src/HOL/Real.thy
changeset 68484 59793df7f853
parent 67399 eab6ce8368fa
child 68527 2f4e2aab190a
equal deleted inserted replaced
68483:087d32a40129 68484:59793df7f853
  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"