src/HOL/Complex/Complex.thy

 by (simp add: complex_inverse_def)
 
 lemma complex_mult_inv_left: "z \<noteq> (0::complex) ==> inverse(z) * z = 1"
 apply (induct z)
 apply (rename_tac x y)
-apply (auto simp add: times_divide_eq complex_mult complex_inverse 
+apply (auto simp add:
              complex_one_def complex_zero_def add_divide_distrib [symmetric] 
              power2_eq_square mult_ac)
 apply (simp_all add: real_sum_squares_not_zero real_sum_squares_not_zero2) 
 done
 

 instance complex :: scaleR ..
 
 defs (overloaded)
   complex_scaleR_def: "r *# x == Complex r 0 * x"
 
-instance complex :: real_algebra_1
+instance complex :: real_field
 proof
   fix a b :: real
   fix x y :: complex
   show "a *# (x + y) = a *# x + a *# y"
     by (simp add: complex_scaleR_def right_distrib)

 by (simp add: i_def complex_of_real_def)
 
 lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
 by (simp add: i_def complex_of_real_def)
 
-(* TODO: generalize and move to Real/RealVector.thy *)
-lemma complex_of_real_inverse [simp]:
+lemma complex_of_real_inverse:
      "complex_of_real(inverse x) = inverse(complex_of_real x)"
-apply (case_tac "x=0", simp)
-apply (simp add: complex_of_real_def divide_inverse power2_eq_square)
-done
-
-(* TODO: generalize and move to Real/RealVector.thy *)
-lemma complex_of_real_divide [simp]:
+by (rule of_real_inverse)
+
+lemma complex_of_real_divide:
       "complex_of_real(x/y) = complex_of_real x / complex_of_real y"
-apply (simp add: complex_divide_def)
-apply (case_tac "y=0", simp)
-apply (simp add: divide_inverse)
-done
+by (rule of_real_divide)
 
 
 subsection{*The Functions @{term Re} and @{term Im}*}
 
 lemma complex_Re_mult_eq: "Re (w * z) = Re w * Re z - Im w * Im z"
-by (induct z, induct w, simp add: complex_mult)
+by (induct z, induct w, simp)
 
 lemma complex_Im_mult_eq: "Im (w * z) = Re w * Im z + Im w * Re z"
-by (induct z, induct w, simp add: complex_mult)
+by (induct z, induct w, simp)
 
 lemma Re_i_times [simp]: "Re(ii * z) = - Im z"
-by (simp add: complex_Re_mult_eq) 
+by (simp add: complex_Re_mult_eq)
 
 lemma Re_times_i [simp]: "Re(z * ii) = - Im z"
-by (simp add: complex_Re_mult_eq) 
+by (simp add: complex_Re_mult_eq)
 
 lemma Im_i_times [simp]: "Im(ii * z) = Re z"
-by (simp add: complex_Im_mult_eq) 
+by (simp add: complex_Im_mult_eq)
 
 lemma Im_times_i [simp]: "Im(z * ii) = Re z"
-by (simp add: complex_Im_mult_eq) 
+by (simp add: complex_Im_mult_eq)
 
 lemma complex_Re_mult: "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)"
 by (simp add: complex_Re_mult_eq)
 
 lemma complex_Re_mult_complex_of_real [simp]:

 lemma complex_cnj_complex_of_real [simp]:
      "cnj (complex_of_real x) = complex_of_real x"
 by (simp add: complex_of_real_def complex_cnj)
 
 lemma complex_cnj_minus: "cnj (-z) = - cnj z"
-by (simp add: cnj_def complex_minus complex_Re_minus complex_Im_minus)
+by (simp add: cnj_def)
 
 lemma complex_cnj_inverse: "cnj(inverse z) = inverse(cnj z)"
-by (induct z, simp add: complex_cnj complex_inverse power2_eq_square)
+by (induct z, simp add: complex_cnj power2_eq_square)
 
 lemma complex_cnj_add: "cnj(w + z) = cnj(w) + cnj(z)"
-by (induct w, induct z, simp add: complex_cnj complex_add)
+by (induct w, induct z, simp add: complex_cnj)
 
 lemma complex_cnj_diff: "cnj(w - z) = cnj(w) - cnj(z)"
 by (simp add: diff_minus complex_cnj_add complex_cnj_minus)
 
 lemma complex_cnj_mult: "cnj(w * z) = cnj(w) * cnj(z)"
-by (induct w, induct z, simp add: complex_cnj complex_mult)
+by (induct w, induct z, simp add: complex_cnj)
 
 lemma complex_cnj_divide: "cnj(w / z) = (cnj w)/(cnj z)"
 by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
 
 lemma complex_cnj_one [simp]: "cnj 1 = 1"
 by (simp add: cnj_def complex_one_def)
 
 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re(z))"
-by (induct z, simp add: complex_add complex_cnj complex_of_real_def)
+by (induct z, simp add: complex_cnj complex_of_real_def)
 
 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im(z)) * ii"
 apply (induct z)
 apply (simp add: complex_add complex_cnj complex_of_real_def diff_minus
                  complex_minus i_def complex_mult)

 
 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
 by (induct z, simp add: complex_zero_def complex_cnj)
 
 lemma complex_mult_cnj: "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)"
-by (induct z,
-    simp add: complex_cnj complex_mult complex_of_real_def power2_eq_square)
+by (induct z, simp add: complex_cnj complex_of_real_def power2_eq_square)
 
 
 subsection{*Modulus*}
 
 instance complex :: norm ..

 
 lemma complex_mod_one [simp]: "cmod(1) = 1"
 by (simp add: cmod_def power2_eq_square)
 
 lemma complex_mod_complex_of_real [simp]: "cmod(complex_of_real x) = abs x"
-by (simp add: complex_of_real_def power2_eq_square complex_mod)
+by (simp add: complex_of_real_def power2_eq_square)
 
 lemma complex_of_real_abs:
      "complex_of_real (abs x) = complex_of_real(cmod(complex_of_real x))"
 by simp
 

 lemma complex_mod_complex_of_real_of_nat [simp]:
      "cmod (complex_of_real(real (n::nat))) = real n"
 by simp
 
 lemma complex_mod_minus [simp]: "cmod (-x) = cmod(x)"
-by (induct x, simp add: complex_mod complex_minus power2_eq_square)
+by (induct x, simp add: power2_eq_square)
 
 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
-by (induct z, simp add: complex_cnj complex_mod power2_eq_square)
+by (induct z, simp add: complex_cnj power2_eq_square)
 
 lemma complex_mod_mult_cnj: "cmod(z * cnj(z)) = cmod(z) ^ 2"
 apply (induct z, simp add: complex_mod complex_cnj complex_mult)
 apply (simp add: power2_eq_square abs_if linorder_not_less real_0_le_add_iff)
 done

 by (simp only: cmod_unit_one complex_mod_mult, simp) 
 
 lemma complex_mod_add_squared_eq:
      "cmod(x + y) ^ 2 = cmod(x) ^ 2 + cmod(y) ^ 2 + 2 * Re(x * cnj y)"
 apply (induct x, induct y)
-apply (auto simp add: complex_add complex_mod_squared complex_mult complex_cnj real_diff_def simp del: realpow_Suc)
+apply (auto simp add: complex_mod_squared complex_cnj real_diff_def simp del: realpow_Suc)
 apply (auto simp add: right_distrib left_distrib power2_eq_square mult_ac add_ac)
 done
 
 lemma complex_Re_mult_cnj_le_cmod [simp]: "Re(x * cnj y) \<le> cmod(x * cnj y)"
 apply (induct x, induct y)
-apply (auto simp add: complex_mod complex_mult complex_cnj real_diff_def simp del: realpow_Suc)
+apply (auto simp add: complex_mod complex_cnj diff_def simp del: realpow_Suc)
 done
 
 lemma complex_Re_mult_cnj_le_cmod2 [simp]: "Re(x * cnj y) \<le> cmod(x * y)"
 by (insert complex_Re_mult_cnj_le_cmod [of x y], simp add: complex_mod_mult)
 

 apply (rule_tac n = 1 in realpow_increasing)
 apply (auto intro:  order_trans [OF _ complex_mod_ge_zero]
             simp add: add_increasing power2_eq_square [symmetric])
 done
 
-instance complex :: real_normed_div_algebra
+instance complex :: real_normed_field
 proof
   fix r :: real
   fix x y :: complex
   show "0 \<le> cmod x"
     by (rule complex_mod_ge_zero)

 lemma complex_mod_mult_less:
      "[| cmod x < r; cmod y < s |] ==> cmod (x * y) < r * s"
 by (auto intro: real_mult_less_mono' simp add: complex_mod_mult)
 
 lemma complex_mod_diff_ineq [simp]: "cmod(a) - cmod(b) \<le> cmod(a + b)"
-apply (rule linorder_cases [of "cmod(a)" "cmod (b)"])
-apply auto
-apply (rule order_trans [of _ 0], rule order_less_imp_le)
-apply (simp add: compare_rls, simp)
-apply (simp add: compare_rls)
-apply (rule complex_mod_minus [THEN subst])
-apply (rule order_trans)
-apply (rule_tac [2] complex_mod_triangle_ineq)
-apply (auto simp add: add_ac)
-done
+proof -
+  have "cmod a - cmod b = cmod a - cmod (- b)" by simp
+  also have "\<dots> \<le> cmod (a - - b)" by (rule norm_triangle_ineq2)
+  also have "\<dots> = cmod (a + b)" by simp
+  finally show ?thesis .
+qed
 
 lemma complex_Re_le_cmod [simp]: "Re z \<le> cmod z"
-by (induct z, simp add: complex_mod del: realpow_Suc)
+by (induct z, simp)
 
 lemma complex_mod_gt_zero: "z \<noteq> 0 ==> 0 < cmod z"
 by (rule zero_less_norm_iff [THEN iffD2])
 
 lemma complex_mod_inverse: "cmod(inverse x) = inverse(cmod x)"
 by (rule norm_inverse)
 
 lemma complex_mod_divide: "cmod(x/y) = cmod(x)/(cmod y)"
-by (simp add: divide_inverse norm_mult norm_inverse)
+by (rule norm_divide)
 
 
 subsection{*Exponentiation*}
 
 primrec

   show "z^(Suc n) = z * (z^n)" by simp
 qed
 
 
 lemma complex_of_real_pow: "complex_of_real (x ^ n) = (complex_of_real x) ^ n"
-apply (induct_tac "n")
-apply (auto simp add: of_real_mult [symmetric])
-done
+by (rule of_real_power)
 
 lemma complex_cnj_pow: "cnj(z ^ n) = cnj(z) ^ n"
 apply (induct_tac "n")
 apply (auto simp add: complex_cnj_mult)
 done
 
 lemma complex_mod_complexpow: "cmod(x ^ n) = cmod(x) ^ n"
-apply (induct_tac "n")
-apply (auto simp add: complex_mod_mult)
-done
+by (rule norm_power)
 
 lemma complexpow_i_squared [simp]: "ii ^ 2 = -(1::complex)"
-by (simp add: i_def complex_mult complex_one_def complex_minus numeral_2_eq_2)
+by (simp add: i_def complex_one_def numeral_2_eq_2)
 
 lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
 by (simp add: i_def complex_zero_def)
 
 

 
 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
 by (simp add: sgn_def)
 
 lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
-by (simp add: i_def complex_of_real_def complex_mult complex_add)
+by (simp add: i_def complex_of_real_def)
 
 lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)"
-by (simp add: i_def complex_one_def complex_mult complex_minus)
+by (simp add: i_def complex_one_def)
 
 lemma complex_eq_cancel_iff2 [simp]:
-     "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
-by (simp add: complex_of_real_def) 
-
-lemma complex_eq_cancel_iff2a [simp]:
      "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
 by (simp add: complex_of_real_def)
 
 lemma Complex_eq_0 [simp]: "(Complex x y = 0) = (x = 0 & y = 0)"
 by (simp add: complex_zero_def)

 
 lemma complex_inverse_complex_split:
      "inverse(complex_of_real x + ii * complex_of_real y) =
       complex_of_real(x/(x ^ 2 + y ^ 2)) -
       ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
-by (simp add: complex_of_real_def i_def complex_mult complex_add
-         diff_minus complex_minus complex_inverse divide_inverse)
+by (simp add: complex_of_real_def i_def diff_minus divide_inverse)
 
 (*----------------------------------------------------------------------------*)
 (* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
 (* many of the theorems are not used - so should they be kept?                *)
 (*----------------------------------------------------------------------------*)
 
-lemma complex_of_real_zero_iff [simp]: "(complex_of_real y = 0) = (y = 0)"
-by (auto simp add: complex_zero_def complex_of_real_def)
+lemma complex_of_real_zero_iff: "(complex_of_real y = 0) = (y = 0)"
+by (rule of_real_eq_0_iff)
 
 lemma cos_arg_i_mult_zero_pos:
    "0 < y ==> cos (arg(Complex 0 y)) = 0"
 apply (simp add: arg_def abs_if)
 apply (rule_tac a = "pi/2" in someI2, auto)

   expi :: "complex => complex"
   "expi z = complex_of_real(exp (Re z)) * cis (Im z)"
 
 lemma complex_split_polar:
      "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
-apply (induct z) 
+apply (induct z)
 apply (auto simp add: polar_Ex complex_of_real_mult_Complex)
 done
 
 lemma rcis_Ex: "\<exists>r a. z = rcis r a"
-apply (induct z) 
+apply (induct z)
 apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex)
 done
 
 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
 by (simp add: rcis_def cis_def)

 by (simp add: rcis_def cis_def)
 
 lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>"
 proof -
   have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)"
-    by (simp only: power_mult_distrib right_distrib) 
+    by (simp only: power_mult_distrib right_distrib)
   thus ?thesis by simp
 qed
 
 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
 by (simp add: rcis_def cis_def sin_cos_squared_add2_mult)
 
 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
 apply (simp add: cmod_def)
 apply (rule real_sqrt_eq_iff [THEN iffD2])
-apply (auto simp add: complex_mult_cnj)
+apply (auto simp add: complex_mult_cnj
+            simp del: of_real_add)
 done
 
 lemma complex_Re_cnj [simp]: "Re(cnj z) = Re z"
 by (induct z, simp add: complex_cnj)
 

 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
 by (simp add: rcis_def)
 
 lemma complex_of_real_minus_one:
    "complex_of_real (-(1::real)) = -(1::complex)"
-by (simp add: complex_of_real_def complex_one_def complex_minus)
+by (simp add: complex_of_real_def complex_one_def)
 
 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
 by (simp add: complex_mult_assoc [symmetric])
 
 

 
 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
 by (simp add: rcis_def power_mult_distrib DeMoivre complex_of_real_pow)
 
 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
-by (simp add: cis_def complex_inverse_complex_split of_real_minus 
-              diff_minus)
+by (simp add: cis_def complex_inverse_complex_split diff_minus)
 
 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
 by (simp add: divide_inverse rcis_def complex_of_real_inverse)
 
 lemma cis_divide: "cis a / cis b = cis (a - b)"

 
 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
 by (auto simp add: DeMoivre)
 
 lemma expi_add: "expi(a + b) = expi(a) * expi(b)"
-by (simp add: expi_def complex_Re_add exp_add complex_Im_add 
-              cis_mult [symmetric] of_real_mult mult_ac)
+by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac)
 
 lemma expi_zero [simp]: "expi (0::complex) = 1"
 by (simp add: expi_def)
 
 lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"

 defs (overloaded)
   complex_number_of_def: "(number_of w :: complex) == of_int w"
     --{*the type constraint is essential!*}
 
 instance complex :: number_ring
-by (intro_classes, simp add: complex_number_of_def) 
+by (intro_classes, simp add: complex_number_of_def)
 
 
 text{*Collapse applications of @{term complex_of_real} to @{term number_of}*}
 lemma complex_number_of [simp]: "complex_of_real (number_of w) = number_of w"
 by (rule of_real_number_of_eq)

    They work for type complex because the reals can be embedded in them.*}
 (* TODO: generalize and move to Real/RealVector.thy *)
 lemma iszero_complex_number_of [simp]:
      "iszero (number_of w :: complex) = iszero (number_of w :: real)"
 by (simp only: complex_of_real_zero_iff complex_number_of [symmetric] 
-               iszero_def)  
+               iszero_def)
 
 lemma complex_number_of_cnj [simp]: "cnj(number_of v :: complex) = number_of v"
 by (simp only: complex_number_of [symmetric] complex_cnj_complex_of_real)
 
 lemma complex_number_of_cmod: 

 FIXME: what do we do about this?
 test "a*(b*c)/(y*z) = d*(b::complex)*(x*a)/z";
 *)
 
 end
-
-