--- a/src/HOL/Analysis/Complex_Transcendental.thy Thu Jan 26 13:59:51 2023 +0000
+++ b/src/HOL/Analysis/Complex_Transcendental.thy Fri Jan 27 13:57:52 2023 +0000
@@ -494,23 +494,8 @@
lemma cos_gt_neg1:
assumes "(t::real) \<in> {-pi<..<pi}"
shows "cos t > -1"
-proof -
- have "cos t \<ge> -1"
- by simp
- moreover have "cos t \<noteq> -1"
- proof
- assume "cos t = -1"
- then obtain n where n: "t = (2 * of_int n + 1) * pi"
- by (subst (asm) cos_eq_minus1) auto
- from assms have "t / pi \<in> {-1<..<1}"
- by (auto simp: field_simps)
- also from n have "t / pi = of_int (2 * n + 1)"
- by auto
- finally show False
- by auto
- qed
- ultimately show ?thesis by linarith
-qed
+ using assms
+ by simp (metis cos_minus cos_monotone_0_pi cos_monotone_minus_pi_0 cos_pi linorder_le_cases)
lemma dist_exp_i_1: "norm(exp(\<i> * of_real t) - 1) = 2 * \<bar>sin(t / 2)\<bar>"
proof -