src/HOL/Probability/Regularity.thy

     note inner = compl(2) and outer = compl(3)
     from compl have [simp]: "B \<in> sets M" by (auto simp: sb borel_eq_closed)
     case 2
     have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)
     also have "\<dots> = (INF K:{K. K \<subseteq> B \<and> compact K}. M (space M) -  M K)"
-      unfolding inner by (subst INF_ereal_cminus) force+
+      unfolding inner by (subst INF_ereal_minus_right) force+
     also have "\<dots> = (INF U:{U. U \<subseteq> B \<and> compact U}. M (space M - U))"
       by (rule INF_cong) (auto simp add: emeasure_compl sb compact_imp_closed)
     also have "\<dots> \<ge> (INF U:{U. U \<subseteq> B \<and> closed U}. M (space M - U))"
       by (rule INF_superset_mono) (auto simp add: compact_imp_closed)
     also have "(INF U:{U. U \<subseteq> B \<and> closed U}. M (space M - U)) =

     ultimately show ?case by (auto intro!: antisym simp: sets_eq_imp_space_eq[OF sb])
 
     case 1
     have "M (space M - B) = M (space M) - emeasure M B" by (auto simp: emeasure_compl)
     also have "\<dots> = (SUP U: {U. B \<subseteq> U \<and> open U}. M (space M) -  M U)"
-      unfolding outer by (subst SUP_ereal_cminus) auto
+      unfolding outer by (subst SUP_ereal_minus_right) auto
     also have "\<dots> = (SUP U:{U. B \<subseteq> U \<and> open U}. M (space M - U))"
       by (rule SUP_cong) (auto simp add: emeasure_compl sb compact_imp_closed)
     also have "\<dots> = (SUP K:{K. K \<subseteq> space M - B \<and> closed K}. emeasure M K)"
       by (subst SUP_image [of "\<lambda>u. space M - u", symmetric, simplified comp_def])
          (rule SUP_cong, auto simp: sU)