author hoelzl Tue Oct 18 12:01:54 2016 +0200 (2016-10-18) changeset 64284 f3b905b2eee2 parent 64283 979cdfdf7a79 child 64285 d7e0123a752b child 64287 d85d88722745
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
 src/HOL/Analysis/Borel_Space.thy file | annotate | diff | revisions src/HOL/Analysis/Complete_Measure.thy file | annotate | diff | revisions src/HOL/Analysis/Set_Integral.thy file | annotate | diff | revisions src/HOL/Analysis/Topology_Euclidean_Space.thy file | annotate | diff | revisions src/HOL/Library/Extended_Nonnegative_Real.thy file | annotate | diff | revisions src/HOL/Library/Permutations.thy file | annotate | diff | revisions src/HOL/Probability/Levy.thy file | annotate | diff | revisions src/HOL/Topological_Spaces.thy file | annotate | diff | revisions
1.1 --- a/src/HOL/Analysis/Borel_Space.thy	Thu Oct 13 18:36:06 2016 +0200
1.2 +++ b/src/HOL/Analysis/Borel_Space.thy	Tue Oct 18 12:01:54 2016 +0200
1.3 @@ -1974,4 +1974,189 @@
1.4  no_notation
1.5    eucl_less (infix "<e" 50)
1.7 +lemma borel_measurable_Max2[measurable (raw)]:
1.8 +  fixes f::"_ \<Rightarrow> _ \<Rightarrow> 'a::{second_countable_topology, dense_linorder, linorder_topology}"
1.9 +  assumes "finite I"
1.10 +    and [measurable]: "\<And>i. f i \<in> borel_measurable M"
1.11 +  shows "(\<lambda>x. Max{f i x |i. i \<in> I}) \<in> borel_measurable M"
1.12 +by (simp add: borel_measurable_Max[OF assms(1), where ?f=f and ?M=M] Setcompr_eq_image)
1.13 +
1.14 +lemma measurable_compose_n [measurable (raw)]:
1.15 +  assumes "T \<in> measurable M M"
1.16 +  shows "(T^^n) \<in> measurable M M"
1.17 +by (induction n, auto simp add: measurable_compose[OF _ assms])
1.18 +
1.19 +lemma measurable_real_imp_nat:
1.20 +  fixes f::"'a \<Rightarrow> nat"
1.21 +  assumes [measurable]: "(\<lambda>x. real(f x)) \<in> borel_measurable M"
1.22 +  shows "f \<in> measurable M (count_space UNIV)"
1.23 +proof -
1.24 +  let ?g = "(\<lambda>x. real(f x))"
1.25 +  have "\<And>(n::nat). ?g-({real n}) \<inter> space M = f-{n} \<inter> space M" by auto
1.26 +  moreover have "\<And>(n::nat). ?g-({real n}) \<inter> space M \<in> sets M" using assms by measurable
1.27 +  ultimately have "\<And>(n::nat). f-{n} \<inter> space M \<in> sets M" by simp
1.28 +  then show ?thesis using measurable_count_space_eq2_countable by blast
1.29 +qed
1.30 +
1.31 +lemma measurable_equality_set [measurable]:
1.32 +  fixes f g::"_\<Rightarrow> 'a::{second_countable_topology, t2_space}"
1.33 +  assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
1.34 +  shows "{x \<in> space M. f x = g x} \<in> sets M"
1.35 +
1.36 +proof -
1.37 +  define A where "A = {x \<in> space M. f x = g x}"
1.38 +  define B where "B = {y. \<exists>x::'a. y = (x,x)}"
1.39 +  have "A = (\<lambda>x. (f x, g x))-B \<inter> space M" unfolding A_def B_def by auto
1.40 +  moreover have "(\<lambda>x. (f x, g x)) \<in> borel_measurable M" by simp
1.41 +  moreover have "B \<in> sets borel" unfolding B_def by (simp add: closed_diagonal)
1.42 +  ultimately have "A \<in> sets M" by simp
1.43 +  then show ?thesis unfolding A_def by simp
1.44 +qed
1.45 +
1.46 +lemma measurable_inequality_set [measurable]:
1.47 +  fixes f g::"_ \<Rightarrow> 'a::{second_countable_topology, linorder_topology}"
1.48 +  assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
1.49 +  shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
1.50 +        "{x \<in> space M. f x < g x} \<in> sets M"
1.51 +        "{x \<in> space M. f x \<ge> g x} \<in> sets M"
1.52 +        "{x \<in> space M. f x > g x} \<in> sets M"
1.53 +proof -
1.54 +  define F where "F = (\<lambda>x. (f x, g x))"
1.55 +  have * [measurable]: "F \<in> borel_measurable M" unfolding F_def by simp
1.56 +
1.57 +  have "{x \<in> space M. f x \<le> g x} = F-{(x, y) | x y. x \<le> y} \<inter> space M" unfolding F_def by auto
1.58 +  moreover have "{(x, y) | x y. x \<le> (y::'a)} \<in> sets borel" using closed_subdiagonal borel_closed by blast
1.59 +  ultimately show "{x \<in> space M. f x \<le> g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
1.60 +
1.61 +  have "{x \<in> space M. f x < g x} = F-{(x, y) | x y. x < y} \<inter> space M" unfolding F_def by auto
1.62 +  moreover have "{(x, y) | x y. x < (y::'a)} \<in> sets borel" using open_subdiagonal borel_open by blast
1.63 +  ultimately show "{x \<in> space M. f x < g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
1.64 +
1.65 +  have "{x \<in> space M. f x \<ge> g x} = F-{(x, y) | x y. x \<ge> y} \<inter> space M" unfolding F_def by auto
1.66 +  moreover have "{(x, y) | x y. x \<ge> (y::'a)} \<in> sets borel" using closed_superdiagonal borel_closed by blast
1.67 +  ultimately show "{x \<in> space M. f x \<ge> g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
1.68 +
1.69 +  have "{x \<in> space M. f x > g x} = F-{(x, y) | x y. x > y} \<inter> space M" unfolding F_def by auto
1.70 +  moreover have "{(x, y) | x y. x > (y::'a)} \<in> sets borel" using open_superdiagonal borel_open by blast
1.71 +  ultimately show "{x \<in> space M. f x > g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
1.72 +qed
1.73 +
1.74 +lemma measurable_limit [measurable]:
1.75 +  fixes f::"nat \<Rightarrow> 'a \<Rightarrow> 'b::first_countable_topology"
1.76 +  assumes [measurable]: "\<And>n::nat. f n \<in> borel_measurable M"
1.77 +  shows "Measurable.pred M (\<lambda>x. (\<lambda>n. f n x) \<longlonglongrightarrow> c)"
1.78 +proof -
1.79 +  obtain A :: "nat \<Rightarrow> 'b set" where A:
1.80 +    "\<And>i. open (A i)"
1.81 +    "\<And>i. c \<in> A i"
1.82 +    "\<And>S. open S \<Longrightarrow> c \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
1.83 +  by (rule countable_basis_at_decseq) blast
1.84 +
1.85 +  have [measurable]: "\<And>N i. (f N)-(A i) \<inter> space M \<in> sets M" using A(1) by auto
1.86 +  then have mes: "(\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. (f N)-(A i) \<inter> space M) \<in> sets M" by blast
1.87 +
1.88 +  have "(u \<longlonglongrightarrow> c) \<longleftrightarrow> (\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)" for u::"nat \<Rightarrow> 'b"
1.89 +  proof
1.90 +    assume "u \<longlonglongrightarrow> c"
1.91 +    then have "eventually (\<lambda>n. u n \<in> A i) sequentially" for i using A(1)[of i] A(2)[of i]
1.92 +      by (simp add: topological_tendstoD)
1.93 +    then show "(\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)" by auto
1.94 +  next
1.95 +    assume H: "(\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)"
1.96 +    show "(u \<longlonglongrightarrow> c)"
1.97 +    proof (rule topological_tendstoI)
1.98 +      fix S assume "open S" "c \<in> S"
1.99 +      with A(3)[OF this] obtain i where "A i \<subseteq> S"
1.100 +        using eventually_False_sequentially eventually_mono by blast
1.101 +      moreover have "eventually (\<lambda>n. u n \<in> A i) sequentially" using H by simp
1.102 +      ultimately show "\<forall>\<^sub>F n in sequentially. u n \<in> S"
1.103 +        by (simp add: eventually_mono subset_eq)
1.104 +    qed
1.105 +  qed
1.106 +  then have "{x. (\<lambda>n. f n x) \<longlonglongrightarrow> c} = (\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. (f N)-(A i))"
1.107 +    by (auto simp add: atLeast_def eventually_at_top_linorder)
1.108 +  then have "{x \<in> space M. (\<lambda>n. f n x) \<longlonglongrightarrow> c} = (\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. (f N)-(A i) \<inter> space M)"
1.109 +    by auto
1.110 +  then have "{x \<in> space M. (\<lambda>n. f n x) \<longlonglongrightarrow> c} \<in> sets M" using mes by simp
1.111 +  then show ?thesis by auto
1.112 +qed
1.114 +lemma measurable_limit2 [measurable]:
1.115 +  fixes u::"nat \<Rightarrow> 'a \<Rightarrow> real"
1.116 +  assumes [measurable]: "\<And>n. u n \<in> borel_measurable M" "v \<in> borel_measurable M"
1.117 +  shows "Measurable.pred M (\<lambda>x. (\<lambda>n. u n x) \<longlonglongrightarrow> v x)"
1.118 +proof -
1.119 +  define w where "w = (\<lambda>n x. u n x - v x)"
1.120 +  have [measurable]: "w n \<in> borel_measurable M" for n unfolding w_def by auto
1.121 +  have "((\<lambda>n. u n x) \<longlonglongrightarrow> v x) \<longleftrightarrow> ((\<lambda>n. w n x) \<longlonglongrightarrow> 0)" for x
1.122 +    unfolding w_def using Lim_null by auto
1.123 +  then show ?thesis using measurable_limit by auto
1.124 +qed
1.126 +lemma measurable_P_restriction [measurable (raw)]:
1.127 +  assumes [measurable]: "Measurable.pred M P" "A \<in> sets M"
1.128 +  shows "{x \<in> A. P x} \<in> sets M"
1.129 +proof -
1.130 +  have "A \<subseteq> space M" using sets.sets_into_space[OF assms(2)].
1.131 +  then have "{x \<in> A. P x} = A \<inter> {x \<in> space M. P x}" by blast
1.132 +  then show ?thesis by auto
1.133 +qed
1.135 +lemma measurable_sum_nat [measurable (raw)]:
1.136 +  fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> nat"
1.137 +  assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> measurable M (count_space UNIV)"
1.138 +  shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> measurable M (count_space UNIV)"
1.139 +proof cases
1.140 +  assume "finite S"
1.141 +  then show ?thesis using assms by induct auto
1.142 +qed simp
1.145 +lemma measurable_abs_powr [measurable]:
1.146 +  fixes p::real
1.147 +  assumes [measurable]: "f \<in> borel_measurable M"
1.148 +  shows "(\<lambda>x. \<bar>f x\<bar> powr p) \<in> borel_measurable M"
1.149 +unfolding powr_def by auto
1.151 +text {* The next one is a variation around \verb+measurable_restrict_space+.*}
1.153 +lemma measurable_restrict_space3:
1.154 +  assumes "f \<in> measurable M N" and
1.155 +          "f \<in> A \<rightarrow> B"
1.156 +  shows "f \<in> measurable (restrict_space M A) (restrict_space N B)"
1.157 +proof -
1.158 +  have "f \<in> measurable (restrict_space M A) N" using assms(1) measurable_restrict_space1 by auto
1.159 +  then show ?thesis by (metis Int_iff funcsetI funcset_mem
1.160 +      measurable_restrict_space2[of f, of "restrict_space M A", of B, of N] assms(2) space_restrict_space)
1.161 +qed
1.163 +text {* The next one is a variation around \verb+measurable_piecewise_restrict+.*}
1.165 +lemma measurable_piecewise_restrict2:
1.166 +  assumes [measurable]: "\<And>n. A n \<in> sets M"
1.167 +      and "space M = (\<Union>(n::nat). A n)"
1.168 +          "\<And>n. \<exists>h \<in> measurable M N. (\<forall>x \<in> A n. f x = h x)"
1.169 +  shows "f \<in> measurable M N"
1.170 +proof (rule measurableI)
1.171 +  fix B assume [measurable]: "B \<in> sets N"
1.172 +  {
1.173 +    fix n::nat
1.174 +    obtain h where [measurable]: "h \<in> measurable M N" and "\<forall>x \<in> A n. f x = h x" using assms(3) by blast
1.175 +    then have *: "f-B \<inter> A n = h-B \<inter> A n" by auto
1.176 +    have "h-B \<inter> A n = h-B \<inter> space M \<inter> A n" using assms(2) sets.sets_into_space by auto
1.177 +    then have "h-B \<inter> A n \<in> sets M" by simp
1.178 +    then have "f-B \<inter> A n \<in> sets M" using * by simp
1.179 +  }
1.180 +  then have "(\<Union>n. f-B \<inter> A n) \<in> sets M" by measurable
1.181 +  moreover have "f-B \<inter> space M = (\<Union>n. f-B \<inter> A n)" using assms(2) by blast
1.182 +  ultimately show "f-B \<inter> space M \<in> sets M" by simp
1.183 +next
1.184 +  fix x assume "x \<in> space M"
1.185 +  then obtain n where "x \<in> A n" using assms(2) by blast
1.186 +  obtain h where [measurable]: "h \<in> measurable M N" and "\<forall>x \<in> A n. f x = h x" using assms(3) by blast
1.187 +  then have "f x = h x" using x \<in> A n by blast
1.188 +  moreover have "h x \<in> space N" by (metis measurable_space x \<in> space M h \<in> measurable M N)
1.189 +  ultimately show "f x \<in> space N" by simp
1.190 +qed
1.192  end
2.1 --- a/src/HOL/Analysis/Complete_Measure.thy	Thu Oct 13 18:36:06 2016 +0200
2.2 +++ b/src/HOL/Analysis/Complete_Measure.thy	Tue Oct 18 12:01:54 2016 +0200
2.3 @@ -791,11 +791,11 @@
2.4      by (auto simp add: emeasure_density measurable_completion nn_integral_completion intro!: nn_integral_cong_AE)
2.5  qed
2.7 -lemma null_sets_subset: "A \<subseteq> B \<Longrightarrow> A \<in> sets M \<Longrightarrow> B \<in> null_sets M \<Longrightarrow> A \<in> null_sets M"
2.8 +lemma null_sets_subset: "B \<in> null_sets M \<Longrightarrow> A \<in> sets M \<Longrightarrow> A \<subseteq> B \<Longrightarrow> A \<in> null_sets M"
2.9    using emeasure_mono[of A B M] by (simp add: null_sets_def)
2.11  lemma (in complete_measure) complete2: "A \<subseteq> B \<Longrightarrow> B \<in> null_sets M \<Longrightarrow> A \<in> null_sets M"
2.12 -  using complete[of A B] null_sets_subset[of A B M] by simp
2.13 +  using complete[of A B] null_sets_subset[of B M A] by simp
2.15  lemma (in complete_measure) AE_iff_null_sets: "(AE x in M. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets M"
2.16    unfolding eventually_ae_filter by (auto intro: complete2)
3.1 --- a/src/HOL/Analysis/Set_Integral.thy	Thu Oct 13 18:36:06 2016 +0200
3.2 +++ b/src/HOL/Analysis/Set_Integral.thy	Tue Oct 18 12:01:54 2016 +0200
3.3 @@ -11,6 +11,1339 @@
3.5  begin
3.7 +lemma bij_inv_eq_iff: "bij p \<Longrightarrow> x = inv p y \<longleftrightarrow> p x = y" (* COPIED FROM Permutations *)
3.8 +  using surj_f_inv_f[of p] by (auto simp add: bij_def)
3.9 +
3.10 +subsection {*Fun.thy*}
3.11 +
3.12 +lemma inj_fn:
3.13 +  fixes f::"'a \<Rightarrow> 'a"
3.14 +  assumes "inj f"
3.15 +  shows "inj (f^^n)"
3.16 +proof (induction n)
3.17 +  case (Suc n)
3.18 +  have "inj (f o (f^^n))"
3.19 +    using inj_comp[OF assms Suc.IH] by simp
3.20 +  then show "inj (f^^(Suc n))"
3.21 +    by auto
3.22 +qed (auto)
3.23 +
3.24 +lemma surj_fn:
3.25 +  fixes f::"'a \<Rightarrow> 'a"
3.26 +  assumes "surj f"
3.27 +  shows "surj (f^^n)"
3.28 +proof (induction n)
3.29 +  case (Suc n)
3.30 +  have "surj (f o (f^^n))"
3.31 +    using assms Suc.IH by (simp add: comp_surj)
3.32 +  then show "surj (f^^(Suc n))"
3.33 +    by auto
3.34 +qed (auto)
3.35 +
3.36 +lemma bij_fn:
3.37 +  fixes f::"'a \<Rightarrow> 'a"
3.38 +  assumes "bij f"
3.39 +  shows "bij (f^^n)"
3.40 +by (rule bijI[OF inj_fn[OF bij_is_inj[OF assms]] surj_fn[OF bij_is_surj[OF assms]]])
3.41 +
3.42 +lemma inv_fn_o_fn_is_id:
3.43 +  fixes f::"'a \<Rightarrow> 'a"
3.44 +  assumes "bij f"
3.45 +  shows "((inv f)^^n) o (f^^n) = (\<lambda>x. x)"
3.46 +proof -
3.47 +  have "((inv f)^^n)((f^^n) x) = x" for x n
3.48 +  proof (induction n)
3.49 +    case (Suc n)
3.50 +    have *: "(inv f) (f y) = y" for y
3.51 +      by (simp add: assms bij_is_inj)
3.52 +    have "(inv f ^^ Suc n) ((f ^^ Suc n) x) = (inv f^^n) (inv f (f ((f^^n) x)))"
3.53 +      by (simp add: funpow_swap1)
3.54 +    also have "... = (inv f^^n) ((f^^n) x)"
3.55 +      using * by auto
3.56 +    also have "... = x" using Suc.IH by auto
3.57 +    finally show ?case by simp
3.58 +  qed (auto)
3.59 +  then show ?thesis unfolding o_def by blast
3.60 +qed
3.61 +
3.62 +lemma fn_o_inv_fn_is_id:
3.63 +  fixes f::"'a \<Rightarrow> 'a"
3.64 +  assumes "bij f"
3.65 +  shows "(f^^n) o ((inv f)^^n) = (\<lambda>x. x)"
3.66 +proof -
3.67 +  have "(f^^n) (((inv f)^^n) x) = x" for x n
3.68 +  proof (induction n)
3.69 +    case (Suc n)
3.70 +    have *: "f(inv f y) = y" for y
3.71 +      using assms by (meson bij_inv_eq_iff)
3.72 +    have "(f ^^ Suc n) ((inv f ^^ Suc n) x) = (f^^n) (f (inv f ((inv f^^n) x)))"
3.73 +      by (simp add: funpow_swap1)
3.74 +    also have "... = (f^^n) ((inv f^^n) x)"
3.75 +      using * by auto
3.76 +    also have "... = x" using Suc.IH by auto
3.77 +    finally show ?case by simp
3.78 +  qed (auto)
3.79 +  then show ?thesis unfolding o_def by blast
3.80 +qed
3.81 +
3.82 +lemma inv_fn:
3.83 +  fixes f::"'a \<Rightarrow> 'a"
3.84 +  assumes "bij f"
3.85 +  shows "inv (f^^n) = ((inv f)^^n)"
3.86 +proof -
3.87 +  have "inv (f^^n) x = ((inv f)^^n) x" for x
3.88 +  apply (rule inv_into_f_eq, auto simp add: inj_fn[OF bij_is_inj[OF assms]])
3.89 +  using fn_o_inv_fn_is_id[OF assms, of n] by (metis comp_apply)
3.90 +  then show ?thesis by auto
3.91 +qed
3.92 +
3.93 +
3.94 +lemma mono_inv:
3.95 +  fixes f::"'a::linorder \<Rightarrow> 'b::linorder"
3.96 +  assumes "mono f" "bij f"
3.97 +  shows "mono (inv f)"
3.98 +proof
3.99 +  fix x y::'b assume "x \<le> y"
3.100 +  then show "inv f x \<le> inv f y"
3.101 +    by (metis (no_types, lifting) assms bij_is_surj eq_iff le_cases mono_def surj_f_inv_f)
3.102 +qed
3.104 +lemma mono_bij_Inf:
3.105 +  fixes f :: "'a::complete_linorder \<Rightarrow> 'b::complete_linorder"
3.106 +  assumes "mono f" "bij f"
3.107 +  shows "f (Inf A) = Inf (fA)"
3.108 +proof -
3.109 +  have "(inv f) (Inf (fA)) \<le> Inf ((inv f)(fA))"
3.110 +    using mono_Inf[OF mono_inv[OF assms], of "fA"] by simp
3.111 +  then have "Inf (fA) \<le> f (Inf ((inv f)(fA)))"
3.112 +    by (metis (no_types, lifting) assms mono_def bij_inv_eq_iff)
3.113 +  also have "... = f(Inf A)"
3.114 +    using assms by (simp add: bij_is_inj)
3.115 +  finally show ?thesis using mono_Inf[OF assms(1), of A] by auto
3.116 +qed
3.119 +lemma Inf_nat_def1:
3.120 +  fixes K::"nat set"
3.121 +  assumes "K \<noteq> {}"
3.122 +  shows "Inf K \<in> K"
3.123 +by (auto simp add: Min_def Inf_nat_def) (meson LeastI assms bot.extremum_unique subsetI)
3.125 +subsection {*Liminf-Limsup.thy*}
3.127 +lemma limsup_shift:
3.128 +  "limsup (\<lambda>n. u (n+1)) = limsup u"
3.129 +proof -
3.130 +  have "(SUP m:{n+1..}. u m) = (SUP m:{n..}. u (m + 1))" for n
3.131 +    apply (rule SUP_eq) using Suc_le_D by auto
3.132 +  then have a: "(INF n. SUP m:{n..}. u (m + 1)) = (INF n. (SUP m:{n+1..}. u m))" by auto
3.133 +  have b: "(INF n. (SUP m:{n+1..}. u m)) = (INF n:{1..}. (SUP m:{n..}. u m))"
3.134 +    apply (rule INF_eq) using Suc_le_D by auto
3.135 +  have "(INF n:{1..}. v n) = (INF n. v n)" if "decseq v" for v::"nat \<Rightarrow> 'a"
3.136 +    apply (rule INF_eq) using decseq v decseq_Suc_iff by auto
3.137 +  moreover have "decseq (\<lambda>n. (SUP m:{n..}. u m))" by (simp add: SUP_subset_mono decseq_def)
3.138 +  ultimately have c: "(INF n:{1..}. (SUP m:{n..}. u m)) = (INF n. (SUP m:{n..}. u m))" by simp
3.139 +  have "(INF n. SUPREMUM {n..} u) = (INF n. SUP m:{n..}. u (m + 1))" using a b c by simp
3.140 +  then show ?thesis by (auto cong: limsup_INF_SUP)
3.141 +qed
3.143 +lemma limsup_shift_k:
3.144 +  "limsup (\<lambda>n. u (n+k)) = limsup u"
3.145 +proof (induction k)
3.146 +  case (Suc k)
3.147 +  have "limsup (\<lambda>n. u (n+k+1)) = limsup (\<lambda>n. u (n+k))" using limsup_shift[where ?u="\<lambda>n. u(n+k)"] by simp
3.148 +  then show ?case using Suc.IH by simp
3.149 +qed (auto)
3.151 +lemma liminf_shift:
3.152 +  "liminf (\<lambda>n. u (n+1)) = liminf u"
3.153 +proof -
3.154 +  have "(INF m:{n+1..}. u m) = (INF m:{n..}. u (m + 1))" for n
3.155 +    apply (rule INF_eq) using Suc_le_D by (auto)
3.156 +  then have a: "(SUP n. INF m:{n..}. u (m + 1)) = (SUP n. (INF m:{n+1..}. u m))" by auto
3.157 +  have b: "(SUP n. (INF m:{n+1..}. u m)) = (SUP n:{1..}. (INF m:{n..}. u m))"
3.158 +    apply (rule SUP_eq) using Suc_le_D by (auto)
3.159 +  have "(SUP n:{1..}. v n) = (SUP n. v n)" if "incseq v" for v::"nat \<Rightarrow> 'a"
3.160 +    apply (rule SUP_eq) using incseq v incseq_Suc_iff by auto
3.161 +  moreover have "incseq (\<lambda>n. (INF m:{n..}. u m))" by (simp add: INF_superset_mono mono_def)
3.162 +  ultimately have c: "(SUP n:{1..}. (INF m:{n..}. u m)) = (SUP n. (INF m:{n..}. u m))" by simp
3.163 +  have "(SUP n. INFIMUM {n..} u) = (SUP n. INF m:{n..}. u (m + 1))" using a b c by simp
3.164 +  then show ?thesis by (auto cong: liminf_SUP_INF)
3.165 +qed
3.167 +lemma liminf_shift_k:
3.168 +  "liminf (\<lambda>n. u (n+k)) = liminf u"
3.169 +proof (induction k)
3.170 +  case (Suc k)
3.171 +  have "liminf (\<lambda>n. u (n+k+1)) = liminf (\<lambda>n. u (n+k))" using liminf_shift[where ?u="\<lambda>n. u(n+k)"] by simp
3.172 +  then show ?case using Suc.IH by simp
3.173 +qed (auto)
3.175 +lemma Limsup_obtain:
3.176 +  fixes u::"_ \<Rightarrow> 'a :: complete_linorder"
3.177 +  assumes "Limsup F u > c"
3.178 +  shows "\<exists>i. u i > c"
3.179 +proof -
3.180 +  have "(INF P:{P. eventually P F}. SUP x:{x. P x}. u x) > c" using assms by (simp add: Limsup_def)
3.181 +  then show ?thesis by (metis eventually_True mem_Collect_eq less_INF_D less_SUP_iff)
3.182 +qed
3.184 +text {* The next lemma is extremely useful, as it often makes it possible to reduce statements
3.185 +about limsups to statements about limits.*}
3.187 +lemma limsup_subseq_lim:
3.188 +  fixes u::"nat \<Rightarrow> 'a :: {complete_linorder, linorder_topology}"
3.189 +  shows "\<exists>r. subseq r \<and> (u o r) \<longlonglongrightarrow> limsup u"
3.190 +proof (cases)
3.191 +  assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. u m \<le> u p"
3.192 +  then have "\<exists>r. \<forall>n. (\<forall>m\<ge>r n. u m \<le> u (r n)) \<and> r n < r (Suc n)"
3.193 +    by (intro dependent_nat_choice) (auto simp: conj_commute)
3.194 +  then obtain r where "subseq r" and mono: "\<And>n m. r n \<le> m \<Longrightarrow> u m \<le> u (r n)"
3.195 +    by (auto simp: subseq_Suc_iff)
3.196 +  define umax where "umax = (\<lambda>n. (SUP m:{n..}. u m))"
3.197 +  have "decseq umax" unfolding umax_def by (simp add: SUP_subset_mono antimono_def)
3.198 +  then have "umax \<longlonglongrightarrow> limsup u" unfolding umax_def by (metis LIMSEQ_INF limsup_INF_SUP)
3.199 +  then have *: "(umax o r) \<longlonglongrightarrow> limsup u" by (simp add: LIMSEQ_subseq_LIMSEQ subseq r)
3.200 +  have "\<And>n. umax(r n) = u(r n)" unfolding umax_def using mono
3.201 +    by (metis SUP_le_iff antisym atLeast_def mem_Collect_eq order_refl)
3.202 +  then have "umax o r = u o r" unfolding o_def by simp
3.203 +  then have "(u o r) \<longlonglongrightarrow> limsup u" using * by simp
3.204 +  then show ?thesis using subseq r by blast
3.205 +next
3.206 +  assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. u m \<le> u p))"
3.207 +  then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. u p < u m" by (force simp: not_le le_less)
3.208 +  have "\<exists>r. \<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<le> u (r (Suc n)))"
3.209 +  proof (rule dependent_nat_choice)
3.210 +    fix x assume "N < x"
3.211 +    then have a: "finite {N<..x}" "{N<..x} \<noteq> {}" by simp_all
3.212 +    have "Max {u i |i. i \<in> {N<..x}} \<in> {u i |i. i \<in> {N<..x}}" apply (rule Max_in) using a by (auto)
3.213 +    then obtain p where "p \<in> {N<..x}" and upmax: "u p = Max{u i |i. i \<in> {N<..x}}" by auto
3.214 +    define U where "U = {m. m > p \<and> u p < u m}"
3.215 +    have "U \<noteq> {}" unfolding U_def using N[of p] p \<in> {N<..x} by auto
3.216 +    define y where "y = Inf U"
3.217 +    then have "y \<in> U" using U \<noteq> {} by (simp add: Inf_nat_def1)
3.218 +    have a: "\<And>i. i \<in> {N<..x} \<Longrightarrow> u i \<le> u p"
3.219 +    proof -
3.220 +      fix i assume "i \<in> {N<..x}"
3.221 +      then have "u i \<in> {u i |i. i \<in> {N<..x}}" by blast
3.222 +      then show "u i \<le> u p" using upmax by simp
3.223 +    qed
3.224 +    moreover have "u p < u y" using y \<in> U U_def by auto
3.225 +    ultimately have "y \<notin> {N<..x}" using not_le by blast
3.226 +    moreover have "y > N" using y \<in> U U_def p \<in> {N<..x} by auto
3.227 +    ultimately have "y > x" by auto
3.229 +    have "\<And>i. i \<in> {N<..y} \<Longrightarrow> u i \<le> u y"
3.230 +    proof -
3.231 +      fix i assume "i \<in> {N<..y}" show "u i \<le> u y"
3.232 +      proof (cases)
3.233 +        assume "i = y"
3.234 +        then show ?thesis by simp
3.235 +      next
3.236 +        assume "\<not>(i=y)"
3.237 +        then have i:"i \<in> {N<..<y}" using i \<in> {N<..y} by simp
3.238 +        have "u i \<le> u p"
3.239 +        proof (cases)
3.240 +          assume "i \<le> x"
3.241 +          then have "i \<in> {N<..x}" using i by simp
3.242 +          then show ?thesis using a by simp
3.243 +        next
3.244 +          assume "\<not>(i \<le> x)"
3.245 +          then have "i > x" by simp
3.246 +          then have *: "i > p" using p \<in> {N<..x} by simp
3.247 +          have "i < Inf U" using i y_def by simp
3.248 +          then have "i \<notin> U" using Inf_nat_def not_less_Least by auto
3.249 +          then show ?thesis using U_def * by auto
3.250 +        qed
3.251 +        then show "u i \<le> u y" using u p < u y by auto
3.252 +      qed
3.253 +    qed
3.254 +    then have "N < y \<and> x < y \<and> (\<forall>i\<in>{N<..y}. u i \<le> u y)" using y > x y > N by auto
3.255 +    then show "\<exists>y>N. x < y \<and> (\<forall>i\<in>{N<..y}. u i \<le> u y)" by auto
3.256 +  qed (auto)
3.257 +  then obtain r where r: "\<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<le> u (r (Suc n)))" by auto
3.258 +  have "subseq r" using r by (auto simp: subseq_Suc_iff)
3.259 +  have "incseq (u o r)" unfolding o_def using r by (simp add: incseq_SucI order.strict_implies_order)
3.260 +  then have "(u o r) \<longlonglongrightarrow> (SUP n. (u o r) n)" using LIMSEQ_SUP by blast
3.261 +  then have "limsup (u o r) = (SUP n. (u o r) n)" by (simp add: lim_imp_Limsup)
3.262 +  moreover have "limsup (u o r) \<le> limsup u" using subseq r by (simp add: limsup_subseq_mono)
3.263 +  ultimately have "(SUP n. (u o r) n) \<le> limsup u" by simp
3.265 +  {
3.266 +    fix i assume i: "i \<in> {N<..}"
3.267 +    obtain n where "i < r (Suc n)" using subseq r using Suc_le_eq seq_suble by blast
3.268 +    then have "i \<in> {N<..r(Suc n)}" using i by simp
3.269 +    then have "u i \<le> u (r(Suc n))" using r by simp
3.270 +    then have "u i \<le> (SUP n. (u o r) n)" unfolding o_def by (meson SUP_upper2 UNIV_I)
3.271 +  }
3.272 +  then have "(SUP i:{N<..}. u i) \<le> (SUP n. (u o r) n)" using SUP_least by blast
3.273 +  then have "limsup u \<le> (SUP n. (u o r) n)" unfolding Limsup_def
3.274 +    by (metis (mono_tags, lifting) INF_lower2 atLeast_Suc_greaterThan atLeast_def eventually_ge_at_top mem_Collect_eq)
3.275 +  then have "limsup u = (SUP n. (u o r) n)" using (SUP n. (u o r) n) \<le> limsup u by simp
3.276 +  then have "(u o r) \<longlonglongrightarrow> limsup u" using (u o r) \<longlonglongrightarrow> (SUP n. (u o r) n) by simp
3.277 +  then show ?thesis using subseq r by auto
3.278 +qed
3.280 +lemma liminf_subseq_lim:
3.281 +  fixes u::"nat \<Rightarrow> 'a :: {complete_linorder, linorder_topology}"
3.282 +  shows "\<exists>r. subseq r \<and> (u o r) \<longlonglongrightarrow> liminf u"
3.283 +proof (cases)
3.284 +  assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. u m \<ge> u p"
3.285 +  then have "\<exists>r. \<forall>n. (\<forall>m\<ge>r n. u m \<ge> u (r n)) \<and> r n < r (Suc n)"
3.286 +    by (intro dependent_nat_choice) (auto simp: conj_commute)
3.287 +  then obtain r where "subseq r" and mono: "\<And>n m. r n \<le> m \<Longrightarrow> u m \<ge> u (r n)"
3.288 +    by (auto simp: subseq_Suc_iff)
3.289 +  define umin where "umin = (\<lambda>n. (INF m:{n..}. u m))"
3.290 +  have "incseq umin" unfolding umin_def by (simp add: INF_superset_mono incseq_def)
3.291 +  then have "umin \<longlonglongrightarrow> liminf u" unfolding umin_def by (metis LIMSEQ_SUP liminf_SUP_INF)
3.292 +  then have *: "(umin o r) \<longlonglongrightarrow> liminf u" by (simp add: LIMSEQ_subseq_LIMSEQ subseq r)
3.293 +  have "\<And>n. umin(r n) = u(r n)" unfolding umin_def using mono
3.294 +    by (metis le_INF_iff antisym atLeast_def mem_Collect_eq order_refl)
3.295 +  then have "umin o r = u o r" unfolding o_def by simp
3.296 +  then have "(u o r) \<longlonglongrightarrow> liminf u" using * by simp
3.297 +  then show ?thesis using subseq r by blast
3.298 +next
3.299 +  assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. u m \<ge> u p))"
3.300 +  then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. u p > u m" by (force simp: not_le le_less)
3.301 +  have "\<exists>r. \<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<ge> u (r (Suc n)))"
3.302 +  proof (rule dependent_nat_choice)
3.303 +    fix x assume "N < x"
3.304 +    then have a: "finite {N<..x}" "{N<..x} \<noteq> {}" by simp_all
3.305 +    have "Min {u i |i. i \<in> {N<..x}} \<in> {u i |i. i \<in> {N<..x}}" apply (rule Min_in) using a by (auto)
3.306 +    then obtain p where "p \<in> {N<..x}" and upmin: "u p = Min{u i |i. i \<in> {N<..x}}" by auto
3.307 +    define U where "U = {m. m > p \<and> u p > u m}"
3.308 +    have "U \<noteq> {}" unfolding U_def using N[of p] p \<in> {N<..x} by auto
3.309 +    define y where "y = Inf U"
3.310 +    then have "y \<in> U" using U \<noteq> {} by (simp add: Inf_nat_def1)
3.311 +    have a: "\<And>i. i \<in> {N<..x} \<Longrightarrow> u i \<ge> u p"
3.312 +    proof -
3.313 +      fix i assume "i \<in> {N<..x}"
3.314 +      then have "u i \<in> {u i |i. i \<in> {N<..x}}" by blast
3.315 +      then show "u i \<ge> u p" using upmin by simp
3.316 +    qed
3.317 +    moreover have "u p > u y" using y \<in> U U_def by auto
3.318 +    ultimately have "y \<notin> {N<..x}" using not_le by blast
3.319 +    moreover have "y > N" using y \<in> U U_def p \<in> {N<..x} by auto
3.320 +    ultimately have "y > x" by auto
3.322 +    have "\<And>i. i \<in> {N<..y} \<Longrightarrow> u i \<ge> u y"
3.323 +    proof -
3.324 +      fix i assume "i \<in> {N<..y}" show "u i \<ge> u y"
3.325 +      proof (cases)
3.326 +        assume "i = y"
3.327 +        then show ?thesis by simp
3.328 +      next
3.329 +        assume "\<not>(i=y)"
3.330 +        then have i:"i \<in> {N<..<y}" using i \<in> {N<..y} by simp
3.331 +        have "u i \<ge> u p"
3.332 +        proof (cases)
3.333 +          assume "i \<le> x"
3.334 +          then have "i \<in> {N<..x}" using i by simp
3.335 +          then show ?thesis using a by simp
3.336 +        next
3.337 +          assume "\<not>(i \<le> x)"
3.338 +          then have "i > x" by simp
3.339 +          then have *: "i > p" using p \<in> {N<..x} by simp
3.340 +          have "i < Inf U" using i y_def by simp
3.341 +          then have "i \<notin> U" using Inf_nat_def not_less_Least by auto
3.342 +          then show ?thesis using U_def * by auto
3.343 +        qed
3.344 +        then show "u i \<ge> u y" using u p > u y by auto
3.345 +      qed
3.346 +    qed
3.347 +    then have "N < y \<and> x < y \<and> (\<forall>i\<in>{N<..y}. u i \<ge> u y)" using y > x y > N by auto
3.348 +    then show "\<exists>y>N. x < y \<and> (\<forall>i\<in>{N<..y}. u i \<ge> u y)" by auto
3.349 +  qed (auto)
3.350 +  then obtain r where r: "\<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<ge> u (r (Suc n)))" by auto
3.351 +  have "subseq r" using r by (auto simp: subseq_Suc_iff)
3.352 +  have "decseq (u o r)" unfolding o_def using r by (simp add: decseq_SucI order.strict_implies_order)
3.353 +  then have "(u o r) \<longlonglongrightarrow> (INF n. (u o r) n)" using LIMSEQ_INF by blast
3.354 +  then have "liminf (u o r) = (INF n. (u o r) n)" by (simp add: lim_imp_Liminf)
3.355 +  moreover have "liminf (u o r) \<ge> liminf u" using subseq r by (simp add: liminf_subseq_mono)
3.356 +  ultimately have "(INF n. (u o r) n) \<ge> liminf u" by simp
3.358 +  {
3.359 +    fix i assume i: "i \<in> {N<..}"
3.360 +    obtain n where "i < r (Suc n)" using subseq r using Suc_le_eq seq_suble by blast
3.361 +    then have "i \<in> {N<..r(Suc n)}" using i by simp
3.362 +    then have "u i \<ge> u (r(Suc n))" using r by simp
3.363 +    then have "u i \<ge> (INF n. (u o r) n)" unfolding o_def by (meson INF_lower2 UNIV_I)
3.364 +  }
3.365 +  then have "(INF i:{N<..}. u i) \<ge> (INF n. (u o r) n)" using INF_greatest by blast
3.366 +  then have "liminf u \<ge> (INF n. (u o r) n)" unfolding Liminf_def
3.367 +    by (metis (mono_tags, lifting) SUP_upper2 atLeast_Suc_greaterThan atLeast_def eventually_ge_at_top mem_Collect_eq)
3.368 +  then have "liminf u = (INF n. (u o r) n)" using (INF n. (u o r) n) \<ge> liminf u by simp
3.369 +  then have "(u o r) \<longlonglongrightarrow> liminf u" using (u o r) \<longlonglongrightarrow> (INF n. (u o r) n) by simp
3.370 +  then show ?thesis using subseq r by auto
3.371 +qed
3.374 +subsection {*Extended-Real.thy*}
3.376 +text{* The proof of this one is copied from \verb+ereal_add_mono+. *}
3.378 +  fixes a b c d :: ereal
3.379 +  assumes "a < b"
3.380 +    and "c < d"
3.381 +  shows "a + c < b + d"
3.382 +using assms
3.383 +apply (cases a)
3.384 +apply (cases rule: ereal3_cases[of b c d], auto)
3.385 +apply (cases rule: ereal3_cases[of b c d], auto)
3.386 +done
3.388 +text {* The next ones are analogues of \verb+mult_mono+ and \verb+mult_mono'+ in ereal.*}
3.390 +lemma ereal_mult_mono:
3.391 +  fixes a b c d::ereal
3.392 +  assumes "b \<ge> 0" "c \<ge> 0" "a \<le> b" "c \<le> d"
3.393 +  shows "a * c \<le> b * d"
3.394 +by (metis ereal_mult_right_mono mult.commute order_trans assms)
3.396 +lemma ereal_mult_mono':
3.397 +  fixes a b c d::ereal
3.398 +  assumes "a \<ge> 0" "c \<ge> 0" "a \<le> b" "c \<le> d"
3.399 +  shows "a * c \<le> b * d"
3.400 +by (metis ereal_mult_right_mono mult.commute order_trans assms)
3.402 +lemma ereal_mult_mono_strict:
3.403 +  fixes a b c d::ereal
3.404 +  assumes "b > 0" "c > 0" "a < b" "c < d"
3.405 +  shows "a * c < b * d"
3.406 +proof -
3.407 +  have "c < \<infinity>" using c < d by auto
3.408 +  then have "a * c < b * c" by (metis ereal_mult_strict_left_mono[OF assms(3) assms(2)] mult.commute)
3.409 +  moreover have "b * c \<le> b * d" using assms(2) assms(4) by (simp add: assms(1) ereal_mult_left_mono less_imp_le)
3.410 +  ultimately show ?thesis by simp
3.411 +qed
3.413 +lemma ereal_mult_mono_strict':
3.414 +  fixes a b c d::ereal
3.415 +  assumes "a > 0" "c > 0" "a < b" "c < d"
3.416 +  shows "a * c < b * d"
3.417 +apply (rule ereal_mult_mono_strict, auto simp add: assms) using assms by auto
3.420 +  fixes a b::ereal
3.421 +  shows "abs(a+b) \<le> abs a + abs b"
3.422 +by (cases rule: ereal2_cases[of a b]) (auto)
3.424 +lemma ereal_abs_diff:
3.425 +  fixes a b::ereal
3.426 +  shows "abs(a-b) \<le> abs a + abs b"
3.427 +by (cases rule: ereal2_cases[of a b]) (auto)
3.429 +lemma sum_constant_ereal:
3.430 +  fixes a::ereal
3.431 +  shows "(\<Sum>i\<in>I. a) = a * card I"
3.432 +apply (cases "finite I", induct set: finite, simp_all)
3.433 +apply (cases a, auto, metis (no_types, hide_lams) add.commute mult.commute semiring_normalization_rules(3))
3.434 +done
3.436 +lemma real_lim_then_eventually_real:
3.437 +  assumes "(u \<longlongrightarrow> ereal l) F"
3.438 +  shows "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) F"
3.439 +proof -
3.440 +  have "ereal l \<in> {-\<infinity><..<(\<infinity>::ereal)}" by simp
3.441 +  moreover have "open {-\<infinity><..<(\<infinity>::ereal)}" by simp
3.442 +  ultimately have "eventually (\<lambda>n. u n \<in> {-\<infinity><..<(\<infinity>::ereal)}) F" using assms tendsto_def by blast
3.443 +  moreover have "\<And>x. x \<in> {-\<infinity><..<(\<infinity>::ereal)} \<Longrightarrow> x = ereal(real_of_ereal x)" using ereal_real by auto
3.444 +  ultimately show ?thesis by (metis (mono_tags, lifting) eventually_mono)
3.445 +qed
3.447 +lemma ereal_Inf_cmult:
3.448 +  assumes "c>(0::real)"
3.449 +  shows "Inf {ereal c * x |x. P x} = ereal c * Inf {x. P x}"
3.450 +proof -
3.451 +  have "(\<lambda>x::ereal. c * x) (Inf {x::ereal. P x}) = Inf ((\<lambda>x::ereal. c * x){x::ereal. P x})"
3.452 +    apply (rule mono_bij_Inf)
3.453 +    apply (simp add: assms ereal_mult_left_mono less_imp_le mono_def)
3.454 +    apply (rule bij_betw_byWitness[of _ "\<lambda>x. (x::ereal) / c"], auto simp add: assms ereal_mult_divide)
3.455 +    using assms ereal_divide_eq apply auto
3.456 +    done
3.457 +  then show ?thesis by (simp only: setcompr_eq_image[symmetric])
3.458 +qed
3.461 +subsubsection {*Continuity of addition*}
3.463 +text {*The next few lemmas remove an unnecessary assumption in \verb+tendsto_add_ereal+, culminating
3.464 +in \verb+tendsto_add_ereal_general+ which essentially says that the addition
3.465 +is continuous on ereal times ereal, except at $(-\infty, \infty)$ and $(\infty, -\infty)$.
3.466 +It is much more convenient in many situations, see for instance the proof of
3.467 +\verb+tendsto_sum_ereal+ below. *}
3.470 +  fixes y :: ereal
3.471 +  assumes y: "y \<noteq> -\<infinity>"
3.472 +  assumes f: "(f \<longlongrightarrow> \<infinity>) F" and g: "(g \<longlongrightarrow> y) F"
3.473 +  shows "((\<lambda>x. f x + g x) \<longlongrightarrow> \<infinity>) F"
3.474 +proof -
3.475 +  have "\<exists>C. eventually (\<lambda>x. g x > ereal C) F"
3.476 +  proof (cases y)
3.477 +    case (real r)
3.478 +    have "y > y-1" using y real by (simp add: ereal_between(1))
3.479 +    then have "eventually (\<lambda>x. g x > y - 1) F" using g y order_tendsto_iff by auto
3.480 +    moreover have "y-1 = ereal(real_of_ereal(y-1))"
3.481 +      by (metis real ereal_eq_1(1) ereal_minus(1) real_of_ereal.simps(1))
3.482 +    ultimately have "eventually (\<lambda>x. g x > ereal(real_of_ereal(y - 1))) F" by simp
3.483 +    then show ?thesis by auto
3.484 +  next
3.485 +    case (PInf)
3.486 +    have "eventually (\<lambda>x. g x > ereal 0) F" using g PInf by (simp add: tendsto_PInfty)
3.487 +    then show ?thesis by auto
3.488 +  qed (simp add: y)
3.489 +  then obtain C::real where ge: "eventually (\<lambda>x. g x > ereal C) F" by auto
3.491 +  {
3.492 +    fix M::real
3.493 +    have "eventually (\<lambda>x. f x > ereal(M - C)) F" using f by (simp add: tendsto_PInfty)
3.494 +    then have "eventually (\<lambda>x. (f x > ereal (M-C)) \<and> (g x > ereal C)) F"
3.495 +      by (auto simp add: ge eventually_conj_iff)
3.496 +    moreover have "\<And>x. ((f x > ereal (M-C)) \<and> (g x > ereal C)) \<Longrightarrow> (f x + g x > ereal M)"
3.497 +      using ereal_add_strict_mono2 by fastforce
3.498 +    ultimately have "eventually (\<lambda>x. f x + g x > ereal M) F" using eventually_mono by force
3.499 +  }
3.500 +  then show ?thesis by (simp add: tendsto_PInfty)
3.501 +qed
3.503 +text{* One would like to deduce the next lemma from the previous one, but the fact
3.504 +that $-(x+y)$ is in general different from $(-x) + (-y)$ in ereal creates difficulties,
3.505 +so it is more efficient to copy the previous proof.*}
3.508 +  fixes y :: ereal
3.509 +  assumes y: "y \<noteq> \<infinity>"
3.510 +  assumes f: "(f \<longlongrightarrow> -\<infinity>) F" and g: "(g \<longlongrightarrow> y) F"
3.511 +  shows "((\<lambda>x. f x + g x) \<longlongrightarrow> -\<infinity>) F"
3.512 +proof -
3.513 +  have "\<exists>C. eventually (\<lambda>x. g x < ereal C) F"
3.514 +  proof (cases y)
3.515 +    case (real r)
3.516 +    have "y < y+1" using y real by (simp add: ereal_between(1))
3.517 +    then have "eventually (\<lambda>x. g x < y + 1) F" using g y order_tendsto_iff by force
3.518 +    moreover have "y+1 = ereal(real_of_ereal (y+1))" by (simp add: real)
3.519 +    ultimately have "eventually (\<lambda>x. g x < ereal(real_of_ereal(y + 1))) F" by simp
3.520 +    then show ?thesis by auto
3.521 +  next
3.522 +    case (MInf)
3.523 +    have "eventually (\<lambda>x. g x < ereal 0) F" using g MInf by (simp add: tendsto_MInfty)
3.524 +    then show ?thesis by auto
3.525 +  qed (simp add: y)
3.526 +  then obtain C::real where ge: "eventually (\<lambda>x. g x < ereal C) F" by auto
3.528 +  {
3.529 +    fix M::real
3.530 +    have "eventually (\<lambda>x. f x < ereal(M - C)) F" using f by (simp add: tendsto_MInfty)
3.531 +    then have "eventually (\<lambda>x. (f x < ereal (M- C)) \<and> (g x < ereal C)) F"
3.532 +      by (auto simp add: ge eventually_conj_iff)
3.533 +    moreover have "\<And>x. ((f x < ereal (M-C)) \<and> (g x < ereal C)) \<Longrightarrow> (f x + g x < ereal M)"
3.534 +      using ereal_add_strict_mono2 by fastforce
3.535 +    ultimately have "eventually (\<lambda>x. f x + g x < ereal M) F" using eventually_mono by force
3.536 +  }
3.537 +  then show ?thesis by (simp add: tendsto_MInfty)
3.538 +qed
3.541 +  fixes x y :: ereal
3.542 +  assumes y: "\<bar>y\<bar> \<noteq> \<infinity>"
3.543 +  assumes f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
3.544 +  shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
3.545 +proof (cases x)
3.546 +  case (real r)
3.547 +  have a: "\<bar>x\<bar> \<noteq> \<infinity>" by (simp add: real)
3.548 +  show ?thesis by (rule tendsto_add_ereal[OF a, OF y, OF f, OF g])
3.549 +next
3.550 +  case PInf
3.551 +  then show ?thesis using tendsto_add_ereal_PInf assms by force
3.552 +next
3.553 +  case MInf
3.554 +  then show ?thesis using tendsto_add_ereal_MInf assms
3.555 +    by (metis abs_ereal.simps(3) ereal_MInfty_eq_plus)
3.556 +qed
3.559 +  fixes x y :: ereal
3.560 +  assumes x: "\<bar>x\<bar> \<noteq> \<infinity>"
3.561 +      and f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
3.562 +  shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
3.563 +proof -
3.564 +  have "((\<lambda>x. g x + f x) \<longlongrightarrow> x + y) F"
3.565 +    using tendsto_add_ereal_general1[OF x, OF g, OF f] add.commute[of "y", of "x"] by simp
3.566 +  moreover have "\<And>x. g x + f x = f x + g x" using add.commute by auto
3.567 +  ultimately show ?thesis by simp
3.568 +qed
3.570 +text {* The next lemma says that the addition is continuous on ereal, except at
3.571 +the pairs $(-\infty, \infty)$ and $(\infty, -\infty)$. *}
3.573 +lemma tendsto_add_ereal_general [tendsto_intros]:
3.574 +  fixes x y :: ereal
3.575 +  assumes "\<not>((x=\<infinity> \<and> y=-\<infinity>) \<or> (x=-\<infinity> \<and> y=\<infinity>))"
3.576 +      and f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
3.577 +  shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
3.578 +proof (cases x)
3.579 +  case (real r)
3.580 +  show ?thesis
3.581 +    apply (rule tendsto_add_ereal_general2) using real assms by auto
3.582 +next
3.583 +  case (PInf)
3.584 +  then have "y \<noteq> -\<infinity>" using assms by simp
3.585 +  then show ?thesis using tendsto_add_ereal_PInf PInf assms by auto
3.586 +next
3.587 +  case (MInf)
3.588 +  then have "y \<noteq> \<infinity>" using assms by simp
3.589 +  then show ?thesis using tendsto_add_ereal_MInf MInf f g by (metis ereal_MInfty_eq_plus)
3.590 +qed
3.592 +subsubsection {*Continuity of multiplication*}
3.594 +text {* In the same way as for addition, we prove that the multiplication is continuous on
3.595 +ereal times ereal, except at $(\infty, 0)$ and $(-\infty, 0)$ and $(0, \infty)$ and $(0, -\infty)$,
3.596 +starting with specific situations.*}
3.598 +lemma tendsto_mult_real_ereal:
3.599 +  assumes "(u \<longlongrightarrow> ereal l) F" "(v \<longlongrightarrow> ereal m) F"
3.600 +  shows "((\<lambda>n. u n * v n) \<longlongrightarrow> ereal l * ereal m) F"
3.601 +proof -
3.602 +  have ureal: "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) F" by (rule real_lim_then_eventually_real[OF assms(1)])
3.603 +  then have "((\<lambda>n. ereal(real_of_ereal(u n))) \<longlongrightarrow> ereal l) F" using assms by auto
3.604 +  then have limu: "((\<lambda>n. real_of_ereal(u n)) \<longlongrightarrow> l) F" by auto
3.605 +  have vreal: "eventually (\<lambda>n. v n = ereal(real_of_ereal(v n))) F" by (rule real_lim_then_eventually_real[OF assms(2)])
3.606 +  then have "((\<lambda>n. ereal(real_of_ereal(v n))) \<longlongrightarrow> ereal m) F" using assms by auto
3.607 +  then have limv: "((\<lambda>n. real_of_ereal(v n)) \<longlongrightarrow> m) F" by auto
3.609 +  {
3.610 +    fix n assume "u n = ereal(real_of_ereal(u n))" "v n = ereal(real_of_ereal(v n))"
3.611 +    then have "ereal(real_of_ereal(u n) * real_of_ereal(v n)) = u n * v n" by (metis times_ereal.simps(1))
3.612 +  }
3.613 +  then have *: "eventually (\<lambda>n. ereal(real_of_ereal(u n) * real_of_ereal(v n)) = u n * v n) F"
3.614 +    using eventually_elim2[OF ureal vreal] by auto
3.616 +  have "((\<lambda>n. real_of_ereal(u n) * real_of_ereal(v n)) \<longlongrightarrow> l * m) F" using tendsto_mult[OF limu limv] by auto
3.617 +  then have "((\<lambda>n. ereal(real_of_ereal(u n)) * real_of_ereal(v n)) \<longlongrightarrow> ereal(l * m)) F" by auto
3.618 +  then show ?thesis using * filterlim_cong by fastforce
3.619 +qed
3.621 +lemma tendsto_mult_ereal_PInf:
3.622 +  fixes f g::"_ \<Rightarrow> ereal"
3.623 +  assumes "(f \<longlongrightarrow> l) F" "l>0" "(g \<longlongrightarrow> \<infinity>) F"
3.624 +  shows "((\<lambda>x. f x * g x) \<longlongrightarrow> \<infinity>) F"
3.625 +proof -
3.626 +  obtain a::real where "0 < ereal a" "a < l" using assms(2) using ereal_dense2 by blast
3.627 +  have *: "eventually (\<lambda>x. f x > a) F" using a < l assms(1) by (simp add: order_tendsto_iff)
3.628 +  {
3.629 +    fix K::real
3.630 +    define M where "M = max K 1"
3.631 +    then have "M > 0" by simp
3.632 +    then have "ereal(M/a) > 0" using ereal a > 0 by simp
3.633 +    then have "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > ereal a * ereal(M/a))"
3.634 +      using ereal_mult_mono_strict'[where ?c = "M/a", OF 0 < ereal a] by auto
3.635 +    moreover have "ereal a * ereal(M/a) = M" using ereal a > 0 by simp
3.636 +    ultimately have "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > M)" by simp
3.637 +    moreover have "M \<ge> K" unfolding M_def by simp
3.638 +    ultimately have Imp: "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > K)"
3.639 +      using ereal_less_eq(3) le_less_trans by blast
3.641 +    have "eventually (\<lambda>x. g x > M/a) F" using assms(3) by (simp add: tendsto_PInfty)
3.642 +    then have "eventually (\<lambda>x. (f x > a) \<and> (g x > M/a)) F"
3.643 +      using * by (auto simp add: eventually_conj_iff)
3.644 +    then have "eventually (\<lambda>x. f x * g x > K) F" using eventually_mono Imp by force
3.645 +  }
3.646 +  then show ?thesis by (auto simp add: tendsto_PInfty)
3.647 +qed
3.649 +lemma tendsto_mult_ereal_pos:
3.650 +  fixes f g::"_ \<Rightarrow> ereal"
3.651 +  assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "l>0" "m>0"
3.652 +  shows "((\<lambda>x. f x * g x) \<longlongrightarrow> l * m) F"
3.653 +proof (cases)
3.654 +  assume *: "l = \<infinity> \<or> m = \<infinity>"
3.655 +  then show ?thesis
3.656 +  proof (cases)
3.657 +    assume "m = \<infinity>"
3.658 +    then show ?thesis using tendsto_mult_ereal_PInf assms by auto
3.659 +  next
3.660 +    assume "\<not>(m = \<infinity>)"
3.661 +    then have "l = \<infinity>" using * by simp
3.662 +    then have "((\<lambda>x. g x * f x) \<longlongrightarrow> l * m) F" using tendsto_mult_ereal_PInf assms by auto
3.663 +    moreover have "\<And>x. g x * f x = f x * g x" using mult.commute by auto
3.664 +    ultimately show ?thesis by simp
3.665 +  qed
3.666 +next
3.667 +  assume "\<not>(l = \<infinity> \<or> m = \<infinity>)"
3.668 +  then have "l < \<infinity>" "m < \<infinity>" by auto
3.669 +  then obtain lr mr where "l = ereal lr" "m = ereal mr"
3.670 +    using l>0 m>0 by (metis ereal_cases ereal_less(6) not_less_iff_gr_or_eq)
3.671 +  then show ?thesis using tendsto_mult_real_ereal assms by auto
3.672 +qed
3.674 +text {*We reduce the general situation to the positive case by multiplying by suitable signs.
3.675 +Unfortunately, as ereal is not a ring, all the neat sign lemmas are not available there. We
3.676 +give the bare minimum we need.*}
3.678 +lemma ereal_sgn_abs:
3.679 +  fixes l::ereal
3.680 +  shows "sgn(l) * l = abs(l)"
3.681 +apply (cases l) by (auto simp add: sgn_if ereal_less_uminus_reorder)
3.683 +lemma sgn_squared_ereal:
3.684 +  assumes "l \<noteq> (0::ereal)"
3.685 +  shows "sgn(l) * sgn(l) = 1"
3.686 +apply (cases l) using assms by (auto simp add: one_ereal_def sgn_if)
3.688 +lemma tendsto_mult_ereal [tendsto_intros]:
3.689 +  fixes f g::"_ \<Rightarrow> ereal"
3.690 +  assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "\<not>((l=0 \<and> abs(m) = \<infinity>) \<or> (m=0 \<and> abs(l) = \<infinity>))"
3.691 +  shows "((\<lambda>x. f x * g x) \<longlongrightarrow> l * m) F"
3.692 +proof (cases)
3.693 +  assume "l=0 \<or> m=0"
3.694 +  then have "abs(l) \<noteq> \<infinity>" "abs(m) \<noteq> \<infinity>" using assms(3) by auto
3.695 +  then obtain lr mr where "l = ereal lr" "m = ereal mr" by auto
3.696 +  then show ?thesis using tendsto_mult_real_ereal assms by auto
3.697 +next
3.698 +  have sgn_finite: "\<And>a::ereal. abs(sgn a) \<noteq> \<infinity>"
3.699 +    by (metis MInfty_neq_ereal(2) PInfty_neq_ereal(2) abs_eq_infinity_cases ereal_times(1) ereal_times(3) ereal_uminus_eq_reorder sgn_ereal.elims)
3.700 +  then have sgn_finite2: "\<And>a b::ereal. abs(sgn a * sgn b) \<noteq> \<infinity>"
3.701 +    by (metis abs_eq_infinity_cases abs_ereal.simps(2) abs_ereal.simps(3) ereal_mult_eq_MInfty ereal_mult_eq_PInfty)
3.702 +  assume "\<not>(l=0 \<or> m=0)"
3.703 +  then have "l \<noteq> 0" "m \<noteq> 0" by auto
3.704 +  then have "abs(l) > 0" "abs(m) > 0"
3.705 +    by (metis abs_ereal_ge0 abs_ereal_less0 abs_ereal_pos ereal_uminus_uminus ereal_uminus_zero less_le not_less)+
3.706 +  then have "sgn(l) * l > 0" "sgn(m) * m > 0" using ereal_sgn_abs by auto
3.707 +  moreover have "((\<lambda>x. sgn(l) * f x) \<longlongrightarrow> (sgn(l) * l)) F"
3.708 +    by (rule tendsto_cmult_ereal, auto simp add: sgn_finite assms(1))
3.709 +  moreover have "((\<lambda>x. sgn(m) * g x) \<longlongrightarrow> (sgn(m) * m)) F"
3.710 +    by (rule tendsto_cmult_ereal, auto simp add: sgn_finite assms(2))
3.711 +  ultimately have *: "((\<lambda>x. (sgn(l) * f x) * (sgn(m) * g x)) \<longlongrightarrow> (sgn(l) * l) * (sgn(m) * m)) F"
3.712 +    using tendsto_mult_ereal_pos by force
3.713 +  have "((\<lambda>x. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x))) \<longlongrightarrow> (sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m))) F"
3.714 +    by (rule tendsto_cmult_ereal, auto simp add: sgn_finite2 *)
3.715 +  moreover have "\<And>x. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x)) = f x * g x"
3.716 +    by (metis mult.left_neutral sgn_squared_ereal[OF l \<noteq> 0] sgn_squared_ereal[OF m \<noteq> 0] mult.assoc mult.commute)
3.717 +  moreover have "(sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m)) = l * m"
3.718 +    by (metis mult.left_neutral sgn_squared_ereal[OF l \<noteq> 0] sgn_squared_ereal[OF m \<noteq> 0] mult.assoc mult.commute)
3.719 +  ultimately show ?thesis by auto
3.720 +qed
3.722 +lemma tendsto_cmult_ereal_general [tendsto_intros]:
3.723 +  fixes f::"_ \<Rightarrow> ereal" and c::ereal
3.724 +  assumes "(f \<longlongrightarrow> l) F" "\<not> (l=0 \<and> abs(c) = \<infinity>)"
3.725 +  shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * l) F"
3.726 +by (cases "c = 0", auto simp add: assms tendsto_mult_ereal)
3.729 +subsubsection {*Continuity of division*}
3.731 +lemma tendsto_inverse_ereal_PInf:
3.732 +  fixes u::"_ \<Rightarrow> ereal"
3.733 +  assumes "(u \<longlongrightarrow> \<infinity>) F"
3.734 +  shows "((\<lambda>x. 1/ u x) \<longlongrightarrow> 0) F"
3.735 +proof -
3.736 +  {
3.737 +    fix e::real assume "e>0"
3.738 +    have "1/e < \<infinity>" by auto
3.739 +    then have "eventually (\<lambda>n. u n > 1/e) F" using assms(1) by (simp add: tendsto_PInfty)
3.740 +    moreover
3.741 +    {
3.742 +      fix z::ereal assume "z>1/e"
3.743 +      then have "z>0" using e>0 using less_le_trans not_le by fastforce
3.744 +      then have "1/z \<ge> 0" by auto
3.745 +      moreover have "1/z < e" using e>0 z>1/e
3.746 +        apply (cases z) apply auto
3.747 +        by (metis (mono_tags, hide_lams) less_ereal.simps(2) less_ereal.simps(4) divide_less_eq ereal_divide_less_pos ereal_less(4)
3.748 +            ereal_less_eq(4) less_le_trans mult_eq_0_iff not_le not_one_less_zero times_ereal.simps(1))
3.749 +      ultimately have "1/z \<ge> 0" "1/z < e" by auto
3.750 +    }
3.751 +    ultimately have "eventually (\<lambda>n. 1/u n<e) F" "eventually (\<lambda>n. 1/u n\<ge>0) F" by (auto simp add: eventually_mono)
3.752 +  } note * = this
3.753 +  show ?thesis
3.754 +  proof (subst order_tendsto_iff, auto)
3.755 +    fix a::ereal assume "a<0"
3.756 +    then show "eventually (\<lambda>n. 1/u n > a) F" using *(2) eventually_mono less_le_trans linordered_field_no_ub by fastforce
3.757 +  next
3.758 +    fix a::ereal assume "a>0"
3.759 +    then obtain e::real where "e>0" "a>e" using ereal_dense2 ereal_less(2) by blast
3.760 +    then have "eventually (\<lambda>n. 1/u n < e) F" using *(1) by auto
3.761 +    then show "eventually (\<lambda>n. 1/u n < a) F" using a>e by (metis (mono_tags, lifting) eventually_mono less_trans)
3.762 +  qed
3.763 +qed
3.765 +text {* The next lemma deserves to exist by itself, as it is so common and useful. *}
3.767 +lemma tendsto_inverse_real [tendsto_intros]:
3.768 +  fixes u::"_ \<Rightarrow> real"
3.769 +  shows "(u \<longlongrightarrow> l) F \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> ((\<lambda>x. 1/ u x) \<longlongrightarrow> 1/l) F"
3.770 +  using tendsto_inverse unfolding inverse_eq_divide .
3.772 +lemma tendsto_inverse_ereal [tendsto_intros]:
3.773 +  fixes u::"_ \<Rightarrow> ereal"
3.774 +  assumes "(u \<longlongrightarrow> l) F" "l \<noteq> 0"
3.775 +  shows "((\<lambda>x. 1/ u x) \<longlongrightarrow> 1/l) F"
3.776 +proof (cases l)
3.777 +  case (real r)
3.778 +  then have "r \<noteq> 0" using assms(2) by auto
3.779 +  then have "1/l = ereal(1/r)" using real by (simp add: one_ereal_def)
3.780 +  define v where "v = (\<lambda>n. real_of_ereal(u n))"
3.781 +  have ureal: "eventually (\<lambda>n. u n = ereal(v n)) F" unfolding v_def using real_lim_then_eventually_real assms(1) real by auto
3.782 +  then have "((\<lambda>n. ereal(v n)) \<longlongrightarrow> ereal r) F" using assms real v_def by auto
3.783 +  then have *: "((\<lambda>n. v n) \<longlongrightarrow> r) F" by auto
3.784 +  then have "((\<lambda>n. 1/v n) \<longlongrightarrow> 1/r) F" using \<open>r \<noteq> 0\<close> tendsto_inverse_real by auto
3.785 +  then have lim: "((\<lambda>n. ereal(1/v n)) \<longlongrightarrow> 1/l) F" using \<open>1/l = ereal(1/r)\<close> by auto
3.787 +  have "r \<in> -{0}" "open (-{(0::real)})" using \<open>r \<noteq> 0\<close> by auto
3.788 +  then have "eventually (\<lambda>n. v n \<in> -{0}) F" using * using topological_tendstoD by blast
3.789 +  then have "eventually (\<lambda>n. v n \<noteq> 0) F" by auto
3.790 +  moreover
3.791 +  {
3.792 +    fix n assume H: "v n \<noteq> 0" "u n = ereal(v n)"
3.793 +    then have "ereal(1/v n) = 1/ereal(v n)" by (simp add: one_ereal_def)
3.794 +    then have "ereal(1/v n) = 1/u n" using H(2) by simp
3.795 +  }
3.796 +  ultimately have "eventually (\<lambda>n. ereal(1/v n) = 1/u n) F" using ureal eventually_elim2 by force
3.797 +  with Lim_transform_eventually[OF this lim] show ?thesis by simp
3.798 +next
3.799 +  case (PInf)
3.800 +  then have "1/l = 0" by auto
3.801 +  then show ?thesis using tendsto_inverse_ereal_PInf assms PInf by auto
3.802 +next
3.803 +  case (MInf)
3.804 +  then have "1/l = 0" by auto
3.805 +  have "1/z = -1/ -z" if "z < 0" for z::ereal
3.806 +    apply (cases z) using divide_ereal_def \<open> z < 0 \<close> by auto
3.807 +  moreover have "eventually (\<lambda>n. u n < 0) F" by (metis (no_types) MInf assms(1) tendsto_MInfty zero_ereal_def)
3.808 +  ultimately have *: "eventually (\<lambda>n. -1/-u n = 1/u n) F" by (simp add: eventually_mono)
3.810 +  define v where "v = (\<lambda>n. - u n)"
3.811 +  have "(v \<longlongrightarrow> \<infinity>) F" unfolding v_def using MInf assms(1) tendsto_uminus_ereal by fastforce
3.812 +  then have "((\<lambda>n. 1/v n) \<longlongrightarrow> 0) F" using tendsto_inverse_ereal_PInf by auto
3.813 +  then have "((\<lambda>n. -1/v n) \<longlongrightarrow> 0) F" using tendsto_uminus_ereal by fastforce
3.814 +  then show ?thesis unfolding v_def using Lim_transform_eventually[OF *] \<open> 1/l = 0 \<close> by auto
3.815 +qed
3.817 +lemma tendsto_divide_ereal [tendsto_intros]:
3.818 +  fixes f g::"_ \<Rightarrow> ereal"
3.819 +  assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "m \<noteq> 0" "\<not>(abs(l) = \<infinity> \<and> abs(m) = \<infinity>)"
3.820 +  shows "((\<lambda>x. f x / g x) \<longlongrightarrow> l / m) F"
3.821 +proof -
3.822 +  define h where "h = (\<lambda>x. 1/ g x)"
3.823 +  have *: "(h \<longlongrightarrow> 1/m) F" unfolding h_def using assms(2) assms(3) tendsto_inverse_ereal by auto
3.824 +  have "((\<lambda>x. f x * h x) \<longlongrightarrow> l * (1/m)) F"
3.825 +    apply (rule tendsto_mult_ereal[OF assms(1) *]) using assms(3) assms(4) by (auto simp add: divide_ereal_def)
3.826 +  moreover have "f x * h x = f x / g x" for x unfolding h_def by (simp add: divide_ereal_def)
3.827 +  moreover have "l * (1/m) = l/m" by (simp add: divide_ereal_def)
3.828 +  ultimately show ?thesis unfolding h_def using Lim_transform_eventually by auto
3.829 +qed
3.832 +subsubsection {*Further limits*}
3.834 +lemma id_nat_ereal_tendsto_PInf [tendsto_intros]:
3.835 +  "(\<lambda> n::nat. real n) \<longlonglongrightarrow> \<infinity>"
3.836 +by (simp add: filterlim_real_sequentially tendsto_PInfty_eq_at_top)
3.838 +lemma tendsto_at_top_pseudo_inverse [tendsto_intros]:
3.839 +  fixes u::"nat \<Rightarrow> nat"
3.840 +  assumes "LIM n sequentially. u n :> at_top"
3.841 +  shows "LIM n sequentially. Inf {N. u N \<ge> n} :> at_top"
3.842 +proof -
3.843 +  {
3.844 +    fix C::nat
3.845 +    define M where "M = Max {u n| n. n \<le> C}+1"
3.846 +    {
3.847 +      fix n assume "n \<ge> M"
3.848 +      have "eventually (\<lambda>N. u N \<ge> n) sequentially" using assms
3.849 +        by (simp add: filterlim_at_top)
3.850 +      then have *: "{N. u N \<ge> n} \<noteq> {}" by force
3.852 +      have "N > C" if "u N \<ge> n" for N
3.853 +      proof (rule ccontr)
3.854 +        assume "\<not>(N > C)"
3.855 +        have "u N \<le> Max {u n| n. n \<le> C}"
3.856 +          apply (rule Max_ge) using \<not>(N > C) by auto
3.857 +        then show False using u N \<ge> n n \<ge> M unfolding M_def by auto
3.858 +      qed
3.859 +      then have **: "{N. u N \<ge> n} \<subseteq> {C..}" by fastforce
3.860 +      have "Inf {N. u N \<ge> n} \<ge> C"
3.861 +        by (metis "*" "**" Inf_nat_def1 atLeast_iff subset_eq)
3.862 +    }
3.863 +    then have "eventually (\<lambda>n. Inf {N. u N \<ge> n} \<ge> C) sequentially"
3.864 +      using eventually_sequentially by auto
3.865 +  }
3.866 +  then show ?thesis using filterlim_at_top by auto
3.867 +qed
3.869 +lemma pseudo_inverse_finite_set:
3.870 +  fixes u::"nat \<Rightarrow> nat"
3.871 +  assumes "LIM n sequentially. u n :> at_top"
3.872 +  shows "finite {N. u N \<le> n}"
3.873 +proof -
3.874 +  fix n
3.875 +  have "eventually (\<lambda>N. u N \<ge> n+1) sequentially" using assms
3.876 +    by (simp add: filterlim_at_top)
3.877 +  then obtain N1 where N1: "\<And>N. N \<ge> N1 \<Longrightarrow> u N \<ge> n + 1"
3.878 +    using eventually_sequentially by auto
3.879 +  have "{N. u N \<le> n} \<subseteq> {..<N1}"
3.880 +    apply auto using N1 by (metis Suc_eq_plus1 not_less not_less_eq_eq)
3.881 +  then show "finite {N. u N \<le> n}" by (simp add: finite_subset)
3.882 +qed
3.884 +lemma tendsto_at_top_pseudo_inverse2 [tendsto_intros]:
3.885 +  fixes u::"nat \<Rightarrow> nat"
3.886 +  assumes "LIM n sequentially. u n :> at_top"
3.887 +  shows "LIM n sequentially. Max {N. u N \<le> n} :> at_top"
3.888 +proof -
3.889 +  {
3.890 +    fix N0::nat
3.891 +    have "N0 \<le> Max {N. u N \<le> n}" if "n \<ge> u N0" for n
3.892 +      apply (rule Max.coboundedI) using pseudo_inverse_finite_set[OF assms] that by auto
3.893 +    then have "eventually (\<lambda>n. N0 \<le> Max {N. u N \<le> n}) sequentially"
3.894 +      using eventually_sequentially by blast
3.895 +  }
3.896 +  then show ?thesis using filterlim_at_top by auto
3.897 +qed
3.899 +lemma ereal_truncation_top [tendsto_intros]:
3.900 +  fixes x::ereal
3.901 +  shows "(\<lambda>n::nat. min x n) \<longlonglongrightarrow> x"
3.902 +proof (cases x)
3.903 +  case (real r)
3.904 +  then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
3.905 +  then have "min x n = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
3.906 +  then have "eventually (\<lambda>n. min x n = x) sequentially" using eventually_at_top_linorder by blast
3.907 +  then show ?thesis by (simp add: Lim_eventually)
3.908 +next
3.909 +  case (PInf)
3.910 +  then have "min x n = n" for n::nat by (auto simp add: min_def)
3.911 +  then show ?thesis using id_nat_ereal_tendsto_PInf PInf by auto
3.912 +next
3.913 +  case (MInf)
3.914 +  then have "min x n = x" for n::nat by (auto simp add: min_def)
3.915 +  then show ?thesis by auto
3.916 +qed
3.918 +lemma ereal_truncation_real_top [tendsto_intros]:
3.919 +  fixes x::ereal
3.920 +  assumes "x \<noteq> - \<infinity>"
3.921 +  shows "(\<lambda>n::nat. real_of_ereal(min x n)) \<longlonglongrightarrow> x"
3.922 +proof (cases x)
3.923 +  case (real r)
3.924 +  then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
3.925 +  then have "min x n = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
3.926 +  then have "real_of_ereal(min x n) = r" if "n \<ge> K" for n using real that by auto
3.927 +  then have "eventually (\<lambda>n. real_of_ereal(min x n) = r) sequentially" using eventually_at_top_linorder by blast
3.928 +  then have "(\<lambda>n. real_of_ereal(min x n)) \<longlonglongrightarrow> r" by (simp add: Lim_eventually)
3.929 +  then show ?thesis using real by auto
3.930 +next
3.931 +  case (PInf)
3.932 +  then have "real_of_ereal(min x n) = n" for n::nat by (auto simp add: min_def)
3.933 +  then show ?thesis using id_nat_ereal_tendsto_PInf PInf by auto
3.934 +qed (simp add: assms)
3.936 +lemma ereal_truncation_bottom [tendsto_intros]:
3.937 +  fixes x::ereal
3.938 +  shows "(\<lambda>n::nat. max x (- real n)) \<longlonglongrightarrow> x"
3.939 +proof (cases x)
3.940 +  case (real r)
3.941 +  then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
3.942 +  then have "max x (-real n) = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
3.943 +  then have "eventually (\<lambda>n. max x (-real n) = x) sequentially" using eventually_at_top_linorder by blast
3.944 +  then show ?thesis by (simp add: Lim_eventually)
3.945 +next
3.946 +  case (MInf)
3.947 +  then have "max x (-real n) = (-1)* ereal(real n)" for n::nat by (auto simp add: max_def)
3.948 +  moreover have "(\<lambda>n. (-1)* ereal(real n)) \<longlonglongrightarrow> -\<infinity>"
3.949 +    using tendsto_cmult_ereal[of "-1", OF _ id_nat_ereal_tendsto_PInf] by (simp add: one_ereal_def)
3.950 +  ultimately show ?thesis using MInf by auto
3.951 +next
3.952 +  case (PInf)
3.953 +  then have "max x (-real n) = x" for n::nat by (auto simp add: max_def)
3.954 +  then show ?thesis by auto
3.955 +qed
3.957 +lemma ereal_truncation_real_bottom [tendsto_intros]:
3.958 +  fixes x::ereal
3.959 +  assumes "x \<noteq> \<infinity>"
3.960 +  shows "(\<lambda>n::nat. real_of_ereal(max x (- real n))) \<longlonglongrightarrow> x"
3.961 +proof (cases x)
3.962 +  case (real r)
3.963 +  then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
3.964 +  then have "max x (-real n) = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
3.965 +  then have "real_of_ereal(max x (-real n)) = r" if "n \<ge> K" for n using real that by auto
3.966 +  then have "eventually (\<lambda>n. real_of_ereal(max x (-real n)) = r) sequentially" using eventually_at_top_linorder by blast
3.967 +  then have "(\<lambda>n. real_of_ereal(max x (-real n))) \<longlonglongrightarrow> r" by (simp add: Lim_eventually)
3.968 +  then show ?thesis using real by auto
3.969 +next
3.970 +  case (MInf)
3.971 +  then have "real_of_ereal(max x (-real n)) = (-1)* ereal(real n)" for n::nat by (auto simp add: max_def)
3.972 +  moreover have "(\<lambda>n. (-1)* ereal(real n)) \<longlonglongrightarrow> -\<infinity>"
3.973 +    using tendsto_cmult_ereal[of "-1", OF _ id_nat_ereal_tendsto_PInf] by (simp add: one_ereal_def)
3.974 +  ultimately show ?thesis using MInf by auto
3.975 +qed (simp add: assms)
3.977 +text {* the next one is copied from \verb+tendsto_sum+. *}
3.978 +lemma tendsto_sum_ereal [tendsto_intros]:
3.979 +  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> ereal"
3.980 +  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> a i) F"
3.981 +          "\<And>i. abs(a i) \<noteq> \<infinity>"
3.982 +  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) \<longlongrightarrow> (\<Sum>i\<in>S. a i)) F"
3.983 +proof (cases "finite S")
3.984 +  assume "finite S" then show ?thesis using assms
3.985 +    by (induct, simp, simp add: tendsto_add_ereal_general2 assms)
3.986 +qed(simp)
3.988 +subsubsection {*Limsups and liminfs*}
3.990 +lemma limsup_finite_then_bounded:
3.991 +  fixes u::"nat \<Rightarrow> real"
3.992 +  assumes "limsup u < \<infinity>"
3.993 +  shows "\<exists>C. \<forall>n. u n \<le> C"
3.994 +proof -
3.995 +  obtain C where C: "limsup u < C" "C < \<infinity>" using assms ereal_dense2 by blast
3.996 +  then have "C = ereal(real_of_ereal C)" using ereal_real by force
3.997 +  have "eventually (\<lambda>n. u n < C) sequentially" using C(1) unfolding Limsup_def
3.998 +    apply (auto simp add: INF_less_iff)
3.999 +    using SUP_lessD eventually_mono by fastforce
3.1000 +  then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> u n < C" using eventually_sequentially by auto
3.1001 +  define D where "D = max (real_of_ereal C) (Max {u n |n. n \<le> N})"
3.1002 +  have "\<And>n. u n \<le> D"
3.1003 +  proof -
3.1004 +    fix n show "u n \<le> D"
3.1005 +    proof (cases)
3.1006 +      assume *: "n \<le> N"
3.1007 +      have "u n \<le> Max {u n |n. n \<le> N}" by (rule Max_ge, auto simp add: *)
3.1008 +      then show "u n \<le> D" unfolding D_def by linarith
3.1009 +    next
3.1010 +      assume "\<not>(n \<le> N)"
3.1011 +      then have "n \<ge> N" by simp
3.1012 +      then have "u n < C" using N by auto
3.1013 +      then have "u n < real_of_ereal C" using C = ereal(real_of_ereal C) less_ereal.simps(1) by fastforce
3.1014 +      then show "u n \<le> D" unfolding D_def by linarith
3.1015 +    qed
3.1016 +  qed
3.1017 +  then show ?thesis by blast
3.1018 +qed
3.1020 +lemma liminf_finite_then_bounded_below:
3.1021 +  fixes u::"nat \<Rightarrow> real"
3.1022 +  assumes "liminf u > -\<infinity>"
3.1023 +  shows "\<exists>C. \<forall>n. u n \<ge> C"
3.1024 +proof -
3.1025 +  obtain C where C: "liminf u > C" "C > -\<infinity>" using assms using ereal_dense2 by blast
3.1026 +  then have "C = ereal(real_of_ereal C)" using ereal_real by force
3.1027 +  have "eventually (\<lambda>n. u n > C) sequentially" using C(1) unfolding Liminf_def
3.1028 +    apply (auto simp add: less_SUP_iff)
3.1029 +    using eventually_elim2 less_INF_D by fastforce
3.1030 +  then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> u n > C" using eventually_sequentially by auto
3.1031 +  define D where "D = min (real_of_ereal C) (Min {u n |n. n \<le> N})"
3.1032 +  have "\<And>n. u n \<ge> D"
3.1033 +  proof -
3.1034 +    fix n show "u n \<ge> D"
3.1035 +    proof (cases)
3.1036 +      assume *: "n \<le> N"
3.1037 +      have "u n \<ge> Min {u n |n. n \<le> N}" by (rule Min_le, auto simp add: *)
3.1038 +      then show "u n \<ge> D" unfolding D_def by linarith
3.1039 +    next
3.1040 +      assume "\<not>(n \<le> N)"
3.1041 +      then have "n \<ge> N" by simp
3.1042 +      then have "u n > C" using N by auto
3.1043 +      then have "u n > real_of_ereal C" using C = ereal(real_of_ereal C) less_ereal.simps(1) by fastforce
3.1044 +      then show "u n \<ge> D" unfolding D_def by linarith
3.1045 +    qed
3.1046 +  qed
3.1047 +  then show ?thesis by blast
3.1048 +qed
3.1050 +lemma liminf_upper_bound:
3.1051 +  fixes u:: "nat \<Rightarrow> ereal"
3.1052 +  assumes "liminf u < l"
3.1053 +  shows "\<exists>N>k. u N < l"
3.1054 +by (metis assms gt_ex less_le_trans liminf_bounded_iff not_less)
3.1056 +text {* The following statement about limsups is reduced to a statement about limits using
3.1057 +subsequences thanks to \verb+limsup_subseq_lim+. The statement for limits follows for instance from
3.1061 +  fixes u v::"nat \<Rightarrow> ereal"
3.1062 +  shows "limsup (\<lambda>n. u n + v n) \<le> limsup u + limsup v"
3.1063 +proof (cases)
3.1064 +  assume "(limsup u = \<infinity>) \<or> (limsup v = \<infinity>)"
3.1065 +  then have "limsup u + limsup v = \<infinity>" by simp
3.1066 +  then show ?thesis by auto
3.1067 +next
3.1068 +  assume "\<not>((limsup u = \<infinity>) \<or> (limsup v = \<infinity>))"
3.1069 +  then have "limsup u < \<infinity>" "limsup v < \<infinity>" by auto
3.1071 +  define w where "w = (\<lambda>n. u n + v n)"
3.1072 +  obtain r where r: "subseq r" "(w o r) \<longlonglongrightarrow> limsup w" using limsup_subseq_lim by auto
3.1073 +  obtain s where s: "subseq s" "(u o r o s) \<longlonglongrightarrow> limsup (u o r)" using limsup_subseq_lim by auto
3.1074 +  obtain t where t: "subseq t" "(v o r o s o t) \<longlonglongrightarrow> limsup (v o r o s)" using limsup_subseq_lim by auto
3.1076 +  define a where "a = r o s o t"
3.1077 +  have "subseq a" using r s t by (simp add: a_def subseq_o)
3.1078 +  have l:"(w o a) \<longlonglongrightarrow> limsup w"
3.1079 +         "(u o a) \<longlonglongrightarrow> limsup (u o r)"
3.1080 +         "(v o a) \<longlonglongrightarrow> limsup (v o r o s)"
3.1081 +  apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
3.1082 +  apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
3.1083 +  apply (metis (no_types, lifting) t(2) a_def comp_assoc)
3.1084 +  done
3.1086 +  have "limsup (u o r) \<le> limsup u" by (simp add: limsup_subseq_mono r(1))
3.1087 +  then have a: "limsup (u o r) \<noteq> \<infinity>" using limsup u < \<infinity> by auto
3.1088 +  have "limsup (v o r o s) \<le> limsup v" by (simp add: comp_assoc limsup_subseq_mono r(1) s(1) subseq_o)
3.1089 +  then have b: "limsup (v o r o s) \<noteq> \<infinity>" using limsup v < \<infinity> by auto
3.1091 +  have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> limsup (u o r) + limsup (v o r o s)"
3.1092 +    using l tendsto_add_ereal_general a b by fastforce
3.1093 +  moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
3.1094 +  ultimately have "(w o a) \<longlonglongrightarrow> limsup (u o r) + limsup (v o r o s)" by simp
3.1095 +  then have "limsup w = limsup (u o r) + limsup (v o r o s)" using l(1) LIMSEQ_unique by blast
3.1096 +  then have "limsup w \<le> limsup u + limsup v"
3.1097 +    using limsup (u o r) \<le> limsup u limsup (v o r o s) \<le> limsup v ereal_add_mono by simp
3.1098 +  then show ?thesis unfolding w_def by simp
3.1099 +qed
3.1101 +text {* There is an asymmetry between liminfs and limsups in ereal, as $\infty + (-\infty) = \infty$.
3.1102 +This explains why there are more assumptions in the next lemma dealing with liminfs that in the
3.1103 +previous one about limsups.*}
3.1106 +  fixes u v::"nat \<Rightarrow> ereal"
3.1107 +  assumes "\<not>((liminf u = \<infinity> \<and> liminf v = -\<infinity>) \<or> (liminf u = -\<infinity> \<and> liminf v = \<infinity>))"
3.1108 +  shows "liminf (\<lambda>n. u n + v n) \<ge> liminf u + liminf v"
3.1109 +proof (cases)
3.1110 +  assume "(liminf u = -\<infinity>) \<or> (liminf v = -\<infinity>)"
3.1111 +  then have *: "liminf u + liminf v = -\<infinity>" using assms by auto
3.1112 +  show ?thesis by (simp add: *)
3.1113 +next
3.1114 +  assume "\<not>((liminf u = -\<infinity>) \<or> (liminf v = -\<infinity>))"
3.1115 +  then have "liminf u > -\<infinity>" "liminf v > -\<infinity>" by auto
3.1117 +  define w where "w = (\<lambda>n. u n + v n)"
3.1118 +  obtain r where r: "subseq r" "(w o r) \<longlonglongrightarrow> liminf w" using liminf_subseq_lim by auto
3.1119 +  obtain s where s: "subseq s" "(u o r o s) \<longlonglongrightarrow> liminf (u o r)" using liminf_subseq_lim by auto
3.1120 +  obtain t where t: "subseq t" "(v o r o s o t) \<longlonglongrightarrow> liminf (v o r o s)" using liminf_subseq_lim by auto
3.1122 +  define a where "a = r o s o t"
3.1123 +  have "subseq a" using r s t by (simp add: a_def subseq_o)
3.1124 +  have l:"(w o a) \<longlonglongrightarrow> liminf w"
3.1125 +         "(u o a) \<longlonglongrightarrow> liminf (u o r)"
3.1126 +         "(v o a) \<longlonglongrightarrow> liminf (v o r o s)"
3.1127 +  apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
3.1128 +  apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
3.1129 +  apply (metis (no_types, lifting) t(2) a_def comp_assoc)
3.1130 +  done
3.1132 +  have "liminf (u o r) \<ge> liminf u" by (simp add: liminf_subseq_mono r(1))
3.1133 +  then have a: "liminf (u o r) \<noteq> -\<infinity>" using liminf u > -\<infinity> by auto
3.1134 +  have "liminf (v o r o s) \<ge> liminf v" by (simp add: comp_assoc liminf_subseq_mono r(1) s(1) subseq_o)
3.1135 +  then have b: "liminf (v o r o s) \<noteq> -\<infinity>" using liminf v > -\<infinity> by auto
3.1137 +  have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> liminf (u o r) + liminf (v o r o s)"
3.1138 +    using l tendsto_add_ereal_general a b by fastforce
3.1139 +  moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
3.1140 +  ultimately have "(w o a) \<longlonglongrightarrow> liminf (u o r) + liminf (v o r o s)" by simp
3.1141 +  then have "liminf w = liminf (u o r) + liminf (v o r o s)" using l(1) LIMSEQ_unique by blast
3.1142 +  then have "liminf w \<ge> liminf u + liminf v"
3.1143 +    using liminf (u o r) \<ge> liminf u liminf (v o r o s) \<ge> liminf v ereal_add_mono by simp
3.1144 +  then show ?thesis unfolding w_def by simp
3.1145 +qed
3.1148 +  fixes u v::"nat \<Rightarrow> ereal"
3.1149 +  assumes "u \<longlonglongrightarrow> a" "abs(a) \<noteq> \<infinity>"
3.1150 +  shows "limsup (\<lambda>n. u n + v n) = a + limsup v"
3.1151 +proof -
3.1152 +  have "limsup u = a" using assms(1) using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
3.1153 +  have "(\<lambda>n. -u n) \<longlonglongrightarrow> -a" using assms(1) by auto
3.1154 +  then have "limsup (\<lambda>n. -u n) = -a" using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
3.1156 +  have "limsup (\<lambda>n. u n + v n) \<le> limsup u + limsup v"
3.1157 +    by (rule ereal_limsup_add_mono)
3.1158 +  then have up: "limsup (\<lambda>n. u n + v n) \<le> a + limsup v" using limsup u = a by simp
3.1160 +  have a: "limsup (\<lambda>n. (u n + v n) + (-u n)) \<le> limsup (\<lambda>n. u n + v n) + limsup (\<lambda>n. -u n)"
3.1161 +    by (rule ereal_limsup_add_mono)
3.1162 +  have "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) sequentially" using assms
3.1163 +    real_lim_then_eventually_real by auto
3.1164 +  moreover have "\<And>x. x = ereal(real_of_ereal(x)) \<Longrightarrow> x + (-x) = 0"
3.1165 +    by (metis plus_ereal.simps(1) right_minus uminus_ereal.simps(1) zero_ereal_def)
3.1166 +  ultimately have "eventually (\<lambda>n. u n + (-u n) = 0) sequentially"
3.1167 +    by (metis (mono_tags, lifting) eventually_mono)
3.1168 +  moreover have "\<And>n. u n + (-u n) = 0 \<Longrightarrow> u n + v n + (-u n) = v n"
3.1170 +  ultimately have "eventually (\<lambda>n. u n + v n + (-u n) = v n) sequentially"
3.1171 +    using eventually_mono by force
3.1172 +  then have "limsup v = limsup (\<lambda>n. u n + v n + (-u n))" using Limsup_eq by force
3.1173 +  then have "limsup v \<le> limsup (\<lambda>n. u n + v n) -a" using a limsup (\<lambda>n. -u n) = -a by (simp add: minus_ereal_def)
3.1174 +  then have "limsup (\<lambda>n. u n + v n) \<ge> a + limsup v" using assms(2) by (metis add.commute ereal_le_minus)
3.1175 +  then show ?thesis using up by simp
3.1176 +qed
3.1178 +lemma ereal_limsup_lim_mult:
3.1179 +  fixes u v::"nat \<Rightarrow> ereal"
3.1180 +  assumes "u \<longlonglongrightarrow> a" "a>0" "a \<noteq> \<infinity>"
3.1181 +  shows "limsup (\<lambda>n. u n * v n) = a * limsup v"
3.1182 +proof -
3.1183 +  define w where "w = (\<lambda>n. u n * v n)"
3.1184 +  obtain r where r: "subseq r" "(v o r) \<longlonglongrightarrow> limsup v" using limsup_subseq_lim by auto
3.1185 +  have "(u o r) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ r by auto
3.1186 +  with tendsto_mult_ereal[OF this r(2)] have "(\<lambda>n. (u o r) n * (v o r) n) \<longlonglongrightarrow> a * limsup v" using assms(2) assms(3) by auto
3.1187 +  moreover have "\<And>n. (w o r) n = (u o r) n * (v o r) n" unfolding w_def by auto
3.1188 +  ultimately have "(w o r) \<longlonglongrightarrow> a * limsup v" unfolding w_def by presburger
3.1189 +  then have "limsup (w o r) = a * limsup v" by (simp add: tendsto_iff_Liminf_eq_Limsup)
3.1190 +  then have I: "limsup w \<ge> a * limsup v" by (metis limsup_subseq_mono r(1))
3.1192 +  obtain s where s: "subseq s" "(w o s) \<longlonglongrightarrow> limsup w" using limsup_subseq_lim by auto
3.1193 +  have *: "(u o s) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ s by auto
3.1194 +  have "eventually (\<lambda>n. (u o s) n > 0) sequentially" using assms(2) * order_tendsto_iff by blast
3.1195 +  moreover have "eventually (\<lambda>n. (u o s) n < \<infinity>) sequentially" using assms(3) * order_tendsto_iff by blast
3.1196 +  moreover have "(w o s) n / (u o s) n = (v o s) n" if "(u o s) n > 0" "(u o s) n < \<infinity>" for n
3.1197 +    unfolding w_def using that by (auto simp add: ereal_divide_eq)
3.1198 +  ultimately have "eventually (\<lambda>n. (w o s) n / (u o s) n = (v o s) n) sequentially" using eventually_elim2 by force
3.1199 +  moreover have "(\<lambda>n. (w o s) n / (u o s) n) \<longlonglongrightarrow> (limsup w) / a"
3.1200 +    apply (rule tendsto_divide_ereal[OF s(2) *]) using assms(2) assms(3) by auto
3.1201 +  ultimately have "(v o s) \<longlonglongrightarrow> (limsup w) / a" using Lim_transform_eventually by fastforce
3.1202 +  then have "limsup (v o s) = (limsup w) / a" by (simp add: tendsto_iff_Liminf_eq_Limsup)
3.1203 +  then have "limsup v \<ge> (limsup w) / a" by (metis limsup_subseq_mono s(1))
3.1204 +  then have "a * limsup v \<ge> limsup w" using assms(2) assms(3) by (simp add: ereal_divide_le_pos)
3.1205 +  then show ?thesis using I unfolding w_def by auto
3.1206 +qed
3.1208 +lemma ereal_liminf_lim_mult:
3.1209 +  fixes u v::"nat \<Rightarrow> ereal"
3.1210 +  assumes "u \<longlonglongrightarrow> a" "a>0" "a \<noteq> \<infinity>"
3.1211 +  shows "liminf (\<lambda>n. u n * v n) = a * liminf v"
3.1212 +proof -
3.1213 +  define w where "w = (\<lambda>n. u n * v n)"
3.1214 +  obtain r where r: "subseq r" "(v o r) \<longlonglongrightarrow> liminf v" using liminf_subseq_lim by auto
3.1215 +  have "(u o r) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ r by auto
3.1216 +  with tendsto_mult_ereal[OF this r(2)] have "(\<lambda>n. (u o r) n * (v o r) n) \<longlonglongrightarrow> a * liminf v" using assms(2) assms(3) by auto
3.1217 +  moreover have "\<And>n. (w o r) n = (u o r) n * (v o r) n" unfolding w_def by auto
3.1218 +  ultimately have "(w o r) \<longlonglongrightarrow> a * liminf v" unfolding w_def by presburger
3.1219 +  then have "liminf (w o r) = a * liminf v" by (simp add: tendsto_iff_Liminf_eq_Limsup)
3.1220 +  then have I: "liminf w \<le> a * liminf v" by (metis liminf_subseq_mono r(1))
3.1222 +  obtain s where s: "subseq s" "(w o s) \<longlonglongrightarrow> liminf w" using liminf_subseq_lim by auto
3.1223 +  have *: "(u o s) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ s by auto
3.1224 +  have "eventually (\<lambda>n. (u o s) n > 0) sequentially" using assms(2) * order_tendsto_iff by blast
3.1225 +  moreover have "eventually (\<lambda>n. (u o s) n < \<infinity>) sequentially" using assms(3) * order_tendsto_iff by blast
3.1226 +  moreover have "(w o s) n / (u o s) n = (v o s) n" if "(u o s) n > 0" "(u o s) n < \<infinity>" for n
3.1227 +    unfolding w_def using that by (auto simp add: ereal_divide_eq)
3.1228 +  ultimately have "eventually (\<lambda>n. (w o s) n / (u o s) n = (v o s) n) sequentially" using eventually_elim2 by force
3.1229 +  moreover have "(\<lambda>n. (w o s) n / (u o s) n) \<longlonglongrightarrow> (liminf w) / a"
3.1230 +    apply (rule tendsto_divide_ereal[OF s(2) *]) using assms(2) assms(3) by auto
3.1231 +  ultimately have "(v o s) \<longlonglongrightarrow> (liminf w) / a" using Lim_transform_eventually by fastforce
3.1232 +  then have "liminf (v o s) = (liminf w) / a" by (simp add: tendsto_iff_Liminf_eq_Limsup)
3.1233 +  then have "liminf v \<le> (liminf w) / a" by (metis liminf_subseq_mono s(1))
3.1234 +  then have "a * liminf v \<le> liminf w" using assms(2) assms(3) by (simp add: ereal_le_divide_pos)
3.1235 +  then show ?thesis using I unfolding w_def by auto
3.1236 +qed
3.1239 +  fixes u v::"nat \<Rightarrow> ereal"
3.1240 +  assumes "u \<longlonglongrightarrow> a" "abs(a) \<noteq> \<infinity>"
3.1241 +  shows "liminf (\<lambda>n. u n + v n) = a + liminf v"
3.1242 +proof -
3.1243 +  have "liminf u = a" using assms(1) tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
3.1244 +  then have *: "abs(liminf u) \<noteq> \<infinity>" using assms(2) by auto
3.1245 +  have "(\<lambda>n. -u n) \<longlonglongrightarrow> -a" using assms(1) by auto
3.1246 +  then have "liminf (\<lambda>n. -u n) = -a" using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
3.1247 +  then have **: "abs(liminf (\<lambda>n. -u n)) \<noteq> \<infinity>" using assms(2) by auto
3.1249 +  have "liminf (\<lambda>n. u n + v n) \<ge> liminf u + liminf v"
3.1250 +    apply (rule ereal_liminf_add_mono) using * by auto
3.1251 +  then have up: "liminf (\<lambda>n. u n + v n) \<ge> a + liminf v" using liminf u = a by simp
3.1253 +  have a: "liminf (\<lambda>n. (u n + v n) + (-u n)) \<ge> liminf (\<lambda>n. u n + v n) + liminf (\<lambda>n. -u n)"
3.1254 +    apply (rule ereal_liminf_add_mono) using ** by auto
3.1255 +  have "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) sequentially" using assms
3.1256 +    real_lim_then_eventually_real by auto
3.1257 +  moreover have "\<And>x. x = ereal(real_of_ereal(x)) \<Longrightarrow> x + (-x) = 0"
3.1258 +    by (metis plus_ereal.simps(1) right_minus uminus_ereal.simps(1) zero_ereal_def)
3.1259 +  ultimately have "eventually (\<lambda>n. u n + (-u n) = 0) sequentially"
3.1260 +    by (metis (mono_tags, lifting) eventually_mono)
3.1261 +  moreover have "\<And>n. u n + (-u n) = 0 \<Longrightarrow> u n + v n + (-u n) = v n"
3.1263 +  ultimately have "eventually (\<lambda>n. u n + v n + (-u n) = v n) sequentially"
3.1264 +    using eventually_mono by force
3.1265 +  then have "liminf v = liminf (\<lambda>n. u n + v n + (-u n))" using Liminf_eq by force
3.1266 +  then have "liminf v \<ge> liminf (\<lambda>n. u n + v n) -a" using a liminf (\<lambda>n. -u n) = -a by (simp add: minus_ereal_def)
3.1267 +  then have "liminf (\<lambda>n. u n + v n) \<le> a + liminf v" using assms(2) by (metis add.commute ereal_minus_le)
3.1268 +  then show ?thesis using up by simp
3.1269 +qed
3.1272 +  fixes u v::"nat \<Rightarrow> ereal"
3.1273 +  shows "liminf (\<lambda>n. u n + v n) \<le> liminf u + limsup v"
3.1274 +proof (cases)
3.1275 +  assume "limsup v = \<infinity> \<or> liminf u = \<infinity>"
3.1276 +  then show ?thesis by auto
3.1277 +next
3.1278 +  assume "\<not>(limsup v = \<infinity> \<or> liminf u = \<infinity>)"
3.1279 +  then have "limsup v < \<infinity>" "liminf u < \<infinity>" by auto
3.1281 +  define w where "w = (\<lambda>n. u n + v n)"
3.1282 +  obtain r where r: "subseq r" "(u o r) \<longlonglongrightarrow> liminf u" using liminf_subseq_lim by auto
3.1283 +  obtain s where s: "subseq s" "(w o r o s) \<longlonglongrightarrow> liminf (w o r)" using liminf_subseq_lim by auto
3.1284 +  obtain t where t: "subseq t" "(v o r o s o t) \<longlonglongrightarrow> limsup (v o r o s)" using limsup_subseq_lim by auto
3.1286 +  define a where "a = r o s o t"
3.1287 +  have "subseq a" using r s t by (simp add: a_def subseq_o)
3.1288 +  have l:"(u o a) \<longlonglongrightarrow> liminf u"
3.1289 +         "(w o a) \<longlonglongrightarrow> liminf (w o r)"
3.1290 +         "(v o a) \<longlonglongrightarrow> limsup (v o r o s)"
3.1291 +  apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
3.1292 +  apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
3.1293 +  apply (metis (no_types, lifting) t(2) a_def comp_assoc)
3.1294 +  done
3.1296 +  have "liminf (w o r) \<ge> liminf w" by (simp add: liminf_subseq_mono r(1))
3.1297 +  have "limsup (v o r o s) \<le> limsup v" by (simp add: comp_assoc limsup_subseq_mono r(1) s(1) subseq_o)
3.1298 +  then have b: "limsup (v o r o s) < \<infinity>" using limsup v < \<infinity> by auto
3.1300 +  have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> liminf u + limsup (v o r o s)"
3.1301 +    apply (rule tendsto_add_ereal_general) using b liminf u < \<infinity> l(1) l(3) by force+
3.1302 +  moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
3.1303 +  ultimately have "(w o a) \<longlonglongrightarrow> liminf u + limsup (v o r o s)" by simp
3.1304 +  then have "liminf (w o r) = liminf u + limsup (v o r o s)" using l(2) using LIMSEQ_unique by blast
3.1305 +  then have "liminf w \<le> liminf u + limsup v"
3.1306 +    using liminf (w o r) \<ge> liminf w limsup (v o r o s) \<le> limsup v
3.1307 +    by (metis add_mono_thms_linordered_semiring(2) le_less_trans not_less)
3.1308 +  then show ?thesis unfolding w_def by simp
3.1309 +qed
3.1311 +lemma ereal_liminf_limsup_minus:
3.1312 +  fixes u v::"nat \<Rightarrow> ereal"
3.1313 +  shows "liminf (\<lambda>n. u n - v n) \<le> limsup u - limsup v"
3.1314 +  unfolding minus_ereal_def
3.1315 +  apply (subst add.commute)
3.1316 +  apply (rule order_trans[OF ereal_liminf_limsup_add])
3.1317 +  using ereal_Limsup_uminus[of sequentially "\<lambda>n. - v n"]
3.1319 +  done
3.1322 +lemma liminf_minus_ennreal:
3.1323 +  fixes u v::"nat \<Rightarrow> ennreal"
3.1324 +  shows "(\<And>n. v n \<le> u n) \<Longrightarrow> liminf (\<lambda>n. u n - v n) \<le> limsup u - limsup v"
3.1325 +  unfolding liminf_SUP_INF limsup_INF_SUP
3.1326 +  including ennreal.lifting
3.1327 +proof (transfer, clarsimp)
3.1328 +  fix v u :: "nat \<Rightarrow> ereal" assume *: "\<forall>x. 0 \<le> v x" "\<forall>x. 0 \<le> u x" "\<And>n. v n \<le> u n"
3.1329 +  moreover have "0 \<le> limsup u - limsup v"
3.1330 +    using * by (intro ereal_diff_positive Limsup_mono always_eventually) simp
3.1331 +  moreover have "0 \<le> (SUPREMUM {x..} v)" for x
3.1332 +    using * by (intro SUP_upper2[of x]) auto
3.1333 +  moreover have "0 \<le> (SUPREMUM {x..} u)" for x
3.1334 +    using * by (intro SUP_upper2[of x]) auto
3.1335 +  ultimately show "(SUP n. INF n:{n..}. max 0 (u n - v n))
3.1336 +            \<le> max 0 ((INF x. max 0 (SUPREMUM {x..} u)) - (INF x. max 0 (SUPREMUM {x..} v)))"
3.1337 +    by (auto simp: * ereal_diff_positive max.absorb2 liminf_SUP_INF[symmetric] limsup_INF_SUP[symmetric] ereal_liminf_limsup_minus)
3.1338 +qed
3.1340  (*
3.1341      Notation
3.1342  *)
3.1343 @@ -755,4 +2088,121 @@
3.1344    then show ?thesis using * by auto
3.1345  qed
3.1347 +text {* The next lemma shows that $L^1$ convergence of a sequence of functions follows from almost
3.1348 +everywhere convergence and the weaker condition of the convergence of the integrated norms (or even
3.1349 +just the nontrivial inequality about them). Useful in a lot of contexts! This statement (or its
3.1350 +variations) are known as Scheffe lemma.
3.1352 +The formalization is more painful as one should jump back and forth between reals and ereals and justify
3.1353 +all the time positivity or integrability (thankfully, measurability is handled more or less automatically).*}
3.1355 +lemma Scheffe_lemma1:
3.1356 +  assumes "\<And>n. integrable M (F n)" "integrable M f"
3.1357 +          "AE x in M. (\<lambda>n. F n x) \<longlonglongrightarrow> f x"
3.1358 +          "limsup (\<lambda>n. \<integral>\<^sup>+ x. norm(F n x) \<partial>M) \<le> (\<integral>\<^sup>+ x. norm(f x) \<partial>M)"
3.1359 +  shows "(\<lambda>n. \<integral>\<^sup>+ x. norm(F n x - f x) \<partial>M) \<longlonglongrightarrow> 0"
3.1360 +proof -
3.1361 +  have [measurable]: "\<And>n. F n \<in> borel_measurable M" "f \<in> borel_measurable M"
3.1362 +    using assms(1) assms(2) by simp_all
3.1363 +  define G where "G = (\<lambda>n x. norm(f x) + norm(F n x) - norm(F n x - f x))"
3.1364 +  have [measurable]: "\<And>n. G n \<in> borel_measurable M" unfolding G_def by simp
3.1365 +  have G_pos[simp]: "\<And>n x. G n x \<ge> 0"
3.1366 +    unfolding G_def by (metis ge_iff_diff_ge_0 norm_minus_commute norm_triangle_ineq4)
3.1367 +  have finint: "(\<integral>\<^sup>+ x. norm(f x) \<partial>M) \<noteq> \<infinity>"
3.1368 +    using has_bochner_integral_implies_finite_norm[OF has_bochner_integral_integrable[OF \<open>integrable M f\<close>]]
3.1369 +    by simp
3.1370 +  then have fin2: "2 * (\<integral>\<^sup>+ x. norm(f x) \<partial>M) \<noteq> \<infinity>"
3.1371 +    by (auto simp: ennreal_mult_eq_top_iff)
3.1373 +  {
3.1374 +    fix x assume *: "(\<lambda>n. F n x) \<longlonglongrightarrow> f x"
3.1375 +    then have "(\<lambda>n. norm(F n x)) \<longlonglongrightarrow> norm(f x)" using tendsto_norm by blast
3.1376 +    moreover have "(\<lambda>n. norm(F n x - f x)) \<longlonglongrightarrow> 0" using * Lim_null tendsto_norm_zero_iff by fastforce
3.1377 +    ultimately have a: "(\<lambda>n. norm(F n x) - norm(F n x - f x)) \<longlonglongrightarrow> norm(f x)" using tendsto_diff by fastforce
3.1378 +    have "(\<lambda>n. norm(f x) + (norm(F n x) - norm(F n x - f x))) \<longlonglongrightarrow> norm(f x) + norm(f x)"
3.1379 +      by (rule tendsto_add) (auto simp add: a)
3.1380 +    moreover have "\<And>n. G n x = norm(f x) + (norm(F n x) - norm(F n x - f x))" unfolding G_def by simp
3.1381 +    ultimately have "(\<lambda>n. G n x) \<longlonglongrightarrow> 2 * norm(f x)" by simp
3.1382 +    then have "(\<lambda>n. ennreal(G n x)) \<longlonglongrightarrow> ennreal(2 * norm(f x))" by simp
3.1383 +    then have "liminf (\<lambda>n. ennreal(G n x)) = ennreal(2 * norm(f x))"
3.1384 +      using sequentially_bot tendsto_iff_Liminf_eq_Limsup by blast
3.1385 +  }
3.1386 +  then have "AE x in M. liminf (\<lambda>n. ennreal(G n x)) = ennreal(2 * norm(f x))" using assms(3) by auto
3.1387 +  then have "(\<integral>\<^sup>+ x. liminf (\<lambda>n. ennreal (G n x)) \<partial>M) = (\<integral>\<^sup>+ x. 2 * ennreal(norm(f x)) \<partial>M)"
3.1388 +    by (simp add: nn_integral_cong_AE ennreal_mult)
3.1389 +  also have "... = 2 * (\<integral>\<^sup>+ x. norm(f x) \<partial>M)" by (rule nn_integral_cmult) auto
3.1390 +  finally have int_liminf: "(\<integral>\<^sup>+ x. liminf (\<lambda>n. ennreal (G n x)) \<partial>M) = 2 * (\<integral>\<^sup>+ x. norm(f x) \<partial>M)"
3.1391 +    by simp
3.1393 +  have "(\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M) = (\<integral>\<^sup>+x. norm(f x) \<partial>M) + (\<integral>\<^sup>+x. norm(F n x) \<partial>M)" for n
3.1394 +    by (rule nn_integral_add) (auto simp add: assms)
3.1395 +  then have "limsup (\<lambda>n. (\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M)) =
3.1396 +      limsup (\<lambda>n. (\<integral>\<^sup>+x. norm(f x) \<partial>M) + (\<integral>\<^sup>+x. norm(F n x) \<partial>M))"
3.1397 +    by simp
3.1398 +  also have "... = (\<integral>\<^sup>+x. norm(f x) \<partial>M) + limsup (\<lambda>n. (\<integral>\<^sup>+x. norm(F n x) \<partial>M))"
3.1399 +    by (rule Limsup_const_add, auto simp add: finint)
3.1400 +  also have "... \<le> (\<integral>\<^sup>+x. norm(f x) \<partial>M) + (\<integral>\<^sup>+x. norm(f x) \<partial>M)"
3.1401 +    using assms(4) by (simp add: add_left_mono)
3.1402 +  also have "... = 2 * (\<integral>\<^sup>+x. norm(f x) \<partial>M)"
3.1403 +    unfolding one_add_one[symmetric] distrib_right by simp
3.1404 +  ultimately have a: "limsup (\<lambda>n. (\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M)) \<le>
3.1405 +    2 * (\<integral>\<^sup>+x. norm(f x) \<partial>M)" by simp
3.1407 +  have le: "ennreal (norm (F n x - f x)) \<le> ennreal (norm (f x)) + ennreal (norm (F n x))" for n x
3.1408 +    by (simp add: norm_minus_commute norm_triangle_ineq4 ennreal_plus[symmetric] ennreal_minus del: ennreal_plus)
3.1409 +  then have le2: "(\<integral>\<^sup>+ x. ennreal (norm (F n x - f x)) \<partial>M) \<le> (\<integral>\<^sup>+ x. ennreal (norm (f x)) + ennreal (norm (F n x)) \<partial>M)" for n
3.1410 +    by (rule nn_integral_mono)
3.1412 +  have "2 * (\<integral>\<^sup>+ x. norm(f x) \<partial>M) = (\<integral>\<^sup>+ x. liminf (\<lambda>n. ennreal (G n x)) \<partial>M)"
3.1413 +    by (simp add: int_liminf)
3.1414 +  also have "\<dots> \<le> liminf (\<lambda>n. (\<integral>\<^sup>+x. G n x \<partial>M))"
3.1415 +    by (rule nn_integral_liminf) auto
3.1416 +  also have "liminf (\<lambda>n. (\<integral>\<^sup>+x. G n x \<partial>M)) =
3.1417 +    liminf (\<lambda>n. (\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M) - (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M))"
3.1418 +  proof (intro arg_cong[where f=liminf] ext)
3.1419 +    fix n
3.1420 +    have "\<And>x. ennreal(G n x) = ennreal(norm(f x)) + ennreal(norm(F n x)) - ennreal(norm(F n x - f x))"
3.1421 +      unfolding G_def by (simp add: ennreal_plus[symmetric] ennreal_minus del: ennreal_plus)
3.1422 +    moreover have "(\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) - ennreal(norm(F n x - f x)) \<partial>M)
3.1423 +            = (\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M) - (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M)"
3.1424 +    proof (rule nn_integral_diff)
3.1425 +      from le show "AE x in M. ennreal (norm (F n x - f x)) \<le> ennreal (norm (f x)) + ennreal (norm (F n x))"
3.1426 +        by simp
3.1427 +      from le2 have "(\<integral>\<^sup>+x. ennreal (norm (F n x - f x)) \<partial>M) < \<infinity>" using assms(1) assms(2)
3.1428 +        by (metis has_bochner_integral_implies_finite_norm integrable.simps Bochner_Integration.integrable_diff)
3.1429 +      then show "(\<integral>\<^sup>+x. ennreal (norm (F n x - f x)) \<partial>M) \<noteq> \<infinity>" by simp
3.1430 +    qed (auto simp add: assms)
3.1431 +    ultimately show "(\<integral>\<^sup>+x. G n x \<partial>M) = (\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M) - (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M)"
3.1432 +      by simp
3.1433 +  qed
3.1434 +  finally have "2 * (\<integral>\<^sup>+ x. norm(f x) \<partial>M) + limsup (\<lambda>n. (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M)) \<le>
3.1435 +    liminf (\<lambda>n. (\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M) - (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M)) +
3.1436 +    limsup (\<lambda>n. (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M))"
3.1437 +    by (intro add_mono) auto
3.1438 +  also have "\<dots> \<le> (limsup (\<lambda>n. \<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M) - limsup (\<lambda>n. \<integral>\<^sup>+x. norm (F n x - f x) \<partial>M)) +
3.1439 +    limsup (\<lambda>n. (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M))"
3.1440 +    by (intro add_mono liminf_minus_ennreal le2) auto
3.1441 +  also have "\<dots> = limsup (\<lambda>n. (\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M))"
3.1442 +    by (intro diff_add_cancel_ennreal Limsup_mono always_eventually allI le2)
3.1443 +  also have "\<dots> \<le> 2 * (\<integral>\<^sup>+x. norm(f x) \<partial>M)"
3.1444 +    by fact
3.1445 +  finally have "limsup (\<lambda>n. (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M)) = 0"
3.1446 +    using fin2 by simp
3.1447 +  then show ?thesis
3.1448 +    by (rule tendsto_0_if_Limsup_eq_0_ennreal)
3.1449 +qed
3.1451 +lemma Scheffe_lemma2:
3.1452 +  fixes F::"nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"
3.1453 +  assumes "\<And> n::nat. F n \<in> borel_measurable M" "integrable M f"
3.1454 +          "AE x in M. (\<lambda>n. F n x) \<longlonglongrightarrow> f x"
3.1455 +          "\<And>n. (\<integral>\<^sup>+ x. norm(F n x) \<partial>M) \<le> (\<integral>\<^sup>+ x. norm(f x) \<partial>M)"
3.1456 +  shows "(\<lambda>n. \<integral>\<^sup>+ x. norm(F n x - f x) \<partial>M) \<longlonglongrightarrow> 0"
3.1457 +proof (rule Scheffe_lemma1)
3.1458 +  fix n::nat
3.1459 +  have "(\<integral>\<^sup>+ x. norm(f x) \<partial>M) < \<infinity>" using assms(2) by (metis has_bochner_integral_implies_finite_norm integrable.cases)
3.1460 +  then have "(\<integral>\<^sup>+ x. norm(F n x) \<partial>M) < \<infinity>" using assms(4)[of n] by auto
3.1461 +  then show "integrable M (F n)" by (subst integrable_iff_bounded, simp add: assms(1)[of n])
3.1462 +qed (auto simp add: assms Limsup_bounded)
3.1464  end
4.1 --- a/src/HOL/Analysis/Topology_Euclidean_Space.thy	Thu Oct 13 18:36:06 2016 +0200
4.2 +++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy	Tue Oct 18 12:01:54 2016 +0200
4.3 @@ -459,6 +459,99 @@
4.4  qed
4.7 +lemma countable_separating_set_linorder1:
4.8 +  shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y))"
4.9 +proof -
4.10 +  obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
4.11 +  define B1 where "B1 = {(LEAST x. x \<in> U)| U. U \<in> A}"
4.12 +  then have "countable B1" using countable A by (simp add: Setcompr_eq_image)
4.13 +  define B2 where "B2 = {(SOME x. x \<in> U)| U. U \<in> A}"
4.14 +  then have "countable B2" using countable A by (simp add: Setcompr_eq_image)
4.15 +  have "\<exists>b \<in> B1 \<union> B2. x < b \<and> b \<le> y" if "x < y" for x y
4.16 +  proof (cases)
4.17 +    assume "\<exists>z. x < z \<and> z < y"
4.18 +    then obtain z where z: "x < z \<and> z < y" by auto
4.19 +    define U where "U = {x<..<y}"
4.20 +    then have "open U" by simp
4.21 +    moreover have "z \<in> U" using z U_def by simp
4.22 +    ultimately obtain V where "V \<in> A" "z \<in> V" "V \<subseteq> U" using topological_basisE[OF topological_basis A] by auto
4.23 +    define w where "w = (SOME x. x \<in> V)"
4.24 +    then have "w \<in> V" using z \<in> V by (metis someI2)
4.25 +    then have "x < w \<and> w \<le> y" using w \<in> V V \<subseteq> U U_def by fastforce
4.26 +    moreover have "w \<in> B1 \<union> B2" using w_def B2_def V \<in> A by auto
4.27 +    ultimately show ?thesis by auto
4.28 +  next
4.29 +    assume "\<not>(\<exists>z. x < z \<and> z < y)"
4.30 +    then have *: "\<And>z. z > x \<Longrightarrow> z \<ge> y" by auto
4.31 +    define U where "U = {x<..}"
4.32 +    then have "open U" by simp
4.33 +    moreover have "y \<in> U" using x < y U_def by simp
4.34 +    ultimately obtain "V" where "V \<in> A" "y \<in> V" "V \<subseteq> U" using topological_basisE[OF topological_basis A] by auto
4.35 +    have "U = {y..}" unfolding U_def using * x < y by auto
4.36 +    then have "V \<subseteq> {y..}" using V \<subseteq> U by simp
4.37 +    then have "(LEAST w. w \<in> V) = y" using y \<in> V by (meson Least_equality atLeast_iff subsetCE)
4.38 +    then have "y \<in> B1 \<union> B2" using V \<in> A B1_def by auto
4.39 +    moreover have "x < y \<and> y \<le> y" using x < y by simp
4.40 +    ultimately show ?thesis by auto
4.41 +  qed
4.42 +  moreover have "countable (B1 \<union> B2)" using countable B1 countable B2 by simp
4.43 +  ultimately show ?thesis by auto
4.44 +qed
4.45 +
4.46 +lemma countable_separating_set_linorder2:
4.47 +  shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x \<le> b \<and> b < y))"
4.48 +proof -
4.49 +  obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
4.50 +  define B1 where "B1 = {(GREATEST x. x \<in> U) | U. U \<in> A}"
4.51 +  then have "countable B1" using countable A by (simp add: Setcompr_eq_image)
4.52 +  define B2 where "B2 = {(SOME x. x \<in> U)| U. U \<in> A}"
4.53 +  then have "countable B2" using countable A by (simp add: Setcompr_eq_image)
4.54 +  have "\<exists>b \<in> B1 \<union> B2. x \<le> b \<and> b < y" if "x < y" for x y
4.55 +  proof (cases)
4.56 +    assume "\<exists>z. x < z \<and> z < y"
4.57 +    then obtain z where z: "x < z \<and> z < y" by auto
4.58 +    define U where "U = {x<..<y}"
4.59 +    then have "open U" by simp
4.60 +    moreover have "z \<in> U" using z U_def by simp
4.61 +    ultimately obtain "V" where "V \<in> A" "z \<in> V" "V \<subseteq> U" using topological_basisE[OF topological_basis A] by auto
4.62 +    define w where "w = (SOME x. x \<in> V)"
4.63 +    then have "w \<in> V" using z \<in> V by (metis someI2)
4.64 +    then have "x \<le> w \<and> w < y" using w \<in> V V \<subseteq> U U_def by fastforce
4.65 +    moreover have "w \<in> B1 \<union> B2" using w_def B2_def V \<in> A by auto
4.66 +    ultimately show ?thesis by auto
4.67 +  next
4.68 +    assume "\<not>(\<exists>z. x < z \<and> z < y)"
4.69 +    then have *: "\<And>z. z < y \<Longrightarrow> z \<le> x" using leI by blast
4.70 +    define U where "U = {..<y}"
4.71 +    then have "open U" by simp
4.72 +    moreover have "x \<in> U" using x < y U_def by simp
4.73 +    ultimately obtain "V" where "V \<in> A" "x \<in> V" "V \<subseteq> U" using topological_basisE[OF topological_basis A] by auto
4.74 +    have "U = {..x}" unfolding U_def using * x < y by auto
4.75 +    then have "V \<subseteq> {..x}" using V \<subseteq> U by simp
4.76 +    then have "(GREATEST x. x \<in> V) = x" using x \<in> V by (meson Greatest_equality atMost_iff subsetCE)
4.77 +    then have "x \<in> B1 \<union> B2" using V \<in> A B1_def by auto
4.78 +    moreover have "x \<le> x \<and> x < y" using x < y by simp
4.79 +    ultimately show ?thesis by auto
4.80 +  qed
4.81 +  moreover have "countable (B1 \<union> B2)" using countable B1 countable B2 by simp
4.82 +  ultimately show ?thesis by auto
4.83 +qed
4.84 +
4.85 +lemma countable_separating_set_dense_linorder:
4.86 +  shows "\<exists>B::('a::{linorder_topology, dense_linorder, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b < y))"
4.87 +proof -
4.88 +  obtain B::"'a set" where B: "countable B" "\<And>x y. x < y \<Longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y)"
4.89 +    using countable_separating_set_linorder1 by auto
4.90 +  have "\<exists>b \<in> B. x < b \<and> b < y" if "x < y" for x y
4.91 +  proof -
4.92 +    obtain z where "x < z" "z < y" using x < y dense by blast
4.93 +    then obtain b where "b \<in> B" "x < b \<and> b \<le> z" using B(2) by auto
4.94 +    then have "x < b \<and> b < y" using z < y by auto
4.95 +    then show ?thesis using b \<in> B by auto
4.96 +  qed
4.97 +  then show ?thesis using B(1) by auto
4.98 +qed
4.99 +
4.100  subsection \<open>Polish spaces\<close>
4.102  text \<open>Textbooks define Polish spaces as completely metrizable.
4.103 @@ -8688,7 +8781,7 @@
4.104      unfolding homeomorphic_def homeomorphism_def
4.105      by (metis equalityI image_subset_iff subsetI)
4.106   qed
4.109  lemma homeomorphicI [intro?]:
4.110     "\<lbrakk>f  S = T; g  T = S;
4.111       continuous_on S f; continuous_on T g;
4.112 @@ -10037,7 +10130,7 @@
4.113      apply (rule openin_Union, clarify)
4.114      apply (metis (full_types) \<open>open U\<close> cont clo openin_trans continuous_openin_preimage_gen)
4.115      done
4.116 -qed
4.117 +qed
4.119  lemma pasting_lemma_exists:
4.120    fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"
5.1 --- a/src/HOL/Library/Extended_Nonnegative_Real.thy	Thu Oct 13 18:36:06 2016 +0200
5.2 +++ b/src/HOL/Library/Extended_Nonnegative_Real.thy	Tue Oct 18 12:01:54 2016 +0200
5.3 @@ -949,6 +949,9 @@
5.4    by (cases "0 \<le> x")
5.5       (auto simp: ennreal_neg ennreal_1[symmetric] simp del: ennreal_1)
5.7 +lemma one_less_ennreal[simp]: "1 < ennreal x \<longleftrightarrow> 1 < x"
5.8 +  by transfer (auto simp: max.absorb2 less_max_iff_disj)
5.9 +
5.10  lemma ennreal_plus[simp]:
5.11    "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> ennreal (a + b) = ennreal a + ennreal b"
5.12    by (transfer fixing: a b) (auto simp: max_absorb2)
6.1 --- a/src/HOL/Library/Permutations.thy	Thu Oct 13 18:36:06 2016 +0200
6.2 +++ b/src/HOL/Library/Permutations.thy	Tue Oct 18 12:01:54 2016 +0200
6.3 @@ -22,6 +22,23 @@
6.4    "Fun.swap a b id x = (if x = a then b else if x = b then a else x)"
6.5    by (simp add: Fun.swap_def)
6.7 +lemma bij_inv_eq_iff: "bij p \<Longrightarrow> x = inv p y \<longleftrightarrow> p x = y"
6.8 +  using surj_f_inv_f[of p] by (auto simp add: bij_def)
6.9 +
6.10 +lemma bij_swap_comp:
6.11 +  assumes bp: "bij p"
6.12 +  shows "Fun.swap a b id \<circ> p = Fun.swap (inv p a) (inv p b) p"
6.13 +  using surj_f_inv_f[OF bij_is_surj[OF bp]]
6.14 +  by (simp add: fun_eq_iff Fun.swap_def bij_inv_eq_iff[OF bp])
6.15 +
6.16 +lemma bij_swap_ompose_bij: "bij p \<Longrightarrow> bij (Fun.swap a b id \<circ> p)"
6.17 +proof -
6.18 +  assume H: "bij p"
6.19 +  show ?thesis
6.20 +    unfolding bij_swap_comp[OF H] bij_swap_iff
6.21 +    using H .
6.22 +qed
6.23 +
6.25  subsection \<open>Basic consequences of the definition\<close>
6.27 @@ -30,7 +47,7 @@
6.29  lemma permutes_in_image: "p permutes S \<Longrightarrow> p x \<in> S \<longleftrightarrow> x \<in> S"
6.30    unfolding permutes_def by metis
6.31 -
6.32 +
6.33  lemma permutes_not_in:
6.34    assumes "f permutes S" "x \<notin> S" shows "f x = x"
6.35    using assms by (auto simp: permutes_def)
6.36 @@ -107,7 +124,7 @@
6.38  (* Next three lemmas contributed by Lukas Bulwahn *)
6.39  lemma permutes_bij_inv_into:
6.40 -  fixes A :: "'a set" and B :: "'b set"
6.41 +  fixes A :: "'a set" and B :: "'b set"
6.42    assumes "p permutes A"
6.43    assumes "bij_betw f A B"
6.44    shows "(\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) permutes B"
6.45 @@ -167,9 +184,9 @@
6.46    unfolding fun_eq_iff permutes_inv_eq[OF pS] permutes_inv_eq[OF permutes_inv[OF pS]]
6.47    by blast
6.49 -lemma permutes_invI:
6.50 +lemma permutes_invI:
6.51    assumes perm: "p permutes S"
6.52 -      and inv:  "\<And>x. x \<in> S \<Longrightarrow> p' (p x) = x"
6.53 +      and inv:  "\<And>x. x \<in> S \<Longrightarrow> p' (p x) = x"
6.54        and outside: "\<And>x. x \<notin> S \<Longrightarrow> p' x = x"
6.55    shows   "inv p = p'"
6.56  proof
6.57 @@ -177,14 +194,14 @@
6.58    proof (cases "x \<in> S")
6.59      assume [simp]: "x \<in> S"
6.60      from assms have "p' x = p' (p (inv p x))" by (simp add: permutes_inverses)
6.61 -    also from permutes_inv[OF perm]
6.62 +    also from permutes_inv[OF perm]
6.63        have "\<dots> = inv p x" by (subst inv) (simp_all add: permutes_in_image)
6.64      finally show "inv p x = p' x" ..
6.65    qed (insert permutes_inv[OF perm], simp_all add: outside permutes_not_in)
6.66  qed
6.68  lemma permutes_vimage: "f permutes A \<Longrightarrow> f - A = A"
6.69 -  by (simp add: bij_vimage_eq_inv_image permutes_bij permutes_image[OF permutes_inv])
6.70 +  by (simp add: bij_vimage_eq_inv_image permutes_bij permutes_image[OF permutes_inv])
6.73  subsection \<open>The number of permutations on a finite set\<close>
6.74 @@ -329,7 +346,7 @@
6.75  lemma finite_permutations:
6.76    assumes fS: "finite S"
6.77    shows "finite {p. p permutes S}"
6.78 -  using card_permutations[OF refl fS]
6.79 +  using card_permutations[OF refl fS]
6.80    by (auto intro: card_ge_0_finite)
6.83 @@ -724,23 +741,6 @@
6.84    qed
6.85  qed
6.87 -lemma bij_inv_eq_iff: "bij p \<Longrightarrow> x = inv p y \<longleftrightarrow> p x = y"
6.88 -  using surj_f_inv_f[of p] by (auto simp add: bij_def)
6.89 -
6.90 -lemma bij_swap_comp:
6.91 -  assumes bp: "bij p"
6.92 -  shows "Fun.swap a b id \<circ> p = Fun.swap (inv p a) (inv p b) p"
6.93 -  using surj_f_inv_f[OF bij_is_surj[OF bp]]
6.94 -  by (simp add: fun_eq_iff Fun.swap_def bij_inv_eq_iff[OF bp])
6.95 -
6.96 -lemma bij_swap_ompose_bij: "bij p \<Longrightarrow> bij (Fun.swap a b id \<circ> p)"
6.97 -proof -
6.98 -  assume H: "bij p"
6.99 -  show ?thesis
6.100 -    unfolding bij_swap_comp[OF H] bij_swap_iff
6.101 -    using H .
6.102 -qed
6.104  lemma permutation_lemma:
6.105    assumes fS: "finite S"
6.106      and p: "bij p"
6.107 @@ -881,7 +881,7 @@
6.108  lemma sign_idempotent: "sign p * sign p = 1"
6.109    by (simp add: sign_def)
6.113  subsection \<open>Permuting a list\<close>
6.115  text \<open>This function permutes a list by applying a permutation to the indices.\<close>
6.116 @@ -889,7 +889,7 @@
6.117  definition permute_list :: "(nat \<Rightarrow> nat) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
6.118    "permute_list f xs = map (\<lambda>i. xs ! (f i)) [0..<length xs]"
6.120 -lemma permute_list_map:
6.121 +lemma permute_list_map:
6.122    assumes "f permutes {..<length xs}"
6.123    shows   "permute_list f (map g xs) = map g (permute_list f xs)"
6.124    using permutes_in_image[OF assms] by (auto simp: permute_list_def)
6.125 @@ -897,7 +897,7 @@
6.126  lemma permute_list_nth:
6.127    assumes "f permutes {..<length xs}" "i < length xs"
6.128    shows   "permute_list f xs ! i = xs ! f i"
6.129 -  using permutes_in_image[OF assms(1)] assms(2)
6.130 +  using permutes_in_image[OF assms(1)] assms(2)
6.131    by (simp add: permute_list_def)
6.133  lemma permute_list_Nil [simp]: "permute_list f [] = []"
6.134 @@ -906,7 +906,7 @@
6.135  lemma length_permute_list [simp]: "length (permute_list f xs) = length xs"
6.136    by (simp add: permute_list_def)
6.138 -lemma permute_list_compose:
6.139 +lemma permute_list_compose:
6.140    assumes "g permutes {..<length xs}"
6.141    shows   "permute_list (f \<circ> g) xs = permute_list g (permute_list f xs)"
6.142    using assms[THEN permutes_in_image] by (auto simp add: permute_list_def)
6.143 @@ -924,7 +924,7 @@
6.144    fix y :: 'a
6.145    from assms have [simp]: "f x < length xs \<longleftrightarrow> x < length xs" for x
6.146      using permutes_in_image[OF assms] by auto
6.147 -  have "count (mset (permute_list f xs)) y =
6.148 +  have "count (mset (permute_list f xs)) y =
6.149            card ((\<lambda>i. xs ! f i) - {y} \<inter> {..<length xs})"
6.150      by (simp add: permute_list_def mset_map count_image_mset atLeast0LessThan)
6.151    also have "(\<lambda>i. xs ! f i) - {y} \<inter> {..<length xs} = f - {i. i < length xs \<and> y = xs ! i}"
6.152 @@ -935,7 +935,7 @@
6.153    finally show "count (mset (permute_list f xs)) y = count (mset xs) y" by simp
6.154  qed
6.156 -lemma set_permute_list [simp]:
6.157 +lemma set_permute_list [simp]:
6.158    assumes "f permutes {..<length xs}"
6.159    shows   "set (permute_list f xs) = set xs"
6.160    by (rule mset_eq_setD[OF mset_permute_list]) fact
6.161 @@ -945,7 +945,7 @@
6.162    shows   "distinct (permute_list f xs) = distinct xs"
6.163    by (simp add: distinct_count_atmost_1 assms)
6.165 -lemma permute_list_zip:
6.166 +lemma permute_list_zip:
6.167    assumes "f permutes A" "A = {..<length xs}"
6.168    assumes [simp]: "length xs = length ys"
6.169    shows   "permute_list f (zip xs ys) = zip (permute_list f xs) (permute_list f ys)"
6.170 @@ -961,7 +961,7 @@
6.171    finally show ?thesis .
6.172  qed
6.174 -lemma map_of_permute:
6.175 +lemma map_of_permute:
6.176    assumes "\<sigma> permutes fst  set xs"
6.177    shows   "map_of xs \<circ> \<sigma> = map_of (map (\<lambda>(x,y). (inv \<sigma> x, y)) xs)" (is "_ = map_of (map ?f _)")
6.178  proof
6.179 @@ -993,7 +993,7 @@
6.180      from this have "count (image_mset f (mset_set (insert x F))) b = Suc (card {a \<in> F. f a = f x})"
6.181        using insert.hyps by auto
6.182      also have "\<dots> = card (insert x {a \<in> F. f a = f x})"
6.183 -      using insert.hyps(1,2) by simp
6.184 +      using insert.hyps(1,2) by simp
6.185      also have "card (insert x {a \<in> F. f a = f x}) = card {a \<in> insert x F. f a = b}"
6.186        using \<open>f x = b\<close> by (auto intro: arg_cong[where f="card"])
6.187      finally show ?thesis using insert by auto
6.188 @@ -1003,7 +1003,7 @@
6.189      with insert A show ?thesis by simp
6.190    qed
6.191  qed
6.194  (* Prove image_mset_eq_implies_permutes *)
6.195  lemma image_mset_eq_implies_permutes:
6.196    fixes f :: "'a \<Rightarrow> 'b"
6.197 @@ -1317,7 +1317,7 @@
6.198  subsection \<open>Constructing permutations from association lists\<close>
6.200  definition list_permutes where
6.201 -  "list_permutes xs A \<longleftrightarrow> set (map fst xs) \<subseteq> A \<and> set (map snd xs) = set (map fst xs) \<and>
6.202 +  "list_permutes xs A \<longleftrightarrow> set (map fst xs) \<subseteq> A \<and> set (map snd xs) = set (map fst xs) \<and>
6.203       distinct (map fst xs) \<and> distinct (map snd xs)"
6.205  lemma list_permutesI [simp]:
6.206 @@ -1349,8 +1349,8 @@
6.207  proof (rule inj_onI)
6.208    fix x y assume xy: "x \<in> set (map fst xs)" "y \<in> set (map fst xs)"
6.209    assume eq: "map_of xs x = map_of xs y"
6.210 -  from xy obtain x' y'
6.211 -    where x'y': "map_of xs x = Some x'" "map_of xs y = Some y'"
6.212 +  from xy obtain x' y'
6.213 +    where x'y': "map_of xs x = Some x'" "map_of xs y = Some y'"
6.214      by (cases "map_of xs x"; cases "map_of xs y")
6.215         (simp_all add: map_of_eq_None_iff)
6.216    moreover from x'y' have *: "(x,x') \<in> set xs" "(y,y') \<in> set xs"
6.217 @@ -1398,7 +1398,7 @@
6.218    also from assms have "?f  set (map fst xs) = (the \<circ> map_of xs)  set (map fst xs)"
6.219      by (intro image_cong refl)
6.220         (auto simp: permutation_of_list_def map_of_eq_None_iff split: option.splits)
6.221 -  also from assms have "\<dots> = set (map fst xs)"
6.222 +  also from assms have "\<dots> = set (map fst xs)"
6.223      by (subst image_map_of') (simp_all add: list_permutes_def)
6.224    finally show "bij_betw ?f (set (map fst xs)) (set (map fst xs))" .
6.225  qed (force simp: permutation_of_list_def dest!: map_of_SomeD split: option.splits)+
7.1 --- a/src/HOL/Probability/Levy.thy	Thu Oct 13 18:36:06 2016 +0200
7.2 +++ b/src/HOL/Probability/Levy.thy	Tue Oct 18 12:01:54 2016 +0200
7.3 @@ -8,11 +8,6 @@
7.4    imports Characteristic_Functions Helly_Selection Sinc_Integral
7.5  begin
7.7 -lemma LIM_zero_cancel:
7.8 -  fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
7.9 -  shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> l) F"
7.10 -unfolding tendsto_iff dist_norm by simp
7.11 -
7.12  subsection \<open>The Levy inversion theorem\<close>
7.14  (* Actually, this is not needed for us -- but it is useful for other purposes. (See Billingsley.) *)
8.1 --- a/src/HOL/Topological_Spaces.thy	Thu Oct 13 18:36:06 2016 +0200
8.2 +++ b/src/HOL/Topological_Spaces.thy	Tue Oct 18 12:01:54 2016 +0200
8.3 @@ -3398,4 +3398,93 @@
8.4  lemma isCont_swap[continuous_intros]: "isCont prod.swap a"
8.5    using continuous_on_eq_continuous_within continuous_on_swap by blast
8.7 +lemma open_diagonal_complement:
8.8 +  "open {(x,y) | x y. x \<noteq> (y::('a::t2_space))}"
8.9 +proof (rule topological_space_class.openI)
8.10 +  fix t assume "t \<in> {(x, y) | x y. x \<noteq> (y::'a)}"
8.11 +  then obtain x y where "t = (x,y)" "x \<noteq> y" by blast
8.12 +  then obtain U V where *: "open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
8.13 +    by (auto simp add: separation_t2)
8.14 +  define T where "T = U \<times> V"
8.15 +  have "open T" using * open_Times T_def by auto
8.16 +  moreover have "t \<in> T" unfolding T_def using t = (x,y) * by auto
8.17 +  moreover have "T \<subseteq> {(x, y) | x y. x \<noteq> y}" unfolding T_def using * by auto
8.18 +  ultimately show "\<exists>T. open T \<and> t \<in> T \<and> T \<subseteq> {(x, y) | x y. x \<noteq> y}" by auto
8.19 +qed
8.20 +
8.21 +lemma closed_diagonal:
8.22 +  "closed {y. \<exists> x::('a::t2_space). y = (x,x)}"
8.23 +proof -
8.24 +  have "{y. \<exists> x::'a. y = (x,x)} = UNIV - {(x,y) | x y. x \<noteq> y}" by auto
8.25 +  then show ?thesis using open_diagonal_complement closed_Diff by auto
8.26 +qed
8.27 +
8.28 +lemma open_superdiagonal:
8.29 +  "open {(x,y) | x y. x > (y::'a::{linorder_topology})}"
8.30 +proof (rule topological_space_class.openI)
8.31 +  fix t assume "t \<in> {(x, y) | x y. y < (x::'a)}"
8.32 +  then obtain x y where "t = (x, y)" "x > y" by blast
8.33 +  show "\<exists>T. open T \<and> t \<in> T \<and> T \<subseteq> {(x, y) | x y. y < x}"
8.34 +  proof (cases)
8.35 +    assume "\<exists>z. y < z \<and> z < x"
8.36 +    then obtain z where z: "y < z \<and> z < x" by blast
8.37 +    define T where "T = {z<..} \<times> {..<z}"
8.38 +    have "open T" unfolding T_def by (simp add: open_Times)
8.39 +    moreover have "t \<in> T" using T_def z t = (x,y) by auto
8.40 +    moreover have "T \<subseteq> {(x, y) | x y. y < x}" unfolding T_def by auto
8.41 +    ultimately show ?thesis by auto
8.42 +  next
8.43 +    assume "\<not>(\<exists>z. y < z \<and> z < x)"
8.44 +    then have *: "{x ..} = {y<..}" "{..< x} = {..y}"
8.45 +      using x > y apply auto using leI by blast
8.46 +    define T where "T = {x ..} \<times> {.. y}"
8.47 +    then have "T = {y<..} \<times> {..< x}" using * by simp
8.48 +    then have "open T" unfolding T_def by (simp add: open_Times)
8.49 +    moreover have "t \<in> T" using T_def t = (x,y) by auto
8.50 +    moreover have "T \<subseteq> {(x, y) | x y. y < x}" unfolding T_def using x > y by auto
8.51 +    ultimately show ?thesis by auto
8.52 +  qed
8.53 +qed
8.54 +
8.55 +lemma closed_subdiagonal:
8.56 +  "closed {(x,y) | x y. x \<le> (y::'a::{linorder_topology})}"
8.57 +proof -
8.58 +  have "{(x,y) | x y. x \<le> (y::'a)} = UNIV - {(x,y) | x y. x > (y::'a)}" by auto
8.59 +  then show ?thesis using open_superdiagonal closed_Diff by auto
8.60 +qed
8.61 +
8.62 +lemma open_subdiagonal:
8.63 +  "open {(x,y) | x y. x < (y::'a::{linorder_topology})}"
8.64 +proof (rule topological_space_class.openI)
8.65 +  fix t assume "t \<in> {(x, y) | x y. y > (x::'a)}"
8.66 +  then obtain x y where "t = (x, y)" "x < y" by blast
8.67 +  show "\<exists>T. open T \<and> t \<in> T \<and> T \<subseteq> {(x, y) | x y. y > x}"
8.68 +  proof (cases)
8.69 +    assume "\<exists>z. y > z \<and> z > x"
8.70 +    then obtain z where z: "y > z \<and> z > x" by blast
8.71 +    define T where "T = {..<z} \<times> {z<..}"
8.72 +    have "open T" unfolding T_def by (simp add: open_Times)
8.73 +    moreover have "t \<in> T" using T_def z t = (x,y) by auto
8.74 +    moreover have "T \<subseteq> {(x, y) |x y. y > x}" unfolding T_def by auto
8.75 +    ultimately show ?thesis by auto
8.76 +  next
8.77 +    assume "\<not>(\<exists>z. y > z \<and> z > x)"
8.78 +    then have *: "{..x} = {..<y}" "{x<..} = {y..}"
8.79 +      using x < y apply auto using leI by blast
8.80 +    define T where "T = {..x} \<times> {y..}"
8.81 +    then have "T = {..<y} \<times> {x<..}" using * by simp
8.82 +    then have "open T" unfolding T_def by (simp add: open_Times)
8.83 +    moreover have "t \<in> T" using T_def t = (x,y) by auto
8.84 +    moreover have "T \<subseteq> {(x, y) |x y. y > x}" unfolding T_def using x < y` by auto
8.85 +    ultimately show ?thesis by auto
8.86 +  qed
8.87 +qed
8.88 +
8.89 +lemma closed_superdiagonal:
8.90 +  "closed {(x,y) | x y. x \<ge> (y::('a::{linorder_topology}))}"
8.91 +proof -
8.92 +  have "{(x,y) | x y. x \<ge> (y::'a)} = UNIV - {(x,y) | x y. x < y}" by auto
8.93 +  then show ?thesis using open_subdiagonal closed_Diff by auto
8.94 +qed
8.95 +
8.96  end