--- a/src/HOL/Complex/Complex.thy Wed Sep 27 03:04:35 2006 +0200
+++ b/src/HOL/Complex/Complex.thy Wed Sep 27 03:05:28 2006 +0200
@@ -159,7 +159,7 @@
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)
@@ -213,7 +213,7 @@
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
@@ -269,41 +269,34 @@
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
+by (rule of_real_inverse)
-(* TODO: generalize and move to Real/RealVector.thy *)
-lemma complex_of_real_divide [simp]:
+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)
@@ -345,19 +338,19 @@
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)
@@ -366,7 +359,7 @@
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)
@@ -381,8 +374,7 @@
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*}
@@ -408,7 +400,7 @@
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))"
@@ -426,10 +418,10 @@
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)
@@ -464,13 +456,13 @@
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)"
@@ -493,7 +485,7 @@
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
@@ -524,19 +516,15 @@
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])
@@ -545,7 +533,7 @@
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*}
@@ -565,9 +553,7 @@
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")
@@ -575,12 +561,10 @@
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)
@@ -610,17 +594,13 @@
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)"
@@ -657,16 +637,15 @@
"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"
@@ -705,12 +684,12 @@
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
@@ -723,7 +702,7 @@
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
@@ -733,7 +712,8 @@
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"
@@ -771,7 +751,7 @@
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])
@@ -790,8 +770,7 @@
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)
@@ -818,8 +797,7 @@
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)
@@ -840,7 +818,7 @@
--{*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}*}
@@ -855,7 +833,7 @@
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)
@@ -908,5 +886,3 @@
*)
end
-
-