (simp add: i_def complex_of_real_def)
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   308
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   309
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
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   310
by (simp add: i_def complex_of_real_def)
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   311
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lemma complex_of_real_one [simp]: "complex_of_real 1 = 1"
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   313
by (simp add: complex_one_def complex_of_real_def)
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   314
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   315
lemma complex_of_real_zero [simp]: "complex_of_real 0 = 0"
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   316
by (simp add: complex_zero_def complex_of_real_def)
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   317
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61de62096768 further tidying of the complex numbers
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parents: 14373
diff changeset
   318
lemma complex_of_real_eq_iff [iff]:
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   319
     "(complex_of_real x = complex_of_real y) = (x = y)"
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   320
by (simp add: complex_of_real_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   321
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   322
lemma complex_of_real_minus [simp]: "complex_of_real(-x) = - complex_of_real x"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   323
by (simp add: complex_of_real_def complex_minus)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   324
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   325
lemma complex_of_real_inverse [simp]:
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   326
     "complex_of_real(inverse x) = inverse(complex_of_real x)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   327
apply (case_tac "x=0", simp)
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   328
apply (simp add: complex_of_real_def divide_inverse power2_eq_square)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   329
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   330
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   331
lemma complex_of_real_add [simp]:
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   332
     "complex_of_real (x + y) = complex_of_real x + complex_of_real y"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   333
by (simp add: complex_add complex_of_real_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   334
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   335
lemma complex_of_real_diff [simp]:
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   336
     "complex_of_real (x - y) = complex_of_real x - complex_of_real y"
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   337
by (simp add: complex_of_real_minus diff_minus)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   338
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   339
lemma complex_of_real_mult [simp]:
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   340
     "complex_of_real (x * y) = complex_of_real x * complex_of_real y"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   341
by (simp add: complex_mult complex_of_real_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   342
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   343
lemma complex_of_real_divide [simp]:
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   344
      "complex_of_real(x/y) = complex_of_real x / complex_of_real y"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   345
apply (simp add: complex_divide_def)
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   346
apply (case_tac "y=0", simp)
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   347
apply (simp add: complex_of_real_mult complex_of_real_inverse 
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   348
                 divide_inverse)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   349
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   350
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   351
lemma complex_mod [simp]: "cmod (Complex x y) = sqrt(x ^ 2 + y ^ 2)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   352
by (simp add: cmod_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   353
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   354
lemma complex_mod_zero [simp]: "cmod(0) = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   355
by (simp add: cmod_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   356
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   357
lemma complex_mod_one [simp]: "cmod(1) = 1"
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   358
by (simp add: cmod_def power2_eq_square)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   359
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   360
lemma complex_mod_complex_of_real [simp]: "cmod(complex_of_real x) = abs x"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   361
by (simp add: complex_of_real_def power2_eq_square complex_mod)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   362
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   363
lemma complex_of_real_abs:
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   364
     "complex_of_real (abs x) = complex_of_real(cmod(complex_of_real x))"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   365
by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   366
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   367
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   368
subsection{*The Functions @{term Re} and @{term Im}*}
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   369
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   370
lemma complex_Re_mult_eq: "Re (w * z) = Re w * Re z - Im w * Im z"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   371
by (induct z, induct w, simp add: complex_mult)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   372
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   373
lemma complex_Im_mult_eq: "Im (w * z) = Re w * Im z + Im w * Re z"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   374
by (induct z, induct w, simp add: complex_mult)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   375
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   376
lemma Re_i_times [simp]: "Re(ii * z) = - Im z"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   377
by (simp add: complex_Re_mult_eq) 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   378
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   379
lemma Re_times_i [simp]: "Re(z * ii) = - Im z"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   380
by (simp add: complex_Re_mult_eq) 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   381
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   382
lemma Im_i_times [simp]: "Im(ii * z) = Re z"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   383
by (simp add: complex_Im_mult_eq) 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   384
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   385
lemma Im_times_i [simp]: "Im(z * ii) = Re z"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   386
by (simp add: complex_Im_mult_eq) 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   387
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   388
lemma complex_Re_mult: "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   389
by (simp add: complex_Re_mult_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   390
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   391
lemma complex_Re_mult_complex_of_real [simp]:
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   392
     "Re (z * complex_of_real c) = Re(z) * c"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   393
by (simp add: complex_Re_mult_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   394
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   395
lemma complex_Im_mult_complex_of_real [simp]:
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   396
     "Im (z * complex_of_real c) = Im(z) * c"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   397
by (simp add: complex_Im_mult_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   398
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   399
lemma complex_Re_mult_complex_of_real2 [simp]:
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   400
     "Re (complex_of_real c * z) = c * Re(z)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   401
by (simp add: complex_Re_mult_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   402
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   403
lemma complex_Im_mult_complex_of_real2 [simp]:
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   404
     "Im (complex_of_real c * z) = c * Im(z)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   405
by (simp add: complex_Im_mult_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   406
 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   407
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   408
subsection{*Conjugation is an Automorphism*}
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   409
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   410
lemma complex_cnj: "cnj (Complex x y) = Complex x (-y)"
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   411
by (simp add: cnj_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   412
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   413
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   414
by (simp add: cnj_def complex_Re_Im_cancel_iff)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   415
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   416
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   417
by (simp add: cnj_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   418
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   419
lemma complex_cnj_complex_of_real [simp]:
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   420
     "cnj (complex_of_real x) = complex_of_real x"
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   421
by (simp add: complex_of_real_def complex_cnj)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   422
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   423
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   424
by (induct z, simp add: complex_cnj complex_mod power2_eq_square)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   425
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   426
lemma complex_cnj_minus: "cnj (-z) = - cnj z"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   427
by (simp add: cnj_def complex_minus complex_Re_minus complex_Im_minus)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   428
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   429
lemma complex_cnj_inverse: "cnj(inverse z) = inverse(cnj z)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   430
by (induct z, simp add: complex_cnj complex_inverse power2_eq_square)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   431
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   432
lemma complex_cnj_add: "cnj(w + z) = cnj(w) + cnj(z)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   433
by (induct w, induct z, simp add: complex_cnj complex_add)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   434
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   435
lemma complex_cnj_diff: "cnj(w - z) = cnj(w) - cnj(z)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   436
by (simp add: diff_minus complex_cnj_add complex_cnj_minus)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   437
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   438
lemma complex_cnj_mult: "cnj(w * z) = cnj(w) * cnj(z)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   439
by (induct w, induct z, simp add: complex_cnj complex_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   440
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   441
lemma complex_cnj_divide: "cnj(w / z) = (cnj w)/(cnj z)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   442
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   443
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   444
lemma complex_cnj_one [simp]: "cnj 1 = 1"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   445
by (simp add: cnj_def complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   446
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   447
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re(z))"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   448
by (induct z, simp add: complex_add complex_cnj complex_of_real_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   449
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   450
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im(z)) * ii"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   451
apply (induct z)
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   452
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
   453
                 complex_minus i_def complex_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   454
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   455
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   456
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   457
by (simp add: cnj_def complex_zero_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   458
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   459
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   460
by (induct z, simp add: complex_zero_def complex_cnj)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   461
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   462
lemma complex_mult_cnj: "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)"
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   463
by (induct z,
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   464
    simp add: complex_cnj complex_mult complex_of_real_def power2_eq_square)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   465
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   466
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   467
subsection{*Modulus*}
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   468
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   469
lemma complex_mod_eq_zero_cancel [simp]: "(cmod x = 0) = (x = 0)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   470
apply (induct x)
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15013
diff changeset
   471
apply (auto iff: real_0_le_add_iff 
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15013
diff changeset
   472
            intro: real_sum_squares_cancel real_sum_squares_cancel2
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   473
            simp add: complex_mod complex_zero_def power2_eq_square)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   474
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   475
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   476
lemma complex_mod_complex_of_real_of_nat [simp]:
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   477
     "cmod (complex_of_real(real (n::nat))) = real n"
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   478
by simp
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   479
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   480
lemma complex_mod_minus [simp]: "cmod (-x) = cmod(x)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   481
by (induct x, simp add: complex_mod complex_minus power2_eq_square)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   482
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   483
lemma complex_mod_mult_cnj: "cmod(z * cnj(z)) = cmod(z) ^ 2"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   484
apply (induct z, simp add: complex_mod complex_cnj complex_mult)
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15013
diff changeset
   485
apply (simp add: power2_eq_square abs_if linorder_not_less real_0_le_add_iff)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   486
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   487
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   488
lemma complex_mod_squared: "cmod(Complex x y) ^ 2 = x ^ 2 + y ^ 2"
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   489
by (simp add: cmod_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   490
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   491
lemma complex_mod_ge_zero [simp]: "0 \<le> cmod x"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   492
by (simp add: cmod_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   493
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   494
lemma abs_cmod_cancel [simp]: "abs(cmod x) = cmod x"
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   495
by (simp add: abs_if linorder_not_less)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   496
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   497
lemma complex_mod_mult: "cmod(x*y) = cmod(x) * cmod(y)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   498
apply (induct x, induct y)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   499
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
   500
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
   501
apply (auto simp add: power2_eq_square [symmetric] simp del: realpow_Suc)
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   502
apply (auto simp add: real_diff_def power2_eq_square right_distrib left_distrib 
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   503
                      add_ac mult_ac)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   504
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   505
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   506
lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   507
by (simp add: cmod_def) 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   508
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   509
lemma cmod_complex_polar [simp]:
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   510
     "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   511
by (simp only: cmod_unit_one complex_mod_mult, simp) 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   512
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   513
lemma complex_mod_add_squared_eq:
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   514
     "cmod(x + y) ^ 2 = cmod(x) ^ 2 + cmod(y) ^ 2 + 2 * Re(x * cnj y)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   515
apply (induct x, induct y)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   516
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
   517
apply (auto simp add: right_distrib left_distrib power2_eq_square mult_ac add_ac)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   518
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   519
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   520
lemma complex_Re_mult_cnj_le_cmod [simp]: "Re(x * cnj y) \<le> cmod(x * cnj y)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   521
apply (induct x, induct y)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   522
apply (auto simp add: complex_mod complex_mult complex_cnj real_diff_def simp del: realpow_Suc)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   523
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   524
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   525
lemma complex_Re_mult_cnj_le_cmod2 [simp]: "Re(x * cnj y) \<le> cmod(x * y)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   526
by (insert complex_Re_mult_cnj_le_cmod [of x y], simp add: complex_mod_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   527
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   528
lemma real_sum_squared_expand:
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   529
     "((x::real) + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   530
by (simp add: left_distrib right_distrib power2_eq_square)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   531
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   532
lemma complex_mod_triangle_squared [simp]:
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   533
     "cmod (x + y) ^ 2 \<le> (cmod(x) + cmod(y)) ^ 2"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   534
by (simp add: real_sum_squared_expand complex_mod_add_squared_eq real_mult_assoc complex_mod_mult [symmetric])
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   535
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   536
lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   537
by (rule order_trans [OF _ complex_mod_ge_zero], simp)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   538
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   539
lemma complex_mod_triangle_ineq [simp]: "cmod (x + y) \<le> cmod(x) + cmod(y)"
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   540
apply (rule_tac n = 1 in realpow_increasing)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   541
apply (auto intro:  order_trans [OF _ complex_mod_ge_zero]
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15013
diff changeset
   542
            simp add: add_increasing power2_eq_square [symmetric])
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   543
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   544
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   545
lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   546
by (insert complex_mod_triangle_ineq [THEN add_right_mono, of b a"-cmod b"], simp)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   547
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   548
lemma complex_mod_diff_commute: "cmod (x - y) = cmod (y - x)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   549
apply (induct x, induct y)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   550
apply (auto simp add: complex_diff complex_mod right_diff_distrib power2_eq_square left_diff_distrib add_ac mult_ac)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   551
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   552
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   553
lemma complex_mod_add_less:
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   554
     "[| cmod x < r; cmod y < s |] ==> cmod (x + y) < r + s"
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   555
by (auto intro: order_le_less_trans complex_mod_triangle_ineq)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   556
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   557
lemma complex_mod_mult_less:
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   558
     "[| cmod x < r; cmod y < s |] ==> cmod (x * y) < r * s"
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   559
by (auto intro: real_mult_less_mono' simp add: complex_mod_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   560
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   561
lemma complex_mod_diff_ineq [simp]: "cmod(a) - cmod(b) \<le> cmod(a + b)"
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   562
apply (rule linorder_cases [of "cmod(a)" "cmod (b)"])
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   563
apply auto
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   564
apply (rule order_trans [of _ 0], rule order_less_imp_le)
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   565
apply (simp add: compare_rls, simp)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   566
apply (simp add: compare_rls)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   567
apply (rule complex_mod_minus [THEN subst])
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   568
apply (rule order_trans)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   569
apply (rule_tac [2] complex_mod_triangle_ineq)
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   570
apply (auto simp add: add_ac)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   571
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   572
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   573
lemma complex_Re_le_cmod [simp]: "Re z \<le> cmod z"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   574
by (induct z, simp add: complex_mod del: realpow_Suc)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   575
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   576
lemma complex_mod_gt_zero: "z \<noteq> 0 ==> 0 < cmod z"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   577
apply (insert complex_mod_ge_zero [of z])
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   578
apply (drule order_le_imp_less_or_eq, auto)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   579
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   580
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   581
20556
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   582
subsection{*The normed division algebra of complex numbers*}
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   583
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   584
instance complex :: norm ..
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   585
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   586
defs (overloaded)
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   587
  complex_norm_def: "norm == cmod"
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   588
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   589
lemma of_real_complex_of_real: "of_real r = complex_of_real r"
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   590
by (simp add: complex_of_real_def of_real_def complex_scaleR_def)
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   591
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   592
instance complex :: real_normed_div_algebra
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   593
proof (intro_classes, unfold complex_norm_def)
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   594
  fix r :: real
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   595
  fix x y :: complex
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   596
  show "0 \<le> cmod x"
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   597
    by (rule complex_mod_ge_zero)
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   598
  show "(cmod x = 0) = (x = 0)"
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   599
    by (rule complex_mod_eq_zero_cancel)
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   600
  show "cmod (x + y) \<le> cmod x + cmod y"
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   601
    by (rule complex_mod_triangle_ineq)
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   602
  show "cmod (of_real r) = abs r"
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   603
    by (simp add: of_real_complex_of_real)
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   604
  show "cmod (x * y) = cmod x * cmod y"
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   605
    by (rule complex_mod_mult)
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   606
  show "cmod 1 = 1"
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   607
    by (rule complex_mod_one)
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   608
qed
2e8227b81bf1 add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents: 20485
diff changeset
   609
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   610
subsection{*A Few More Theorems*}
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   611
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   612
lemma complex_mod_inverse: "cmod(inverse x) = inverse(cmod x)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   613
apply (case_tac "x=0", simp)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   614
apply (rule_tac c1 = "cmod x" in real_mult_left_cancel [THEN iffD1])
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   615
apply (auto simp add: complex_mod_mult [symmetric])
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   616
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   617
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   618
lemma complex_mod_divide: "cmod(x/y) = cmod(x)/(cmod y)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   619
by (simp add: complex_divide_def divide_inverse complex_mod_mult complex_mod_inverse)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   620
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   621
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   622
subsection{*Exponentiation*}
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   623
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   624
primrec
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   625
     complexpow_0:   "z ^ 0       = 1"
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   626
     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
   627
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   628
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14691
diff changeset
   629
instance complex :: recpower
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   630
proof
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   631
  fix z :: complex
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   632
  fix n :: nat
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   633
  show "z^0 = 1" by simp
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   634
  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
   635
qed
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   636
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   637
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   638
lemma complex_of_real_pow: "complex_of_real (x ^ n) = (complex_of_real x) ^ n"
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   639
apply (induct_tac "n")
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   640
apply (auto simp add: complex_of_real_mult [symmetric])
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   641
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   642
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   643
lemma complex_cnj_pow: "cnj(z ^ n) = cnj(z) ^ n"
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   644
apply (induct_tac "n")
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   645
apply (auto simp add: complex_cnj_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   646
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   647
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   648
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
   649
apply (induct_tac "n")
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   650
apply (auto simp add: complex_mod_mult)
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   651
done
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   652
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   653
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
   654
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
   655
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   656
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   657
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
   658
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   659
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   660
subsection{*The Function @{term sgn}*}
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   661
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   662
lemma sgn_zero [simp]: "sgn 0 = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   663
by (simp add: sgn_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   664
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   665
lemma sgn_one [simp]: "sgn 1 = 1"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   666
by (simp add: sgn_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   667
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   668
lemma sgn_minus: "sgn (-z) = - sgn(z)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   669
by (simp add: sgn_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   670
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   671
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   672
by (simp add: sgn_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   673
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   674
lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   675
by (simp add: i_def complex_of_real_def complex_mult complex_add)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   676
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   677
lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   678
by (simp add: i_def complex_one_def complex_mult complex_minus)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   679
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   680
lemma complex_eq_cancel_iff2 [simp]:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   681
     "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   682
by (simp add: complex_of_real_def) 
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   683
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   684
lemma complex_eq_cancel_iff2a [simp]:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   685
     "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   686
by (simp add: complex_of_real_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   687
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   688
lemma Complex_eq_0 [simp]: "(Complex x y = 0) = (x = 0 & y = 0)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   689
by (simp add: complex_zero_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   690
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   691
lemma Complex_eq_1 [simp]: "(Complex x y = 1) = (x = 1 & y = 0)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   692
by (simp add: complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   693
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   694
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 & y = 1)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   695
by (simp add: i_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   696
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   697
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   698
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   699
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   700
proof (induct z)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   701
  case (Complex x y)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   702
    have "sqrt (x\<twosuperior> + y\<twosuperior>) * inverse (x\<twosuperior> + y\<twosuperior>) = inverse (sqrt (x\<twosuperior> + y\<twosuperior>))"
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   703
      by (simp add: divide_inverse [symmetric] sqrt_divide_self_eq)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   704
    thus "Re (sgn (Complex x y)) = Re (Complex x y) /cmod (Complex x y)"
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   705
       by (simp add: sgn_def complex_of_real_def divide_inverse)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   706
qed
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   707
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   708
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   709
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   710
proof (induct z)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   711
  case (Complex x y)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   712
    have "sqrt (x\<twosuperior> + y\<twosuperior>) * inverse (x\<twosuperior> + y\<twosuperior>) = inverse (sqrt (x\<twosuperior> + y\<twosuperior>))"
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   713
      by (simp add: divide_inverse [symmetric] sqrt_divide_self_eq)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   714
    thus "Im (sgn (Complex x y)) = Im (Complex x y) /cmod (Complex x y)"
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   715
       by (simp add: sgn_def complex_of_real_def divide_inverse)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   716
qed
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   717
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   718
lemma complex_inverse_complex_split:
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   719
     "inverse(complex_of_real x + ii * complex_of_real y) =
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   720
      complex_of_real(x/(x ^ 2 + y ^ 2)) -
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   721
      ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   722
by (simp add: complex_of_real_def i_def complex_mult complex_add
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   723
         diff_minus complex_minus complex_inverse divide_inverse)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   724
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   725
(*----------------------------------------------------------------------------*)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   726
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   727
(* many of the theorems are not used - so should they be kept?                *)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   728
(*----------------------------------------------------------------------------*)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   729
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   730
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
   731
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
   732
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   733
lemma cos_arg_i_mult_zero_pos:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   734
   "0 < y ==> cos (arg(Complex 0 y)) = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   735
apply (simp add: arg_def abs_if)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   736
apply (rule_tac a = "pi/2" in someI2, auto)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   737
apply (rule order_less_trans [of _ 0], auto)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   738
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   739
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   740
lemma cos_arg_i_mult_zero_neg:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   741
   "y < 0 ==> cos (arg(Complex 0 y)) = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   742
apply (simp add: arg_def abs_if)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   743
apply (rule_tac a = "- pi/2" in someI2, auto)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   744
apply (rule order_trans [of _ 0], auto)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   745
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   746
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   747
lemma cos_arg_i_mult_zero [simp]:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   748
     "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   749
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   750
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   751
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   752
subsection{*Finally! Polar Form for Complex Numbers*}
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   753
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   754
lemma complex_split_polar:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   755
     "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   756
apply (induct z) 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   757
apply (auto simp add: polar_Ex complex_of_real_mult_Complex)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   758
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   759
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   760
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   761
apply (induct z) 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   762
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   763
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   764
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   765
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   766
by (simp add: rcis_def cis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   767
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   768
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   769
by (simp add: rcis_def cis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   770
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   771
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   772
proof -
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   773
  have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   774
    by (simp only: power_mult_distrib right_distrib) 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   775
  thus ?thesis by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   776
qed
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   777
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   778
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   779
by (simp add: rcis_def cis_def sin_cos_squared_add2_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   780
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   781
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   782
apply (simp add: cmod_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   783
apply (rule real_sqrt_eq_iff [THEN iffD2])
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   784
apply (auto simp add: complex_mult_cnj)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   785
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   786
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   787
lemma complex_Re_cnj [simp]: "Re(cnj z) = Re z"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   788
by (induct z, simp add: complex_cnj)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   789
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   790
lemma complex_Im_cnj [simp]: "Im(cnj z) = - Im z"
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   791
by (induct z, simp add: complex_cnj)
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   792
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   793
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   794
by (induct z, simp add: complex_cnj complex_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   795
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   796
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   797
(*---------------------------------------------------------------------------*)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   798
(*  (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b)                      *)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   799
(*---------------------------------------------------------------------------*)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   800
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   801
lemma cis_rcis_eq: "cis a = rcis 1 a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   802
by (simp add: rcis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   803
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   804
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   805
by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   806
              complex_of_real_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   807
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   808
lemma cis_mult: "cis a * cis b = cis (a + b)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   809
by (simp add: cis_rcis_eq rcis_mult)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   810
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   811
lemma cis_zero [simp]: "cis 0 = 1"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   812
by (simp add: cis_def complex_one_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   813
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   814
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   815
by (simp add: rcis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   816
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   817
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   818
by (simp add: rcis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   819
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   820
lemma complex_of_real_minus_one:
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   821
   "complex_of_real (-(1::real)) = -(1::complex)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   822
by (simp add: complex_of_real_def complex_one_def complex_minus)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   823
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   824
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   825
by (simp add: complex_mult_assoc [symmetric])
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   826
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   827
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   828
lemma cis_real_of_nat_Suc_mult:
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   829
   "cis (real (Suc n) * a) = cis a * cis (real n * a)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14374
diff changeset
   830
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   831
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   832
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   833
apply (induct_tac "n")
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   834
apply (auto simp add: cis_real_of_nat_Suc_mult)
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   835
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   836
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   837
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   838
by (simp add: rcis_def power_mult_distrib DeMoivre complex_of_real_pow)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   839
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   840
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   841
by (simp add: cis_def complex_inverse_complex_split complex_of_real_minus 
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   842
              diff_minus)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   843
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   844
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   845
by (simp add: divide_inverse rcis_def complex_of_real_inverse)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   846
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   847
lemma cis_divide: "cis a / cis b = cis (a - b)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   848
by (simp add: complex_divide_def cis_mult real_diff_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   849
14354
988aa4648597 types complex and hcomplex are now instances of class ringpower:
paulson
parents: 14353
diff changeset
   850
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   851
apply (simp add: complex_divide_def)
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   852
apply (case_tac "r2=0", simp)
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   853
apply (simp add: rcis_inverse rcis_mult real_diff_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   854
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   855
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   856
lemma Re_cis [simp]: "Re(cis a) = cos a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   857
by (simp add: cis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   858
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   859
lemma Im_cis [simp]: "Im(cis a) = sin a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   860
by (simp add: cis_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   861
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   862
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   863
by (auto simp add: DeMoivre)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   864
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   865
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   866
by (auto simp add: DeMoivre)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   867
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   868
lemma expi_add: "expi(a + b) = expi(a) * expi(b)"
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   869
by (simp add: expi_def complex_Re_add exp_add complex_Im_add 
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   870
              cis_mult [symmetric] complex_of_real_mult mult_ac)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   871
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   872
lemma expi_zero [simp]: "expi (0::complex) = 1"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   873
by (simp add: expi_def)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   874
14374
61de62096768 further tidying of the complex numbers
paulson
parents: 14373
diff changeset
   875
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
14373
67a628beb981 tidying of the complex numbers
paulson
parents: 14354
diff changeset
   876
apply (insert rcis_Ex [of z])
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   877
apply (auto simp add: expi_def rcis_def complex_mult_assoc [symmetric] complex_of_real_mult)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14323
diff changeset
   878
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
14323
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   879
done
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   880
27724f528f82 converting Complex/Complex.ML to Isar
paulson
parents: 13957
diff changeset
   881
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   882
subsection{*Numerals and Arithmetic*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   883
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   884
instance complex :: number ..
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   885
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   886
defs (overloaded)
20485
3078fd2eec7b got rid of Numeral.bin type
haftmann
parents: 19765
diff changeset
   887
  complex_number_of_def: "(number_of w :: complex) == of_int w"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   888
    --{*the type constraint is essential!*}
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   889
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   890
instance complex :: number_ring
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   891
by (intro_classes, simp add: complex_number_of_def) 
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   892
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   893
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   894
lemma complex_of_real_of_nat [simp]: "complex_of_real (of_nat n) = of_nat n"
19765
dfe940911617 misc cleanup;
wenzelm
parents: 15481
diff changeset
   895
  by (induct n) simp_all
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   896
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   897
lemma complex_of_real_of_int [simp]: "complex_of_real (of_int z) = of_int z"
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   898
proof (cases z)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   899
  case (1 n)
19765
dfe940911617 misc cleanup;
wenzelm
parents: 15481
diff changeset
   900
  thus ?thesis by simp
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   901
next
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   902
  case (2 n)
19765
dfe940911617 misc cleanup;
wenzelm
parents: 15481
diff changeset
   903
  thus ?thesis 
dfe940911617 misc cleanup;
wenzelm
parents: 15481
diff changeset
   904
    by (simp only: of_int_minus complex_of_real_minus, simp)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   905
qed
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   906
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   907
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   908
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
   909
lemma complex_number_of [simp]: "complex_of_real (number_of w) = number_of w"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   910
by (simp add: complex_number_of_def real_number_of_def) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   911
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   912
text{*This theorem is necessary because theorems such as
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   913
   @{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
   914
   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
   915
   They work for type complex because the reals can be embedded in them.*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   916
lemma iszero_complex_number_of [simp]:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   917
     "iszero (number_of w :: complex) = iszero (number_of w :: real)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   918
by (simp only: complex_of_real_zero_iff complex_number_of [symmetric] 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   919
               iszero_def)  
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   920
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   921
lemma complex_number_of_cnj [simp]: "cnj(number_of v :: complex) = number_of v"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15234
diff changeset
   922
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
   923
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   924
lemma complex_number_of_cmod: 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   925
      "cmod(number_of v :: complex) = abs (number_of v :: real)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   926
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
   927
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   928
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
   929
by (simp only: complex_number_of [symmetric] Re_complex_of_real)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   930
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   931
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
   932
by (simp only: complex_number_of [symmetric] Im_complex_of_real)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   933
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   934
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
   935
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
   936
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   937
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   938
(*examples:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   939
print_depth 22
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   940
set timing;
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   941
set trace_simp;
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   942
fun test s = (Goal s, by (Simp_tac 1)); 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   943
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   944
test "23 * ii + 45 * ii= (x::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   945
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   946
test "5 * ii + 12 - 45 * ii= (x::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   947
test "5 * ii + 40 - 12 * ii + 9 = (x::complex) + 89 * ii";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   948
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
   949
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   950
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
   951
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
   952
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   953
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   954
fun test s = (Goal s; by (Asm_simp_tac 1)); 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   955
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   956
test "x*k = k*(y::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   957
test "k = k*(y::complex)"; 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   958
test "a*(b*c) = (b::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   959
test "a*(b*c) = d*(b::complex)*(x*a)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   960
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   961
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   962
test "(x*k) / (k*(y::complex)) = (uu::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   963
test "(k) / (k*(y::complex)) = (uu::complex)"; 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   964
test "(a*(b*c)) / ((b::complex)) = (uu::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   965
test "(a*(b*c)) / (d*(b::complex)*(x*a)) = (uu::complex)";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   966
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14691
diff changeset
   967
FIXME: what do we do about this?
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   968
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
   969
*)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   970
13957
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
   971
end
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
   972
10dbf16be15f new session Complex for the complex numbers
paulson
parents:
diff changeset
   973