# HG changeset patch
# User huffman
# Date 1315237130 25200
# Node ID 0b900a9d8023435403fe5d67ec1dd2f536a10894
# Parent  f3a8c19708c8dbebdc66281d08d6a5e09f4c0c9a
tuned indentation

diff -r f3a8c19708c8 -r 0b900a9d8023 src/HOL/Complex.thy
--- a/src/HOL/Complex.thy	Mon Sep 05 14:42:31 2011 +0200
+++ b/src/HOL/Complex.thy	Mon Sep 05 08:38:50 2011 -0700
@@ -12,15 +12,11 @@
 
 datatype complex = Complex real real
 
-primrec
-  Re :: "complex \<Rightarrow> real"
-where
-  Re: "Re (Complex x y) = x"
+primrec Re :: "complex \<Rightarrow> real"
+  where Re: "Re (Complex x y) = x"
 
-primrec
-  Im :: "complex \<Rightarrow> real"
-where
-  Im: "Im (Complex x y) = y"
+primrec Im :: "complex \<Rightarrow> real"
+  where Im: "Im (Complex x y) = y"
 
 lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"
   by (induct z) simp
@@ -37,17 +33,17 @@
 instantiation complex :: ab_group_add
 begin
 
-definition
-  complex_zero_def: "0 = Complex 0 0"
+definition complex_zero_def:
+  "0 = Complex 0 0"
 
-definition
-  complex_add_def: "x + y = Complex (Re x + Re y) (Im x + Im y)"
+definition complex_add_def:
+  "x + y = Complex (Re x + Re y) (Im x + Im y)"
 
-definition
-  complex_minus_def: "- x = Complex (- Re x) (- Im x)"
+definition complex_minus_def:
+  "- x = Complex (- Re x) (- Im x)"
 
-definition
-  complex_diff_def: "x - (y\<Colon>complex) = x + - y"
+definition complex_diff_def:
+  "x - (y\<Colon>complex) = x + - y"
 
 lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
   by (simp add: complex_zero_def)
@@ -94,25 +90,23 @@
 end
 
 
-
 subsection {* Multiplication and Division *}
 
 instantiation complex :: field_inverse_zero
 begin
 
-definition
-  complex_one_def: "1 = Complex 1 0"
+definition complex_one_def:
+  "1 = Complex 1 0"
 
-definition
-  complex_mult_def: "x * y =
-    Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
+definition complex_mult_def:
+  "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
 
-definition
-  complex_inverse_def: "inverse x =
+definition complex_inverse_def:
+  "inverse x =
     Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))"
 
-definition
-  complex_divide_def: "x / (y\<Colon>complex) = x * inverse y"
+definition complex_divide_def:
+  "x / (y\<Colon>complex) = x * inverse y"
 
 lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"
   by (simp add: complex_one_def)
@@ -147,10 +141,10 @@
 
 instance
   by intro_classes (simp_all add: complex_mult_def
-  right_distrib left_distrib right_diff_distrib left_diff_distrib
-  complex_inverse_def complex_divide_def
-  power2_eq_square add_divide_distrib [symmetric]
-  complex_eq_iff)
+    right_distrib left_distrib right_diff_distrib left_diff_distrib
+    complex_inverse_def complex_divide_def
+    power2_eq_square add_divide_distrib [symmetric]
+    complex_eq_iff)
 
 end
 
@@ -160,8 +154,8 @@
 instantiation complex :: number_ring
 begin
 
-definition number_of_complex where
-  complex_number_of_def: "number_of w = (of_int w \<Colon> complex)"
+definition complex_number_of_def:
+  "number_of w = (of_int w \<Colon> complex)"
 
 instance
   by intro_classes (simp only: complex_number_of_def)
@@ -169,26 +163,26 @@
 end
 
 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
-by (induct n) simp_all
+  by (induct n) simp_all
 
 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
-by (induct n) simp_all
+  by (induct n) simp_all
 
 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
-by (cases z rule: int_diff_cases) simp
+  by (cases z rule: int_diff_cases) simp
 
 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
-by (cases z rule: int_diff_cases) simp
+  by (cases z rule: int_diff_cases) simp
 
 lemma complex_Re_number_of [simp]: "Re (number_of v) = number_of v"
-unfolding number_of_eq by (rule complex_Re_of_int)
+  unfolding number_of_eq by (rule complex_Re_of_int)
 
 lemma complex_Im_number_of [simp]: "Im (number_of v) = 0"
-unfolding number_of_eq by (rule complex_Im_of_int)
+  unfolding number_of_eq by (rule complex_Im_of_int)
 
 lemma Complex_eq_number_of [simp]:
   "(Complex a b = number_of w) = (a = number_of w \<and> b = 0)"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 
 subsection {* Scalar Multiplication *}
@@ -196,8 +190,8 @@
 instantiation complex :: real_field
 begin
 
-definition
-  complex_scaleR_def: "scaleR r x = Complex (r * Re x) (r * Im x)"
+definition complex_scaleR_def:
+  "scaleR r x = Complex (r * Re x) (r * Im x)"
 
 lemma complex_scaleR [simp]:
   "scaleR r (Complex a b) = Complex (r * a) (r * b)"
@@ -231,34 +225,33 @@
 
 subsection{* Properties of Embedding from Reals *}
 
-abbreviation
-  complex_of_real :: "real \<Rightarrow> complex" where
-    "complex_of_real \<equiv> of_real"
+abbreviation complex_of_real :: "real \<Rightarrow> complex"
+  where "complex_of_real \<equiv> of_real"
 
 lemma complex_of_real_def: "complex_of_real r = Complex r 0"
-by (simp add: of_real_def complex_scaleR_def)
+  by (simp add: of_real_def complex_scaleR_def)
 
 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
-by (simp add: complex_of_real_def)
+  by (simp add: complex_of_real_def)
 
 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
-by (simp add: complex_of_real_def)
+  by (simp add: complex_of_real_def)
 
 lemma Complex_add_complex_of_real [simp]:
-     "Complex x y + complex_of_real r = Complex (x+r) y"
-by (simp add: complex_of_real_def)
+  shows "Complex x y + complex_of_real r = Complex (x+r) y"
+  by (simp add: complex_of_real_def)
 
 lemma complex_of_real_add_Complex [simp]:
-     "complex_of_real r + Complex x y = Complex (r+x) y"
-by (simp add: complex_of_real_def)
+  shows "complex_of_real r + Complex x y = Complex (r+x) y"
+  by (simp add: complex_of_real_def)
 
 lemma Complex_mult_complex_of_real:
-     "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
-by (simp add: complex_of_real_def)
+  shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
+  by (simp add: complex_of_real_def)
 
 lemma complex_of_real_mult_Complex:
-     "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
-by (simp add: complex_of_real_def)
+  shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
+  by (simp add: complex_of_real_def)
 
 
 subsection {* Vector Norm *}
@@ -269,9 +262,8 @@
 definition complex_norm_def:
   "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
 
-abbreviation
-  cmod :: "complex \<Rightarrow> real" where
-  "cmod \<equiv> norm"
+abbreviation cmod :: "complex \<Rightarrow> real"
+  where "cmod \<equiv> norm"
 
 definition complex_sgn_def:
   "sgn x = x /\<^sub>R cmod x"
@@ -313,29 +305,30 @@
 end
 
 lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1"
-by simp
+  by simp
 
 lemma cmod_complex_polar [simp]:
-     "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
-by (simp add: norm_mult)
+  "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
+  by (simp add: norm_mult)
 
 lemma complex_Re_le_cmod: "Re x \<le> cmod x"
-unfolding complex_norm_def
-by (rule real_sqrt_sum_squares_ge1)
+  unfolding complex_norm_def
+  by (rule real_sqrt_sum_squares_ge1)
 
 lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x"
-by (rule order_trans [OF _ norm_ge_zero], simp)
+  by (rule order_trans [OF _ norm_ge_zero], simp)
 
 lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a"
-by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
+  by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
 
 lemmas real_sum_squared_expand = power2_sum [where 'a=real]
 
 lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
-by (cases x) simp
+  by (cases x) simp
 
 lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
-by (cases x) simp
+  by (cases x) simp
+
 
 subsection {* Completeness of the Complexes *}
 
@@ -357,25 +350,25 @@
 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
 
 lemma tendsto_Complex [tendsto_intros]:
-  assumes "(f ---> a) net" and "(g ---> b) net"
-  shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) net"
+  assumes "(f ---> a) F" and "(g ---> b) F"
+  shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
 proof (rule tendstoI)
   fix r :: real assume "0 < r"
   hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)
-  have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) net"
-    using `(f ---> a) net` and `0 < r / sqrt 2` by (rule tendstoD)
+  have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F"
+    using `(f ---> a) F` and `0 < r / sqrt 2` by (rule tendstoD)
   moreover
-  have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) net"
-    using `(g ---> b) net` and `0 < r / sqrt 2` by (rule tendstoD)
+  have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F"
+    using `(g ---> b) F` and `0 < r / sqrt 2` by (rule tendstoD)
   ultimately
-  show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) net"
+  show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F"
     by (rule eventually_elim2)
        (simp add: dist_norm real_sqrt_sum_squares_less)
 qed
 
 lemma LIMSEQ_Complex:
   "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) ----> Complex a b"
-by (rule tendsto_Complex)
+  by (rule tendsto_Complex)
 
 instance complex :: banach
 proof
@@ -394,133 +387,131 @@
 
 subsection {* The Complex Number @{term "\<i>"} *}
 
-definition
-  "ii" :: complex  ("\<i>") where
-  i_def: "ii \<equiv> Complex 0 1"
+definition "ii" :: complex  ("\<i>")
+  where i_def: "ii \<equiv> Complex 0 1"
 
 lemma complex_Re_i [simp]: "Re ii = 0"
-by (simp add: i_def)
+  by (simp add: i_def)
 
 lemma complex_Im_i [simp]: "Im ii = 1"
-by (simp add: i_def)
+  by (simp add: i_def)
 
 lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
-by (simp add: i_def)
+  by (simp add: i_def)
 
 lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_i_not_one [simp]: "ii \<noteq> 1"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
-by (simp add: i_def complex_of_real_def)
+  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)
+  by (simp add: i_def complex_of_real_def)
 
 lemma i_squared [simp]: "ii * ii = -1"
-by (simp add: i_def)
+  by (simp add: i_def)
 
 lemma power2_i [simp]: "ii\<twosuperior> = -1"
-by (simp add: power2_eq_square)
+  by (simp add: power2_eq_square)
 
 lemma inverse_i [simp]: "inverse ii = - ii"
-by (rule inverse_unique, simp)
+  by (rule inverse_unique, simp)
 
 
 subsection {* Complex Conjugation *}
 
-definition
-  cnj :: "complex \<Rightarrow> complex" where
+definition cnj :: "complex \<Rightarrow> complex" where
   "cnj z = Complex (Re z) (- Im z)"
 
 lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
-by (simp add: cnj_def)
+  by (simp add: cnj_def)
 
 lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
-by (simp add: cnj_def)
+  by (simp add: cnj_def)
 
 lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
-by (simp add: cnj_def)
+  by (simp add: cnj_def)
 
 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
-by (simp add: cnj_def)
+  by (simp add: cnj_def)
 
 lemma complex_cnj_zero [simp]: "cnj 0 = 0"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_cnj_minus: "cnj (- x) = - cnj x"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_cnj_one [simp]: "cnj 1 = 1"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
-by (simp add: complex_inverse_def)
+  by (simp add: complex_inverse_def)
 
 lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
-by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
+  by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
 
 lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
-by (induct n, simp_all add: complex_cnj_mult)
+  by (induct n, simp_all add: complex_cnj_mult)
 
 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
-by (simp add: complex_norm_def)
+  by (simp add: complex_norm_def)
 
 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_cnj_i [simp]: "cnj ii = - ii"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
-by (simp add: complex_eq_iff)
+  by (simp add: complex_eq_iff)
 
 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
-by (simp add: complex_eq_iff power2_eq_square)
+  by (simp add: complex_eq_iff power2_eq_square)
 
 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
-by (simp add: norm_mult power2_eq_square)
+  by (simp add: norm_mult power2_eq_square)
 
 lemma bounded_linear_cnj: "bounded_linear cnj"
   using complex_cnj_add complex_cnj_scaleR
@@ -537,34 +528,33 @@
 
 text {*------------ Argand -------------*}
 
-definition
-  arg :: "complex => real" where
+definition arg :: "complex => real" where
   "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)"
 
 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
-by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute)
+  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
 
 lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
-by (simp add: i_def complex_of_real_def)
+  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)
+  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)
+  shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
+  by (simp add: complex_of_real_def)
 
 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
-by (simp add: complex_sgn_def divide_inverse)
+  by (simp add: complex_sgn_def divide_inverse)
 
 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
-by (simp add: complex_sgn_def divide_inverse)
+  by (simp add: complex_sgn_def divide_inverse)
 
 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 diff_minus 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 *)
@@ -638,10 +628,10 @@
 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 Im_rcis [simp]: "Im(rcis r a) = r * sin 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 -
@@ -651,44 +641,44 @@
 qed
 
 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
-by (simp add: rcis_def cis_def sin_cos_squared_add2_mult)
+  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))"
-by (simp add: cmod_def power2_eq_square)
+  by (simp add: cmod_def power2_eq_square)
 
 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
-by simp
+  by simp
 
 lemma cis_rcis_eq: "cis a = rcis 1 a"
-by (simp add: rcis_def)
+  by (simp add: rcis_def)
 
 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
-by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib
-              complex_of_real_def)
+  by (simp add: rcis_def cis_def cos_add sin_add right_distrib
+    right_diff_distrib complex_of_real_def)
 
 lemma cis_mult: "cis a * cis b = cis (a + b)"
-by (simp add: cis_rcis_eq rcis_mult)
+  by (simp add: cis_rcis_eq rcis_mult)
 
 lemma cis_zero [simp]: "cis 0 = 1"
-by (simp add: cis_def complex_one_def)
+  by (simp add: cis_def complex_one_def)
 
 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
-by (simp add: rcis_def)
+  by (simp add: rcis_def)
 
 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
-by (simp add: rcis_def)
+  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)
+  by (simp add: complex_of_real_def complex_one_def)
 
 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
-by (simp add: mult_assoc [symmetric])
+  by (simp add: mult_assoc [symmetric])
 
 
 lemma cis_real_of_nat_Suc_mult:
    "cis (real (Suc n) * a) = cis a * cis (real n * a)"
-by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib)
+  by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib)
 
 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
 apply (induct_tac "n")
@@ -696,16 +686,16 @@
 done
 
 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
-by (simp add: rcis_def power_mult_distrib DeMoivre)
+  by (simp add: rcis_def power_mult_distrib DeMoivre)
 
 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
-by (simp add: cis_def complex_inverse_complex_split 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)
+  by (simp add: divide_inverse rcis_def)
 
 lemma cis_divide: "cis a / cis b = cis (a - b)"
-by (simp add: complex_divide_def cis_mult diff_minus)
+  by (simp add: complex_divide_def cis_mult diff_minus)
 
 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
 apply (simp add: complex_divide_def)
@@ -714,16 +704,16 @@
 done
 
 lemma Re_cis [simp]: "Re(cis a) = cos a"
-by (simp add: cis_def)
+  by (simp add: cis_def)
 
 lemma Im_cis [simp]: "Im(cis a) = sin a"
-by (simp add: cis_def)
+  by (simp add: cis_def)
 
 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
-by (auto simp add: DeMoivre)
+  by (auto simp add: DeMoivre)
 
 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
-by (auto simp add: DeMoivre)
+  by (auto simp add: DeMoivre)
 
 lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
 apply (insert rcis_Ex [of z])
@@ -732,7 +722,7 @@
 done
 
 lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
-by (simp add: expi_def cis_def)
+  by (simp add: expi_def cis_def)
 
 text {* Legacy theorem names *}