diff -r d45b78cb86cf -r 6aba668aea78 src/HOL/Analysis/Regularity.thy --- a/src/HOL/Analysis/Regularity.thy Wed Apr 25 21:29:02 2018 +0100 +++ b/src/HOL/Analysis/Regularity.thy Thu Apr 26 19:51:32 2018 +0200 @@ -107,7 +107,7 @@ finally have "measure M (space M) \ measure M K + e" using \0 < e\ by simp hence "emeasure M (space M) \ emeasure M K + e" - using \0 < e\ by (simp add: emeasure_eq_measure measure_nonneg ennreal_plus[symmetric] del: ennreal_plus) + using \0 < e\ by (simp add: emeasure_eq_measure reorient: ennreal_plus) moreover have "compact K" unfolding compact_eq_totally_bounded proof safe @@ -139,9 +139,9 @@ also have "\ \ measure M (space M) - measure M K" by (simp add: emeasure_eq_measure sU sb finite_measure_mono) also have "\ \ e" - using K \0 < e\ by (simp add: emeasure_eq_measure ennreal_plus[symmetric] measure_nonneg del: ennreal_plus) + using K \0 < e\ by (simp add: emeasure_eq_measure reorient: ennreal_plus) finally have "emeasure M A \ emeasure M (A \ K) + ennreal e" - using \0 by (simp add: emeasure_eq_measure algebra_simps ennreal_plus[symmetric] measure_nonneg del: ennreal_plus) + using \0 by (simp add: emeasure_eq_measure algebra_simps reorient: ennreal_plus) moreover have "A \ K \ A" "compact (A \ K)" using \closed A\ \compact K\ by auto ultimately show "\K \ A. compact K \ emeasure M A \ emeasure M K + ennreal e" by blast @@ -301,7 +301,7 @@ have "measure M (\i. D i) < (\ii (\i0 < e\ - by (auto intro: sum_mono simp: emeasure_eq_measure measure_nonneg ennreal_plus[symmetric] simp del: ennreal_plus) + by (auto intro: sum_mono simp: emeasure_eq_measure reorient: ennreal_plus) also have "\ = (\ii \ (\i0 < e\ @@ -310,7 +310,7 @@ have "measure M (\i. D i) < (\ii. D i) < M ?K + e" - using \0 by (auto simp: mK emeasure_eq_measure measure_nonneg sum_nonneg ennreal_less_iff ennreal_plus[symmetric] simp del: ennreal_plus) + using \0 by (auto simp: mK emeasure_eq_measure sum_nonneg ennreal_less_iff reorient: ennreal_plus) moreover have "?K \ (\i. D i)" using K by auto moreover @@ -332,9 +332,9 @@ from INF_approx_ennreal[OF \0 < e/(2 powr Suc i)\ this] show "\U. D i \ U \ open U \ e/(2 powr Suc i) > emeasure M U - emeasure M (D i)" using \0 - by (auto simp: emeasure_eq_measure measure_nonneg sum_nonneg ennreal_less_iff ennreal_plus[symmetric] ennreal_minus + by (auto simp: emeasure_eq_measure sum_nonneg ennreal_less_iff ennreal_minus finite_measure_mono sb - simp del: ennreal_plus) + reorient: ennreal_plus) qed then obtain U where U: "\i. D i \ U i" "\i. open (U i)" "\i. e/(2 powr Suc i) > emeasure M (U i) - emeasure M (D i)" @@ -367,7 +367,7 @@ also have "\ = ennreal e" by (subst suminf_ennreal_eq[OF zero_le_power power_half_series]) auto finally have "emeasure M ?U \ emeasure M (\i. D i) + ennreal e" - using \0 by (simp add: emeasure_eq_measure ennreal_plus[symmetric] measure_nonneg del: ennreal_plus) + using \0 by (simp add: emeasure_eq_measure reorient: ennreal_plus) moreover have "(\i. D i) \ ?U" using U by auto moreover