src/HOL/Complex/Complex.thy
author nipkow
Wed Aug 18 11:09:40 2004 +0200 (2004-08-18)
changeset 15140 322485b816ac
parent 15131 c69542757a4d
child 15234 ec91a90c604e
permissions -rw-r--r--
import -> imports
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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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constdefs
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  (*----------- modulus ------------*)
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  cmod :: "complex => real"
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  "cmod z == sqrt(Re(z) ^ 2 + Im(z) ^ 2)"
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  (*----- injection from reals -----*)
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  complex_of_real :: "real => complex"
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  "complex_of_real r == Complex r 0"
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  (*------- complex conjugate ------*)
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  cnj :: "complex => complex"
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  "cnj z == Complex (Re z) (-Im z)"
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  (*------------ Argand -------------*)
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  sgn :: "complex => complex"
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  "sgn z == z / complex_of_real(cmod z)"
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  arg :: "complex => real"
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  "arg z == @a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi"
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defs (overloaded)
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  complex_zero_def:
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  "0 == Complex 0 0"
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  complex_one_def:
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  "1 == Complex 1 0"
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  i_def: "ii == Complex 0 1"
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  complex_minus_def: "- z == Complex (- Re z) (- Im z)"
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  complex_inverse_def:
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   "inverse z ==
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    Complex (Re z / ((Re z)\<twosuperior> + (Im z)\<twosuperior>)) (- Im z / ((Re z)\<twosuperior> + (Im z)\<twosuperior>))"
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  complex_add_def:
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    "z + w == Complex (Re z + Re w) (Im z + Im w)"
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  complex_diff_def:
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    "z - w == z + - (w::complex)"
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  complex_mult_def:
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    "z * w == Complex (Re z * Re w - Im z * Im w) (Re z * Im w + Im z * Re w)"
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  complex_divide_def: "w / (z::complex) == w * inverse z"
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constdefs
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  (* abbreviation for (cos a + i sin a) *)
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  cis :: "real => complex"
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  "cis a == Complex (cos a) (sin a)"
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  (* abbreviation for r*(cos a + i sin a) *)
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  rcis :: "[real, real] => complex"
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  "rcis r a == complex_of_real r * cis a"
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  (* e ^ (x + iy) *)
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  expi :: "complex => complex"
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  "expi z == complex_of_real(exp (Re z)) * cis (Im z)"
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lemma complex_equality [intro?]: "Re z = Re w ==> Im z = Im w ==> z = w"
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  by (induct z, induct w) simp
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lemma Re [simp]: "Re(Complex x y) = x"
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by simp
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lemma Im [simp]: "Im(Complex x y) = y"
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by simp
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lemma complex_Re_Im_cancel_iff: "(w=z) = (Re(w) = Re(z) & Im(w) = Im(z))"
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by (induct w, induct z, simp)
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lemma complex_Re_zero [simp]: "Re 0 = 0"
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by (simp add: complex_zero_def)
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lemma complex_Im_zero [simp]: "Im 0 = 0"
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by (simp add: complex_zero_def)
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lemma complex_Re_one [simp]: "Re 1 = 1"
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by (simp add: complex_one_def)
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lemma complex_Im_one [simp]: "Im 1 = 0"
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by (simp add: complex_one_def)
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lemma complex_Re_i [simp]: "Re(ii) = 0"
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by (simp add: i_def)
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lemma complex_Im_i [simp]: "Im(ii) = 1"
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by (simp add: i_def)
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lemma Re_complex_of_real [simp]: "Re(complex_of_real z) = z"
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by (simp add: complex_of_real_def)
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lemma Im_complex_of_real [simp]: "Im(complex_of_real z) = 0"
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by (simp add: complex_of_real_def)
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subsection{*Unary Minus*}
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lemma complex_minus [simp]: "- (Complex x y) = Complex (-x) (-y)"
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by (simp add: complex_minus_def)
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lemma complex_Re_minus [simp]: "Re (-z) = - Re z"
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by (simp add: complex_minus_def)
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lemma complex_Im_minus [simp]: "Im (-z) = - Im z"
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by (simp add: complex_minus_def)
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subsection{*Addition*}
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lemma complex_add [simp]:
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     "Complex x1 y1 + Complex x2 y2 = Complex (x1+x2) (y1+y2)"
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by (simp add: complex_add_def)
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lemma complex_Re_add [simp]: "Re(x + y) = Re(x) + Re(y)"
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by (simp add: complex_add_def)
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lemma complex_Im_add [simp]: "Im(x + y) = Im(x) + Im(y)"
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by (simp add: complex_add_def)
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lemma complex_add_commute: "(u::complex) + v = v + u"
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by (simp add: complex_add_def add_commute)
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lemma complex_add_assoc: "((u::complex) + v) + w = u + (v + w)"
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by (simp add: complex_add_def add_assoc)
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lemma complex_add_zero_left: "(0::complex) + z = z"
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by (simp add: complex_add_def complex_zero_def)
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lemma complex_add_zero_right: "z + (0::complex) = z"
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by (simp add: complex_add_def complex_zero_def)
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lemma complex_add_minus_left: "-z + z = (0::complex)"
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by (simp add: complex_add_def complex_minus_def complex_zero_def)
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lemma complex_diff:
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      "Complex x1 y1 - Complex x2 y2 = Complex (x1-x2) (y1-y2)"
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by (simp add: complex_add_def complex_minus_def complex_diff_def)
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lemma complex_Re_diff [simp]: "Re(x - y) = Re(x) - Re(y)"
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by (simp add: complex_diff_def)
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lemma complex_Im_diff [simp]: "Im(x - y) = Im(x) - Im(y)"
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by (simp add: complex_diff_def)
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subsection{*Multiplication*}
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lemma complex_mult [simp]:
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     "Complex x1 y1 * Complex x2 y2 = Complex (x1*x2 - y1*y2) (x1*y2 + y1*x2)"
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by (simp add: complex_mult_def)
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lemma complex_mult_commute: "(w::complex) * z = z * w"
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by (simp add: complex_mult_def mult_commute add_commute)
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lemma complex_mult_assoc: "((u::complex) * v) * w = u * (v * w)"
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by (simp add: complex_mult_def mult_ac add_ac
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              right_diff_distrib right_distrib left_diff_distrib left_distrib)
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lemma complex_mult_one_left: "(1::complex) * z = z"
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by (simp add: complex_mult_def complex_one_def)
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lemma complex_mult_one_right: "z * (1::complex) = z"
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by (simp add: complex_mult_def complex_one_def)
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subsection{*Inverse*}
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lemma complex_inverse [simp]:
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     "inverse (Complex x y) = Complex (x/(x ^ 2 + y ^ 2)) (-y/(x ^ 2 + y ^ 2))"
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by (simp add: complex_inverse_def)
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lemma complex_mult_inv_left: "z \<noteq> (0::complex) ==> inverse(z) * z = 1"
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apply (induct z)
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apply (rename_tac x y)
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apply (auto simp add: complex_mult complex_inverse complex_one_def
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      complex_zero_def add_divide_distrib [symmetric] power2_eq_square mult_ac)
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apply (drule_tac y = y in real_sum_squares_not_zero)
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apply (drule_tac [2] x = x in real_sum_squares_not_zero2, auto)
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done
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subsection {* The field of complex numbers *}
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instance complex :: field
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proof
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  fix z u v w :: complex
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  show "(u + v) + w = u + (v + w)"
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    by (rule complex_add_assoc)
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  show "z + w = w + z"
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    by (rule complex_add_commute)
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  show "0 + z = z"
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    by (rule complex_add_zero_left)
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  show "-z + z = 0"
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    by (rule complex_add_minus_left)
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  show "z - w = z + -w"
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    by (simp add: complex_diff_def)
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  show "(u * v) * w = u * (v * w)"
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    by (rule complex_mult_assoc)
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  show "z * w = w * z"
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    by (rule complex_mult_commute)
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  show "1 * z = z"
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    by (rule complex_mult_one_left)
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  show "0 \<noteq> (1::complex)"
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    by (simp add: complex_zero_def complex_one_def)
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  show "(u + v) * w = u * w + v * w"
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    by (simp add: complex_mult_def complex_add_def left_distrib 
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                  diff_minus add_ac)
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  show "z / w = z * inverse w"
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    by (simp add: complex_divide_def)
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  assume "w \<noteq> 0"
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  thus "inverse w * w = 1"
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    by (simp add: complex_mult_inv_left)
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qed
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instance complex :: division_by_zero
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proof
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  show "inverse 0 = (0::complex)"
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    by (simp add: complex_inverse_def complex_zero_def)
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qed
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subsection{*Embedding Properties for @{term complex_of_real} Map*}
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lemma Complex_add_complex_of_real [simp]:
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     "Complex x y + complex_of_real r = Complex (x+r) y"
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by (simp add: complex_of_real_def)
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lemma complex_of_real_add_Complex [simp]:
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     "complex_of_real r + Complex x y = Complex (r+x) y"
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by (simp add: i_def complex_of_real_def)
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lemma Complex_mult_complex_of_real:
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     "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
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by (simp add: complex_of_real_def)
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lemma complex_of_real_mult_Complex:
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     "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
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by (simp add: i_def complex_of_real_def)
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lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
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by (simp add: i_def complex_of_real_def)
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lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
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by (simp add: i_def complex_of_real_def)
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lemma complex_of_real_one [simp]: "complex_of_real 1 = 1"
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by (simp add: complex_one_def complex_of_real_def)
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lemma complex_of_real_zero [simp]: "complex_of_real 0 = 0"
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by (simp add: complex_zero_def complex_of_real_def)
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lemma complex_of_real_eq_iff [iff]:
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     "(complex_of_real x = complex_of_real y) = (x = y)"
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by (simp add: complex_of_real_def)
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lemma complex_of_real_minus [simp]: "complex_of_real(-x) = - complex_of_real x"
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by (simp add: complex_of_real_def complex_minus)
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lemma complex_of_real_inverse [simp]:
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     "complex_of_real(inverse x) = inverse(complex_of_real x)"
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apply (case_tac "x=0", simp)
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apply (simp add: complex_of_real_def divide_inverse power2_eq_square)
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done
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lemma complex_of_real_add [simp]:
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     "complex_of_real (x + y) = complex_of_real x + complex_of_real y"
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by (simp add: complex_add complex_of_real_def)
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lemma complex_of_real_diff [simp]:
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     "complex_of_real (x - y) = complex_of_real x - complex_of_real y"
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by (simp add: complex_of_real_minus diff_minus)
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lemma complex_of_real_mult [simp]:
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     "complex_of_real (x * y) = complex_of_real x * complex_of_real y"
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by (simp add: complex_mult complex_of_real_def)
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lemma complex_of_real_divide [simp]:
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      "complex_of_real(x/y) = complex_of_real x / complex_of_real y"
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apply (simp add: complex_divide_def)
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apply (case_tac "y=0", simp)
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apply (simp add: complex_of_real_mult complex_of_real_inverse 
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                 divide_inverse)
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done
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lemma complex_mod [simp]: "cmod (Complex x y) = sqrt(x ^ 2 + y ^ 2)"
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by (simp add: cmod_def)
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lemma complex_mod_zero [simp]: "cmod(0) = 0"
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by (simp add: cmod_def)
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lemma complex_mod_one [simp]: "cmod(1) = 1"
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by (simp add: cmod_def power2_eq_square)
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lemma complex_mod_complex_of_real [simp]: "cmod(complex_of_real x) = abs x"
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by (simp add: complex_of_real_def power2_eq_square complex_mod)
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lemma complex_of_real_abs:
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     "complex_of_real (abs x) = complex_of_real(cmod(complex_of_real x))"
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by simp
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paulson@14323
   341
paulson@14377
   342
subsection{*The Functions @{term Re} and @{term Im}*}
paulson@14377
   343
paulson@14377
   344
lemma complex_Re_mult_eq: "Re (w * z) = Re w * Re z - Im w * Im z"
paulson@14377
   345
by (induct z, induct w, simp add: complex_mult)
paulson@14377
   346
paulson@14377
   347
lemma complex_Im_mult_eq: "Im (w * z) = Re w * Im z + Im w * Re z"
paulson@14377
   348
by (induct z, induct w, simp add: complex_mult)
paulson@14377
   349
paulson@14377
   350
lemma Re_i_times [simp]: "Re(ii * z) = - Im z"
paulson@14377
   351
by (simp add: complex_Re_mult_eq) 
paulson@14377
   352
paulson@14377
   353
lemma Re_times_i [simp]: "Re(z * ii) = - Im z"
paulson@14377
   354
by (simp add: complex_Re_mult_eq) 
paulson@14377
   355
paulson@14377
   356
lemma Im_i_times [simp]: "Im(ii * z) = Re z"
paulson@14377
   357
by (simp add: complex_Im_mult_eq) 
paulson@14377
   358
paulson@14377
   359
lemma Im_times_i [simp]: "Im(z * ii) = Re z"
paulson@14377
   360
by (simp add: complex_Im_mult_eq) 
paulson@14377
   361
paulson@14377
   362
lemma complex_Re_mult: "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)"
paulson@14377
   363
by (simp add: complex_Re_mult_eq)
paulson@14377
   364
paulson@14377
   365
lemma complex_Re_mult_complex_of_real [simp]:
paulson@14377
   366
     "Re (z * complex_of_real c) = Re(z) * c"
paulson@14377
   367
by (simp add: complex_Re_mult_eq)
paulson@14377
   368
paulson@14377
   369
lemma complex_Im_mult_complex_of_real [simp]:
paulson@14377
   370
     "Im (z * complex_of_real c) = Im(z) * c"
paulson@14377
   371
by (simp add: complex_Im_mult_eq)
paulson@14377
   372
paulson@14377
   373
lemma complex_Re_mult_complex_of_real2 [simp]:
paulson@14377
   374
     "Re (complex_of_real c * z) = c * Re(z)"
paulson@14377
   375
by (simp add: complex_Re_mult_eq)
paulson@14377
   376
paulson@14377
   377
lemma complex_Im_mult_complex_of_real2 [simp]:
paulson@14377
   378
     "Im (complex_of_real c * z) = c * Im(z)"
paulson@14377
   379
by (simp add: complex_Im_mult_eq)
paulson@14377
   380
 
paulson@14377
   381
paulson@14323
   382
subsection{*Conjugation is an Automorphism*}
paulson@14323
   383
paulson@14373
   384
lemma complex_cnj: "cnj (Complex x y) = Complex x (-y)"
paulson@14373
   385
by (simp add: cnj_def)
paulson@14323
   386
paulson@14374
   387
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
paulson@14373
   388
by (simp add: cnj_def complex_Re_Im_cancel_iff)
paulson@14323
   389
paulson@14374
   390
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
paulson@14373
   391
by (simp add: cnj_def)
paulson@14323
   392
paulson@14374
   393
lemma complex_cnj_complex_of_real [simp]:
paulson@14373
   394
     "cnj (complex_of_real x) = complex_of_real x"
paulson@14373
   395
by (simp add: complex_of_real_def complex_cnj)
paulson@14323
   396
paulson@14374
   397
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
paulson@14373
   398
by (induct z, simp add: complex_cnj complex_mod power2_eq_square)
paulson@14323
   399
paulson@14323
   400
lemma complex_cnj_minus: "cnj (-z) = - cnj z"
paulson@14373
   401
by (simp add: cnj_def complex_minus complex_Re_minus complex_Im_minus)
paulson@14323
   402
paulson@14323
   403
lemma complex_cnj_inverse: "cnj(inverse z) = inverse(cnj z)"
paulson@14373
   404
by (induct z, simp add: complex_cnj complex_inverse power2_eq_square)
paulson@14323
   405
paulson@14323
   406
lemma complex_cnj_add: "cnj(w + z) = cnj(w) + cnj(z)"
paulson@14373
   407
by (induct w, induct z, simp add: complex_cnj complex_add)
paulson@14323
   408
paulson@14323
   409
lemma complex_cnj_diff: "cnj(w - z) = cnj(w) - cnj(z)"
paulson@15013
   410
by (simp add: diff_minus complex_cnj_add complex_cnj_minus)
paulson@14323
   411
paulson@14323
   412
lemma complex_cnj_mult: "cnj(w * z) = cnj(w) * cnj(z)"
paulson@14373
   413
by (induct w, induct z, simp add: complex_cnj complex_mult)
paulson@14323
   414
paulson@14323
   415
lemma complex_cnj_divide: "cnj(w / z) = (cnj w)/(cnj z)"
paulson@14373
   416
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
paulson@14323
   417
paulson@14374
   418
lemma complex_cnj_one [simp]: "cnj 1 = 1"
paulson@14373
   419
by (simp add: cnj_def complex_one_def)
paulson@14323
   420
paulson@14323
   421
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re(z))"
paulson@14373
   422
by (induct z, simp add: complex_add complex_cnj complex_of_real_def)
paulson@14323
   423
paulson@14323
   424
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im(z)) * ii"
paulson@14373
   425
apply (induct z)
paulson@15013
   426
apply (simp add: complex_add complex_cnj complex_of_real_def diff_minus
paulson@14354
   427
                 complex_minus i_def complex_mult)
paulson@14323
   428
done
paulson@14323
   429
paulson@14354
   430
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
paulson@14334
   431
by (simp add: cnj_def complex_zero_def)
paulson@14323
   432
paulson@14374
   433
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
paulson@14373
   434
by (induct z, simp add: complex_zero_def complex_cnj)
paulson@14323
   435
paulson@14323
   436
lemma complex_mult_cnj: "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)"
paulson@14374
   437
by (induct z,
paulson@14374
   438
    simp add: complex_cnj complex_mult complex_of_real_def power2_eq_square)
paulson@14323
   439
paulson@14323
   440
paulson@14323
   441
subsection{*Modulus*}
paulson@14323
   442
paulson@14374
   443
lemma complex_mod_eq_zero_cancel [simp]: "(cmod x = 0) = (x = 0)"
paulson@14373
   444
apply (induct x)
paulson@15085
   445
apply (auto iff: real_0_le_add_iff 
paulson@15085
   446
            intro: real_sum_squares_cancel real_sum_squares_cancel2
paulson@14373
   447
            simp add: complex_mod complex_zero_def power2_eq_square)
paulson@14323
   448
done
paulson@14323
   449
paulson@14374
   450
lemma complex_mod_complex_of_real_of_nat [simp]:
paulson@14373
   451
     "cmod (complex_of_real(real (n::nat))) = real n"
paulson@14373
   452
by simp
paulson@14323
   453
paulson@14374
   454
lemma complex_mod_minus [simp]: "cmod (-x) = cmod(x)"
paulson@14373
   455
by (induct x, simp add: complex_mod complex_minus power2_eq_square)
paulson@14323
   456
paulson@14323
   457
lemma complex_mod_mult_cnj: "cmod(z * cnj(z)) = cmod(z) ^ 2"
paulson@14373
   458
apply (induct z, simp add: complex_mod complex_cnj complex_mult)
paulson@15085
   459
apply (simp add: power2_eq_square abs_if linorder_not_less real_0_le_add_iff)
paulson@14323
   460
done
paulson@14323
   461
paulson@14373
   462
lemma complex_mod_squared: "cmod(Complex x y) ^ 2 = x ^ 2 + y ^ 2"
paulson@14373
   463
by (simp add: cmod_def)
paulson@14323
   464
paulson@14374
   465
lemma complex_mod_ge_zero [simp]: "0 \<le> cmod x"
paulson@14373
   466
by (simp add: cmod_def)
paulson@14323
   467
paulson@14374
   468
lemma abs_cmod_cancel [simp]: "abs(cmod x) = cmod x"
paulson@14374
   469
by (simp add: abs_if linorder_not_less)
paulson@14323
   470
paulson@14323
   471
lemma complex_mod_mult: "cmod(x*y) = cmod(x) * cmod(y)"
paulson@14373
   472
apply (induct x, induct y)
paulson@14377
   473
apply (auto simp add: complex_mult complex_mod real_sqrt_mult_distrib2[symmetric])
paulson@14348
   474
apply (rule_tac n = 1 in power_inject_base)
paulson@14353
   475
apply (auto simp add: power2_eq_square [symmetric] simp del: realpow_Suc)
paulson@14374
   476
apply (auto simp add: real_diff_def power2_eq_square right_distrib left_distrib 
paulson@14374
   477
                      add_ac mult_ac)
paulson@14323
   478
done
paulson@14323
   479
paulson@14377
   480
lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1"
paulson@14377
   481
by (simp add: cmod_def) 
paulson@14377
   482
paulson@14377
   483
lemma cmod_complex_polar [simp]:
paulson@14377
   484
     "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
paulson@14377
   485
by (simp only: cmod_unit_one complex_mod_mult, simp) 
paulson@14377
   486
paulson@14374
   487
lemma complex_mod_add_squared_eq:
paulson@14374
   488
     "cmod(x + y) ^ 2 = cmod(x) ^ 2 + cmod(y) ^ 2 + 2 * Re(x * cnj y)"
paulson@14373
   489
apply (induct x, induct y)
paulson@14323
   490
apply (auto simp add: complex_add complex_mod_squared complex_mult complex_cnj real_diff_def simp del: realpow_Suc)
paulson@14353
   491
apply (auto simp add: right_distrib left_distrib power2_eq_square mult_ac add_ac)
paulson@14323
   492
done
paulson@14323
   493
paulson@14374
   494
lemma complex_Re_mult_cnj_le_cmod [simp]: "Re(x * cnj y) \<le> cmod(x * cnj y)"
paulson@14373
   495
apply (induct x, induct y)
paulson@14323
   496
apply (auto simp add: complex_mod complex_mult complex_cnj real_diff_def simp del: realpow_Suc)
paulson@14323
   497
done
paulson@14323
   498
paulson@14374
   499
lemma complex_Re_mult_cnj_le_cmod2 [simp]: "Re(x * cnj y) \<le> cmod(x * y)"
paulson@14373
   500
by (insert complex_Re_mult_cnj_le_cmod [of x y], simp add: complex_mod_mult)
paulson@14323
   501
paulson@14374
   502
lemma real_sum_squared_expand:
paulson@14374
   503
     "((x::real) + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y"
paulson@14373
   504
by (simp add: left_distrib right_distrib power2_eq_square)
paulson@14323
   505
paulson@14374
   506
lemma complex_mod_triangle_squared [simp]:
paulson@14374
   507
     "cmod (x + y) ^ 2 \<le> (cmod(x) + cmod(y)) ^ 2"
paulson@14373
   508
by (simp add: real_sum_squared_expand complex_mod_add_squared_eq real_mult_assoc complex_mod_mult [symmetric])
paulson@14323
   509
paulson@14374
   510
lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x"
paulson@14373
   511
by (rule order_trans [OF _ complex_mod_ge_zero], simp)
paulson@14323
   512
paulson@14374
   513
lemma complex_mod_triangle_ineq [simp]: "cmod (x + y) \<le> cmod(x) + cmod(y)"
paulson@14334
   514
apply (rule_tac n = 1 in realpow_increasing)
paulson@14323
   515
apply (auto intro:  order_trans [OF _ complex_mod_ge_zero]
paulson@15085
   516
            simp add: add_increasing power2_eq_square [symmetric])
paulson@14323
   517
done
paulson@14323
   518
paulson@14374
   519
lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a"
paulson@14373
   520
by (insert complex_mod_triangle_ineq [THEN add_right_mono, of b a"-cmod b"], simp)
paulson@14323
   521
paulson@14323
   522
lemma complex_mod_diff_commute: "cmod (x - y) = cmod (y - x)"
paulson@14373
   523
apply (induct x, induct y)
paulson@14353
   524
apply (auto simp add: complex_diff complex_mod right_diff_distrib power2_eq_square left_diff_distrib add_ac mult_ac)
paulson@14323
   525
done
paulson@14323
   526
paulson@14374
   527
lemma complex_mod_add_less:
paulson@14374
   528
     "[| cmod x < r; cmod y < s |] ==> cmod (x + y) < r + s"
paulson@14334
   529
by (auto intro: order_le_less_trans complex_mod_triangle_ineq)
paulson@14323
   530
paulson@14374
   531
lemma complex_mod_mult_less:
paulson@14374
   532
     "[| cmod x < r; cmod y < s |] ==> cmod (x * y) < r * s"
paulson@14334
   533
by (auto intro: real_mult_less_mono' simp add: complex_mod_mult)
paulson@14323
   534
paulson@14374
   535
lemma complex_mod_diff_ineq [simp]: "cmod(a) - cmod(b) \<le> cmod(a + b)"
paulson@14323
   536
apply (rule linorder_cases [of "cmod(a)" "cmod (b)"])
paulson@14323
   537
apply auto
paulson@14334
   538
apply (rule order_trans [of _ 0], rule order_less_imp_le)
paulson@14374
   539
apply (simp add: compare_rls, simp)
paulson@14323
   540
apply (simp add: compare_rls)
paulson@14323
   541
apply (rule complex_mod_minus [THEN subst])
paulson@14323
   542
apply (rule order_trans)
paulson@14323
   543
apply (rule_tac [2] complex_mod_triangle_ineq)
paulson@14373
   544
apply (auto simp add: add_ac)
paulson@14323
   545
done
paulson@14323
   546
paulson@14374
   547
lemma complex_Re_le_cmod [simp]: "Re z \<le> cmod z"
paulson@14373
   548
by (induct z, simp add: complex_mod del: realpow_Suc)
paulson@14323
   549
paulson@14354
   550
lemma complex_mod_gt_zero: "z \<noteq> 0 ==> 0 < cmod z"
paulson@14373
   551
apply (insert complex_mod_ge_zero [of z])
paulson@14334
   552
apply (drule order_le_imp_less_or_eq, auto)
paulson@14323
   553
done
paulson@14323
   554
paulson@14323
   555
paulson@14323
   556
subsection{*A Few More Theorems*}
paulson@14323
   557
paulson@14323
   558
lemma complex_mod_inverse: "cmod(inverse x) = inverse(cmod x)"
paulson@14373
   559
apply (case_tac "x=0", simp)
paulson@14323
   560
apply (rule_tac c1 = "cmod x" in real_mult_left_cancel [THEN iffD1])
paulson@14323
   561
apply (auto simp add: complex_mod_mult [symmetric])
paulson@14323
   562
done
paulson@14323
   563
paulson@14373
   564
lemma complex_mod_divide: "cmod(x/y) = cmod(x)/(cmod y)"
paulson@15013
   565
by (simp add: complex_divide_def divide_inverse complex_mod_mult complex_mod_inverse)
paulson@14323
   566
paulson@14354
   567
paulson@14354
   568
subsection{*Exponentiation*}
paulson@14354
   569
paulson@14354
   570
primrec
paulson@14354
   571
     complexpow_0:   "z ^ 0       = 1"
paulson@14354
   572
     complexpow_Suc: "z ^ (Suc n) = (z::complex) * (z ^ n)"
paulson@14354
   573
paulson@14354
   574
paulson@15003
   575
instance complex :: recpower
paulson@14354
   576
proof
paulson@14354
   577
  fix z :: complex
paulson@14354
   578
  fix n :: nat
paulson@14354
   579
  show "z^0 = 1" by simp
paulson@14354
   580
  show "z^(Suc n) = z * (z^n)" by simp
paulson@14354
   581
qed
paulson@14323
   582
paulson@14323
   583
paulson@14354
   584
lemma complex_of_real_pow: "complex_of_real (x ^ n) = (complex_of_real x) ^ n"
paulson@14323
   585
apply (induct_tac "n")
paulson@14354
   586
apply (auto simp add: complex_of_real_mult [symmetric])
paulson@14323
   587
done
paulson@14323
   588
paulson@14354
   589
lemma complex_cnj_pow: "cnj(z ^ n) = cnj(z) ^ n"
paulson@14323
   590
apply (induct_tac "n")
paulson@14354
   591
apply (auto simp add: complex_cnj_mult)
paulson@14323
   592
done
paulson@14323
   593
paulson@14354
   594
lemma complex_mod_complexpow: "cmod(x ^ n) = cmod(x) ^ n"
paulson@14354
   595
apply (induct_tac "n")
paulson@14354
   596
apply (auto simp add: complex_mod_mult)
paulson@14354
   597
done
paulson@14354
   598
paulson@14354
   599
lemma complexpow_i_squared [simp]: "ii ^ 2 = -(1::complex)"
paulson@14354
   600
by (simp add: i_def complex_mult complex_one_def complex_minus numeral_2_eq_2)
paulson@14354
   601
paulson@14354
   602
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
paulson@14373
   603
by (simp add: i_def complex_zero_def)
paulson@14354
   604
paulson@14354
   605
paulson@14354
   606
subsection{*The Function @{term sgn}*}
paulson@14323
   607
paulson@14374
   608
lemma sgn_zero [simp]: "sgn 0 = 0"
paulson@14373
   609
by (simp add: sgn_def)
paulson@14323
   610
paulson@14374
   611
lemma sgn_one [simp]: "sgn 1 = 1"
paulson@14373
   612
by (simp add: sgn_def)
paulson@14323
   613
paulson@14323
   614
lemma sgn_minus: "sgn (-z) = - sgn(z)"
paulson@14373
   615
by (simp add: sgn_def)
paulson@14323
   616
paulson@14374
   617
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
paulson@14377
   618
by (simp add: sgn_def)
paulson@14323
   619
paulson@14323
   620
lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
paulson@14373
   621
by (simp add: i_def complex_of_real_def complex_mult complex_add)
paulson@14323
   622
paulson@14374
   623
lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)"
paulson@14373
   624
by (simp add: i_def complex_one_def complex_mult complex_minus)
paulson@14323
   625
paulson@14374
   626
lemma complex_eq_cancel_iff2 [simp]:
paulson@14377
   627
     "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
paulson@14377
   628
by (simp add: complex_of_real_def) 
paulson@14323
   629
paulson@14374
   630
lemma complex_eq_cancel_iff2a [simp]:
paulson@14377
   631
     "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
paulson@14377
   632
by (simp add: complex_of_real_def)
paulson@14323
   633
paulson@14377
   634
lemma Complex_eq_0 [simp]: "(Complex x y = 0) = (x = 0 & y = 0)"
paulson@14377
   635
by (simp add: complex_zero_def)
paulson@14323
   636
paulson@14377
   637
lemma Complex_eq_1 [simp]: "(Complex x y = 1) = (x = 1 & y = 0)"
paulson@14377
   638
by (simp add: complex_one_def)
paulson@14323
   639
paulson@14377
   640
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 & y = 1)"
paulson@14377
   641
by (simp add: i_def)
paulson@14323
   642
paulson@15013
   643
paulson@15013
   644
paulson@14374
   645
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
paulson@15013
   646
proof (induct z)
paulson@15013
   647
  case (Complex x y)
paulson@15013
   648
    have "sqrt (x\<twosuperior> + y\<twosuperior>) * inverse (x\<twosuperior> + y\<twosuperior>) = inverse (sqrt (x\<twosuperior> + y\<twosuperior>))"
paulson@15013
   649
      by (simp add: divide_inverse [symmetric] sqrt_divide_self_eq)
paulson@15013
   650
    thus "Re (sgn (Complex x y)) = Re (Complex x y) /cmod (Complex x y)"
paulson@15013
   651
       by (simp add: sgn_def complex_of_real_def divide_inverse)
paulson@15013
   652
qed
paulson@15013
   653
paulson@14323
   654
paulson@14374
   655
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
paulson@15013
   656
proof (induct z)
paulson@15013
   657
  case (Complex x y)
paulson@15013
   658
    have "sqrt (x\<twosuperior> + y\<twosuperior>) * inverse (x\<twosuperior> + y\<twosuperior>) = inverse (sqrt (x\<twosuperior> + y\<twosuperior>))"
paulson@15013
   659
      by (simp add: divide_inverse [symmetric] sqrt_divide_self_eq)
paulson@15013
   660
    thus "Im (sgn (Complex x y)) = Im (Complex x y) /cmod (Complex x y)"
paulson@15013
   661
       by (simp add: sgn_def complex_of_real_def divide_inverse)
paulson@15013
   662
qed
paulson@14323
   663
paulson@14323
   664
lemma complex_inverse_complex_split:
paulson@14323
   665
     "inverse(complex_of_real x + ii * complex_of_real y) =
paulson@14323
   666
      complex_of_real(x/(x ^ 2 + y ^ 2)) -
paulson@14323
   667
      ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
paulson@14374
   668
by (simp add: complex_of_real_def i_def complex_mult complex_add
paulson@15013
   669
         diff_minus complex_minus complex_inverse divide_inverse)
paulson@14323
   670
paulson@14323
   671
(*----------------------------------------------------------------------------*)
paulson@14323
   672
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
paulson@14323
   673
(* many of the theorems are not used - so should they be kept?                *)
paulson@14323
   674
(*----------------------------------------------------------------------------*)
paulson@14323
   675
paulson@14354
   676
lemma complex_of_real_zero_iff [simp]: "(complex_of_real y = 0) = (y = 0)"
paulson@14354
   677
by (auto simp add: complex_zero_def complex_of_real_def)
paulson@14354
   678
paulson@14354
   679
lemma cos_arg_i_mult_zero_pos:
paulson@14377
   680
   "0 < y ==> cos (arg(Complex 0 y)) = 0"
paulson@14373
   681
apply (simp add: arg_def abs_if)
paulson@14334
   682
apply (rule_tac a = "pi/2" in someI2, auto)
paulson@14334
   683
apply (rule order_less_trans [of _ 0], auto)
paulson@14323
   684
done
paulson@14323
   685
paulson@14354
   686
lemma cos_arg_i_mult_zero_neg:
paulson@14377
   687
   "y < 0 ==> cos (arg(Complex 0 y)) = 0"
paulson@14373
   688
apply (simp add: arg_def abs_if)
paulson@14334
   689
apply (rule_tac a = "- pi/2" in someI2, auto)
paulson@14334
   690
apply (rule order_trans [of _ 0], auto)
paulson@14323
   691
done
paulson@14323
   692
paulson@14374
   693
lemma cos_arg_i_mult_zero [simp]:
paulson@14377
   694
     "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
paulson@14377
   695
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
paulson@14323
   696
paulson@14323
   697
paulson@14323
   698
subsection{*Finally! Polar Form for Complex Numbers*}
paulson@14323
   699
paulson@14374
   700
lemma complex_split_polar:
paulson@14377
   701
     "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
paulson@14377
   702
apply (induct z) 
paulson@14377
   703
apply (auto simp add: polar_Ex complex_of_real_mult_Complex)
paulson@14323
   704
done
paulson@14323
   705
paulson@14354
   706
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
paulson@14377
   707
apply (induct z) 
paulson@14377
   708
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex)
paulson@14323
   709
done
paulson@14323
   710
paulson@14374
   711
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
paulson@14373
   712
by (simp add: rcis_def cis_def)
paulson@14323
   713
paulson@14348
   714
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
paulson@14373
   715
by (simp add: rcis_def cis_def)
paulson@14323
   716
paulson@14377
   717
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>"
paulson@14377
   718
proof -
paulson@14377
   719
  have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)"
paulson@14377
   720
    by (simp only: power_mult_distrib right_distrib) 
paulson@14377
   721
  thus ?thesis by simp
paulson@14377
   722
qed
paulson@14323
   723
paulson@14374
   724
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
paulson@14377
   725
by (simp add: rcis_def cis_def sin_cos_squared_add2_mult)
paulson@14323
   726
paulson@14323
   727
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
paulson@14373
   728
apply (simp add: cmod_def)
paulson@14323
   729
apply (rule real_sqrt_eq_iff [THEN iffD2])
paulson@14323
   730
apply (auto simp add: complex_mult_cnj)
paulson@14323
   731
done
paulson@14323
   732
paulson@14374
   733
lemma complex_Re_cnj [simp]: "Re(cnj z) = Re z"
paulson@14373
   734
by (induct z, simp add: complex_cnj)
paulson@14323
   735
paulson@14374
   736
lemma complex_Im_cnj [simp]: "Im(cnj z) = - Im z"
paulson@14374
   737
by (induct z, simp add: complex_cnj)
paulson@14374
   738
paulson@14374
   739
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
paulson@14373
   740
by (induct z, simp add: complex_cnj complex_mult)
paulson@14323
   741
paulson@14323
   742
paulson@14323
   743
(*---------------------------------------------------------------------------*)
paulson@14323
   744
(*  (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b)                      *)
paulson@14323
   745
(*---------------------------------------------------------------------------*)
paulson@14323
   746
paulson@14323
   747
lemma cis_rcis_eq: "cis a = rcis 1 a"
paulson@14373
   748
by (simp add: rcis_def)
paulson@14323
   749
paulson@14374
   750
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
paulson@15013
   751
by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib
paulson@15013
   752
              complex_of_real_def)
paulson@14323
   753
paulson@14323
   754
lemma cis_mult: "cis a * cis b = cis (a + b)"
paulson@14373
   755
by (simp add: cis_rcis_eq rcis_mult)
paulson@14323
   756
paulson@14374
   757
lemma cis_zero [simp]: "cis 0 = 1"
paulson@14377
   758
by (simp add: cis_def complex_one_def)
paulson@14323
   759
paulson@14374
   760
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
paulson@14373
   761
by (simp add: rcis_def)
paulson@14323
   762
paulson@14374
   763
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
paulson@14373
   764
by (simp add: rcis_def)
paulson@14323
   765
paulson@14323
   766
lemma complex_of_real_minus_one:
paulson@14323
   767
   "complex_of_real (-(1::real)) = -(1::complex)"
paulson@14377
   768
by (simp add: complex_of_real_def complex_one_def complex_minus)
paulson@14323
   769
paulson@14374
   770
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
paulson@14373
   771
by (simp add: complex_mult_assoc [symmetric])
paulson@14323
   772
paulson@14323
   773
paulson@14323
   774
lemma cis_real_of_nat_Suc_mult:
paulson@14323
   775
   "cis (real (Suc n) * a) = cis a * cis (real n * a)"
paulson@14377
   776
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib)
paulson@14323
   777
paulson@14323
   778
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
paulson@14323
   779
apply (induct_tac "n")
paulson@14323
   780
apply (auto simp add: cis_real_of_nat_Suc_mult)
paulson@14323
   781
done
paulson@14323
   782
paulson@14374
   783
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
paulson@14374
   784
by (simp add: rcis_def power_mult_distrib DeMoivre complex_of_real_pow)
paulson@14323
   785
paulson@14374
   786
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
paulson@14374
   787
by (simp add: cis_def complex_inverse_complex_split complex_of_real_minus 
paulson@15013
   788
              diff_minus)
paulson@14323
   789
paulson@14323
   790
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
paulson@14430
   791
by (simp add: divide_inverse rcis_def complex_of_real_inverse)
paulson@14323
   792
paulson@14323
   793
lemma cis_divide: "cis a / cis b = cis (a - b)"
paulson@14373
   794
by (simp add: complex_divide_def cis_mult real_diff_def)
paulson@14323
   795
paulson@14354
   796
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
paulson@14373
   797
apply (simp add: complex_divide_def)
paulson@14373
   798
apply (case_tac "r2=0", simp)
paulson@14373
   799
apply (simp add: rcis_inverse rcis_mult real_diff_def)
paulson@14323
   800
done
paulson@14323
   801
paulson@14374
   802
lemma Re_cis [simp]: "Re(cis a) = cos a"
paulson@14373
   803
by (simp add: cis_def)
paulson@14323
   804
paulson@14374
   805
lemma Im_cis [simp]: "Im(cis a) = sin a"
paulson@14373
   806
by (simp add: cis_def)
paulson@14323
   807
paulson@14323
   808
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
paulson@14334
   809
by (auto simp add: DeMoivre)
paulson@14323
   810
paulson@14323
   811
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
paulson@14334
   812
by (auto simp add: DeMoivre)
paulson@14323
   813
paulson@14323
   814
lemma expi_add: "expi(a + b) = expi(a) * expi(b)"
paulson@14374
   815
by (simp add: expi_def complex_Re_add exp_add complex_Im_add 
paulson@14374
   816
              cis_mult [symmetric] complex_of_real_mult mult_ac)
paulson@14323
   817
paulson@14374
   818
lemma expi_zero [simp]: "expi (0::complex) = 1"
paulson@14373
   819
by (simp add: expi_def)
paulson@14323
   820
paulson@14374
   821
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
paulson@14373
   822
apply (insert rcis_Ex [of z])
paulson@14323
   823
apply (auto simp add: expi_def rcis_def complex_mult_assoc [symmetric] complex_of_real_mult)
paulson@14334
   824
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
paulson@14323
   825
done
paulson@14323
   826
paulson@14323
   827
paulson@14387
   828
subsection{*Numerals and Arithmetic*}
paulson@14387
   829
paulson@14387
   830
instance complex :: number ..
paulson@14387
   831
paulson@15013
   832
defs (overloaded)
paulson@15013
   833
  complex_number_of_def: "(number_of w :: complex) == of_int (Rep_Bin w)"
paulson@15013
   834
    --{*the type constraint is essential!*}
paulson@14387
   835
paulson@14387
   836
instance complex :: number_ring
paulson@15013
   837
by (intro_classes, simp add: complex_number_of_def) 
paulson@15013
   838
paulson@15013
   839
paulson@15013
   840
lemma complex_of_real_of_nat [simp]: "complex_of_real (of_nat n) = of_nat n"
paulson@15013
   841
by (induct n, simp_all) 
paulson@15013
   842
paulson@15013
   843
lemma complex_of_real_of_int [simp]: "complex_of_real (of_int z) = of_int z"
paulson@15013
   844
proof (cases z)
paulson@15013
   845
  case (1 n)
paulson@15013
   846
    thus ?thesis by simp
paulson@15013
   847
next
paulson@15013
   848
  case (2 n)
paulson@15013
   849
    thus ?thesis 
paulson@15013
   850
      by (simp only: of_int_minus complex_of_real_minus, simp)
paulson@14387
   851
qed
paulson@14387
   852
paulson@14387
   853
paulson@14387
   854
text{*Collapse applications of @{term complex_of_real} to @{term number_of}*}
paulson@14387
   855
lemma complex_number_of [simp]: "complex_of_real (number_of w) = number_of w"
paulson@15013
   856
by (simp add: complex_number_of_def real_number_of_def) 
paulson@14387
   857
paulson@14387
   858
text{*This theorem is necessary because theorems such as
paulson@14387
   859
   @{text iszero_number_of_0} only hold for ordered rings. They cannot
paulson@14387
   860
   be generalized to fields in general because they fail for finite fields.
paulson@14387
   861
   They work for type complex because the reals can be embedded in them.*}
paulson@14387
   862
lemma iszero_complex_number_of [simp]:
paulson@14387
   863
     "iszero (number_of w :: complex) = iszero (number_of w :: real)"
paulson@14387
   864
by (simp only: complex_of_real_zero_iff complex_number_of [symmetric] 
paulson@14387
   865
               iszero_def)  
paulson@14387
   866
paulson@14387
   867
lemma complex_number_of_cnj [simp]: "cnj(number_of v :: complex) = number_of v"
paulson@14387
   868
apply (subst complex_number_of [symmetric])
paulson@14387
   869
apply (rule complex_cnj_complex_of_real)
paulson@14387
   870
done
paulson@14387
   871
paulson@14387
   872
lemma complex_number_of_cmod: 
paulson@14387
   873
      "cmod(number_of v :: complex) = abs (number_of v :: real)"
paulson@14387
   874
by (simp only: complex_number_of [symmetric] complex_mod_complex_of_real)
paulson@14387
   875
paulson@14387
   876
lemma complex_number_of_Re [simp]: "Re(number_of v :: complex) = number_of v"
paulson@14387
   877
by (simp only: complex_number_of [symmetric] Re_complex_of_real)
paulson@14387
   878
paulson@14387
   879
lemma complex_number_of_Im [simp]: "Im(number_of v :: complex) = 0"
paulson@14387
   880
by (simp only: complex_number_of [symmetric] Im_complex_of_real)
paulson@14387
   881
paulson@14387
   882
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
paulson@14387
   883
by (simp add: expi_def complex_Re_mult_eq complex_Im_mult_eq cis_def)
paulson@14387
   884
paulson@14387
   885
paulson@14387
   886
(*examples:
paulson@14387
   887
print_depth 22
paulson@14387
   888
set timing;
paulson@14387
   889
set trace_simp;
paulson@14387
   890
fun test s = (Goal s, by (Simp_tac 1)); 
paulson@14387
   891
paulson@14387
   892
test "23 * ii + 45 * ii= (x::complex)";
paulson@14387
   893
paulson@14387
   894
test "5 * ii + 12 - 45 * ii= (x::complex)";
paulson@14387
   895
test "5 * ii + 40 - 12 * ii + 9 = (x::complex) + 89 * ii";
paulson@14387
   896
test "5 * ii + 40 - 12 * ii + 9 - 78 = (x::complex) + 89 * ii";
paulson@14387
   897
paulson@14387
   898
test "l + 10 * ii + 90 + 3*l +  9 + 45 * ii= (x::complex)";
paulson@14387
   899
test "87 + 10 * ii + 90 + 3*7 +  9 + 45 * ii= (x::complex)";
paulson@14387
   900
paulson@14387
   901
paulson@14387
   902
fun test s = (Goal s; by (Asm_simp_tac 1)); 
paulson@14387
   903
paulson@14387
   904
test "x*k = k*(y::complex)";
paulson@14387
   905
test "k = k*(y::complex)"; 
paulson@14387
   906
test "a*(b*c) = (b::complex)";
paulson@14387
   907
test "a*(b*c) = d*(b::complex)*(x*a)";
paulson@14387
   908
paulson@14387
   909
paulson@14387
   910
test "(x*k) / (k*(y::complex)) = (uu::complex)";
paulson@14387
   911
test "(k) / (k*(y::complex)) = (uu::complex)"; 
paulson@14387
   912
test "(a*(b*c)) / ((b::complex)) = (uu::complex)";
paulson@14387
   913
test "(a*(b*c)) / (d*(b::complex)*(x*a)) = (uu::complex)";
paulson@14387
   914
paulson@15003
   915
FIXME: what do we do about this?
paulson@14387
   916
test "a*(b*c)/(y*z) = d*(b::complex)*(x*a)/z";
paulson@14387
   917
*)
paulson@14387
   918
paulson@14323
   919
paulson@14323
   920
ML
paulson@14323
   921
{*
paulson@14323
   922
val complex_zero_def = thm"complex_zero_def";
paulson@14323
   923
val complex_one_def = thm"complex_one_def";
paulson@14323
   924
val complex_minus_def = thm"complex_minus_def";
paulson@14323
   925
val complex_divide_def = thm"complex_divide_def";
paulson@14323
   926
val complex_mult_def = thm"complex_mult_def";
paulson@14323
   927
val complex_add_def = thm"complex_add_def";
paulson@14323
   928
val complex_of_real_def = thm"complex_of_real_def";
paulson@14323
   929
val i_def = thm"i_def";
paulson@14323
   930
val expi_def = thm"expi_def";
paulson@14323
   931
val cis_def = thm"cis_def";
paulson@14323
   932
val rcis_def = thm"rcis_def";
paulson@14323
   933
val cmod_def = thm"cmod_def";
paulson@14323
   934
val cnj_def = thm"cnj_def";
paulson@14323
   935
val sgn_def = thm"sgn_def";
paulson@14323
   936
val arg_def = thm"arg_def";
paulson@14323
   937
val complexpow_0 = thm"complexpow_0";
paulson@14323
   938
val complexpow_Suc = thm"complexpow_Suc";
paulson@14323
   939
paulson@14323
   940
val Re = thm"Re";
paulson@14323
   941
val Im = thm"Im";
paulson@14323
   942
val complex_Re_Im_cancel_iff = thm"complex_Re_Im_cancel_iff";
paulson@14323
   943
val complex_Re_zero = thm"complex_Re_zero";
paulson@14323
   944
val complex_Im_zero = thm"complex_Im_zero";
paulson@14323
   945
val complex_Re_one = thm"complex_Re_one";
paulson@14323
   946
val complex_Im_one = thm"complex_Im_one";
paulson@14323
   947
val complex_Re_i = thm"complex_Re_i";
paulson@14323
   948
val complex_Im_i = thm"complex_Im_i";
paulson@14323
   949
val Re_complex_of_real = thm"Re_complex_of_real";
paulson@14323
   950
val Im_complex_of_real = thm"Im_complex_of_real";
paulson@14323
   951
val complex_minus = thm"complex_minus";
paulson@14323
   952
val complex_Re_minus = thm"complex_Re_minus";
paulson@14323
   953
val complex_Im_minus = thm"complex_Im_minus";
paulson@14323
   954
val complex_add = thm"complex_add";
paulson@14323
   955
val complex_Re_add = thm"complex_Re_add";
paulson@14323
   956
val complex_Im_add = thm"complex_Im_add";
paulson@14323
   957
val complex_add_commute = thm"complex_add_commute";
paulson@14323
   958
val complex_add_assoc = thm"complex_add_assoc";
paulson@14323
   959
val complex_add_zero_left = thm"complex_add_zero_left";
paulson@14323
   960
val complex_add_zero_right = thm"complex_add_zero_right";
paulson@14323
   961
val complex_diff = thm"complex_diff";
paulson@14323
   962
val complex_mult = thm"complex_mult";
paulson@14323
   963
val complex_mult_one_left = thm"complex_mult_one_left";
paulson@14323
   964
val complex_mult_one_right = thm"complex_mult_one_right";
paulson@14323
   965
val complex_inverse = thm"complex_inverse";
paulson@14323
   966
val complex_of_real_one = thm"complex_of_real_one";
paulson@14323
   967
val complex_of_real_zero = thm"complex_of_real_zero";
paulson@14323
   968
val complex_of_real_eq_iff = thm"complex_of_real_eq_iff";
paulson@14323
   969
val complex_of_real_minus = thm"complex_of_real_minus";
paulson@14323
   970
val complex_of_real_inverse = thm"complex_of_real_inverse";
paulson@14323
   971
val complex_of_real_add = thm"complex_of_real_add";
paulson@14323
   972
val complex_of_real_diff = thm"complex_of_real_diff";
paulson@14323
   973
val complex_of_real_mult = thm"complex_of_real_mult";
paulson@14323
   974
val complex_of_real_divide = thm"complex_of_real_divide";
paulson@14323
   975
val complex_of_real_pow = thm"complex_of_real_pow";
paulson@14323
   976
val complex_mod = thm"complex_mod";
paulson@14323
   977
val complex_mod_zero = thm"complex_mod_zero";
paulson@14323
   978
val complex_mod_one = thm"complex_mod_one";
paulson@14323
   979
val complex_mod_complex_of_real = thm"complex_mod_complex_of_real";
paulson@14323
   980
val complex_of_real_abs = thm"complex_of_real_abs";
paulson@14323
   981
val complex_cnj = thm"complex_cnj";
paulson@14323
   982
val complex_cnj_cancel_iff = thm"complex_cnj_cancel_iff";
paulson@14323
   983
val complex_cnj_cnj = thm"complex_cnj_cnj";
paulson@14323
   984
val complex_cnj_complex_of_real = thm"complex_cnj_complex_of_real";
paulson@14323
   985
val complex_mod_cnj = thm"complex_mod_cnj";
paulson@14323
   986
val complex_cnj_minus = thm"complex_cnj_minus";
paulson@14323
   987
val complex_cnj_inverse = thm"complex_cnj_inverse";
paulson@14323
   988
val complex_cnj_add = thm"complex_cnj_add";
paulson@14323
   989
val complex_cnj_diff = thm"complex_cnj_diff";
paulson@14323
   990
val complex_cnj_mult = thm"complex_cnj_mult";
paulson@14323
   991
val complex_cnj_divide = thm"complex_cnj_divide";
paulson@14323
   992
val complex_cnj_one = thm"complex_cnj_one";
paulson@14323
   993
val complex_cnj_pow = thm"complex_cnj_pow";
paulson@14323
   994
val complex_add_cnj = thm"complex_add_cnj";
paulson@14323
   995
val complex_diff_cnj = thm"complex_diff_cnj";
paulson@14323
   996
val complex_cnj_zero = thm"complex_cnj_zero";
paulson@14323
   997
val complex_cnj_zero_iff = thm"complex_cnj_zero_iff";
paulson@14323
   998
val complex_mult_cnj = thm"complex_mult_cnj";
paulson@14323
   999
val complex_mod_eq_zero_cancel = thm"complex_mod_eq_zero_cancel";
paulson@14323
  1000
val complex_mod_complex_of_real_of_nat = thm"complex_mod_complex_of_real_of_nat";
paulson@14323
  1001
val complex_mod_minus = thm"complex_mod_minus";
paulson@14323
  1002
val complex_mod_mult_cnj = thm"complex_mod_mult_cnj";
paulson@14323
  1003
val complex_mod_squared = thm"complex_mod_squared";
paulson@14323
  1004
val complex_mod_ge_zero = thm"complex_mod_ge_zero";
paulson@14323
  1005
val abs_cmod_cancel = thm"abs_cmod_cancel";
paulson@14323
  1006
val complex_mod_mult = thm"complex_mod_mult";
paulson@14323
  1007
val complex_mod_add_squared_eq = thm"complex_mod_add_squared_eq";
paulson@14323
  1008
val complex_Re_mult_cnj_le_cmod = thm"complex_Re_mult_cnj_le_cmod";
paulson@14323
  1009
val complex_Re_mult_cnj_le_cmod2 = thm"complex_Re_mult_cnj_le_cmod2";
paulson@14323
  1010
val real_sum_squared_expand = thm"real_sum_squared_expand";
paulson@14323
  1011
val complex_mod_triangle_squared = thm"complex_mod_triangle_squared";
paulson@14323
  1012
val complex_mod_minus_le_complex_mod = thm"complex_mod_minus_le_complex_mod";
paulson@14323
  1013
val complex_mod_triangle_ineq = thm"complex_mod_triangle_ineq";
paulson@14323
  1014
val complex_mod_triangle_ineq2 = thm"complex_mod_triangle_ineq2";
paulson@14323
  1015
val complex_mod_diff_commute = thm"complex_mod_diff_commute";
paulson@14323
  1016
val complex_mod_add_less = thm"complex_mod_add_less";
paulson@14323
  1017
val complex_mod_mult_less = thm"complex_mod_mult_less";
paulson@14323
  1018
val complex_mod_diff_ineq = thm"complex_mod_diff_ineq";
paulson@14323
  1019
val complex_Re_le_cmod = thm"complex_Re_le_cmod";
paulson@14323
  1020
val complex_mod_gt_zero = thm"complex_mod_gt_zero";
paulson@14323
  1021
val complex_mod_complexpow = thm"complex_mod_complexpow";
paulson@14323
  1022
val complex_mod_inverse = thm"complex_mod_inverse";
paulson@14323
  1023
val complex_mod_divide = thm"complex_mod_divide";
paulson@14323
  1024
val complexpow_i_squared = thm"complexpow_i_squared";
paulson@14323
  1025
val complex_i_not_zero = thm"complex_i_not_zero";
paulson@14323
  1026
val sgn_zero = thm"sgn_zero";
paulson@14323
  1027
val sgn_one = thm"sgn_one";
paulson@14323
  1028
val sgn_minus = thm"sgn_minus";
paulson@14323
  1029
val sgn_eq = thm"sgn_eq";
paulson@14323
  1030
val i_mult_eq = thm"i_mult_eq";
paulson@14323
  1031
val i_mult_eq2 = thm"i_mult_eq2";
paulson@14323
  1032
val Re_sgn = thm"Re_sgn";
paulson@14323
  1033
val Im_sgn = thm"Im_sgn";
paulson@14323
  1034
val complex_inverse_complex_split = thm"complex_inverse_complex_split";
paulson@14323
  1035
val cos_arg_i_mult_zero = thm"cos_arg_i_mult_zero";
paulson@14323
  1036
val complex_of_real_zero_iff = thm"complex_of_real_zero_iff";
paulson@14323
  1037
val rcis_Ex = thm"rcis_Ex";
paulson@14323
  1038
val Re_rcis = thm"Re_rcis";
paulson@14323
  1039
val Im_rcis = thm"Im_rcis";
paulson@14323
  1040
val complex_mod_rcis = thm"complex_mod_rcis";
paulson@14323
  1041
val complex_mod_sqrt_Re_mult_cnj = thm"complex_mod_sqrt_Re_mult_cnj";
paulson@14323
  1042
val complex_Re_cnj = thm"complex_Re_cnj";
paulson@14323
  1043
val complex_Im_cnj = thm"complex_Im_cnj";
paulson@14323
  1044
val complex_In_mult_cnj_zero = thm"complex_In_mult_cnj_zero";
paulson@14323
  1045
val complex_Re_mult = thm"complex_Re_mult";
paulson@14323
  1046
val complex_Re_mult_complex_of_real = thm"complex_Re_mult_complex_of_real";
paulson@14323
  1047
val complex_Im_mult_complex_of_real = thm"complex_Im_mult_complex_of_real";
paulson@14323
  1048
val complex_Re_mult_complex_of_real2 = thm"complex_Re_mult_complex_of_real2";
paulson@14323
  1049
val complex_Im_mult_complex_of_real2 = thm"complex_Im_mult_complex_of_real2";
paulson@14323
  1050
val cis_rcis_eq = thm"cis_rcis_eq";
paulson@14323
  1051
val rcis_mult = thm"rcis_mult";
paulson@14323
  1052
val cis_mult = thm"cis_mult";
paulson@14323
  1053
val cis_zero = thm"cis_zero";
paulson@14323
  1054
val rcis_zero_mod = thm"rcis_zero_mod";
paulson@14323
  1055
val rcis_zero_arg = thm"rcis_zero_arg";
paulson@14323
  1056
val complex_of_real_minus_one = thm"complex_of_real_minus_one";
paulson@14323
  1057
val complex_i_mult_minus = thm"complex_i_mult_minus";
paulson@14323
  1058
val cis_real_of_nat_Suc_mult = thm"cis_real_of_nat_Suc_mult";
paulson@14323
  1059
val DeMoivre = thm"DeMoivre";
paulson@14323
  1060
val DeMoivre2 = thm"DeMoivre2";
paulson@14323
  1061
val cis_inverse = thm"cis_inverse";
paulson@14323
  1062
val rcis_inverse = thm"rcis_inverse";
paulson@14323
  1063
val cis_divide = thm"cis_divide";
paulson@14323
  1064
val rcis_divide = thm"rcis_divide";
paulson@14323
  1065
val Re_cis = thm"Re_cis";
paulson@14323
  1066
val Im_cis = thm"Im_cis";
paulson@14323
  1067
val cos_n_Re_cis_pow_n = thm"cos_n_Re_cis_pow_n";
paulson@14323
  1068
val sin_n_Im_cis_pow_n = thm"sin_n_Im_cis_pow_n";
paulson@14323
  1069
val expi_add = thm"expi_add";
paulson@14323
  1070
val expi_zero = thm"expi_zero";
paulson@14323
  1071
val complex_Re_mult_eq = thm"complex_Re_mult_eq";
paulson@14323
  1072
val complex_Im_mult_eq = thm"complex_Im_mult_eq";
paulson@14323
  1073
val complex_expi_Ex = thm"complex_expi_Ex";
paulson@14323
  1074
*}
paulson@14323
  1075
paulson@13957
  1076
end
paulson@13957
  1077
paulson@13957
  1078