--- a/src/HOL/Analysis/Bochner_Integration.thy Wed Jun 06 13:04:52 2018 +0200
+++ b/src/HOL/Analysis/Bochner_Integration.thy Wed Jun 06 18:19:55 2018 +0200
@@ -476,7 +476,7 @@
(auto intro: f simple_function_compose1 elim: simple_bochner_integrable.cases
intro!: sum.cong ennreal_cong_mult
simp: ac_simps ennreal_mult
- reorient: sum_ennreal)
+ simp flip: sum_ennreal)
also have "\<dots> = (\<integral>\<^sup>+x. f x \<partial>M)"
using f
by (intro nn_integral_eq_simple_integral[symmetric])
@@ -504,7 +504,7 @@
using simple_bochner_integrable_compose2[of "\<lambda>x y. norm (x - y)" M "s" "t"] s t
by (auto intro!: simple_bochner_integral_eq_nn_integral)
also have "\<dots> \<le> (\<integral>\<^sup>+x. ennreal (norm (f x - s x)) + ennreal (norm (f x - t x)) \<partial>M)"
- by (auto intro!: nn_integral_mono reorient: ennreal_plus)
+ by (auto intro!: nn_integral_mono simp flip: ennreal_plus)
(metis (erased, hide_lams) add_diff_cancel_left add_diff_eq diff_add_eq order_trans
norm_minus_commute norm_triangle_ineq4 order_refl)
also have "\<dots> = ?S + ?T"
@@ -594,7 +594,7 @@
proof (intro always_eventually allI)
fix i have "?f i \<le> (\<integral>\<^sup>+ x. (norm (f x - sf i x)) + ennreal (norm (g x - sg i x)) \<partial>M)"
by (auto intro!: nn_integral_mono norm_diff_triangle_ineq
- reorient: ennreal_plus)
+ simp flip: ennreal_plus)
also have "\<dots> = ?g i"
by (intro nn_integral_add) auto
finally show "?f i \<le> ?g i" .
@@ -747,7 +747,7 @@
finally have s_fin: "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) < \<infinity>" .
have "(\<integral>\<^sup>+ x. norm (f x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ennreal (norm (f x - s i x)) + ennreal (norm (s i x)) \<partial>M)"
- by (auto intro!: nn_integral_mono reorient: ennreal_plus)
+ by (auto intro!: nn_integral_mono simp flip: ennreal_plus)
(metis add.commute norm_triangle_sub)
also have "\<dots> = (\<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) + (\<integral>\<^sup>+x. norm (s i x) \<partial>M)"
by (rule nn_integral_add) auto
@@ -783,7 +783,7 @@
by (intro simple_bochner_integral_eq_nn_integral)
(auto intro: s simple_bochner_integrable_compose2)
also have "\<dots> \<le> (\<integral>\<^sup>+x. ennreal (norm (f x - s n x)) + norm (f x) \<partial>M)"
- by (auto intro!: nn_integral_mono reorient: ennreal_plus)
+ by (auto intro!: nn_integral_mono simp flip: ennreal_plus)
(metis add.commute norm_minus_commute norm_triangle_sub)
also have "\<dots> = ?t n"
by (rule nn_integral_add) auto
@@ -828,7 +828,7 @@
using tendsto_add[OF \<open>?S \<longlonglongrightarrow> 0\<close> \<open>?T \<longlonglongrightarrow> 0\<close>] by simp
qed
then have "(\<lambda>i. norm (?s i - ?t i)) \<longlonglongrightarrow> 0"
- by (simp reorient: ennreal_0)
+ by (simp flip: ennreal_0)
ultimately have "norm (x - y) = 0"
by (rule LIMSEQ_unique)
then show "x = y" by simp
@@ -1174,7 +1174,7 @@
by (intro simple_bochner_integral_bounded s f)
also have "\<dots> < ennreal (e / 2) + e / 2"
by (intro add_strict_mono M n m)
- also have "\<dots> = e" using \<open>0<e\<close> by (simp reorient: ennreal_plus)
+ also have "\<dots> = e" using \<open>0<e\<close> by (simp flip: ennreal_plus)
finally show "dist (?s n) (?s m) < e"
using \<open>0<e\<close> by (simp add: dist_norm ennreal_less_iff)
qed
@@ -1219,7 +1219,7 @@
fix x assume "(\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
from tendsto_diff[OF tendsto_const[of "u' x"] this]
show "(\<lambda>i. ennreal (norm (u' x - u i x))) \<longlonglongrightarrow> 0"
- by (simp add: tendsto_norm_zero_iff reorient: ennreal_0)
+ by (simp add: tendsto_norm_zero_iff flip: ennreal_0)
qed
qed (insert bnd w_nonneg, auto)
then show ?thesis by simp
@@ -2117,7 +2117,7 @@
by auto
qed
then have "((\<lambda>n. norm((\<integral>x. u n x \<partial>M) - (\<integral>x. f x \<partial>M))) \<longlongrightarrow> 0) F"
- by (simp reorient: ennreal_0)
+ by (simp flip: ennreal_0)
then have "((\<lambda>n. ((\<integral>x. u n x \<partial>M) - (\<integral>x. f x \<partial>M))) \<longlongrightarrow> 0) F" using tendsto_norm_zero_iff by blast
then show ?thesis using Lim_null by auto
qed
@@ -2215,7 +2215,7 @@
ultimately have "(\<lambda>n. ennreal (norm(u (r n) x))) \<longlonglongrightarrow> 0"
using tendsto_Limsup[of sequentially "\<lambda>n. ennreal (norm(u (r n) x))"] by auto
then have "(\<lambda>n. norm(u (r n) x)) \<longlonglongrightarrow> 0"
- by (simp reorient: ennreal_0)
+ by (simp flip: ennreal_0)
then have "(\<lambda>n. u (r n) x) \<longlonglongrightarrow> 0"
by (simp add: tendsto_norm_zero_iff)
}