equal
deleted
inserted
replaced
1385 |
1385 |
1386 lemma real_lbound_gt_zero: "0 < d1 \<Longrightarrow> 0 < d2 \<Longrightarrow> \<exists>e. 0 < e \<and> e < d1 \<and> e < d2" |
1386 lemma real_lbound_gt_zero: "0 < d1 \<Longrightarrow> 0 < d2 \<Longrightarrow> \<exists>e. 0 < e \<and> e < d1 \<and> e < d2" |
1387 for d1 d2 :: real |
1387 for d1 d2 :: real |
1388 by (rule exI [where x = "min d1 d2 / 2"]) (simp add: min_def) |
1388 by (rule exI [where x = "min d1 d2 / 2"]) (simp add: min_def) |
1389 |
1389 |
1390 text \<open>Similar results are proved in @{theory Fields}\<close> |
1390 text \<open>Similar results are proved in @{theory HOL.Fields}\<close> |
1391 lemma real_less_half_sum: "x < y \<Longrightarrow> x < (x + y) / 2" |
1391 lemma real_less_half_sum: "x < y \<Longrightarrow> x < (x + y) / 2" |
1392 for x y :: real |
1392 for x y :: real |
1393 by auto |
1393 by auto |
1394 |
1394 |
1395 lemma real_gt_half_sum: "x < y \<Longrightarrow> (x + y) / 2 < y" |
1395 lemma real_gt_half_sum: "x < y \<Longrightarrow> (x + y) / 2 < y" |