equal
deleted
inserted
replaced
3146 by (metis False divide_self_if eq power_divide power_one) |
3146 by (metis False divide_self_if eq power_divide power_one) |
3147 then obtain j where j: "w / z = exp (2 * of_real pi * \<i> * j / n)" and "j < n" |
3147 then obtain j where j: "w / z = exp (2 * of_real pi * \<i> * j / n)" and "j < n" |
3148 using Suc_leI assms \<open>2 \<le> n\<close> complex_roots_unity [THEN eqset_imp_iff, of n "w/z"] |
3148 using Suc_leI assms \<open>2 \<le> n\<close> complex_roots_unity [THEN eqset_imp_iff, of n "w/z"] |
3149 by force |
3149 by force |
3150 have "cmod (w/z - 1) < 2 * sin (pi / real n)" |
3150 have "cmod (w/z - 1) < 2 * sin (pi / real n)" |
3151 using lt assms \<open>z \<noteq> 0\<close> by (simp add: divide_simps norm_divide) |
3151 using lt assms \<open>z \<noteq> 0\<close> by (simp add: field_split_simps norm_divide) |
3152 then have "cmod (exp (\<i> * of_real (2 * pi * j / n)) - 1) < 2 * sin (pi / real n)" |
3152 then have "cmod (exp (\<i> * of_real (2 * pi * j / n)) - 1) < 2 * sin (pi / real n)" |
3153 by (simp add: j field_simps) |
3153 by (simp add: j field_simps) |
3154 then have "2 * \<bar>sin((2 * pi * j / n) / 2)\<bar> < 2 * sin (pi / real n)" |
3154 then have "2 * \<bar>sin((2 * pi * j / n) / 2)\<bar> < 2 * sin (pi / real n)" |
3155 by (simp only: dist_exp_i_1) |
3155 by (simp only: dist_exp_i_1) |
3156 then have sin_less: "sin((pi * j / n)) < sin (pi / real n)" |
3156 then have sin_less: "sin((pi * j / n)) < sin (pi / real n)" |