more symbols;
authorwenzelm
Tue Dec 29 23:04:53 2015 +0100 (2015-12-29)
changeset 61969e01015e49041
parent 61968 e13e70f32407
child 61970 6226261144d7
more symbols;
NEWS
src/HOL/Complex.thy
src/HOL/Decision_Procs/Approximation.thy
src/HOL/Library/BigO.thy
src/HOL/Library/Extended_Real.thy
src/HOL/Library/Formal_Power_Series.thy
src/HOL/Library/Indicator_Function.thy
src/HOL/Library/Liminf_Limsup.thy
src/HOL/Library/Lub_Glb.thy
src/HOL/Library/Product_Vector.thy
src/HOL/Limits.thy
src/HOL/Metis_Examples/Clausification.thy
src/HOL/Multivariate_Analysis/Bounded_Continuous_Function.thy
src/HOL/Multivariate_Analysis/Bounded_Linear_Function.thy
src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy
src/HOL/Multivariate_Analysis/Complex_Transcendental.thy
src/HOL/Multivariate_Analysis/Derivative.thy
src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy
src/HOL/Multivariate_Analysis/Integration.thy
src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
src/HOL/Multivariate_Analysis/Uniform_Limit.thy
src/HOL/NSA/HSEQ.thy
src/HOL/NthRoot.thy
src/HOL/Probability/Bochner_Integration.thy
src/HOL/Probability/Borel_Space.thy
src/HOL/Probability/Caratheodory.thy
src/HOL/Probability/Distributions.thy
src/HOL/Probability/Fin_Map.thy
src/HOL/Probability/Infinite_Product_Measure.thy
src/HOL/Probability/Interval_Integral.thy
src/HOL/Probability/Lebesgue_Measure.thy
src/HOL/Probability/Measure_Space.thy
src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy
src/HOL/Probability/Projective_Family.thy
src/HOL/Probability/Projective_Limit.thy
src/HOL/Probability/Radon_Nikodym.thy
src/HOL/Probability/Regularity.thy
src/HOL/Probability/Set_Integral.thy
src/HOL/Real_Vector_Spaces.thy
src/HOL/Series.thy
src/HOL/Topological_Spaces.thy
src/HOL/Transcendental.thy
     1.1 --- a/NEWS	Tue Dec 29 22:41:22 2015 +0100
     1.2 +++ b/NEWS	Tue Dec 29 23:04:53 2015 +0100
     1.3 @@ -502,6 +502,8 @@
     1.4    notation Preorder.equiv ("op ~~")
     1.5      and Preorder.equiv ("(_/ ~~ _)" [51, 51] 50)
     1.6  
     1.7 +  notation (in topological_space) LIMSEQ ("((_)/ ----> (_))" [60, 60] 60)
     1.8 +
     1.9  * The alternative notation "\<Colon>" for type and sort constraints has been
    1.10  removed: in LaTeX document output it looks the same as "::".
    1.11  INCOMPATIBILITY, use plain "::" instead.
     2.1 --- a/src/HOL/Complex.thy	Tue Dec 29 22:41:22 2015 +0100
     2.2 +++ b/src/HOL/Complex.thy	Tue Dec 29 23:04:53 2015 +0100
     2.3 @@ -415,7 +415,7 @@
     2.4  proof
     2.5    fix X :: "nat \<Rightarrow> complex"
     2.6    assume X: "Cauchy X"
     2.7 -  then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
     2.8 +  then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) \<longlonglongrightarrow> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
     2.9      by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)
    2.10    then show "convergent X"
    2.11      unfolding complex.collapse by (rule convergentI)
     3.1 --- a/src/HOL/Decision_Procs/Approximation.thy	Tue Dec 29 22:41:22 2015 +0100
     3.2 +++ b/src/HOL/Decision_Procs/Approximation.thy	Tue Dec 29 23:04:53 2015 +0100
     3.3 @@ -2103,7 +2103,7 @@
     3.4      using ln_series[of "x + 1"] \<open>0 \<le> x\<close> \<open>x < 1\<close> by auto
     3.5  
     3.6    have "norm x < 1" using assms by auto
     3.7 -  have "?a ----> 0" unfolding Suc_eq_plus1[symmetric] inverse_eq_divide[symmetric]
     3.8 +  have "?a \<longlonglongrightarrow> 0" unfolding Suc_eq_plus1[symmetric] inverse_eq_divide[symmetric]
     3.9      using tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF \<open>norm x < 1\<close>]]] by auto
    3.10    have "0 \<le> ?a n" for n
    3.11      by (rule mult_nonneg_nonneg) (auto simp: \<open>0 \<le> x\<close>)
    3.12 @@ -2117,7 +2117,7 @@
    3.13        by (rule mult_left_mono, fact less_imp_le[OF \<open>x < 1\<close>]) (auto simp: \<open>0 \<le> x\<close>)
    3.14      thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
    3.15    qed auto
    3.16 -  from summable_Leibniz'(2,4)[OF \<open>?a ----> 0\<close> \<open>\<And>n. 0 \<le> ?a n\<close>, OF \<open>\<And>n. ?a (Suc n) \<le> ?a n\<close>, unfolded ln_eq]
    3.17 +  from summable_Leibniz'(2,4)[OF \<open>?a \<longlonglongrightarrow> 0\<close> \<open>\<And>n. 0 \<le> ?a n\<close>, OF \<open>\<And>n. ?a (Suc n) \<le> ?a n\<close>, unfolded ln_eq]
    3.18    show ?lb and ?ub
    3.19      unfolding atLeast0LessThan by auto
    3.20  qed
     4.1 --- a/src/HOL/Library/BigO.thy	Tue Dec 29 22:41:22 2015 +0100
     4.2 +++ b/src/HOL/Library/BigO.thy	Tue Dec 29 23:04:53 2015 +0100
     4.3 @@ -840,7 +840,7 @@
     4.4    apply (auto split: split_max simp add: func_plus)
     4.5    done
     4.6  
     4.7 -lemma bigo_LIMSEQ1: "f =o O(g) \<Longrightarrow> g ----> 0 \<Longrightarrow> f ----> (0::real)"
     4.8 +lemma bigo_LIMSEQ1: "f =o O(g) \<Longrightarrow> g \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> (0::real)"
     4.9    apply (simp add: LIMSEQ_iff bigo_alt_def)
    4.10    apply clarify
    4.11    apply (drule_tac x = "r / c" in spec)
    4.12 @@ -863,7 +863,7 @@
    4.13    apply simp
    4.14    done
    4.15  
    4.16 -lemma bigo_LIMSEQ2: "f =o g +o O(h) \<Longrightarrow> h ----> 0 \<Longrightarrow> f ----> a \<Longrightarrow> g ----> (a::real)"
    4.17 +lemma bigo_LIMSEQ2: "f =o g +o O(h) \<Longrightarrow> h \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> g \<longlonglongrightarrow> (a::real)"
    4.18    apply (drule set_plus_imp_minus)
    4.19    apply (drule bigo_LIMSEQ1)
    4.20    apply assumption
     5.1 --- a/src/HOL/Library/Extended_Real.thy	Tue Dec 29 22:41:22 2015 +0100
     5.2 +++ b/src/HOL/Library/Extended_Real.thy	Tue Dec 29 23:04:53 2015 +0100
     5.3 @@ -40,10 +40,10 @@
     5.4    show ?thesis
     5.5      unfolding continuous_within
     5.6    proof (intro tendsto_at_left_sequentially[of bot])
     5.7 -    fix S :: "nat \<Rightarrow> 'a" assume S: "incseq S" and S_x: "S ----> x"
     5.8 +    fix S :: "nat \<Rightarrow> 'a" assume S: "incseq S" and S_x: "S \<longlonglongrightarrow> x"
     5.9      from S_x have x_eq: "x = (SUP i. S i)"
    5.10        by (rule LIMSEQ_unique) (intro LIMSEQ_SUP S)
    5.11 -    show "(\<lambda>n. f (S n)) ----> f x"
    5.12 +    show "(\<lambda>n. f (S n)) \<longlonglongrightarrow> f x"
    5.13        unfolding x_eq sup_continuousD[OF f S]
    5.14        using S sup_continuous_mono[OF f] by (intro LIMSEQ_SUP) (auto simp: mono_def)
    5.15    qed (insert x, auto simp: bot_less)
    5.16 @@ -77,10 +77,10 @@
    5.17    show ?thesis
    5.18      unfolding continuous_within
    5.19    proof (intro tendsto_at_right_sequentially[of _ top])
    5.20 -    fix S :: "nat \<Rightarrow> 'a" assume S: "decseq S" and S_x: "S ----> x"
    5.21 +    fix S :: "nat \<Rightarrow> 'a" assume S: "decseq S" and S_x: "S \<longlonglongrightarrow> x"
    5.22      from S_x have x_eq: "x = (INF i. S i)"
    5.23        by (rule LIMSEQ_unique) (intro LIMSEQ_INF S)
    5.24 -    show "(\<lambda>n. f (S n)) ----> f x"
    5.25 +    show "(\<lambda>n. f (S n)) \<longlonglongrightarrow> f x"
    5.26        unfolding x_eq inf_continuousD[OF f S]
    5.27        using S inf_continuous_mono[OF f] by (intro LIMSEQ_INF) (auto simp: mono_def antimono_def)
    5.28    qed (insert x, auto simp: less_top)
    5.29 @@ -2118,10 +2118,10 @@
    5.30  lemma countable_approach:
    5.31    fixes x :: ereal
    5.32    assumes "x \<noteq> -\<infinity>"
    5.33 -  shows "\<exists>f. incseq f \<and> (\<forall>i::nat. f i < x) \<and> (f ----> x)"
    5.34 +  shows "\<exists>f. incseq f \<and> (\<forall>i::nat. f i < x) \<and> (f \<longlonglongrightarrow> x)"
    5.35  proof (cases x)
    5.36    case (real r)
    5.37 -  moreover have "(\<lambda>n. r - inverse (real (Suc n))) ----> r - 0"
    5.38 +  moreover have "(\<lambda>n. r - inverse (real (Suc n))) \<longlonglongrightarrow> r - 0"
    5.39      by (intro tendsto_intros LIMSEQ_inverse_real_of_nat)
    5.40    ultimately show ?thesis
    5.41      by (intro exI[of _ "\<lambda>n. x - inverse (Suc n)"]) (auto simp: incseq_def)
    5.42 @@ -2141,7 +2141,7 @@
    5.43      by (auto intro!: exI[of _ "\<lambda>_. -\<infinity>"] simp: bot_ereal_def)
    5.44  next
    5.45    assume "Sup A \<noteq> -\<infinity>"
    5.46 -  then obtain l where "incseq l" and l: "\<And>i::nat. l i < Sup A" and l_Sup: "l ----> Sup A"
    5.47 +  then obtain l where "incseq l" and l: "\<And>i::nat. l i < Sup A" and l_Sup: "l \<longlonglongrightarrow> Sup A"
    5.48      by (auto dest: countable_approach)
    5.49  
    5.50    have "\<exists>f. \<forall>n. (f n \<in> A \<and> l n \<le> f n) \<and> (f n \<le> f (Suc n))"
    5.51 @@ -2161,9 +2161,9 @@
    5.52    moreover
    5.53    have "(SUP i. f i) = Sup A"
    5.54    proof (rule tendsto_unique)
    5.55 -    show "f ----> (SUP i. f i)"
    5.56 +    show "f \<longlonglongrightarrow> (SUP i. f i)"
    5.57        by (rule LIMSEQ_SUP \<open>incseq f\<close>)+
    5.58 -    show "f ----> Sup A"
    5.59 +    show "f \<longlonglongrightarrow> Sup A"
    5.60        using l f
    5.61        by (intro tendsto_sandwich[OF _ _ l_Sup tendsto_const])
    5.62           (auto simp: Sup_upper)
    5.63 @@ -2454,8 +2454,8 @@
    5.64    assumes "convergent a"
    5.65    shows "convergent (\<lambda>n. ereal (a n))" and "lim (\<lambda>n. ereal (a n)) = ereal (lim a)"
    5.66  proof -
    5.67 -  from assms obtain L where L: "a ----> L" unfolding convergent_def ..
    5.68 -  hence lim: "(\<lambda>n. ereal (a n)) ----> ereal L" using lim_ereal by auto
    5.69 +  from assms obtain L where L: "a \<longlonglongrightarrow> L" unfolding convergent_def ..
    5.70 +  hence lim: "(\<lambda>n. ereal (a n)) \<longlonglongrightarrow> ereal L" using lim_ereal by auto
    5.71    thus "convergent (\<lambda>n. ereal (a n))" unfolding convergent_def ..
    5.72    thus "lim (\<lambda>n. ereal (a n)) = ereal (lim a)" using lim L limI by metis
    5.73  qed
    5.74 @@ -2495,7 +2495,7 @@
    5.75      by auto
    5.76  qed
    5.77  
    5.78 -lemma Lim_PInfty: "f ----> \<infinity> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n \<ge> ereal B)"
    5.79 +lemma Lim_PInfty: "f \<longlonglongrightarrow> \<infinity> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n \<ge> ereal B)"
    5.80    unfolding tendsto_PInfty eventually_sequentially
    5.81  proof safe
    5.82    fix r
    5.83 @@ -2508,7 +2508,7 @@
    5.84      by (blast intro: less_le_trans)
    5.85  qed (blast intro: less_imp_le)
    5.86  
    5.87 -lemma Lim_MInfty: "f ----> -\<infinity> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. ereal B \<ge> f n)"
    5.88 +lemma Lim_MInfty: "f \<longlonglongrightarrow> -\<infinity> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. ereal B \<ge> f n)"
    5.89    unfolding tendsto_MInfty eventually_sequentially
    5.90  proof safe
    5.91    fix r
    5.92 @@ -2521,24 +2521,24 @@
    5.93      by (blast intro: le_less_trans)
    5.94  qed (blast intro: less_imp_le)
    5.95  
    5.96 -lemma Lim_bounded_PInfty: "f ----> l \<Longrightarrow> (\<And>n. f n \<le> ereal B) \<Longrightarrow> l \<noteq> \<infinity>"
    5.97 +lemma Lim_bounded_PInfty: "f \<longlonglongrightarrow> l \<Longrightarrow> (\<And>n. f n \<le> ereal B) \<Longrightarrow> l \<noteq> \<infinity>"
    5.98    using LIMSEQ_le_const2[of f l "ereal B"] by auto
    5.99  
   5.100 -lemma Lim_bounded_MInfty: "f ----> l \<Longrightarrow> (\<And>n. ereal B \<le> f n) \<Longrightarrow> l \<noteq> -\<infinity>"
   5.101 +lemma Lim_bounded_MInfty: "f \<longlonglongrightarrow> l \<Longrightarrow> (\<And>n. ereal B \<le> f n) \<Longrightarrow> l \<noteq> -\<infinity>"
   5.102    using LIMSEQ_le_const[of f l "ereal B"] by auto
   5.103  
   5.104  lemma tendsto_explicit:
   5.105 -  "f ----> f0 \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> f0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> S))"
   5.106 +  "f \<longlonglongrightarrow> f0 \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> f0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> S))"
   5.107    unfolding tendsto_def eventually_sequentially by auto
   5.108  
   5.109 -lemma Lim_bounded_PInfty2: "f ----> l \<Longrightarrow> \<forall>n\<ge>N. f n \<le> ereal B \<Longrightarrow> l \<noteq> \<infinity>"
   5.110 +lemma Lim_bounded_PInfty2: "f \<longlonglongrightarrow> l \<Longrightarrow> \<forall>n\<ge>N. f n \<le> ereal B \<Longrightarrow> l \<noteq> \<infinity>"
   5.111    using LIMSEQ_le_const2[of f l "ereal B"] by fastforce
   5.112  
   5.113 -lemma Lim_bounded_ereal: "f ----> (l :: 'a::linorder_topology) \<Longrightarrow> \<forall>n\<ge>M. f n \<le> C \<Longrightarrow> l \<le> C"
   5.114 +lemma Lim_bounded_ereal: "f \<longlonglongrightarrow> (l :: 'a::linorder_topology) \<Longrightarrow> \<forall>n\<ge>M. f n \<le> C \<Longrightarrow> l \<le> C"
   5.115    by (intro LIMSEQ_le_const2) auto
   5.116  
   5.117  lemma Lim_bounded2_ereal:
   5.118 -  assumes lim:"f ----> (l :: 'a::linorder_topology)"
   5.119 +  assumes lim:"f \<longlonglongrightarrow> (l :: 'a::linorder_topology)"
   5.120      and ge: "\<forall>n\<ge>N. f n \<ge> C"
   5.121    shows "l \<ge> C"
   5.122    using ge
   5.123 @@ -2696,7 +2696,7 @@
   5.124    fixes x :: ereal
   5.125    assumes "\<bar>x\<bar> \<noteq> \<infinity>"
   5.126      and "\<And>r. 0 < r \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r"
   5.127 -  shows "u ----> x"
   5.128 +  shows "u \<longlonglongrightarrow> x"
   5.129  proof (rule topological_tendstoI, unfold eventually_sequentially)
   5.130    obtain rx where rx: "x = ereal rx"
   5.131      using assms by (cases x) auto
   5.132 @@ -2732,7 +2732,7 @@
   5.133  qed
   5.134  
   5.135  lemma tendsto_obtains_N:
   5.136 -  assumes "f ----> f0"
   5.137 +  assumes "f \<longlonglongrightarrow> f0"
   5.138    assumes "open S"
   5.139      and "f0 \<in> S"
   5.140    obtains N where "\<forall>n\<ge>N. f n \<in> S"
   5.141 @@ -2742,10 +2742,10 @@
   5.142  lemma ereal_LimI_finite_iff:
   5.143    fixes x :: ereal
   5.144    assumes "\<bar>x\<bar> \<noteq> \<infinity>"
   5.145 -  shows "u ----> x \<longleftrightarrow> (\<forall>r. 0 < r \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r))"
   5.146 +  shows "u \<longlonglongrightarrow> x \<longleftrightarrow> (\<forall>r. 0 < r \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r))"
   5.147    (is "?lhs \<longleftrightarrow> ?rhs")
   5.148  proof
   5.149 -  assume lim: "u ----> x"
   5.150 +  assume lim: "u \<longlonglongrightarrow> x"
   5.151    {
   5.152      fix r :: ereal
   5.153      assume "r > 0"
   5.154 @@ -2762,7 +2762,7 @@
   5.155      by auto
   5.156  next
   5.157    assume ?rhs
   5.158 -  then show "u ----> x"
   5.159 +  then show "u \<longlonglongrightarrow> x"
   5.160      using ereal_LimI_finite[of x] assms by auto
   5.161  qed
   5.162  
   5.163 @@ -2924,7 +2924,7 @@
   5.164    shows "suminf f \<le> x"
   5.165  proof (rule Lim_bounded_ereal)
   5.166    have "summable f" using pos[THEN summable_ereal_pos] .
   5.167 -  then show "(\<lambda>N. \<Sum>n<N. f n) ----> suminf f"
   5.168 +  then show "(\<lambda>N. \<Sum>n<N. f n) \<longlonglongrightarrow> suminf f"
   5.169      by (auto dest!: summable_sums simp: sums_def atLeast0LessThan)
   5.170    show "\<forall>n\<ge>0. setsum f {..<n} \<le> x"
   5.171      using assms by auto
   5.172 @@ -3195,11 +3195,11 @@
   5.173    assumes f: "\<And>i. 0 \<le> f i"
   5.174    shows "(\<Sum>i. f (i + k)) \<le> suminf f"
   5.175  proof -
   5.176 -  have "(\<lambda>n. \<Sum>i<n. f (i + k)) ----> (\<Sum>i. f (i + k))"
   5.177 +  have "(\<lambda>n. \<Sum>i<n. f (i + k)) \<longlonglongrightarrow> (\<Sum>i. f (i + k))"
   5.178      using summable_sums[OF summable_ereal_pos] by (simp add: sums_def atLeast0LessThan f)
   5.179 -  moreover have "(\<lambda>n. \<Sum>i<n. f i) ----> (\<Sum>i. f i)"
   5.180 +  moreover have "(\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> (\<Sum>i. f i)"
   5.181      using summable_sums[OF summable_ereal_pos] by (simp add: sums_def atLeast0LessThan f)
   5.182 -  then have "(\<lambda>n. \<Sum>i<n + k. f i) ----> (\<Sum>i. f i)"
   5.183 +  then have "(\<lambda>n. \<Sum>i<n + k. f i) \<longlonglongrightarrow> (\<Sum>i. f i)"
   5.184      by (rule LIMSEQ_ignore_initial_segment)
   5.185    ultimately show ?thesis
   5.186    proof (rule LIMSEQ_le, safe intro!: exI[of _ k])
   5.187 @@ -3430,7 +3430,7 @@
   5.188  lemma limsup_le_liminf_real:
   5.189    fixes X :: "nat \<Rightarrow> real" and L :: real
   5.190    assumes 1: "limsup X \<le> L" and 2: "L \<le> liminf X"
   5.191 -  shows "X ----> L"
   5.192 +  shows "X \<longlonglongrightarrow> L"
   5.193  proof -
   5.194    from 1 2 have "limsup X \<le> liminf X" by auto
   5.195    hence 3: "limsup X = liminf X"
   5.196 @@ -3442,7 +3442,7 @@
   5.197      by (rule convergent_limsup_cl)
   5.198    also from 1 2 3 have "limsup X = L" by auto
   5.199    finally have "lim (\<lambda>n. ereal(X n)) = L" ..
   5.200 -  hence "(\<lambda>n. ereal (X n)) ----> L"
   5.201 +  hence "(\<lambda>n. ereal (X n)) \<longlonglongrightarrow> L"
   5.202      apply (elim subst)
   5.203      by (subst convergent_LIMSEQ_iff [symmetric], rule 4)
   5.204    thus ?thesis by simp
   5.205 @@ -3450,33 +3450,33 @@
   5.206  
   5.207  lemma liminf_PInfty:
   5.208    fixes X :: "nat \<Rightarrow> ereal"
   5.209 -  shows "X ----> \<infinity> \<longleftrightarrow> liminf X = \<infinity>"
   5.210 +  shows "X \<longlonglongrightarrow> \<infinity> \<longleftrightarrow> liminf X = \<infinity>"
   5.211    by (metis Liminf_PInfty trivial_limit_sequentially)
   5.212  
   5.213  lemma limsup_MInfty:
   5.214    fixes X :: "nat \<Rightarrow> ereal"
   5.215 -  shows "X ----> -\<infinity> \<longleftrightarrow> limsup X = -\<infinity>"
   5.216 +  shows "X \<longlonglongrightarrow> -\<infinity> \<longleftrightarrow> limsup X = -\<infinity>"
   5.217    by (metis Limsup_MInfty trivial_limit_sequentially)
   5.218  
   5.219  lemma ereal_lim_mono:
   5.220    fixes X Y :: "nat \<Rightarrow> 'a::linorder_topology"
   5.221    assumes "\<And>n. N \<le> n \<Longrightarrow> X n \<le> Y n"
   5.222 -    and "X ----> x"
   5.223 -    and "Y ----> y"
   5.224 +    and "X \<longlonglongrightarrow> x"
   5.225 +    and "Y \<longlonglongrightarrow> y"
   5.226    shows "x \<le> y"
   5.227    using assms(1) by (intro LIMSEQ_le[OF assms(2,3)]) auto
   5.228  
   5.229  lemma incseq_le_ereal:
   5.230    fixes X :: "nat \<Rightarrow> 'a::linorder_topology"
   5.231    assumes inc: "incseq X"
   5.232 -    and lim: "X ----> L"
   5.233 +    and lim: "X \<longlonglongrightarrow> L"
   5.234    shows "X N \<le> L"
   5.235    using inc
   5.236    by (intro ereal_lim_mono[of N, OF _ tendsto_const lim]) (simp add: incseq_def)
   5.237  
   5.238  lemma decseq_ge_ereal:
   5.239    assumes dec: "decseq X"
   5.240 -    and lim: "X ----> (L::'a::linorder_topology)"
   5.241 +    and lim: "X \<longlonglongrightarrow> (L::'a::linorder_topology)"
   5.242    shows "X N \<ge> L"
   5.243    using dec by (intro ereal_lim_mono[of N, OF _ lim tendsto_const]) (simp add: decseq_def)
   5.244  
   5.245 @@ -3491,21 +3491,21 @@
   5.246  lemma ereal_Sup_lim:
   5.247    fixes a :: "'a::{complete_linorder,linorder_topology}"
   5.248    assumes "\<And>n. b n \<in> s"
   5.249 -    and "b ----> a"
   5.250 +    and "b \<longlonglongrightarrow> a"
   5.251    shows "a \<le> Sup s"
   5.252    by (metis Lim_bounded_ereal assms complete_lattice_class.Sup_upper)
   5.253  
   5.254  lemma ereal_Inf_lim:
   5.255    fixes a :: "'a::{complete_linorder,linorder_topology}"
   5.256    assumes "\<And>n. b n \<in> s"
   5.257 -    and "b ----> a"
   5.258 +    and "b \<longlonglongrightarrow> a"
   5.259    shows "Inf s \<le> a"
   5.260    by (metis Lim_bounded2_ereal assms complete_lattice_class.Inf_lower)
   5.261  
   5.262  lemma SUP_Lim_ereal:
   5.263    fixes X :: "nat \<Rightarrow> 'a::{complete_linorder,linorder_topology}"
   5.264    assumes inc: "incseq X"
   5.265 -    and l: "X ----> l"
   5.266 +    and l: "X \<longlonglongrightarrow> l"
   5.267    shows "(SUP n. X n) = l"
   5.268    using LIMSEQ_SUP[OF inc] tendsto_unique[OF trivial_limit_sequentially l]
   5.269    by simp
   5.270 @@ -3513,25 +3513,25 @@
   5.271  lemma INF_Lim_ereal:
   5.272    fixes X :: "nat \<Rightarrow> 'a::{complete_linorder,linorder_topology}"
   5.273    assumes dec: "decseq X"
   5.274 -    and l: "X ----> l"
   5.275 +    and l: "X \<longlonglongrightarrow> l"
   5.276    shows "(INF n. X n) = l"
   5.277    using LIMSEQ_INF[OF dec] tendsto_unique[OF trivial_limit_sequentially l]
   5.278    by simp
   5.279  
   5.280  lemma SUP_eq_LIMSEQ:
   5.281    assumes "mono f"
   5.282 -  shows "(SUP n. ereal (f n)) = ereal x \<longleftrightarrow> f ----> x"
   5.283 +  shows "(SUP n. ereal (f n)) = ereal x \<longleftrightarrow> f \<longlonglongrightarrow> x"
   5.284  proof
   5.285    have inc: "incseq (\<lambda>i. ereal (f i))"
   5.286      using \<open>mono f\<close> unfolding mono_def incseq_def by auto
   5.287    {
   5.288 -    assume "f ----> x"
   5.289 -    then have "(\<lambda>i. ereal (f i)) ----> ereal x"
   5.290 +    assume "f \<longlonglongrightarrow> x"
   5.291 +    then have "(\<lambda>i. ereal (f i)) \<longlonglongrightarrow> ereal x"
   5.292        by auto
   5.293      from SUP_Lim_ereal[OF inc this] show "(SUP n. ereal (f n)) = ereal x" .
   5.294    next
   5.295      assume "(SUP n. ereal (f n)) = ereal x"
   5.296 -    with LIMSEQ_SUP[OF inc] show "f ----> x" by auto
   5.297 +    with LIMSEQ_SUP[OF inc] show "f \<longlonglongrightarrow> x" by auto
   5.298    }
   5.299  qed
   5.300  
     6.1 --- a/src/HOL/Library/Formal_Power_Series.thy	Tue Dec 29 22:41:22 2015 +0100
     6.2 +++ b/src/HOL/Library/Formal_Power_Series.thy	Tue Dec 29 23:04:53 2015 +0100
     6.3 @@ -898,8 +898,8 @@
     6.4    apply (simp add: setsum.delta')
     6.5    done
     6.6  
     6.7 -lemma fps_notation: "(\<lambda>n. setsum (\<lambda>i. fps_const(a$i) * X^i) {0..n}) ----> a"
     6.8 -  (is "?s ----> a")
     6.9 +lemma fps_notation: "(\<lambda>n. setsum (\<lambda>i. fps_const(a$i) * X^i) {0..n}) \<longlonglongrightarrow> a"
    6.10 +  (is "?s \<longlonglongrightarrow> a")
    6.11  proof -
    6.12    have "\<exists>n0. \<forall>n \<ge> n0. dist (?s n) a < r" if "r > 0" for r
    6.13    proof -
    6.14 @@ -4413,11 +4413,11 @@
    6.15  
    6.16    show "convergent X"
    6.17    proof (rule convergentI)
    6.18 -    show "X ----> Abs_fps (\<lambda>i. X (M i) $ i)"
    6.19 +    show "X \<longlonglongrightarrow> Abs_fps (\<lambda>i. X (M i) $ i)"
    6.20        unfolding tendsto_iff
    6.21      proof safe
    6.22        fix e::real assume e: "0 < e"
    6.23 -      have "(\<lambda>n. inverse (2 ^ n) :: real) ----> 0" by (rule LIMSEQ_inverse_realpow_zero) simp_all
    6.24 +      have "(\<lambda>n. inverse (2 ^ n) :: real) \<longlonglongrightarrow> 0" by (rule LIMSEQ_inverse_realpow_zero) simp_all
    6.25        from this and e have "eventually (\<lambda>i. inverse (2 ^ i) < e) sequentially"
    6.26          by (rule order_tendstoD)
    6.27        then obtain i where "inverse (2 ^ i) < e"
     7.1 --- a/src/HOL/Library/Indicator_Function.thy	Tue Dec 29 22:41:22 2015 +0100
     7.2 +++ b/src/HOL/Library/Indicator_Function.thy	Tue Dec 29 23:04:53 2015 +0100
     7.3 @@ -83,7 +83,7 @@
     7.4  
     7.5  lemma LIMSEQ_indicator_incseq:
     7.6    assumes "incseq A"
     7.7 -  shows "(\<lambda>i. indicator (A i) x :: 'a :: {topological_space, one, zero}) ----> indicator (\<Union>i. A i) x"
     7.8 +  shows "(\<lambda>i. indicator (A i) x :: 'a :: {topological_space, one, zero}) \<longlonglongrightarrow> indicator (\<Union>i. A i) x"
     7.9  proof cases
    7.10    assume "\<exists>i. x \<in> A i"
    7.11    then obtain i where "x \<in> A i"
    7.12 @@ -97,9 +97,9 @@
    7.13  qed (auto simp: indicator_def)
    7.14  
    7.15  lemma LIMSEQ_indicator_UN:
    7.16 -  "(\<lambda>k. indicator (\<Union>i<k. A i) x :: 'a :: {topological_space, one, zero}) ----> indicator (\<Union>i. A i) x"
    7.17 +  "(\<lambda>k. indicator (\<Union>i<k. A i) x :: 'a :: {topological_space, one, zero}) \<longlonglongrightarrow> indicator (\<Union>i. A i) x"
    7.18  proof -
    7.19 -  have "(\<lambda>k. indicator (\<Union>i<k. A i) x::'a) ----> indicator (\<Union>k. \<Union>i<k. A i) x"
    7.20 +  have "(\<lambda>k. indicator (\<Union>i<k. A i) x::'a) \<longlonglongrightarrow> indicator (\<Union>k. \<Union>i<k. A i) x"
    7.21      by (intro LIMSEQ_indicator_incseq) (auto simp: incseq_def intro: less_le_trans)
    7.22    also have "(\<Union>k. \<Union>i<k. A i) = (\<Union>i. A i)"
    7.23      by auto
    7.24 @@ -108,7 +108,7 @@
    7.25  
    7.26  lemma LIMSEQ_indicator_decseq:
    7.27    assumes "decseq A"
    7.28 -  shows "(\<lambda>i. indicator (A i) x :: 'a :: {topological_space, one, zero}) ----> indicator (\<Inter>i. A i) x"
    7.29 +  shows "(\<lambda>i. indicator (A i) x :: 'a :: {topological_space, one, zero}) \<longlonglongrightarrow> indicator (\<Inter>i. A i) x"
    7.30  proof cases
    7.31    assume "\<exists>i. x \<notin> A i"
    7.32    then obtain i where "x \<notin> A i"
    7.33 @@ -122,9 +122,9 @@
    7.34  qed (auto simp: indicator_def)
    7.35  
    7.36  lemma LIMSEQ_indicator_INT:
    7.37 -  "(\<lambda>k. indicator (\<Inter>i<k. A i) x :: 'a :: {topological_space, one, zero}) ----> indicator (\<Inter>i. A i) x"
    7.38 +  "(\<lambda>k. indicator (\<Inter>i<k. A i) x :: 'a :: {topological_space, one, zero}) \<longlonglongrightarrow> indicator (\<Inter>i. A i) x"
    7.39  proof -
    7.40 -  have "(\<lambda>k. indicator (\<Inter>i<k. A i) x::'a) ----> indicator (\<Inter>k. \<Inter>i<k. A i) x"
    7.41 +  have "(\<lambda>k. indicator (\<Inter>i<k. A i) x::'a) \<longlonglongrightarrow> indicator (\<Inter>k. \<Inter>i<k. A i) x"
    7.42      by (intro LIMSEQ_indicator_decseq) (auto simp: decseq_def intro: less_le_trans)
    7.43    also have "(\<Inter>k. \<Inter>i<k. A i) = (\<Inter>i. A i)"
    7.44      by auto
     8.1 --- a/src/HOL/Library/Liminf_Limsup.thy	Tue Dec 29 22:41:22 2015 +0100
     8.2 +++ b/src/HOL/Library/Liminf_Limsup.thy	Tue Dec 29 23:04:53 2015 +0100
     8.3 @@ -365,9 +365,9 @@
     8.4  
     8.5  lemma lim_increasing_cl:
     8.6    assumes "\<And>n m. n \<ge> m \<Longrightarrow> f n \<ge> f m"
     8.7 -  obtains l where "f ----> (l::'a::{complete_linorder,linorder_topology})"
     8.8 +  obtains l where "f \<longlonglongrightarrow> (l::'a::{complete_linorder,linorder_topology})"
     8.9  proof
    8.10 -  show "f ----> (SUP n. f n)"
    8.11 +  show "f \<longlonglongrightarrow> (SUP n. f n)"
    8.12      using assms
    8.13      by (intro increasing_tendsto)
    8.14         (auto simp: SUP_upper eventually_sequentially less_SUP_iff intro: less_le_trans)
    8.15 @@ -375,9 +375,9 @@
    8.16  
    8.17  lemma lim_decreasing_cl:
    8.18    assumes "\<And>n m. n \<ge> m \<Longrightarrow> f n \<le> f m"
    8.19 -  obtains l where "f ----> (l::'a::{complete_linorder,linorder_topology})"
    8.20 +  obtains l where "f \<longlonglongrightarrow> (l::'a::{complete_linorder,linorder_topology})"
    8.21  proof
    8.22 -  show "f ----> (INF n. f n)"
    8.23 +  show "f \<longlonglongrightarrow> (INF n. f n)"
    8.24      using assms
    8.25      by (intro decreasing_tendsto)
    8.26         (auto simp: INF_lower eventually_sequentially INF_less_iff intro: le_less_trans)
    8.27 @@ -385,7 +385,7 @@
    8.28  
    8.29  lemma compact_complete_linorder:
    8.30    fixes X :: "nat \<Rightarrow> 'a::{complete_linorder,linorder_topology}"
    8.31 -  shows "\<exists>l r. subseq r \<and> (X \<circ> r) ----> l"
    8.32 +  shows "\<exists>l r. subseq r \<and> (X \<circ> r) \<longlonglongrightarrow> l"
    8.33  proof -
    8.34    obtain r where "subseq r" and mono: "monoseq (X \<circ> r)"
    8.35      using seq_monosub[of X]
    8.36 @@ -393,7 +393,7 @@
    8.37      by auto
    8.38    then have "(\<forall>n m. m \<le> n \<longrightarrow> (X \<circ> r) m \<le> (X \<circ> r) n) \<or> (\<forall>n m. m \<le> n \<longrightarrow> (X \<circ> r) n \<le> (X \<circ> r) m)"
    8.39      by (auto simp add: monoseq_def)
    8.40 -  then obtain l where "(X \<circ> r) ----> l"
    8.41 +  then obtain l where "(X \<circ> r) \<longlonglongrightarrow> l"
    8.42       using lim_increasing_cl[of "X \<circ> r"] lim_decreasing_cl[of "X \<circ> r"]
    8.43       by auto
    8.44    then show ?thesis
     9.1 --- a/src/HOL/Library/Lub_Glb.thy	Tue Dec 29 22:41:22 2015 +0100
     9.2 +++ b/src/HOL/Library/Lub_Glb.thy	Tue Dec 29 23:04:53 2015 +0100
     9.3 @@ -223,9 +223,9 @@
     9.4    fixes X :: "nat \<Rightarrow> real"
     9.5    assumes u: "isLub UNIV {x. \<exists>n. X n = x} u" (* FIXME: use 'range X' *)
     9.6    assumes X: "\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n"
     9.7 -  shows "X ----> u"
     9.8 +  shows "X \<longlonglongrightarrow> u"
     9.9  proof -
    9.10 -  have "X ----> (SUP i. X i)"
    9.11 +  have "X \<longlonglongrightarrow> (SUP i. X i)"
    9.12      using u[THEN isLubD1] X
    9.13      by (intro LIMSEQ_incseq_SUP) (auto simp: incseq_def image_def eq_commute bdd_above_setle)
    9.14    also have "(SUP i. X i) = u"
    10.1 --- a/src/HOL/Library/Product_Vector.thy	Tue Dec 29 22:41:22 2015 +0100
    10.2 +++ b/src/HOL/Library/Product_Vector.thy	Tue Dec 29 23:04:53 2015 +0100
    10.3 @@ -383,13 +383,13 @@
    10.4  instance prod :: (complete_space, complete_space) complete_space
    10.5  proof
    10.6    fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
    10.7 -  have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
    10.8 +  have 1: "(\<lambda>n. fst (X n)) \<longlonglongrightarrow> lim (\<lambda>n. fst (X n))"
    10.9      using Cauchy_fst [OF \<open>Cauchy X\<close>]
   10.10      by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
   10.11 -  have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
   10.12 +  have 2: "(\<lambda>n. snd (X n)) \<longlonglongrightarrow> lim (\<lambda>n. snd (X n))"
   10.13      using Cauchy_snd [OF \<open>Cauchy X\<close>]
   10.14      by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
   10.15 -  have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
   10.16 +  have "X \<longlonglongrightarrow> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
   10.17      using tendsto_Pair [OF 1 2] by simp
   10.18    then show "convergent X"
   10.19      by (rule convergentI)
    11.1 --- a/src/HOL/Limits.thy	Tue Dec 29 22:41:22 2015 +0100
    11.2 +++ b/src/HOL/Limits.thy	Tue Dec 29 23:04:53 2015 +0100
    11.3 @@ -329,7 +329,7 @@
    11.4  
    11.5  (* TODO: delete *)
    11.6  (* FIXME: one use in NSA/HSEQ.thy *)
    11.7 -lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
    11.8 +lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X \<longlonglongrightarrow> L)"
    11.9    apply (rule_tac x="X m" in exI)
   11.10    apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
   11.11    unfolding eventually_sequentially
   11.12 @@ -1468,25 +1468,25 @@
   11.13  
   11.14  subsection \<open>Limits of Sequences\<close>
   11.15  
   11.16 -lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
   11.17 +lemma [trans]: "X=Y ==> Y \<longlonglongrightarrow> z ==> X \<longlonglongrightarrow> z"
   11.18    by simp
   11.19  
   11.20  lemma LIMSEQ_iff:
   11.21    fixes L :: "'a::real_normed_vector"
   11.22 -  shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
   11.23 +  shows "(X \<longlonglongrightarrow> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
   11.24  unfolding lim_sequentially dist_norm ..
   11.25  
   11.26  lemma LIMSEQ_I:
   11.27    fixes L :: "'a::real_normed_vector"
   11.28 -  shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
   11.29 +  shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X \<longlonglongrightarrow> L"
   11.30  by (simp add: LIMSEQ_iff)
   11.31  
   11.32  lemma LIMSEQ_D:
   11.33    fixes L :: "'a::real_normed_vector"
   11.34 -  shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
   11.35 +  shows "\<lbrakk>X \<longlonglongrightarrow> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
   11.36  by (simp add: LIMSEQ_iff)
   11.37  
   11.38 -lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
   11.39 +lemma LIMSEQ_linear: "\<lbrakk> X \<longlonglongrightarrow> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) \<longlonglongrightarrow> x"
   11.40    unfolding tendsto_def eventually_sequentially
   11.41    by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)
   11.42  
   11.43 @@ -1499,7 +1499,7 @@
   11.44  
   11.45  lemma Bseq_inverse:
   11.46    fixes a :: "'a::real_normed_div_algebra"
   11.47 -  shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
   11.48 +  shows "\<lbrakk>X \<longlonglongrightarrow> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
   11.49    by (rule Bfun_inverse)
   11.50  
   11.51  text\<open>Transformation of limit.\<close>
   11.52 @@ -1607,7 +1607,7 @@
   11.53  text\<open>An unbounded sequence's inverse tends to 0\<close>
   11.54  
   11.55  lemma LIMSEQ_inverse_zero:
   11.56 -  "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
   11.57 +  "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) \<longlonglongrightarrow> 0"
   11.58    apply (rule filterlim_compose[OF tendsto_inverse_0])
   11.59    apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
   11.60    apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
   11.61 @@ -1615,7 +1615,7 @@
   11.62  
   11.63  text\<open>The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity\<close>
   11.64  
   11.65 -lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
   11.66 +lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) \<longlonglongrightarrow> 0"
   11.67    by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
   11.68              filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
   11.69  
   11.70 @@ -1623,16 +1623,16 @@
   11.71  infinity is now easily proved\<close>
   11.72  
   11.73  lemma LIMSEQ_inverse_real_of_nat_add:
   11.74 -     "(%n. r + inverse(real(Suc n))) ----> r"
   11.75 +     "(%n. r + inverse(real(Suc n))) \<longlonglongrightarrow> r"
   11.76    using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
   11.77  
   11.78  lemma LIMSEQ_inverse_real_of_nat_add_minus:
   11.79 -     "(%n. r + -inverse(real(Suc n))) ----> r"
   11.80 +     "(%n. r + -inverse(real(Suc n))) \<longlonglongrightarrow> r"
   11.81    using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
   11.82    by auto
   11.83  
   11.84  lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
   11.85 -     "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
   11.86 +     "(%n. r*( 1 + -inverse(real(Suc n)))) \<longlonglongrightarrow> r"
   11.87    using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
   11.88    by auto
   11.89  
   11.90 @@ -1649,24 +1649,24 @@
   11.91  lemma lim_inverse_n: "((\<lambda>n. inverse(of_nat n)) ---> (0::'a::real_normed_field)) sequentially"
   11.92    using lim_1_over_n by (simp add: inverse_eq_divide)
   11.93  
   11.94 -lemma LIMSEQ_Suc_n_over_n: "(\<lambda>n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) ----> 1"
   11.95 +lemma LIMSEQ_Suc_n_over_n: "(\<lambda>n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
   11.96  proof (rule Lim_transform_eventually)
   11.97    show "eventually (\<lambda>n. 1 + inverse (of_nat n :: 'a) = of_nat (Suc n) / of_nat n) sequentially"
   11.98      using eventually_gt_at_top[of "0::nat"] by eventually_elim (simp add: field_simps)
   11.99 -  have "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) ----> 1 + 0"
  11.100 +  have "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1 + 0"
  11.101      by (intro tendsto_add tendsto_const lim_inverse_n)
  11.102 -  thus "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) ----> 1" by simp
  11.103 +  thus "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1" by simp
  11.104  qed
  11.105  
  11.106 -lemma LIMSEQ_n_over_Suc_n: "(\<lambda>n. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) ----> 1"
  11.107 +lemma LIMSEQ_n_over_Suc_n: "(\<lambda>n. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
  11.108  proof (rule Lim_transform_eventually)
  11.109    show "eventually (\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a) = 
  11.110                          of_nat n / of_nat (Suc n)) sequentially"
  11.111      using eventually_gt_at_top[of "0::nat"] 
  11.112      by eventually_elim (simp add: field_simps del: of_nat_Suc)
  11.113 -  have "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) ----> inverse 1"
  11.114 +  have "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> inverse 1"
  11.115      by (intro tendsto_inverse LIMSEQ_Suc_n_over_n) simp_all
  11.116 -  thus "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) ----> 1" by simp
  11.117 +  thus "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> 1" by simp
  11.118  qed
  11.119  
  11.120  subsection \<open>Convergence on sequences\<close>
  11.121 @@ -1733,8 +1733,8 @@
  11.122    assumes "convergent f"
  11.123    shows   "convergent (\<lambda>n. norm (f n))"
  11.124  proof -
  11.125 -  from assms have "f ----> lim f" by (simp add: convergent_LIMSEQ_iff)
  11.126 -  hence "(\<lambda>n. norm (f n)) ----> norm (lim f)" by (rule tendsto_norm)
  11.127 +  from assms have "f \<longlonglongrightarrow> lim f" by (simp add: convergent_LIMSEQ_iff)
  11.128 +  hence "(\<lambda>n. norm (f n)) \<longlonglongrightarrow> norm (lim f)" by (rule tendsto_norm)
  11.129    thus ?thesis by (auto simp: convergent_def)
  11.130  qed
  11.131  
  11.132 @@ -1794,7 +1794,7 @@
  11.133    fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
  11.134    assumes u: "bdd_above (range X)"
  11.135    assumes X: "incseq X"
  11.136 -  shows "X ----> (SUP i. X i)"
  11.137 +  shows "X \<longlonglongrightarrow> (SUP i. X i)"
  11.138    by (rule order_tendstoI)
  11.139       (auto simp: eventually_sequentially u less_cSUP_iff intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
  11.140  
  11.141 @@ -1802,7 +1802,7 @@
  11.142    fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
  11.143    assumes u: "bdd_below (range X)"
  11.144    assumes X: "decseq X"
  11.145 -  shows "X ----> (INF i. X i)"
  11.146 +  shows "X \<longlonglongrightarrow> (INF i. X i)"
  11.147    by (rule order_tendstoI)
  11.148       (auto simp: eventually_sequentially u cINF_less_iff intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
  11.149  
  11.150 @@ -1845,24 +1845,24 @@
  11.151  lemma incseq_convergent:
  11.152    fixes X :: "nat \<Rightarrow> real"
  11.153    assumes "incseq X" and "\<forall>i. X i \<le> B"
  11.154 -  obtains L where "X ----> L" "\<forall>i. X i \<le> L"
  11.155 +  obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. X i \<le> L"
  11.156  proof atomize_elim
  11.157    from incseq_bounded[OF assms] \<open>incseq X\<close> Bseq_monoseq_convergent[of X]
  11.158 -  obtain L where "X ----> L"
  11.159 +  obtain L where "X \<longlonglongrightarrow> L"
  11.160      by (auto simp: convergent_def monoseq_def incseq_def)
  11.161 -  with \<open>incseq X\<close> show "\<exists>L. X ----> L \<and> (\<forall>i. X i \<le> L)"
  11.162 +  with \<open>incseq X\<close> show "\<exists>L. X \<longlonglongrightarrow> L \<and> (\<forall>i. X i \<le> L)"
  11.163      by (auto intro!: exI[of _ L] incseq_le)
  11.164  qed
  11.165  
  11.166  lemma decseq_convergent:
  11.167    fixes X :: "nat \<Rightarrow> real"
  11.168    assumes "decseq X" and "\<forall>i. B \<le> X i"
  11.169 -  obtains L where "X ----> L" "\<forall>i. L \<le> X i"
  11.170 +  obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. L \<le> X i"
  11.171  proof atomize_elim
  11.172    from decseq_bounded[OF assms] \<open>decseq X\<close> Bseq_monoseq_convergent[of X]
  11.173 -  obtain L where "X ----> L"
  11.174 +  obtain L where "X \<longlonglongrightarrow> L"
  11.175      by (auto simp: convergent_def monoseq_def decseq_def)
  11.176 -  with \<open>decseq X\<close> show "\<exists>L. X ----> L \<and> (\<forall>i. L \<le> X i)"
  11.177 +  with \<open>decseq X\<close> show "\<exists>L. X \<longlonglongrightarrow> L \<and> (\<forall>i. L \<le> X i)"
  11.178      by (auto intro!: exI[of _ L] decseq_le)
  11.179  qed
  11.180  
  11.181 @@ -1901,39 +1901,39 @@
  11.182    "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
  11.183  by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
  11.184  
  11.185 -lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
  11.186 +lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) \<longlonglongrightarrow> 0"
  11.187    by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
  11.188  
  11.189  lemma LIMSEQ_realpow_zero:
  11.190 -  "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
  11.191 +  "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
  11.192  proof cases
  11.193    assume "0 \<le> x" and "x \<noteq> 0"
  11.194    hence x0: "0 < x" by simp
  11.195    assume x1: "x < 1"
  11.196    from x0 x1 have "1 < inverse x"
  11.197      by (rule one_less_inverse)
  11.198 -  hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
  11.199 +  hence "(\<lambda>n. inverse (inverse x ^ n)) \<longlonglongrightarrow> 0"
  11.200      by (rule LIMSEQ_inverse_realpow_zero)
  11.201    thus ?thesis by (simp add: power_inverse)
  11.202  qed (rule LIMSEQ_imp_Suc, simp)
  11.203  
  11.204  lemma LIMSEQ_power_zero:
  11.205    fixes x :: "'a::{real_normed_algebra_1}"
  11.206 -  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
  11.207 +  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
  11.208  apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
  11.209  apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
  11.210  apply (simp add: power_abs norm_power_ineq)
  11.211  done
  11.212  
  11.213 -lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) ----> 0"
  11.214 +lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) \<longlonglongrightarrow> 0"
  11.215    by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
  11.216  
  11.217  text\<open>Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}\<close>
  11.218  
  11.219 -lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) ----> 0"
  11.220 +lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) \<longlonglongrightarrow> 0"
  11.221    by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
  11.222  
  11.223 -lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) ----> 0"
  11.224 +lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) \<longlonglongrightarrow> 0"
  11.225    by (rule LIMSEQ_power_zero) simp
  11.226  
  11.227  
  11.228 @@ -2162,8 +2162,8 @@
  11.229  subsection \<open>Nested Intervals and Bisection -- Needed for Compactness\<close>
  11.230  
  11.231  lemma nested_sequence_unique:
  11.232 -  assumes "\<forall>n. f n \<le> f (Suc n)" "\<forall>n. g (Suc n) \<le> g n" "\<forall>n. f n \<le> g n" "(\<lambda>n. f n - g n) ----> 0"
  11.233 -  shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f ----> l) \<and> ((\<forall>n. l \<le> g n) \<and> g ----> l)"
  11.234 +  assumes "\<forall>n. f n \<le> f (Suc n)" "\<forall>n. g (Suc n) \<le> g n" "\<forall>n. f n \<le> g n" "(\<lambda>n. f n - g n) \<longlonglongrightarrow> 0"
  11.235 +  shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f \<longlonglongrightarrow> l) \<and> ((\<forall>n. l \<le> g n) \<and> g \<longlonglongrightarrow> l)"
  11.236  proof -
  11.237    have "incseq f" unfolding incseq_Suc_iff by fact
  11.238    have "decseq g" unfolding decseq_Suc_iff by fact
  11.239 @@ -2171,15 +2171,15 @@
  11.240    { fix n
  11.241      from \<open>decseq g\<close> have "g n \<le> g 0" by (rule decseqD) simp
  11.242      with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f n \<le> g 0" by auto }
  11.243 -  then obtain u where "f ----> u" "\<forall>i. f i \<le> u"
  11.244 +  then obtain u where "f \<longlonglongrightarrow> u" "\<forall>i. f i \<le> u"
  11.245      using incseq_convergent[OF \<open>incseq f\<close>] by auto
  11.246    moreover
  11.247    { fix n
  11.248      from \<open>incseq f\<close> have "f 0 \<le> f n" by (rule incseqD) simp
  11.249      with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f 0 \<le> g n" by simp }
  11.250 -  then obtain l where "g ----> l" "\<forall>i. l \<le> g i"
  11.251 +  then obtain l where "g \<longlonglongrightarrow> l" "\<forall>i. l \<le> g i"
  11.252      using decseq_convergent[OF \<open>decseq g\<close>] by auto
  11.253 -  moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF \<open>f ----> u\<close> \<open>g ----> l\<close>]]
  11.254 +  moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF \<open>f \<longlonglongrightarrow> u\<close> \<open>g \<longlonglongrightarrow> l\<close>]]
  11.255    ultimately show ?thesis by auto
  11.256  qed
  11.257  
  11.258 @@ -2198,14 +2198,14 @@
  11.259  
  11.260    { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
  11.261  
  11.262 -  have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l ----> x) \<and> ((\<forall>n. x \<le> u n) \<and> u ----> x)"
  11.263 +  have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l \<longlonglongrightarrow> x) \<and> ((\<forall>n. x \<le> u n) \<and> u \<longlonglongrightarrow> x)"
  11.264    proof (safe intro!: nested_sequence_unique)
  11.265      fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
  11.266    next
  11.267      { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
  11.268 -    then show "(\<lambda>n. l n - u n) ----> 0" by (simp add: LIMSEQ_divide_realpow_zero)
  11.269 +    then show "(\<lambda>n. l n - u n) \<longlonglongrightarrow> 0" by (simp add: LIMSEQ_divide_realpow_zero)
  11.270    qed fact
  11.271 -  then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l ----> x" "u ----> x" by auto
  11.272 +  then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l \<longlonglongrightarrow> x" "u \<longlonglongrightarrow> x" by auto
  11.273    obtain d where "0 < d" and d: "\<And>a b. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> b - a < d \<Longrightarrow> P a b"
  11.274      using \<open>l 0 \<le> x\<close> \<open>x \<le> u 0\<close> local[of x] by auto
  11.275  
  11.276 @@ -2218,9 +2218,9 @@
  11.277        qed (simp add: \<open>\<not> P a b\<close>) }
  11.278      moreover
  11.279      { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
  11.280 -        using \<open>0 < d\<close> \<open>l ----> x\<close> by (intro order_tendstoD[of _ x]) auto
  11.281 +        using \<open>0 < d\<close> \<open>l \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
  11.282        moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
  11.283 -        using \<open>0 < d\<close> \<open>u ----> x\<close> by (intro order_tendstoD[of _ x]) auto
  11.284 +        using \<open>0 < d\<close> \<open>u \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
  11.285        ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
  11.286        proof eventually_elim
  11.287          fix n assume "x - d / 2 < l n" "u n < x + d / 2"
    12.1 --- a/src/HOL/Metis_Examples/Clausification.thy	Tue Dec 29 22:41:22 2015 +0100
    12.2 +++ b/src/HOL/Metis_Examples/Clausification.thy	Tue Dec 29 23:04:53 2015 +0100
    12.3 @@ -115,7 +115,7 @@
    12.4  
    12.5  lemma
    12.6    fixes x :: real
    12.7 -  assumes fn_le: "!!n. f n \<le> x" and 1: "f ----> lim f"
    12.8 +  assumes fn_le: "!!n. f n \<le> x" and 1: "f \<longlonglongrightarrow> lim f"
    12.9    shows "lim f \<le> x"
   12.10  by (metis 1 LIMSEQ_le_const2 fn_le)
   12.11  
    13.1 --- a/src/HOL/Multivariate_Analysis/Bounded_Continuous_Function.thy	Tue Dec 29 22:41:22 2015 +0100
    13.2 +++ b/src/HOL/Multivariate_Analysis/Bounded_Continuous_Function.thy	Tue Dec 29 23:04:53 2015 +0100
    13.3 @@ -138,7 +138,7 @@
    13.4    unfolding closed_sequential_limits
    13.5  proof safe
    13.6    fix f l
    13.7 -  assume seq: "\<forall>n. f n \<in> Abs_bcontfun ` (Pi I X \<inter> bcontfun)" and lim: "f ----> l"
    13.8 +  assume seq: "\<forall>n. f n \<in> Abs_bcontfun ` (Pi I X \<inter> bcontfun)" and lim: "f \<longlonglongrightarrow> l"
    13.9    have lim_fun: "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x. dist (Rep_bcontfun (f n) x) (Rep_bcontfun l x) < e"
   13.10      using LIMSEQ_imp_Cauchy[OF lim, simplified Cauchy_def] metric_LIMSEQ_D[OF lim]
   13.11      by (intro uniformly_cauchy_imp_uniformly_convergent[where P="\<lambda>_. True", simplified])
   13.12 @@ -156,7 +156,7 @@
   13.13          by (auto simp: Abs_bcontfun_inverse)
   13.14      qed
   13.15      moreover note sequentially_bot
   13.16 -    moreover have "(\<lambda>n. Rep_bcontfun (f n) x) ----> Rep_bcontfun l x"
   13.17 +    moreover have "(\<lambda>n. Rep_bcontfun (f n) x) \<longlonglongrightarrow> Rep_bcontfun l x"
   13.18        using lim_fun by (blast intro!: metric_LIMSEQ_I)
   13.19      ultimately show "Rep_bcontfun l x \<in> X x"
   13.20        by (rule Lim_in_closed_set)
    14.1 --- a/src/HOL/Multivariate_Analysis/Bounded_Linear_Function.thy	Tue Dec 29 22:41:22 2015 +0100
    14.2 +++ b/src/HOL/Multivariate_Analysis/Bounded_Linear_Function.thy	Tue Dec 29 23:04:53 2015 +0100
    14.3 @@ -249,7 +249,7 @@
    14.4        by (subst xy) (simp add: blinfun.bilinear_simps)
    14.5      finally have "convergent (\<lambda>n. X n x)" .
    14.6    }
    14.7 -  then obtain v where v: "\<And>x. (\<lambda>n. X n x) ----> v x"
    14.8 +  then obtain v where v: "\<And>x. (\<lambda>n. X n x) \<longlonglongrightarrow> v x"
    14.9      unfolding convergent_def
   14.10      by metis
   14.11  
   14.12 @@ -269,7 +269,7 @@
   14.13        finally show "norm (norm (X m) - norm (X n)) < e" .
   14.14      qed
   14.15    qed
   14.16 -  then obtain K where K: "(\<lambda>n. norm (X n)) ----> K"
   14.17 +  then obtain K where K: "(\<lambda>n. norm (X n)) \<longlonglongrightarrow> K"
   14.18      unfolding Cauchy_convergent_iff convergent_def
   14.19      by metis
   14.20  
   14.21 @@ -290,10 +290,10 @@
   14.22          by (simp add: ac_simps)
   14.23      qed
   14.24    qed
   14.25 -  hence Bv: "\<And>x. (\<lambda>n. X n x) ----> Blinfun v x"
   14.26 +  hence Bv: "\<And>x. (\<lambda>n. X n x) \<longlonglongrightarrow> Blinfun v x"
   14.27      by (auto simp: bounded_linear_Blinfun_apply v)
   14.28  
   14.29 -  have "X ----> Blinfun v"
   14.30 +  have "X \<longlonglongrightarrow> Blinfun v"
   14.31    proof (rule LIMSEQ_I)
   14.32      fix r::real assume "r > 0"
   14.33      def r' \<equiv> "r / 2"
   14.34 @@ -320,7 +320,7 @@
   14.35              using M[OF n elim] by (simp add: mult_right_mono)
   14.36            finally show ?case .
   14.37          qed
   14.38 -        have tendsto_v: "(\<lambda>m. norm (X n x - X m x)) ----> norm (X n x - Blinfun v x)"
   14.39 +        have tendsto_v: "(\<lambda>m. norm (X n x - X m x)) \<longlonglongrightarrow> norm (X n x - Blinfun v x)"
   14.40            by (auto intro!: tendsto_intros Bv)
   14.41          show "norm ((X n - Blinfun v) x) \<le> r' * norm x"
   14.42            by (auto intro!: tendsto_ge_const tendsto_v ev_le simp: blinfun.bilinear_simps)
    15.1 --- a/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy	Tue Dec 29 22:41:22 2015 +0100
    15.2 +++ b/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy	Tue Dec 29 23:04:53 2015 +0100
    15.3 @@ -838,7 +838,7 @@
    15.4    proof (rule has_derivative_sequence [OF cvs _ _ x])
    15.5      show "\<forall>n. \<forall>x\<in>s. (f n has_derivative (op * (f' n x))) (at x within s)"
    15.6        by (metis has_field_derivative_def df)
    15.7 -  next show "(\<lambda>n. f n x) ----> l"
    15.8 +  next show "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
    15.9      by (rule tf)
   15.10    next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
   15.11      by (blast intro: **)
    16.1 --- a/src/HOL/Multivariate_Analysis/Complex_Transcendental.thy	Tue Dec 29 22:41:22 2015 +0100
    16.2 +++ b/src/HOL/Multivariate_Analysis/Complex_Transcendental.thy	Tue Dec 29 23:04:53 2015 +0100
    16.3 @@ -1636,7 +1636,7 @@
    16.4    have "\<forall>n>0. 3 \<le> n \<longrightarrow> 1 \<le> ln (real_of_nat n)"
    16.5      using ln3_gt_1
    16.6      by (force intro: order_trans [of _ "ln 3"] ln3_gt_1)
    16.7 -  moreover have "(\<lambda>n. cmod (Ln (of_nat n) / of_nat n powr s)) ----> 0"
    16.8 +  moreover have "(\<lambda>n. cmod (Ln (of_nat n) / of_nat n powr s)) \<longlonglongrightarrow> 0"
    16.9      using lim_Ln_over_power [OF assms]
   16.10      by (metis tendsto_norm_zero_iff)
   16.11    ultimately show ?thesis
    17.1 --- a/src/HOL/Multivariate_Analysis/Derivative.thy	Tue Dec 29 22:41:22 2015 +0100
    17.2 +++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Tue Dec 29 23:04:53 2015 +0100
    17.3 @@ -1957,7 +1957,7 @@
    17.4        qed
    17.5      qed
    17.6    qed
    17.7 -  then obtain g where g: "\<forall>x\<in>s. (\<lambda>n. f n x) ----> g x" ..
    17.8 +  then obtain g where g: "\<forall>x\<in>s. (\<lambda>n. f n x) \<longlonglongrightarrow> g x" ..
    17.9    have lem2: "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm ((f n x - f n y) - (g x - g y)) \<le> e * norm (x - y)"
   17.10    proof (rule, rule)
   17.11      fix e :: real
    18.1 --- a/src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy	Tue Dec 29 22:41:22 2015 +0100
    18.2 +++ b/src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy	Tue Dec 29 23:04:53 2015 +0100
    18.3 @@ -366,10 +366,10 @@
    18.4  instance vec :: (complete_space, finite) complete_space
    18.5  proof
    18.6    fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"
    18.7 -  have "\<And>i. (\<lambda>n. X n $ i) ----> lim (\<lambda>n. X n $ i)"
    18.8 +  have "\<And>i. (\<lambda>n. X n $ i) \<longlonglongrightarrow> lim (\<lambda>n. X n $ i)"
    18.9      using Cauchy_vec_nth [OF \<open>Cauchy X\<close>]
   18.10      by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
   18.11 -  hence "X ----> vec_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"
   18.12 +  hence "X \<longlonglongrightarrow> vec_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"
   18.13      by (simp add: vec_tendstoI)
   18.14    then show "convergent X"
   18.15      by (rule convergentI)
    19.1 --- a/src/HOL/Multivariate_Analysis/Integration.thy	Tue Dec 29 22:41:22 2015 +0100
    19.2 +++ b/src/HOL/Multivariate_Analysis/Integration.thy	Tue Dec 29 23:04:53 2015 +0100
    19.3 @@ -4580,7 +4580,7 @@
    19.4          by (auto intro!: triangle3)
    19.5      qed
    19.6    qed
    19.7 -  then obtain s where s: "i ----> s"
    19.8 +  then obtain s where s: "i \<longlonglongrightarrow> s"
    19.9      using convergent_eq_cauchy[symmetric] by blast
   19.10    show ?thesis
   19.11      unfolding integrable_on_def has_integral
   19.12 @@ -10201,8 +10201,8 @@
   19.13    fixes f :: "nat \<Rightarrow> 'a::euclidean_space \<Rightarrow> real"
   19.14    assumes f: "\<And>k. (f k has_integral x k) s"
   19.15    assumes "\<And>k x. x \<in> s \<Longrightarrow> f k x \<le> f (Suc k) x"
   19.16 -  assumes "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>k. f k x) ----> g x"
   19.17 -  assumes "x ----> x'"
   19.18 +  assumes "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
   19.19 +  assumes "x \<longlonglongrightarrow> x'"
   19.20    shows "(g has_integral x') s"
   19.21  proof -
   19.22    have x_eq: "x = (\<lambda>i. integral s (f i))"
   19.23 @@ -10210,13 +10210,13 @@
   19.24    then have x: "{integral s (f k) |k. True} = range x"
   19.25      by auto
   19.26  
   19.27 -  have "g integrable_on s \<and> (\<lambda>k. integral s (f k)) ----> integral s g"
   19.28 +  have "g integrable_on s \<and> (\<lambda>k. integral s (f k)) \<longlonglongrightarrow> integral s g"
   19.29    proof (intro monotone_convergence_increasing allI ballI assms)
   19.30      show "bounded {integral s (f k) |k. True}"
   19.31        unfolding x by (rule convergent_imp_bounded) fact
   19.32    qed (auto intro: f)
   19.33    moreover then have "integral s g = x'"
   19.34 -    by (intro LIMSEQ_unique[OF _ \<open>x ----> x'\<close>]) (simp add: x_eq)
   19.35 +    by (intro LIMSEQ_unique[OF _ \<open>x \<longlonglongrightarrow> x'\<close>]) (simp add: x_eq)
   19.36    ultimately show ?thesis
   19.37      by (simp add: has_integral_integral)
   19.38  qed
   19.39 @@ -12087,7 +12087,7 @@
   19.40      show "Inf {f j x |j. k \<le> j} \<le> Inf {f j x |j. Suc k \<le> j}"
   19.41        by (intro cInf_superset_mono) (auto simp: \<open>x\<in>s\<close>)
   19.42  
   19.43 -    show "(\<lambda>k::nat. Inf {f j x |j. k \<le> j}) ----> g x"
   19.44 +    show "(\<lambda>k::nat. Inf {f j x |j. k \<le> j}) \<longlonglongrightarrow> g x"
   19.45      proof (rule LIMSEQ_I, goal_cases)
   19.46        case r: (1 r)
   19.47        then have "0<r/2"
   19.48 @@ -12183,8 +12183,8 @@
   19.49  lemma has_integral_dominated_convergence:
   19.50    fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
   19.51    assumes "\<And>k. (f k has_integral y k) s" "h integrable_on s"
   19.52 -    "\<And>k. \<forall>x\<in>s. norm (f k x) \<le> h x" "\<forall>x\<in>s. (\<lambda>k. f k x) ----> g x"
   19.53 -    and x: "y ----> x"
   19.54 +    "\<And>k. \<forall>x\<in>s. norm (f k x) \<le> h x" "\<forall>x\<in>s. (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
   19.55 +    and x: "y \<longlonglongrightarrow> x"
   19.56    shows "(g has_integral x) s"
   19.57  proof -
   19.58    have int_f: "\<And>k. (f k) integrable_on s"
   19.59 @@ -12193,9 +12193,9 @@
   19.60      by (intro integrable_integral dominated_convergence[OF int_f assms(2)]) fact+
   19.61    moreover have "integral s g = x"
   19.62    proof (rule LIMSEQ_unique)
   19.63 -    show "(\<lambda>i. integral s (f i)) ----> x"
   19.64 +    show "(\<lambda>i. integral s (f i)) \<longlonglongrightarrow> x"
   19.65        using integral_unique[OF assms(1)] x by simp
   19.66 -    show "(\<lambda>i. integral s (f i)) ----> integral s g"
   19.67 +    show "(\<lambda>i. integral s (f i)) \<longlonglongrightarrow> integral s g"
   19.68        by (intro dominated_convergence[OF int_f assms(2)]) fact+
   19.69    qed
   19.70    ultimately show ?thesis
    20.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Tue Dec 29 22:41:22 2015 +0100
    20.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Tue Dec 29 23:04:53 2015 +0100
    20.3 @@ -39,7 +39,7 @@
    20.4    using dist_triangle[of y z x] by (simp add: dist_commute)
    20.5  
    20.6  (* LEGACY *)
    20.7 -lemma lim_subseq: "subseq r \<Longrightarrow> s ----> l \<Longrightarrow> (s \<circ> r) ----> l"
    20.8 +lemma lim_subseq: "subseq r \<Longrightarrow> s \<longlonglongrightarrow> l \<Longrightarrow> (s \<circ> r) \<longlonglongrightarrow> l"
    20.9    by (rule LIMSEQ_subseq_LIMSEQ)
   20.10  
   20.11  lemma countable_PiE:
   20.12 @@ -2365,7 +2365,7 @@
   20.13  
   20.14  lemma Lim_within_LIMSEQ:
   20.15    fixes a :: "'a::first_countable_topology"
   20.16 -  assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
   20.17 +  assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
   20.18    shows "(X ---> L) (at a within T)"
   20.19    using assms unfolding tendsto_def [where l=L]
   20.20    by (simp add: sequentially_imp_eventually_within)
   20.21 @@ -2430,7 +2430,7 @@
   20.22      then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto
   20.23    }
   20.24    then have "\<forall>n. f n \<in> S - {x}" by auto
   20.25 -  moreover have "(\<lambda>n. f n) ----> x"
   20.26 +  moreover have "(\<lambda>n. f n) \<longlonglongrightarrow> x"
   20.27    proof (rule topological_tendstoI)
   20.28      fix S
   20.29      assume "open S" "x \<in> S"
   20.30 @@ -2441,7 +2441,7 @@
   20.31    ultimately show ?rhs by fast
   20.32  next
   20.33    assume ?rhs
   20.34 -  then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x"
   20.35 +  then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f \<longlonglongrightarrow> x"
   20.36      by auto
   20.37    show ?lhs
   20.38      unfolding islimpt_def
   20.39 @@ -3515,7 +3515,7 @@
   20.40  lemma acc_point_range_imp_convergent_subsequence:
   20.41    fixes l :: "'a :: first_countable_topology"
   20.42    assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"
   20.43 -  shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"
   20.44 +  shows "\<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
   20.45  proof -
   20.46    from countable_basis_at_decseq[of l]
   20.47    obtain A where A:
   20.48 @@ -3542,7 +3542,7 @@
   20.49    have "subseq r"
   20.50      by (auto simp: r_def s subseq_Suc_iff)
   20.51    moreover
   20.52 -  have "(\<lambda>n. f (r n)) ----> l"
   20.53 +  have "(\<lambda>n. f (r n)) \<longlonglongrightarrow> l"
   20.54    proof (rule topological_tendstoI)
   20.55      fix S
   20.56      assume "open S" "l \<in> S"
   20.57 @@ -3560,7 +3560,7 @@
   20.58      ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"
   20.59        by eventually_elim auto
   20.60    qed
   20.61 -  ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"
   20.62 +  ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
   20.63      by (auto simp: convergent_def comp_def)
   20.64  qed
   20.65  
   20.66 @@ -3658,7 +3658,7 @@
   20.67  lemma islimpt_range_imp_convergent_subsequence:
   20.68    fixes l :: "'a :: {t1_space, first_countable_topology}"
   20.69    assumes l: "l islimpt (range f)"
   20.70 -  shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"
   20.71 +  shows "\<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
   20.72    using l unfolding islimpt_eq_acc_point
   20.73    by (rule acc_point_range_imp_convergent_subsequence)
   20.74  
   20.75 @@ -3987,11 +3987,11 @@
   20.76    using assms unfolding seq_compact_def by fast
   20.77  
   20.78  lemma closed_sequentially: (* TODO: move upwards *)
   20.79 -  assumes "closed s" and "\<forall>n. f n \<in> s" and "f ----> l"
   20.80 +  assumes "closed s" and "\<forall>n. f n \<in> s" and "f \<longlonglongrightarrow> l"
   20.81    shows "l \<in> s"
   20.82  proof (rule ccontr)
   20.83    assume "l \<notin> s"
   20.84 -  with \<open>closed s\<close> and \<open>f ----> l\<close> have "eventually (\<lambda>n. f n \<in> - s) sequentially"
   20.85 +  with \<open>closed s\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "eventually (\<lambda>n. f n \<in> - s) sequentially"
   20.86      by (fast intro: topological_tendstoD)
   20.87    with \<open>\<forall>n. f n \<in> s\<close> show "False"
   20.88      by simp
   20.89 @@ -4005,13 +4005,13 @@
   20.90    hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"
   20.91      by simp_all
   20.92    from \<open>seq_compact s\<close> and \<open>\<forall>n. f n \<in> s\<close>
   20.93 -  obtain l r where "l \<in> s" and r: "subseq r" and l: "(f \<circ> r) ----> l"
   20.94 +  obtain l r where "l \<in> s" and r: "subseq r" and l: "(f \<circ> r) \<longlonglongrightarrow> l"
   20.95      by (rule seq_compactE)
   20.96    from \<open>\<forall>n. f n \<in> t\<close> have "\<forall>n. (f \<circ> r) n \<in> t"
   20.97      by simp
   20.98    from \<open>closed t\<close> and this and l have "l \<in> t"
   20.99      by (rule closed_sequentially)
  20.100 -  with \<open>l \<in> s\<close> and r and l show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"
  20.101 +  with \<open>l \<in> s\<close> and r and l show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
  20.102      by fast
  20.103  qed
  20.104  
  20.105 @@ -4027,7 +4027,7 @@
  20.106  proof (safe intro!: countably_compactI)
  20.107    fix A
  20.108    assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"
  20.109 -  have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) ----> x"
  20.110 +  have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) \<longlonglongrightarrow> x"
  20.111      using \<open>seq_compact U\<close> by (fastforce simp: seq_compact_def subset_eq)
  20.112    show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
  20.113    proof cases
  20.114 @@ -4048,7 +4048,7 @@
  20.115          using \<open>A \<noteq> {}\<close> unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)
  20.116        then have "range X \<subseteq> U"
  20.117          by auto
  20.118 -      with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x"
  20.119 +      with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) \<longlonglongrightarrow> x"
  20.120          by auto
  20.121        from \<open>x\<in>U\<close> \<open>U \<subseteq> \<Union>A\<close> from_nat_into_surj[OF \<open>countable A\<close>]
  20.122        obtain n where "x \<in> from_nat_into A n" by auto
  20.123 @@ -4115,7 +4115,7 @@
  20.124    have "subseq r"
  20.125      by (auto simp: r_def s subseq_Suc_iff)
  20.126    moreover
  20.127 -  have "(\<lambda>n. X (r n)) ----> x"
  20.128 +  have "(\<lambda>n. X (r n)) \<longlonglongrightarrow> x"
  20.129    proof (rule topological_tendstoI)
  20.130      fix S
  20.131      assume "open S" "x \<in> S"
  20.132 @@ -4133,7 +4133,7 @@
  20.133      ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"
  20.134        by eventually_elim auto
  20.135    qed
  20.136 -  ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) ----> x"
  20.137 +  ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) \<longlonglongrightarrow> x"
  20.138      using \<open>x \<in> U\<close> by (auto simp: convergent_def comp_def)
  20.139  qed
  20.140  
  20.141 @@ -4191,9 +4191,9 @@
  20.142          using pigeonhole_infinite[OF _ True] by auto
  20.143        then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"
  20.144          using infinite_enumerate by blast
  20.145 -      then have "subseq r \<and> (f \<circ> r) ----> f l"
  20.146 +      then have "subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> f l"
  20.147          by (simp add: fr o_def)
  20.148 -      with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"
  20.149 +      with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
  20.150          by auto
  20.151      next
  20.152        case False
  20.153 @@ -4326,7 +4326,7 @@
  20.154  
  20.155  lemma compact_def:
  20.156    "compact (S :: 'a::metric_space set) \<longleftrightarrow>
  20.157 -   (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) ----> l))"
  20.158 +   (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
  20.159    unfolding compact_eq_seq_compact_metric seq_compact_def by auto
  20.160  
  20.161  subsubsection \<open>Complete the chain of compactness variants\<close>
  20.162 @@ -4405,7 +4405,7 @@
  20.163  (* TODO: is this lemma necessary? *)
  20.164  lemma bounded_increasing_convergent:
  20.165    fixes s :: "nat \<Rightarrow> real"
  20.166 -  shows "bounded {s n| n. True} \<Longrightarrow> \<forall>n. s n \<le> s (Suc n) \<Longrightarrow> \<exists>l. s ----> l"
  20.167 +  shows "bounded {s n| n. True} \<Longrightarrow> \<forall>n. s n \<le> s (Suc n) \<Longrightarrow> \<exists>l. s \<longlonglongrightarrow> l"
  20.168    using Bseq_mono_convergent[of s] incseq_Suc_iff[of s]
  20.169    by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)
  20.170  
  20.171 @@ -4417,7 +4417,7 @@
  20.172      unfolding comp_def by (metis seq_monosub)
  20.173    then have "Bseq (f \<circ> r)"
  20.174      unfolding Bseq_eq_bounded using f by (force intro: bounded_subset)
  20.175 -  with r show "\<exists>l r. subseq r \<and> (f \<circ> r) ----> l"
  20.176 +  with r show "\<exists>l r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
  20.177      using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)
  20.178  qed
  20.179  
  20.180 @@ -4531,7 +4531,7 @@
  20.181      by (rule bounded_fst)
  20.182    then have s1: "bounded (range (fst \<circ> f))"
  20.183      by (simp add: image_comp)
  20.184 -  obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) ----> l1"
  20.185 +  obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) \<longlonglongrightarrow> l1"
  20.186      using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast
  20.187    from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"
  20.188      by (auto simp add: image_comp intro: bounded_snd bounded_subset)
  20.189 @@ -4551,16 +4551,16 @@
  20.190  subsubsection \<open>Completeness\<close>
  20.191  
  20.192  definition complete :: "'a::metric_space set \<Rightarrow> bool"
  20.193 -  where "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>s. f ----> l))"
  20.194 +  where "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>s. f \<longlonglongrightarrow> l))"
  20.195  
  20.196  lemma completeI:
  20.197 -  assumes "\<And>f. \<forall>n. f n \<in> s \<Longrightarrow> Cauchy f \<Longrightarrow> \<exists>l\<in>s. f ----> l"
  20.198 +  assumes "\<And>f. \<forall>n. f n \<in> s \<Longrightarrow> Cauchy f \<Longrightarrow> \<exists>l\<in>s. f \<longlonglongrightarrow> l"
  20.199    shows "complete s"
  20.200    using assms unfolding complete_def by fast
  20.201  
  20.202  lemma completeE:
  20.203    assumes "complete s" and "\<forall>n. f n \<in> s" and "Cauchy f"
  20.204 -  obtains l where "l \<in> s" and "f ----> l"
  20.205 +  obtains l where "l \<in> s" and "f \<longlonglongrightarrow> l"
  20.206    using assms unfolding complete_def by fast
  20.207  
  20.208  lemma compact_imp_complete:
  20.209 @@ -4570,7 +4570,7 @@
  20.210    {
  20.211      fix f
  20.212      assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
  20.213 -    from as(1) obtain l r where lr: "l\<in>s" "subseq r" "(f \<circ> r) ----> l"
  20.214 +    from as(1) obtain l r where lr: "l\<in>s" "subseq r" "(f \<circ> r) \<longlonglongrightarrow> l"
  20.215        using assms unfolding compact_def by blast
  20.216  
  20.217      note lr' = seq_suble [OF lr(2)]
  20.218 @@ -4706,7 +4706,7 @@
  20.219            by (simp add: dist_commute)
  20.220        qed
  20.221  
  20.222 -      ultimately show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"
  20.223 +      ultimately show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
  20.224          using assms unfolding complete_def by blast
  20.225      qed
  20.226    qed
  20.227 @@ -4789,19 +4789,19 @@
  20.228  proof (rule completeI)
  20.229    fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
  20.230    then have "convergent f" by (rule Cauchy_convergent)
  20.231 -  then show "\<exists>l\<in>UNIV. f ----> l" unfolding convergent_def by simp
  20.232 +  then show "\<exists>l\<in>UNIV. f \<longlonglongrightarrow> l" unfolding convergent_def by simp
  20.233  qed
  20.234  
  20.235  lemma complete_imp_closed:
  20.236    assumes "complete s"
  20.237    shows "closed s"
  20.238  proof (unfold closed_sequential_limits, clarify)
  20.239 -  fix f x assume "\<forall>n. f n \<in> s" and "f ----> x"
  20.240 -  from \<open>f ----> x\<close> have "Cauchy f"
  20.241 +  fix f x assume "\<forall>n. f n \<in> s" and "f \<longlonglongrightarrow> x"
  20.242 +  from \<open>f \<longlonglongrightarrow> x\<close> have "Cauchy f"
  20.243      by (rule LIMSEQ_imp_Cauchy)
  20.244 -  with \<open>complete s\<close> and \<open>\<forall>n. f n \<in> s\<close> obtain l where "l \<in> s" and "f ----> l"
  20.245 +  with \<open>complete s\<close> and \<open>\<forall>n. f n \<in> s\<close> obtain l where "l \<in> s" and "f \<longlonglongrightarrow> l"
  20.246      by (rule completeE)
  20.247 -  from \<open>f ----> x\<close> and \<open>f ----> l\<close> have "x = l"
  20.248 +  from \<open>f \<longlonglongrightarrow> x\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "x = l"
  20.249      by (rule LIMSEQ_unique)
  20.250    with \<open>l \<in> s\<close> show "x \<in> s"
  20.251      by simp
  20.252 @@ -4814,11 +4814,11 @@
  20.253    fix f assume "\<forall>n. f n \<in> s \<inter> t" and "Cauchy f"
  20.254    then have "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"
  20.255      by simp_all
  20.256 -  from \<open>complete s\<close> obtain l where "l \<in> s" and "f ----> l"
  20.257 +  from \<open>complete s\<close> obtain l where "l \<in> s" and "f \<longlonglongrightarrow> l"
  20.258      using \<open>\<forall>n. f n \<in> s\<close> and \<open>Cauchy f\<close> by (rule completeE)
  20.259 -  from \<open>closed t\<close> and \<open>\<forall>n. f n \<in> t\<close> and \<open>f ----> l\<close> have "l \<in> t"
  20.260 +  from \<open>closed t\<close> and \<open>\<forall>n. f n \<in> t\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "l \<in> t"
  20.261      by (rule closed_sequentially)
  20.262 -  with \<open>l \<in> s\<close> and \<open>f ----> l\<close> show "\<exists>l\<in>s \<inter> t. f ----> l"
  20.263 +  with \<open>l \<in> s\<close> and \<open>f \<longlonglongrightarrow> l\<close> show "\<exists>l\<in>s \<inter> t. f \<longlonglongrightarrow> l"
  20.264      by fast
  20.265  qed
  20.266  
  20.267 @@ -4887,7 +4887,7 @@
  20.268      using choice[of "\<lambda>n x. x \<in> s n"] by auto
  20.269    from assms(4,1) have "seq_compact (s 0)"
  20.270      by (simp add: bounded_closed_imp_seq_compact)
  20.271 -  then obtain l r where lr: "l \<in> s 0" "subseq r" "(x \<circ> r) ----> l"
  20.272 +  then obtain l r where lr: "l \<in> s 0" "subseq r" "(x \<circ> r) \<longlonglongrightarrow> l"
  20.273      using x and assms(3) unfolding seq_compact_def by blast
  20.274    have "\<forall>n. l \<in> s n"
  20.275    proof
  20.276 @@ -4898,7 +4898,7 @@
  20.277        using x and assms(3) and lr(2) [THEN seq_suble] by auto
  20.278      then have "\<forall>i. (x \<circ> r) (i + n) \<in> s n"
  20.279        using assms(3) by (fast intro!: le_add2)
  20.280 -    moreover have "(\<lambda>i. (x \<circ> r) (i + n)) ----> l"
  20.281 +    moreover have "(\<lambda>i. (x \<circ> r) (i + n)) \<longlonglongrightarrow> l"
  20.282        using lr(3) by (rule LIMSEQ_ignore_initial_segment)
  20.283      ultimately show "l \<in> s n"
  20.284        by (rule closed_sequentially)
  20.285 @@ -5497,9 +5497,9 @@
  20.286    } note le = this
  20.287    {
  20.288      fix x y
  20.289 -    assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"
  20.290 -    assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"
  20.291 -    have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"
  20.292 +    assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) \<longlonglongrightarrow> 0"
  20.293 +    assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) \<longlonglongrightarrow> 0"
  20.294 +    have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) \<longlonglongrightarrow> 0"
  20.295        by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
  20.296          simp add: le)
  20.297    }
  20.298 @@ -5589,8 +5589,8 @@
  20.299  
  20.300  lemma continuous_imp_tendsto:
  20.301    assumes "continuous (at x0) f"
  20.302 -    and "x ----> x0"
  20.303 -  shows "(f \<circ> x) ----> (f x0)"
  20.304 +    and "x \<longlonglongrightarrow> x0"
  20.305 +  shows "(f \<circ> x) \<longlonglongrightarrow> (f x0)"
  20.306  proof (rule topological_tendstoI)
  20.307    fix S
  20.308    assume "open S" "f x0 \<in> S"
    21.1 --- a/src/HOL/Multivariate_Analysis/Uniform_Limit.thy	Tue Dec 29 22:41:22 2015 +0100
    21.2 +++ b/src/HOL/Multivariate_Analysis/Uniform_Limit.thy	Tue Dec 29 23:04:53 2015 +0100
    21.3 @@ -207,7 +207,7 @@
    21.4      show "\<forall>x\<in>X. dist (f n x) (?f x) < e"
    21.5      proof
    21.6        fix x assume x: "x \<in> X"
    21.7 -      with assms have "(\<lambda>n. f n x) ----> ?f x" 
    21.8 +      with assms have "(\<lambda>n. f n x) \<longlonglongrightarrow> ?f x" 
    21.9          by (auto dest!: Cauchy_convergent uniformly_Cauchy_imp_Cauchy simp: convergent_LIMSEQ_iff)
   21.10        with \<open>e/2 > 0\<close> have "eventually (\<lambda>m. m \<ge> N \<and> dist (f m x) (?f x) < e/2) sequentially"
   21.11          by (intro tendstoD eventually_conj eventually_ge_at_top)
    22.1 --- a/src/HOL/NSA/HSEQ.thy	Tue Dec 29 22:41:22 2015 +0100
    22.2 +++ b/src/HOL/NSA/HSEQ.thy	Tue Dec 29 23:04:53 2015 +0100
    22.3 @@ -162,7 +162,7 @@
    22.4  subsubsection {* Equivalence of @{term LIMSEQ} and @{term NSLIMSEQ} *}
    22.5  
    22.6  lemma LIMSEQ_NSLIMSEQ:
    22.7 -  assumes X: "X ----> L" shows "X ----NS> L"
    22.8 +  assumes X: "X \<longlonglongrightarrow> L" shows "X ----NS> L"
    22.9  proof (rule NSLIMSEQ_I)
   22.10    fix N assume N: "N \<in> HNatInfinite"
   22.11    have "starfun X N - star_of L \<in> Infinitesimal"
   22.12 @@ -180,7 +180,7 @@
   22.13  qed
   22.14  
   22.15  lemma NSLIMSEQ_LIMSEQ:
   22.16 -  assumes X: "X ----NS> L" shows "X ----> L"
   22.17 +  assumes X: "X ----NS> L" shows "X \<longlonglongrightarrow> L"
   22.18  proof (rule LIMSEQ_I)
   22.19    fix r::real assume r: "0 < r"
   22.20    have "\<exists>no. \<forall>n\<ge>no. hnorm (starfun X n - star_of L) < star_of r"
   22.21 @@ -199,7 +199,7 @@
   22.22      by transfer
   22.23  qed
   22.24  
   22.25 -theorem LIMSEQ_NSLIMSEQ_iff: "(f ----> L) = (f ----NS> L)"
   22.26 +theorem LIMSEQ_NSLIMSEQ_iff: "(f \<longlonglongrightarrow> L) = (f ----NS> L)"
   22.27  by (blast intro: LIMSEQ_NSLIMSEQ NSLIMSEQ_LIMSEQ)
   22.28  
   22.29  subsubsection {* Derived theorems about @{term NSLIMSEQ} *}
    23.1 --- a/src/HOL/NthRoot.thy	Tue Dec 29 22:41:22 2015 +0100
    23.2 +++ b/src/HOL/NthRoot.thy	Tue Dec 29 23:04:53 2015 +0100
    23.3 @@ -649,12 +649,12 @@
    23.4    apply auto
    23.5    done
    23.6  
    23.7 -lemma LIMSEQ_root: "(\<lambda>n. root n n) ----> 1"
    23.8 +lemma LIMSEQ_root: "(\<lambda>n. root n n) \<longlonglongrightarrow> 1"
    23.9  proof -
   23.10    def x \<equiv> "\<lambda>n. root n n - 1"
   23.11 -  have "x ----> sqrt 0"
   23.12 +  have "x \<longlonglongrightarrow> sqrt 0"
   23.13    proof (rule tendsto_sandwich[OF _ _ tendsto_const])
   23.14 -    show "(\<lambda>x. sqrt (2 / x)) ----> sqrt 0"
   23.15 +    show "(\<lambda>x. sqrt (2 / x)) \<longlonglongrightarrow> sqrt 0"
   23.16        by (intro tendsto_intros tendsto_divide_0[OF tendsto_const] filterlim_mono[OF filterlim_real_sequentially])
   23.17           (simp_all add: at_infinity_eq_at_top_bot)
   23.18      { fix n :: nat assume "2 < n"
   23.19 @@ -686,13 +686,13 @@
   23.20  
   23.21  lemma LIMSEQ_root_const:
   23.22    assumes "0 < c"
   23.23 -  shows "(\<lambda>n. root n c) ----> 1"
   23.24 +  shows "(\<lambda>n. root n c) \<longlonglongrightarrow> 1"
   23.25  proof -
   23.26    { fix c :: real assume "1 \<le> c"
   23.27      def x \<equiv> "\<lambda>n. root n c - 1"
   23.28 -    have "x ----> 0"
   23.29 +    have "x \<longlonglongrightarrow> 0"
   23.30      proof (rule tendsto_sandwich[OF _ _ tendsto_const])
   23.31 -      show "(\<lambda>n. c / n) ----> 0"
   23.32 +      show "(\<lambda>n. c / n) \<longlonglongrightarrow> 0"
   23.33          by (intro tendsto_divide_0[OF tendsto_const] filterlim_mono[OF filterlim_real_sequentially])
   23.34             (simp_all add: at_infinity_eq_at_top_bot)
   23.35        { fix n :: nat assume "1 < n"
   23.36 @@ -713,7 +713,7 @@
   23.37        show "eventually (\<lambda>n. 0 \<le> x n) sequentially"
   23.38          using \<open>1 \<le> c\<close> by (auto intro!: exI[of _ 1] simp: eventually_sequentially le_diff_eq x_def)
   23.39      qed
   23.40 -    from tendsto_add[OF this tendsto_const[of 1]] have "(\<lambda>n. root n c) ----> 1"
   23.41 +    from tendsto_add[OF this tendsto_const[of 1]] have "(\<lambda>n. root n c) \<longlonglongrightarrow> 1"
   23.42        by (simp add: x_def) }
   23.43    note ge_1 = this
   23.44  
   23.45 @@ -724,7 +724,7 @@
   23.46      assume "\<not> 1 \<le> c"
   23.47      with \<open>0 < c\<close> have "1 \<le> 1 / c"
   23.48        by simp
   23.49 -    then have "(\<lambda>n. 1 / root n (1 / c)) ----> 1 / 1"
   23.50 +    then have "(\<lambda>n. 1 / root n (1 / c)) \<longlonglongrightarrow> 1 / 1"
   23.51        by (intro tendsto_divide tendsto_const ge_1 \<open>1 \<le> 1 / c\<close> one_neq_zero)
   23.52      then show ?thesis
   23.53        by (rule filterlim_cong[THEN iffD1, rotated 3])
    24.1 --- a/src/HOL/Probability/Bochner_Integration.thy	Tue Dec 29 22:41:22 2015 +0100
    24.2 +++ b/src/HOL/Probability/Bochner_Integration.thy	Tue Dec 29 23:04:53 2015 +0100
    24.3 @@ -18,7 +18,7 @@
    24.4  lemma borel_measurable_implies_sequence_metric:
    24.5    fixes f :: "'a \<Rightarrow> 'b :: {metric_space, second_countable_topology}"
    24.6    assumes [measurable]: "f \<in> borel_measurable M"
    24.7 -  shows "\<exists>F. (\<forall>i. simple_function M (F i)) \<and> (\<forall>x\<in>space M. (\<lambda>i. F i x) ----> f x) \<and>
    24.8 +  shows "\<exists>F. (\<forall>i. simple_function M (F i)) \<and> (\<forall>x\<in>space M. (\<lambda>i. F i x) \<longlonglongrightarrow> f x) \<and>
    24.9      (\<forall>i. \<forall>x\<in>space M. dist (F i x) z \<le> 2 * dist (f x) z)"
   24.10  proof -
   24.11    obtain D :: "'b set" where "countable D" and D: "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d\<in>D. d \<in> X"
   24.12 @@ -101,7 +101,7 @@
   24.13      note * = this
   24.14  
   24.15      fix x assume "x \<in> space M"
   24.16 -    show "(\<lambda>i. F i x) ----> f x"
   24.17 +    show "(\<lambda>i. F i x) \<longlonglongrightarrow> f x"
   24.18      proof cases
   24.19        assume "f x = z"
   24.20        then have "\<And>i n. x \<notin> A i n"
   24.21 @@ -173,7 +173,7 @@
   24.22    assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
   24.23    assumes mult: "\<And>u c. 0 \<le> c \<Longrightarrow> u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)"
   24.24    assumes add: "\<And>u v. u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> v \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> u x = 0 \<or> v x = 0) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
   24.25 -  assumes seq: "\<And>U. (\<And>i. U i \<in> borel_measurable M) \<Longrightarrow> (\<And>i x. 0 \<le> U i x) \<Longrightarrow> (\<And>i. P (U i)) \<Longrightarrow> incseq U \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. U i x) ----> u x) \<Longrightarrow> P u"
   24.26 +  assumes seq: "\<And>U. (\<And>i. U i \<in> borel_measurable M) \<Longrightarrow> (\<And>i x. 0 \<le> U i x) \<Longrightarrow> (\<And>i. P (U i)) \<Longrightarrow> incseq U \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. U i x) \<longlonglongrightarrow> u x) \<Longrightarrow> P u"
   24.27    shows "P u"
   24.28  proof -
   24.29    have "(\<lambda>x. ereal (u x)) \<in> borel_measurable M" using u by auto
   24.30 @@ -199,13 +199,13 @@
   24.31                 intro!: real_of_ereal_positive_mono)
   24.32    next
   24.33      fix x assume x: "x \<in> space M"
   24.34 -    have "(\<lambda>i. U i x) ----> (SUP i. U i x)"
   24.35 +    have "(\<lambda>i. U i x) \<longlonglongrightarrow> (SUP i. U i x)"
   24.36        using U(2) by (intro LIMSEQ_SUP) (auto simp: incseq_def le_fun_def)
   24.37      moreover have "(\<lambda>i. U i x) = (\<lambda>i. ereal (U' i x))"
   24.38        using x nn U(3) by (auto simp: fun_eq_iff U'_def ereal_real image_iff eq_commute)
   24.39      moreover have "(SUP i. U i x) = ereal (u x)"
   24.40        using sup u(2) by (simp add: max_def)
   24.41 -    ultimately show "(\<lambda>i. U' i x) ----> u x" 
   24.42 +    ultimately show "(\<lambda>i. U' i x) \<longlonglongrightarrow> u x" 
   24.43        by simp
   24.44    next
   24.45      fix i
   24.46 @@ -516,8 +516,8 @@
   24.47    for M f x where
   24.48    "f \<in> borel_measurable M \<Longrightarrow>
   24.49      (\<And>i. simple_bochner_integrable M (s i)) \<Longrightarrow>
   24.50 -    (\<lambda>i. \<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) ----> 0 \<Longrightarrow>
   24.51 -    (\<lambda>i. simple_bochner_integral M (s i)) ----> x \<Longrightarrow>
   24.52 +    (\<lambda>i. \<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) \<longlonglongrightarrow> 0 \<Longrightarrow>
   24.53 +    (\<lambda>i. simple_bochner_integral M (s i)) \<longlonglongrightarrow> x \<Longrightarrow>
   24.54      has_bochner_integral M f x"
   24.55  
   24.56  lemma has_bochner_integral_cong:
   24.57 @@ -530,7 +530,7 @@
   24.58    "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (AE x in M. f x = g x) \<Longrightarrow>
   24.59      has_bochner_integral M f x \<longleftrightarrow> has_bochner_integral M g x"
   24.60    unfolding has_bochner_integral.simps
   24.61 -  by (intro arg_cong[where f=Ex] ext conj_cong rev_conj_cong refl arg_cong[where f="\<lambda>x. x ----> 0"]
   24.62 +  by (intro arg_cong[where f=Ex] ext conj_cong rev_conj_cong refl arg_cong[where f="\<lambda>x. x \<longlonglongrightarrow> 0"]
   24.63              nn_integral_cong_AE)
   24.64       auto
   24.65  
   24.66 @@ -572,8 +572,8 @@
   24.67      has_bochner_integral M (\<lambda>x. f x + g x) (x + y)"
   24.68  proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
   24.69    fix sf sg
   24.70 -  assume f_sf: "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - sf i x) \<partial>M) ----> 0"
   24.71 -  assume g_sg: "(\<lambda>i. \<integral>\<^sup>+ x. norm (g x - sg i x) \<partial>M) ----> 0"
   24.72 +  assume f_sf: "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - sf i x) \<partial>M) \<longlonglongrightarrow> 0"
   24.73 +  assume g_sg: "(\<lambda>i. \<integral>\<^sup>+ x. norm (g x - sg i x) \<partial>M) \<longlonglongrightarrow> 0"
   24.74  
   24.75    assume sf: "\<forall>i. simple_bochner_integrable M (sf i)"
   24.76      and sg: "\<forall>i. simple_bochner_integrable M (sg i)"
   24.77 @@ -584,10 +584,10 @@
   24.78    show "\<And>i. simple_bochner_integrable M (\<lambda>x. sf i x + sg i x)"
   24.79      using sf sg by (simp add: simple_bochner_integrable_compose2)
   24.80  
   24.81 -  show "(\<lambda>i. \<integral>\<^sup>+ x. (norm (f x + g x - (sf i x + sg i x))) \<partial>M) ----> 0"
   24.82 -    (is "?f ----> 0")
   24.83 +  show "(\<lambda>i. \<integral>\<^sup>+ x. (norm (f x + g x - (sf i x + sg i x))) \<partial>M) \<longlonglongrightarrow> 0"
   24.84 +    (is "?f \<longlonglongrightarrow> 0")
   24.85    proof (rule tendsto_sandwich)
   24.86 -    show "eventually (\<lambda>n. 0 \<le> ?f n) sequentially" "(\<lambda>_. 0) ----> 0"
   24.87 +    show "eventually (\<lambda>n. 0 \<le> ?f n) sequentially" "(\<lambda>_. 0) \<longlonglongrightarrow> 0"
   24.88        by (auto simp: nn_integral_nonneg)
   24.89      show "eventually (\<lambda>i. ?f i \<le> (\<integral>\<^sup>+ x. (norm (f x - sf i x)) \<partial>M) + \<integral>\<^sup>+ x. (norm (g x - sg i x)) \<partial>M) sequentially"
   24.90        (is "eventually (\<lambda>i. ?f i \<le> ?g i) sequentially")
   24.91 @@ -598,7 +598,7 @@
   24.92          by (intro nn_integral_add) auto
   24.93        finally show "?f i \<le> ?g i" .
   24.94      qed
   24.95 -    show "?g ----> 0"
   24.96 +    show "?g \<longlonglongrightarrow> 0"
   24.97        using tendsto_add_ereal[OF _ _ f_sf g_sg] by simp
   24.98    qed
   24.99  qed (auto simp: simple_bochner_integral_add tendsto_add)
  24.100 @@ -614,7 +614,7 @@
  24.101    then show "(\<lambda>x. T (f x)) \<in> borel_measurable M"
  24.102      by auto
  24.103  
  24.104 -  fix s assume f_s: "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - s i x) \<partial>M) ----> 0"
  24.105 +  fix s assume f_s: "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - s i x) \<partial>M) \<longlonglongrightarrow> 0"
  24.106    assume s: "\<forall>i. simple_bochner_integrable M (s i)"
  24.107    then show "\<And>i. simple_bochner_integrable M (\<lambda>x. T (s i x))"
  24.108      by (auto intro: simple_bochner_integrable_compose2 T.zero)
  24.109 @@ -625,10 +625,10 @@
  24.110    obtain K where K: "K > 0" "\<And>x i. norm (T (f x) - T (s i x)) \<le> norm (f x - s i x) * K"
  24.111      using T.pos_bounded by (auto simp: T.diff[symmetric])
  24.112  
  24.113 -  show "(\<lambda>i. \<integral>\<^sup>+ x. norm (T (f x) - T (s i x)) \<partial>M) ----> 0"
  24.114 -    (is "?f ----> 0")
  24.115 +  show "(\<lambda>i. \<integral>\<^sup>+ x. norm (T (f x) - T (s i x)) \<partial>M) \<longlonglongrightarrow> 0"
  24.116 +    (is "?f \<longlonglongrightarrow> 0")
  24.117    proof (rule tendsto_sandwich)
  24.118 -    show "eventually (\<lambda>n. 0 \<le> ?f n) sequentially" "(\<lambda>_. 0) ----> 0"
  24.119 +    show "eventually (\<lambda>n. 0 \<le> ?f n) sequentially" "(\<lambda>_. 0) \<longlonglongrightarrow> 0"
  24.120        by (auto simp: nn_integral_nonneg)
  24.121  
  24.122      show "eventually (\<lambda>i. ?f i \<le> K * (\<integral>\<^sup>+ x. norm (f x - s i x) \<partial>M)) sequentially"
  24.123 @@ -640,12 +640,12 @@
  24.124          using K by (intro nn_integral_cmult) auto
  24.125        finally show "?f i \<le> ?g i" .
  24.126      qed
  24.127 -    show "?g ----> 0"
  24.128 +    show "?g \<longlonglongrightarrow> 0"
  24.129        using tendsto_cmult_ereal[OF _ f_s, of "ereal K"] by simp
  24.130    qed
  24.131  
  24.132 -  assume "(\<lambda>i. simple_bochner_integral M (s i)) ----> x"
  24.133 -  with s show "(\<lambda>i. simple_bochner_integral M (\<lambda>x. T (s i x))) ----> T x"
  24.134 +  assume "(\<lambda>i. simple_bochner_integral M (s i)) \<longlonglongrightarrow> x"
  24.135 +  with s show "(\<lambda>i. simple_bochner_integral M (\<lambda>x. T (s i x))) \<longlonglongrightarrow> T x"
  24.136      by (auto intro!: T.tendsto simp: T.simple_bochner_integral_linear)
  24.137  qed
  24.138  
  24.139 @@ -724,7 +724,7 @@
  24.140  proof (elim has_bochner_integral.cases)
  24.141    fix s v
  24.142    assume [measurable]: "f \<in> borel_measurable M" and s: "\<And>i. simple_bochner_integrable M (s i)" and
  24.143 -    lim_0: "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0"
  24.144 +    lim_0: "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) \<longlonglongrightarrow> 0"
  24.145    from order_tendstoD[OF lim_0, of "\<infinity>"]
  24.146    obtain i where f_s_fin: "(\<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) < \<infinity>"
  24.147      by (metis (mono_tags, lifting) eventually_False_sequentially eventually_mono
  24.148 @@ -760,9 +760,9 @@
  24.149    shows "norm x \<le> (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
  24.150  using assms proof
  24.151    fix s assume
  24.152 -    x: "(\<lambda>i. simple_bochner_integral M (s i)) ----> x" (is "?s ----> x") and
  24.153 +    x: "(\<lambda>i. simple_bochner_integral M (s i)) \<longlonglongrightarrow> x" (is "?s \<longlonglongrightarrow> x") and
  24.154      s[simp]: "\<And>i. simple_bochner_integrable M (s i)" and
  24.155 -    lim: "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0" and
  24.156 +    lim: "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) \<longlonglongrightarrow> 0" and
  24.157      f[measurable]: "f \<in> borel_measurable M"
  24.158  
  24.159    have [measurable]: "\<And>i. s i \<in> borel_measurable M"
  24.160 @@ -770,7 +770,7 @@
  24.161  
  24.162    show "norm x \<le> (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
  24.163    proof (rule LIMSEQ_le)
  24.164 -    show "(\<lambda>i. ereal (norm (?s i))) ----> norm x"
  24.165 +    show "(\<lambda>i. ereal (norm (?s i))) \<longlonglongrightarrow> norm x"
  24.166        using x by (intro tendsto_intros lim_ereal[THEN iffD2])
  24.167      show "\<exists>N. \<forall>n\<ge>N. norm (?s n) \<le> (\<integral>\<^sup>+x. norm (f x - s n x) \<partial>M) + (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
  24.168        (is "\<exists>N. \<forall>n\<ge>N. _ \<le> ?t n")
  24.169 @@ -788,10 +788,10 @@
  24.170          by (rule nn_integral_add) auto
  24.171        finally show "norm (?s n) \<le> ?t n" .
  24.172      qed
  24.173 -    have "?t ----> 0 + (\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M)"
  24.174 +    have "?t \<longlonglongrightarrow> 0 + (\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M)"
  24.175        using has_bochner_integral_implies_finite_norm[OF i]
  24.176        by (intro tendsto_add_ereal tendsto_const lim) auto
  24.177 -    then show "?t ----> \<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M"
  24.178 +    then show "?t \<longlonglongrightarrow> \<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M"
  24.179        by simp
  24.180    qed
  24.181  qed
  24.182 @@ -802,8 +802,8 @@
  24.183    assume f[measurable]: "f \<in> borel_measurable M"
  24.184  
  24.185    fix s t
  24.186 -  assume "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - s i x) \<partial>M) ----> 0" (is "?S ----> 0")
  24.187 -  assume "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - t i x) \<partial>M) ----> 0" (is "?T ----> 0")
  24.188 +  assume "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - s i x) \<partial>M) \<longlonglongrightarrow> 0" (is "?S \<longlonglongrightarrow> 0")
  24.189 +  assume "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - t i x) \<partial>M) \<longlonglongrightarrow> 0" (is "?T \<longlonglongrightarrow> 0")
  24.190    assume s: "\<And>i. simple_bochner_integrable M (s i)"
  24.191    assume t: "\<And>i. simple_bochner_integrable M (t i)"
  24.192  
  24.193 @@ -812,22 +812,22 @@
  24.194  
  24.195    let ?s = "\<lambda>i. simple_bochner_integral M (s i)"
  24.196    let ?t = "\<lambda>i. simple_bochner_integral M (t i)"
  24.197 -  assume "?s ----> x" "?t ----> y"
  24.198 -  then have "(\<lambda>i. norm (?s i - ?t i)) ----> norm (x - y)"
  24.199 +  assume "?s \<longlonglongrightarrow> x" "?t \<longlonglongrightarrow> y"
  24.200 +  then have "(\<lambda>i. norm (?s i - ?t i)) \<longlonglongrightarrow> norm (x - y)"
  24.201      by (intro tendsto_intros)
  24.202    moreover
  24.203 -  have "(\<lambda>i. ereal (norm (?s i - ?t i))) ----> ereal 0"
  24.204 +  have "(\<lambda>i. ereal (norm (?s i - ?t i))) \<longlonglongrightarrow> ereal 0"
  24.205    proof (rule tendsto_sandwich)
  24.206 -    show "eventually (\<lambda>i. 0 \<le> ereal (norm (?s i - ?t i))) sequentially" "(\<lambda>_. 0) ----> ereal 0"
  24.207 +    show "eventually (\<lambda>i. 0 \<le> ereal (norm (?s i - ?t i))) sequentially" "(\<lambda>_. 0) \<longlonglongrightarrow> ereal 0"
  24.208        by (auto simp: nn_integral_nonneg zero_ereal_def[symmetric])
  24.209  
  24.210      show "eventually (\<lambda>i. norm (?s i - ?t i) \<le> ?S i + ?T i) sequentially"
  24.211        by (intro always_eventually allI simple_bochner_integral_bounded s t f)
  24.212 -    show "(\<lambda>i. ?S i + ?T i) ----> ereal 0"
  24.213 -      using tendsto_add_ereal[OF _ _ \<open>?S ----> 0\<close> \<open>?T ----> 0\<close>]
  24.214 +    show "(\<lambda>i. ?S i + ?T i) \<longlonglongrightarrow> ereal 0"
  24.215 +      using tendsto_add_ereal[OF _ _ \<open>?S \<longlonglongrightarrow> 0\<close> \<open>?T \<longlonglongrightarrow> 0\<close>]
  24.216        by (simp add: zero_ereal_def[symmetric])
  24.217    qed
  24.218 -  then have "(\<lambda>i. norm (?s i - ?t i)) ----> 0"
  24.219 +  then have "(\<lambda>i. norm (?s i - ?t i)) \<longlonglongrightarrow> 0"
  24.220      by simp
  24.221    ultimately have "norm (x - y) = 0"
  24.222      by (rule LIMSEQ_unique)
  24.223 @@ -841,11 +841,11 @@
  24.224    shows "has_bochner_integral M g x"
  24.225    using f
  24.226  proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
  24.227 -  fix s assume "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0"
  24.228 +  fix s assume "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) \<longlonglongrightarrow> 0"
  24.229    also have "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) = (\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (g x - s i x)) \<partial>M)"
  24.230      using ae
  24.231      by (intro ext nn_integral_cong_AE, eventually_elim) simp
  24.232 -  finally show "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (g x - s i x)) \<partial>M) ----> 0" .
  24.233 +  finally show "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (g x - s i x)) \<partial>M) \<longlonglongrightarrow> 0" .
  24.234  qed (auto intro: g)
  24.235  
  24.236  lemma has_bochner_integral_eq_AE:
  24.237 @@ -1144,12 +1144,12 @@
  24.238    fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
  24.239    assumes f[measurable]: "f \<in> borel_measurable M"
  24.240    assumes s: "\<And>i. simple_bochner_integrable M (s i)"
  24.241 -  assumes lim: "(\<lambda>i. \<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) ----> 0" (is "?S ----> 0")
  24.242 +  assumes lim: "(\<lambda>i. \<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) \<longlonglongrightarrow> 0" (is "?S \<longlonglongrightarrow> 0")
  24.243    shows "integrable M f"
  24.244  proof -
  24.245    let ?s = "\<lambda>n. simple_bochner_integral M (s n)"
  24.246  
  24.247 -  have "\<exists>x. ?s ----> x"
  24.248 +  have "\<exists>x. ?s \<longlonglongrightarrow> x"
  24.249      unfolding convergent_eq_cauchy
  24.250    proof (rule metric_CauchyI)
  24.251      fix e :: real assume "0 < e"
  24.252 @@ -1172,7 +1172,7 @@
  24.253          by (simp add: dist_norm)
  24.254      qed
  24.255    qed
  24.256 -  then obtain x where "?s ----> x" ..
  24.257 +  then obtain x where "?s \<longlonglongrightarrow> x" ..
  24.258    show ?thesis
  24.259      by (rule, rule) fact+
  24.260  qed
  24.261 @@ -1183,14 +1183,14 @@
  24.262         "\<And>i. u i \<in> borel_measurable M" "u' \<in> borel_measurable M" "w \<in> borel_measurable M"
  24.263      and bound: "\<And>j. AE x in M. norm (u j x) \<le> w x"
  24.264      and w: "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>"
  24.265 -    and u': "AE x in M. (\<lambda>i. u i x) ----> u' x"
  24.266 -  shows "(\<lambda>i. (\<integral>\<^sup>+x. norm (u' x - u i x) \<partial>M)) ----> 0"
  24.267 +    and u': "AE x in M. (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
  24.268 +  shows "(\<lambda>i. (\<integral>\<^sup>+x. norm (u' x - u i x) \<partial>M)) \<longlonglongrightarrow> 0"
  24.269  proof -
  24.270    have "AE x in M. \<forall>j. norm (u j x) \<le> w x"
  24.271      unfolding AE_all_countable by rule fact
  24.272    with u' have bnd: "AE x in M. \<forall>j. norm (u' x - u j x) \<le> 2 * w x"
  24.273    proof (eventually_elim, intro allI)
  24.274 -    fix i x assume "(\<lambda>i. u i x) ----> u' x" "\<forall>j. norm (u j x) \<le> w x" "\<forall>j. norm (u j x) \<le> w x"
  24.275 +    fix i x assume "(\<lambda>i. u i x) \<longlonglongrightarrow> u' x" "\<forall>j. norm (u j x) \<le> w x" "\<forall>j. norm (u j x) \<le> w x"
  24.276      then have "norm (u' x) \<le> w x" "norm (u i x) \<le> w x"
  24.277        by (auto intro: LIMSEQ_le_const2 tendsto_norm)
  24.278      then have "norm (u' x) + norm (u i x) \<le> 2 * w x"
  24.279 @@ -1200,16 +1200,16 @@
  24.280      finally (xtrans) show "norm (u' x - u i x) \<le> 2 * w x" .
  24.281    qed
  24.282    
  24.283 -  have "(\<lambda>i. (\<integral>\<^sup>+x. norm (u' x - u i x) \<partial>M)) ----> (\<integral>\<^sup>+x. 0 \<partial>M)"
  24.284 +  have "(\<lambda>i. (\<integral>\<^sup>+x. norm (u' x - u i x) \<partial>M)) \<longlonglongrightarrow> (\<integral>\<^sup>+x. 0 \<partial>M)"
  24.285    proof (rule nn_integral_dominated_convergence)  
  24.286      show "(\<integral>\<^sup>+x. 2 * w x \<partial>M) < \<infinity>"
  24.287        by (rule nn_integral_mult_bounded_inf[OF _ w, of 2]) auto
  24.288 -    show "AE x in M. (\<lambda>i. ereal (norm (u' x - u i x))) ----> 0"
  24.289 +    show "AE x in M. (\<lambda>i. ereal (norm (u' x - u i x))) \<longlonglongrightarrow> 0"
  24.290        using u' 
  24.291      proof eventually_elim
  24.292 -      fix x assume "(\<lambda>i. u i x) ----> u' x"
  24.293 +      fix x assume "(\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
  24.294        from tendsto_diff[OF tendsto_const[of "u' x"] this]
  24.295 -      show "(\<lambda>i. ereal (norm (u' x - u i x))) ----> 0"
  24.296 +      show "(\<lambda>i. ereal (norm (u' x - u i x))) \<longlonglongrightarrow> 0"
  24.297          by (simp add: zero_ereal_def tendsto_norm_zero_iff)
  24.298      qed
  24.299    qed (insert bnd, auto)
  24.300 @@ -1223,7 +1223,7 @@
  24.301  proof -
  24.302    from borel_measurable_implies_sequence_metric[OF f, of 0] obtain s where
  24.303      s: "\<And>i. simple_function M (s i)" and
  24.304 -    pointwise: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) ----> f x" and
  24.305 +    pointwise: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) \<longlonglongrightarrow> f x" and
  24.306      bound: "\<And>i x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> 2 * norm (f x)"
  24.307      by (simp add: norm_conv_dist) metis
  24.308    
  24.309 @@ -1241,7 +1241,7 @@
  24.310      show "\<And>i. simple_bochner_integrable M (s i)"
  24.311        by (rule simple_bochner_integrableI_bounded) fact+
  24.312  
  24.313 -    show "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0"
  24.314 +    show "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) \<longlonglongrightarrow> 0"
  24.315      proof (rule nn_integral_dominated_convergence_norm)
  24.316        show "\<And>j. AE x in M. norm (s j x) \<le> 2 * norm (f x)"
  24.317          using bound by auto
  24.318 @@ -1249,7 +1249,7 @@
  24.319          using s by (auto intro: borel_measurable_simple_function)
  24.320        show "(\<integral>\<^sup>+ x. ereal (2 * norm (f x)) \<partial>M) < \<infinity>"
  24.321          using fin unfolding times_ereal.simps(1)[symmetric] by (subst nn_integral_cmult) auto
  24.322 -      show "AE x in M. (\<lambda>i. s i x) ----> f x"
  24.323 +      show "AE x in M. (\<lambda>i. s i x) \<longlonglongrightarrow> f x"
  24.324          using pointwise by auto
  24.325      qed fact
  24.326    qed fact
  24.327 @@ -1456,11 +1456,11 @@
  24.328  lemma
  24.329    fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and w :: "'a \<Rightarrow> real"
  24.330    assumes "f \<in> borel_measurable M" "\<And>i. s i \<in> borel_measurable M" "integrable M w"
  24.331 -  assumes lim: "AE x in M. (\<lambda>i. s i x) ----> f x"
  24.332 +  assumes lim: "AE x in M. (\<lambda>i. s i x) \<longlonglongrightarrow> f x"
  24.333    assumes bound: "\<And>i. AE x in M. norm (s i x) \<le> w x"
  24.334    shows integrable_dominated_convergence: "integrable M f"
  24.335      and integrable_dominated_convergence2: "\<And>i. integrable M (s i)"
  24.336 -    and integral_dominated_convergence: "(\<lambda>i. integral\<^sup>L M (s i)) ----> integral\<^sup>L M f"
  24.337 +    and integral_dominated_convergence: "(\<lambda>i. integral\<^sup>L M (s i)) \<longlonglongrightarrow> integral\<^sup>L M f"
  24.338  proof -
  24.339    have "AE x in M. 0 \<le> w x"
  24.340      using bound[of 0] by eventually_elim (auto intro: norm_ge_zero order_trans)
  24.341 @@ -1487,16 +1487,16 @@
  24.342      have "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) \<le> (\<integral>\<^sup>+x. w x \<partial>M)"
  24.343        using all_bound lim
  24.344      proof (intro nn_integral_mono_AE, eventually_elim)
  24.345 -      fix x assume "\<forall>i. norm (s i x) \<le> w x" "(\<lambda>i. s i x) ----> f x"
  24.346 +      fix x assume "\<forall>i. norm (s i x) \<le> w x" "(\<lambda>i. s i x) \<longlonglongrightarrow> f x"
  24.347        then show "ereal (norm (f x)) \<le> ereal (w x)"
  24.348          by (intro LIMSEQ_le_const2[where X="\<lambda>i. ereal (norm (s i x))"] tendsto_intros lim_ereal[THEN iffD2]) auto
  24.349      qed
  24.350      with w show "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>" by auto
  24.351    qed fact
  24.352  
  24.353 -  have "(\<lambda>n. ereal (norm (integral\<^sup>L M (s n) - integral\<^sup>L M f))) ----> ereal 0" (is "?d ----> ereal 0")
  24.354 +  have "(\<lambda>n. ereal (norm (integral\<^sup>L M (s n) - integral\<^sup>L M f))) \<longlonglongrightarrow> ereal 0" (is "?d \<longlonglongrightarrow> ereal 0")
  24.355    proof (rule tendsto_sandwich)
  24.356 -    show "eventually (\<lambda>n. ereal 0 \<le> ?d n) sequentially" "(\<lambda>_. ereal 0) ----> ereal 0" by auto
  24.357 +    show "eventually (\<lambda>n. ereal 0 \<le> ?d n) sequentially" "(\<lambda>_. ereal 0) \<longlonglongrightarrow> ereal 0" by auto
  24.358      show "eventually (\<lambda>n. ?d n \<le> (\<integral>\<^sup>+x. norm (s n x - f x) \<partial>M)) sequentially"
  24.359      proof (intro always_eventually allI)
  24.360        fix n
  24.361 @@ -1506,7 +1506,7 @@
  24.362          by (intro int_f int_s integrable_diff integral_norm_bound_ereal)
  24.363        finally show "?d n \<le> (\<integral>\<^sup>+x. norm (s n x - f x) \<partial>M)" .
  24.364      qed
  24.365 -    show "(\<lambda>n. \<integral>\<^sup>+x. norm (s n x - f x) \<partial>M) ----> ereal 0"
  24.366 +    show "(\<lambda>n. \<integral>\<^sup>+x. norm (s n x - f x) \<partial>M) \<longlonglongrightarrow> ereal 0"
  24.367        unfolding zero_ereal_def[symmetric]
  24.368        apply (subst norm_minus_commute)
  24.369      proof (rule nn_integral_dominated_convergence_norm[where w=w])
  24.370 @@ -1514,10 +1514,10 @@
  24.371          using int_s unfolding integrable_iff_bounded by auto
  24.372      qed fact+
  24.373    qed
  24.374 -  then have "(\<lambda>n. integral\<^sup>L M (s n) - integral\<^sup>L M f) ----> 0"
  24.375 +  then have "(\<lambda>n. integral\<^sup>L M (s n) - integral\<^sup>L M f) \<longlonglongrightarrow> 0"
  24.376      unfolding lim_ereal tendsto_norm_zero_iff .
  24.377    from tendsto_add[OF this tendsto_const[of "integral\<^sup>L M f"]]
  24.378 -  show "(\<lambda>i. integral\<^sup>L M (s i)) ----> integral\<^sup>L M f"  by simp
  24.379 +  show "(\<lambda>i. integral\<^sup>L M (s i)) \<longlonglongrightarrow> integral\<^sup>L M f"  by simp
  24.380  qed
  24.381  
  24.382  context
  24.383 @@ -1535,15 +1535,15 @@
  24.384    obtain N where w: "\<And>n. N \<le> n \<Longrightarrow> AE x in M. norm (s (X n) x) \<le> w x"
  24.385      by (auto simp: eventually_sequentially)
  24.386  
  24.387 -  show "(\<lambda>n. integral\<^sup>L M (s (X n))) ----> integral\<^sup>L M f"
  24.388 +  show "(\<lambda>n. integral\<^sup>L M (s (X n))) \<longlonglongrightarrow> integral\<^sup>L M f"
  24.389    proof (rule LIMSEQ_offset, rule integral_dominated_convergence)
  24.390      show "AE x in M. norm (s (X (n + N)) x) \<le> w x" for n
  24.391        by (rule w) auto
  24.392 -    show "AE x in M. (\<lambda>n. s (X (n + N)) x) ----> f x"
  24.393 +    show "AE x in M. (\<lambda>n. s (X (n + N)) x) \<longlonglongrightarrow> f x"
  24.394        using lim
  24.395      proof eventually_elim
  24.396        fix x assume "((\<lambda>i. s i x) ---> f x) at_top"
  24.397 -      then show "(\<lambda>n. s (X (n + N)) x) ----> f x"
  24.398 +      then show "(\<lambda>n. s (X (n + N)) x) \<longlonglongrightarrow> f x"
  24.399          by (intro LIMSEQ_ignore_initial_segment filterlim_compose[OF _ X])
  24.400      qed
  24.401    qed fact+
  24.402 @@ -1557,11 +1557,11 @@
  24.403    proof (rule integrable_dominated_convergence)
  24.404      show "AE x in M. norm (s (N + i) x) \<le> w x" for i :: nat
  24.405        by (intro w) auto
  24.406 -    show "AE x in M. (\<lambda>i. s (N + real i) x) ----> f x"
  24.407 +    show "AE x in M. (\<lambda>i. s (N + real i) x) \<longlonglongrightarrow> f x"
  24.408        using lim
  24.409      proof eventually_elim
  24.410        fix x assume "((\<lambda>i. s i x) ---> f x) at_top"
  24.411 -      then show "(\<lambda>n. s (N + n) x) ----> f x"
  24.412 +      then show "(\<lambda>n. s (N + n) x) \<longlonglongrightarrow> f x"
  24.413          by (rule filterlim_compose)
  24.414             (auto intro!: filterlim_tendsto_add_at_top filterlim_real_sequentially)
  24.415      qed
  24.416 @@ -1606,7 +1606,7 @@
  24.417  
  24.418        show ?case
  24.419        proof (rule LIMSEQ_unique)
  24.420 -        show "(\<lambda>i. ereal (integral\<^sup>L M (s i))) ----> ereal (integral\<^sup>L M f)"
  24.421 +        show "(\<lambda>i. ereal (integral\<^sup>L M (s i))) \<longlonglongrightarrow> ereal (integral\<^sup>L M f)"
  24.422            unfolding lim_ereal
  24.423          proof (rule integral_dominated_convergence[where w=f])
  24.424            show "integrable M f" by fact
  24.425 @@ -1615,13 +1615,13 @@
  24.426          qed (insert seq, auto)
  24.427          have int_s: "\<And>i. integrable M (s i)"
  24.428            using seq f s_le_f by (intro integrable_bound[OF f(3)]) auto
  24.429 -        have "(\<lambda>i. \<integral>\<^sup>+ x. s i x \<partial>M) ----> \<integral>\<^sup>+ x. f x \<partial>M"
  24.430 +        have "(\<lambda>i. \<integral>\<^sup>+ x. s i x \<partial>M) \<longlonglongrightarrow> \<integral>\<^sup>+ x. f x \<partial>M"
  24.431            using seq s_le_f f
  24.432            by (intro nn_integral_dominated_convergence[where w=f])
  24.433               (auto simp: integrable_iff_bounded)
  24.434          also have "(\<lambda>i. \<integral>\<^sup>+x. s i x \<partial>M) = (\<lambda>i. \<integral>x. s i x \<partial>M)"
  24.435            using seq int_s by simp
  24.436 -        finally show "(\<lambda>i. \<integral>x. s i x \<partial>M) ----> \<integral>\<^sup>+x. f x \<partial>M"
  24.437 +        finally show "(\<lambda>i. \<integral>x. s i x \<partial>M) \<longlonglongrightarrow> \<integral>\<^sup>+x. f x \<partial>M"
  24.438            by simp
  24.439        qed
  24.440      qed }
  24.441 @@ -1660,7 +1660,7 @@
  24.442        by (simp add: suminf_ereal' sums)
  24.443    qed simp
  24.444  
  24.445 -  have 2: "AE x in M. (\<lambda>n. \<Sum>i<n. f i x) ----> (\<Sum>i. f i x)"
  24.446 +  have 2: "AE x in M. (\<lambda>n. \<Sum>i<n. f i x) \<longlonglongrightarrow> (\<Sum>i. f i x)"
  24.447      using summable by eventually_elim (auto intro: summable_LIMSEQ summable_norm_cancel)
  24.448  
  24.449    have 3: "\<And>j. AE x in M. norm (\<Sum>i<j. f i x) \<le> (\<Sum>i. norm (f i x))"
  24.450 @@ -1742,14 +1742,14 @@
  24.451    assumes base: "\<And>A c. A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow> P (\<lambda>x. indicator A x *\<^sub>R c)"
  24.452    assumes add: "\<And>f g. integrable M f \<Longrightarrow> P f \<Longrightarrow> integrable M g \<Longrightarrow> P g \<Longrightarrow> P (\<lambda>x. f x + g x)"
  24.453    assumes lim: "\<And>f s. (\<And>i. integrable M (s i)) \<Longrightarrow> (\<And>i. P (s i)) \<Longrightarrow>
  24.454 -   (\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) ----> f x) \<Longrightarrow>
  24.455 +   (\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) \<longlonglongrightarrow> f x) \<Longrightarrow>
  24.456     (\<And>i x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> 2 * norm (f x)) \<Longrightarrow> integrable M f \<Longrightarrow> P f"
  24.457    shows "P f"
  24.458  proof -
  24.459    from \<open>integrable M f\<close> have f: "f \<in> borel_measurable M" "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
  24.460      unfolding integrable_iff_bounded by auto
  24.461    from borel_measurable_implies_sequence_metric[OF f(1)]
  24.462 -  obtain s where s: "\<And>i. simple_function M (s i)" "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) ----> f x"
  24.463 +  obtain s where s: "\<And>i. simple_function M (s i)" "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) \<longlonglongrightarrow> f x"
  24.464      "\<And>i x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> 2 * norm (f x)"
  24.465      unfolding norm_conv_dist by metis
  24.466  
  24.467 @@ -1799,7 +1799,7 @@
  24.468      then show "P (s' i)"
  24.469        by (subst s'_eq) (auto intro!: setsum base)
  24.470  
  24.471 -    fix x assume "x \<in> space M" with s show "(\<lambda>i. s' i x) ----> f x"
  24.472 +    fix x assume "x \<in> space M" with s show "(\<lambda>i. s' i x) \<longlonglongrightarrow> f x"
  24.473        by (simp add: s'_eq_s)
  24.474      show "norm (s' i x) \<le> 2 * norm (f x)"
  24.475        using \<open>x \<in> space M\<close> s by (simp add: s'_eq_s)
  24.476 @@ -1923,10 +1923,10 @@
  24.477      case (lim f s)
  24.478      show ?case
  24.479      proof (rule LIMSEQ_unique)
  24.480 -      show "(\<lambda>i. integral\<^sup>L (restrict_space M \<Omega>) (s i)) ----> integral\<^sup>L (restrict_space M \<Omega>) f"
  24.481 +      show "(\<lambda>i. integral\<^sup>L (restrict_space M \<Omega>) (s i)) \<longlonglongrightarrow> integral\<^sup>L (restrict_space M \<Omega>) f"
  24.482          using lim by (intro integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"]) simp_all
  24.483        
  24.484 -      show "(\<lambda>i. integral\<^sup>L (restrict_space M \<Omega>) (s i)) ----> (\<integral> x. indicator \<Omega> x *\<^sub>R f x \<partial>M)"
  24.485 +      show "(\<lambda>i. integral\<^sup>L (restrict_space M \<Omega>) (s i)) \<longlonglongrightarrow> (\<integral> x. indicator \<Omega> x *\<^sub>R f x \<partial>M)"
  24.486          unfolding lim
  24.487          using lim
  24.488          by (intro integral_dominated_convergence[where w="\<lambda>x. 2 * norm (indicator \<Omega> x *\<^sub>R f x)"])
  24.489 @@ -1998,16 +1998,16 @@
  24.490    
  24.491      show ?case
  24.492      proof (rule LIMSEQ_unique)
  24.493 -      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. g x *\<^sub>R s i x)) ----> integral\<^sup>L M (\<lambda>x. g x *\<^sub>R f x)"
  24.494 +      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. g x *\<^sub>R s i x)) \<longlonglongrightarrow> integral\<^sup>L M (\<lambda>x. g x *\<^sub>R f x)"
  24.495        proof (rule integral_dominated_convergence)
  24.496          show "integrable M (\<lambda>x. 2 * norm (g x *\<^sub>R f x))"
  24.497            by (intro integrable_mult_right integrable_norm integrable_density[THEN iffD1] lim g) auto
  24.498 -        show "AE x in M. (\<lambda>i. g x *\<^sub>R s i x) ----> g x *\<^sub>R f x"
  24.499 +        show "AE x in M. (\<lambda>i. g x *\<^sub>R s i x) \<longlonglongrightarrow> g x *\<^sub>R f x"
  24.500            using lim(3) by (auto intro!: tendsto_scaleR AE_I2[of M])
  24.501          show "\<And>i. AE x in M. norm (g x *\<^sub>R s i x) \<le> 2 * norm (g x *\<^sub>R f x)"
  24.502            using lim(4) g by (auto intro!: AE_I2[of M] mult_left_mono simp: field_simps)
  24.503        qed auto
  24.504 -      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. g x *\<^sub>R s i x)) ----> integral\<^sup>L (density M g) f"
  24.505 +      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. g x *\<^sub>R s i x)) \<longlonglongrightarrow> integral\<^sup>L (density M g) f"
  24.506          unfolding lim(2)[symmetric]
  24.507          by (rule integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"])
  24.508             (insert lim(3-5), auto)
  24.509 @@ -2077,16 +2077,16 @@
  24.510    
  24.511      show ?case
  24.512      proof (rule LIMSEQ_unique)
  24.513 -      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. s i (g x))) ----> integral\<^sup>L M (\<lambda>x. f (g x))"
  24.514 +      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. s i (g x))) \<longlonglongrightarrow> integral\<^sup>L M (\<lambda>x. f (g x))"
  24.515        proof (rule integral_dominated_convergence)
  24.516          show "integrable M (\<lambda>x. 2 * norm (f (g x)))"
  24.517            using lim by (auto simp: integrable_distr_eq) 
  24.518 -        show "AE x in M. (\<lambda>i. s i (g x)) ----> f (g x)"
  24.519 +        show "AE x in M. (\<lambda>i. s i (g x)) \<longlonglongrightarrow> f (g x)"
  24.520            using lim(3) g[THEN measurable_space] by auto
  24.521          show "\<And>i. AE x in M. norm (s i (g x)) \<le> 2 * norm (f (g x))"
  24.522            using lim(4) g[THEN measurable_space] by auto
  24.523        qed auto
  24.524 -      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. s i (g x))) ----> integral\<^sup>L (distr M N g) f"
  24.525 +      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. s i (g x))) \<longlonglongrightarrow> integral\<^sup>L (distr M N g) f"
  24.526          unfolding lim(2)[symmetric]
  24.527          by (rule integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"])
  24.528             (insert lim(3-5), auto)
  24.529 @@ -2266,8 +2266,8 @@
  24.530    fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real"
  24.531    assumes i: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
  24.532      and pos: "\<And>i. AE x in M. 0 \<le> f i x"
  24.533 -    and lim: "AE x in M. (\<lambda>i. f i x) ----> u x"
  24.534 -    and ilim: "(\<lambda>i. integral\<^sup>L M (f i)) ----> x"
  24.535 +    and lim: "AE x in M. (\<lambda>i. f i x) \<longlonglongrightarrow> u x"
  24.536 +    and ilim: "(\<lambda>i. integral\<^sup>L M (f i)) \<longlonglongrightarrow> x"
  24.537      and u: "u \<in> borel_measurable M"
  24.538    shows "integrable M u"
  24.539    and "integral\<^sup>L M u = x"
  24.540 @@ -2295,7 +2295,7 @@
  24.541        using u by auto
  24.542      from mono pos[of 0] lim show "AE x in M. ereal (- u x) \<le> 0"
  24.543      proof eventually_elim
  24.544 -      fix x assume "mono (\<lambda>n. f n x)" "0 \<le> f 0 x" "(\<lambda>i. f i x) ----> u x"
  24.545 +      fix x assume "mono (\<lambda>n. f n x)" "0 \<le> f 0 x" "(\<lambda>i. f i x) \<longlonglongrightarrow> u x"
  24.546        then show "ereal (- u x) \<le> 0"
  24.547          using incseq_le[of "\<lambda>n. f n x" "u x" 0] by (simp add: mono_def incseq_def)
  24.548      qed
  24.549 @@ -2307,8 +2307,8 @@
  24.550  lemma
  24.551    fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real"
  24.552    assumes f: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
  24.553 -  and lim: "AE x in M. (\<lambda>i. f i x) ----> u x"
  24.554 -  and ilim: "(\<lambda>i. integral\<^sup>L M (f i)) ----> x"
  24.555 +  and lim: "AE x in M. (\<lambda>i. f i x) \<longlonglongrightarrow> u x"
  24.556 +  and ilim: "(\<lambda>i. integral\<^sup>L M (f i)) \<longlonglongrightarrow> x"
  24.557    and u: "u \<in> borel_measurable M"
  24.558    shows integrable_monotone_convergence: "integrable M u"
  24.559      and integral_monotone_convergence: "integral\<^sup>L M u = x"
  24.560 @@ -2320,9 +2320,9 @@
  24.561      using mono by (auto simp: mono_def le_fun_def)
  24.562    have 3: "\<And>n. AE x in M. 0 \<le> f n x - f 0 x"
  24.563      using mono by (auto simp: field_simps mono_def le_fun_def)
  24.564 -  have 4: "AE x in M. (\<lambda>i. f i x - f 0 x) ----> u x - f 0 x"
  24.565 +  have 4: "AE x in M. (\<lambda>i. f i x - f 0 x) \<longlonglongrightarrow> u x - f 0 x"
  24.566      using lim by (auto intro!: tendsto_diff)
  24.567 -  have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x \<partial>M)) ----> x - integral\<^sup>L M (f 0)"
  24.568 +  have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x \<partial>M)) \<longlonglongrightarrow> x - integral\<^sup>L M (f 0)"
  24.569      using f ilim by (auto intro!: tendsto_diff)
  24.570    have 6: "(\<lambda>x. u x - f 0 x) \<in> borel_measurable M"
  24.571      using f[of 0] u by auto
  24.572 @@ -2462,17 +2462,17 @@
  24.573    shows "((\<lambda>y. \<integral> x. indicator {.. y} x *\<^sub>R f x \<partial>M) ---> \<integral> x. f x \<partial>M) at_top"
  24.574  proof (rule tendsto_at_topI_sequentially)
  24.575    fix X :: "nat \<Rightarrow> real" assume "filterlim X at_top sequentially"
  24.576 -  show "(\<lambda>n. \<integral>x. indicator {..X n} x *\<^sub>R f x \<partial>M) ----> integral\<^sup>L M f"
  24.577 +  show "(\<lambda>n. \<integral>x. indicator {..X n} x *\<^sub>R f x \<partial>M) \<longlonglongrightarrow> integral\<^sup>L M f"
  24.578    proof (rule integral_dominated_convergence)
  24.579      show "integrable M (\<lambda>x. norm (f x))"
  24.580        by (rule integrable_norm) fact
  24.581 -    show "AE x in M. (\<lambda>n. indicator {..X n} x *\<^sub>R f x) ----> f x"
  24.582 +    show "AE x in M. (\<lambda>n. indicator {..X n} x *\<^sub>R f x) \<longlonglongrightarrow> f x"
  24.583      proof
  24.584        fix x
  24.585        from \<open>filterlim X at_top sequentially\<close> 
  24.586        have "eventually (\<lambda>n. x \<le> X n) sequentially"
  24.587          unfolding filterlim_at_top_ge[where c=x] by auto
  24.588 -      then show "(\<lambda>n. indicator {..X n} x *\<^sub>R f x) ----> f x"
  24.589 +      then show "(\<lambda>n. indicator {..X n} x *\<^sub>R f x) \<longlonglongrightarrow> f x"
  24.590          by (intro Lim_eventually) (auto split: split_indicator elim!: eventually_mono)
  24.591      qed
  24.592      fix n show "AE x in M. norm (indicator {..X n} x *\<^sub>R f x) \<le> norm (f x)"
  24.593 @@ -2497,9 +2497,9 @@
  24.594        by (rule eventually_sequentiallyI[of "nat \<lceil>x\<rceil>"])
  24.595           (auto split: split_indicator simp: nat_le_iff ceiling_le_iff) }
  24.596    from filterlim_cong[OF refl refl this]
  24.597 -  have "AE x in M. (\<lambda>i. f x * indicator {..real i} x) ----> f x"
  24.598 +  have "AE x in M. (\<lambda>i. f x * indicator {..real i} x) \<longlonglongrightarrow> f x"
  24.599      by simp
  24.600 -  have "(\<lambda>i. \<integral> x. f x * indicator {..real i} x \<partial>M) ----> x"
  24.601 +  have "(\<lambda>i. \<integral> x. f x * indicator {..real i} x \<partial>M) \<longlonglongrightarrow> x"
  24.602      using conv filterlim_real_sequentially by (rule filterlim_compose)
  24.603    have M_measure[simp]: "borel_measurable M = borel_measurable borel"
  24.604      using M by (simp add: sets_eq_imp_space_eq measurable_def)
  24.605 @@ -2539,7 +2539,7 @@
  24.606  proof -
  24.607    from borel_measurable_implies_sequence_metric[OF f, of 0] guess s ..
  24.608    then have s: "\<And>i. simple_function (N \<Otimes>\<^sub>M M) (s i)"
  24.609 -    "\<And>x y. x \<in> space N \<Longrightarrow> y \<in> space M \<Longrightarrow> (\<lambda>i. s i (x, y)) ----> f x y"
  24.610 +    "\<And>x y. x \<in> space N \<Longrightarrow> y \<in> space M \<Longrightarrow> (\<lambda>i. s i (x, y)) \<longlonglongrightarrow> f x y"
  24.611      "\<And>i x y. x \<in> space N \<Longrightarrow> y \<in> space M \<Longrightarrow> norm (s i (x, y)) \<le> 2 * norm (f x y)"
  24.612      by (auto simp: space_pair_measure norm_conv_dist)
  24.613  
  24.614 @@ -2569,12 +2569,12 @@
  24.615      { assume int_f: "integrable M (f x)"
  24.616        have int_2f: "integrable M (\<lambda>y. 2 * norm (f x y))"
  24.617          by (intro integrable_norm integrable_mult_right int_f)
  24.618 -      have "(\<lambda>i. integral\<^sup>L M (\<lambda>y. s i (x, y))) ----> integral\<^sup>L M (f x)"
  24.619 +      have "(\<lambda>i. integral\<^sup>L M (\<lambda>y. s i (x, y))) \<longlonglongrightarrow> integral\<^sup>L M (f x)"
  24.620        proof (rule integral_dominated_convergence)
  24.621          from int_f show "f x \<in> borel_measurable M" by auto
  24.622          show "\<And>i. (\<lambda>y. s i (x, y)) \<in> borel_measurable M"
  24.623            using x by simp
  24.624 -        show "AE xa in M. (\<lambda>i. s i (x, xa)) ----> f x xa"
  24.625 +        show "AE xa in M. (\<lambda>i. s i (x, xa)) \<longlonglongrightarrow> f x xa"
  24.626            using x s(2) by auto
  24.627          show "\<And>i. AE xa in M. norm (s i (x, xa)) \<le> 2 * norm (f x xa)"
  24.628            using x s(3) by auto
  24.629 @@ -2597,10 +2597,10 @@
  24.630          qed
  24.631          then have "integral\<^sup>L M (\<lambda>y. s i (x, y)) = simple_bochner_integral M (\<lambda>y. s i (x, y))"
  24.632            by (rule simple_bochner_integrable_eq_integral[symmetric]) }
  24.633 -      ultimately have "(\<lambda>i. simple_bochner_integral M (\<lambda>y. s i (x, y))) ----> integral\<^sup>L M (f x)"
  24.634 +      ultimately have "(\<lambda>i. simple_bochner_integral M (\<lambda>y. s i (x, y))) \<longlonglongrightarrow> integral\<^sup>L M (f x)"
  24.635          by simp }
  24.636      then 
  24.637 -    show "(\<lambda>i. f' i x) ----> integral\<^sup>L M (f x)"
  24.638 +    show "(\<lambda>i. f' i x) \<longlonglongrightarrow> integral\<^sup>L M (f x)"
  24.639        unfolding f'_def
  24.640        by (cases "integrable M (f x)") (simp_all add: not_integrable_integral_eq)
  24.641    qed
  24.642 @@ -2779,25 +2779,25 @@
  24.643    
  24.644    show ?case
  24.645    proof (rule LIMSEQ_unique)
  24.646 -    show "(\<lambda>i. integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) (s i)) ----> integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f"
  24.647 +    show "(\<lambda>i. integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) (s i)) \<longlonglongrightarrow> integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f"
  24.648      proof (rule integral_dominated_convergence)
  24.649        show "integrable (M1 \<Otimes>\<^sub>M M2) (\<lambda>x. 2 * norm (f x))"
  24.650          using lim(5) by auto
  24.651      qed (insert lim, auto)
  24.652 -    have "(\<lambda>i. \<integral> x. \<integral> y. s i (x, y) \<partial>M2 \<partial>M1) ----> \<integral> x. \<integral> y. f (x, y) \<partial>M2 \<partial>M1"
  24.653 +    have "(\<lambda>i. \<integral> x. \<integral> y. s i (x, y) \<partial>M2 \<partial>M1) \<longlonglongrightarrow> \<integral> x. \<integral> y. f (x, y) \<partial>M2 \<partial>M1"
  24.654      proof (rule integral_dominated_convergence)
  24.655        have "AE x in M1. \<forall>i. integrable M2 (\<lambda>y. s i (x, y))"
  24.656          unfolding AE_all_countable using AE_integrable_fst'[OF lim(1)] ..
  24.657        with AE_space AE_integrable_fst'[OF lim(5)]
  24.658 -      show "AE x in M1. (\<lambda>i. \<integral> y. s i (x, y) \<partial>M2) ----> \<integral> y. f (x, y) \<partial>M2"
  24.659 +      show "AE x in M1. (\<lambda>i. \<integral> y. s i (x, y) \<partial>M2) \<longlonglongrightarrow> \<integral> y. f (x, y) \<partial>M2"
  24.660        proof eventually_elim
  24.661          fix x assume x: "x \<in> space M1" and
  24.662            s: "\<forall>i. integrable M2 (\<lambda>y. s i (x, y))" and f: "integrable M2 (\<lambda>y. f (x, y))"
  24.663 -        show "(\<lambda>i. \<integral> y. s i (x, y) \<partial>M2) ----> \<integral> y. f (x, y) \<partial>M2"
  24.664 +        show "(\<lambda>i. \<integral> y. s i (x, y) \<partial>M2) \<longlonglongrightarrow> \<integral> y. f (x, y) \<partial>M2"
  24.665          proof (rule integral_dominated_convergence)
  24.666            show "integrable M2 (\<lambda>y. 2 * norm (f (x, y)))"
  24.667               using f by auto
  24.668 -          show "AE xa in M2. (\<lambda>i. s i (x, xa)) ----> f (x, xa)"
  24.669 +          show "AE xa in M2. (\<lambda>i. s i (x, xa)) \<longlonglongrightarrow> f (x, xa)"
  24.670              using x lim(3) by (auto simp: space_pair_measure)
  24.671            show "\<And>i. AE xa in M2. norm (s i (x, xa)) \<le> 2 * norm (f (x, xa))"
  24.672              using x lim(4) by (auto simp: space_pair_measure)
  24.673 @@ -2820,7 +2820,7 @@
  24.674            by simp
  24.675        qed
  24.676      qed simp_all
  24.677 -    then show "(\<lambda>i. integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) (s i)) ----> \<integral> x. \<integral> y. f (x, y) \<partial>M2 \<partial>M1"
  24.678 +    then show "(\<lambda>i. integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) (s i)) \<longlonglongrightarrow> \<integral> x. \<integral> y. f (x, y) \<partial>M2 \<partial>M1"
  24.679        using lim by simp
  24.680    qed
  24.681  qed
  24.682 @@ -2991,12 +2991,12 @@
  24.683        using integrable_subalgebra[OF _ N, of f] integrable_subalgebra[OF _ N, of "s i" for i] by simp_all
  24.684      show ?case
  24.685      proof (intro LIMSEQ_unique)
  24.686 -      show "(\<lambda>i. integral\<^sup>L N (s i)) ----> integral\<^sup>L N f"
  24.687 +      show "(\<lambda>i. integral\<^sup>L N (s i)) \<longlonglongrightarrow> integral\<^sup>L N f"
  24.688          apply (rule integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"])
  24.689          using lim
  24.690          apply auto
  24.691          done
  24.692 -      show "(\<lambda>i. integral\<^sup>L N (s i)) ----> integral\<^sup>L M f"
  24.693 +      show "(\<lambda>i. integral\<^sup>L N (s i)) \<longlonglongrightarrow> integral\<^sup>L M f"
  24.694          unfolding lim
  24.695          apply (rule integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"])
  24.696          using lim M N(2)
    25.1 --- a/src/HOL/Probability/Borel_Space.thy	Tue Dec 29 22:41:22 2015 +0100
    25.2 +++ b/src/HOL/Probability/Borel_Space.thy	Tue Dec 29 23:04:53 2015 +0100
    25.3 @@ -1300,7 +1300,7 @@
    25.4  
    25.5  lemma borel_measurable_ereal_LIMSEQ:
    25.6    fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
    25.7 -  assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
    25.8 +  assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
    25.9    and u: "\<And>i. u i \<in> borel_measurable M"
   25.10    shows "u' \<in> borel_measurable M"
   25.11  proof -
   25.12 @@ -1319,7 +1319,7 @@
   25.13  
   25.14  lemma borel_measurable_LIMSEQ:
   25.15    fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
   25.16 -  assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
   25.17 +  assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
   25.18    and u: "\<And>i. u i \<in> borel_measurable M"
   25.19    shows "u' \<in> borel_measurable M"
   25.20  proof -
   25.21 @@ -1333,7 +1333,7 @@
   25.22  lemma borel_measurable_LIMSEQ_metric:
   25.23    fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b :: metric_space"
   25.24    assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
   25.25 -  assumes lim: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. f i x) ----> g x"
   25.26 +  assumes lim: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. f i x) \<longlonglongrightarrow> g x"
   25.27    shows "g \<in> borel_measurable M"
   25.28    unfolding borel_eq_closed
   25.29  proof (safe intro!: measurable_measure_of)
   25.30 @@ -1341,7 +1341,7 @@
   25.31  
   25.32    have [measurable]: "(\<lambda>x. infdist (g x) A) \<in> borel_measurable M"
   25.33    proof (rule borel_measurable_LIMSEQ)
   25.34 -    show "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. infdist (f i x) A) ----> infdist (g x) A"
   25.35 +    show "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. infdist (f i x) A) \<longlonglongrightarrow> infdist (g x) A"
   25.36        by (intro tendsto_infdist lim)
   25.37      show "\<And>i. (\<lambda>x. infdist (f i x) A) \<in> borel_measurable M"
   25.38        by (intro borel_measurable_continuous_on[where f="\<lambda>x. infdist x A"]
   25.39 @@ -1381,7 +1381,7 @@
   25.40      have "convergent (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)"
   25.41        by (cases "Cauchy (\<lambda>i. f i x)")
   25.42           (auto simp add: convergent_eq_cauchy[symmetric] convergent_def)
   25.43 -    then show "(\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0) ----> u' x"
   25.44 +    then show "(\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0) \<longlonglongrightarrow> u' x"
   25.45        unfolding u'_def 
   25.46        by (rule convergent_LIMSEQ_iff[THEN iffD1])
   25.47    qed measurable
    26.1 --- a/src/HOL/Probability/Caratheodory.thy	Tue Dec 29 22:41:22 2015 +0100
    26.2 +++ b/src/HOL/Probability/Caratheodory.thy	Tue Dec 29 23:04:53 2015 +0100
    26.3 @@ -540,7 +540,7 @@
    26.4  
    26.5  lemma (in ring_of_sets) caratheodory_empty_continuous:
    26.6    assumes f: "positive M f" "additive M f" and fin: "\<And>A. A \<in> M \<Longrightarrow> f A \<noteq> \<infinity>"
    26.7 -  assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0"
    26.8 +  assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
    26.9    shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
   26.10  proof (intro caratheodory' empty_continuous_imp_countably_additive f)
   26.11    show "\<forall>A\<in>M. f A \<noteq> \<infinity>" using fin by auto
    27.1 --- a/src/HOL/Probability/Distributions.thy	Tue Dec 29 22:41:22 2015 +0100
    27.2 +++ b/src/HOL/Probability/Distributions.thy	Tue Dec 29 23:04:53 2015 +0100
    27.3 @@ -116,7 +116,7 @@
    27.4    shows "(\<integral>\<^sup>+x. ereal (x^k * exp (-x)) * indicator {0 ..} x \<partial>lborel) = real_of_nat (fact k)"
    27.5  proof (rule LIMSEQ_unique)
    27.6    let ?X = "\<lambda>n. \<integral>\<^sup>+ x. ereal (x^k * exp (-x)) * indicator {0 .. real n} x \<partial>lborel"
    27.7 -  show "?X ----> (\<integral>\<^sup>+x. ereal (x^k * exp (-x)) * indicator {0 ..} x \<partial>lborel)"
    27.8 +  show "?X \<longlonglongrightarrow> (\<integral>\<^sup>+x. ereal (x^k * exp (-x)) * indicator {0 ..} x \<partial>lborel)"
    27.9      apply (intro nn_integral_LIMSEQ)
   27.10      apply (auto simp: incseq_def le_fun_def eventually_sequentially
   27.11                  split: split_indicator intro!: Lim_eventually)
   27.12 @@ -126,7 +126,7 @@
   27.13    have "((\<lambda>x::real. (1 - (\<Sum>n\<le>k. (x ^ n / exp x) / (fact n))) * fact k) --->
   27.14          (1 - (\<Sum>n\<le>k. 0 / (fact n))) * fact k) at_top"
   27.15      by (intro tendsto_intros tendsto_power_div_exp_0) simp
   27.16 -  then show "?X ----> real_of_nat (fact k)"
   27.17 +  then show "?X \<longlonglongrightarrow> real_of_nat (fact k)"
   27.18      by (subst nn_intergal_power_times_exp_Icc)
   27.19         (auto simp: exp_minus field_simps intro!: filterlim_compose[OF _ filterlim_real_sequentially])
   27.20  qed
    28.1 --- a/src/HOL/Probability/Fin_Map.thy	Tue Dec 29 22:41:22 2015 +0100
    28.2 +++ b/src/HOL/Probability/Fin_Map.thy	Tue Dec 29 23:04:53 2015 +0100
    28.3 @@ -420,8 +420,8 @@
    28.4  lemma tendsto_finmap:
    28.5    fixes f::"nat \<Rightarrow> ('i \<Rightarrow>\<^sub>F ('a::metric_space))"
    28.6    assumes ind_f:  "\<And>n. domain (f n) = domain g"
    28.7 -  assumes proj_g:  "\<And>i. i \<in> domain g \<Longrightarrow> (\<lambda>n. (f n) i) ----> g i"
    28.8 -  shows "f ----> g"
    28.9 +  assumes proj_g:  "\<And>i. i \<in> domain g \<Longrightarrow> (\<lambda>n. (f n) i) \<longlonglongrightarrow> g i"
   28.10 +  shows "f \<longlonglongrightarrow> g"
   28.11    unfolding tendsto_iff
   28.12  proof safe
   28.13    fix e::real assume "0 < e"
   28.14 @@ -472,9 +472,9 @@
   28.15        qed
   28.16      qed
   28.17      hence "convergent (p i)" by (metis Cauchy_convergent_iff)
   28.18 -    hence "p i ----> q i" unfolding q_def convergent_def by (metis limI)
   28.19 +    hence "p i \<longlonglongrightarrow> q i" unfolding q_def convergent_def by (metis limI)
   28.20    } note p = this
   28.21 -  have "P ----> Q"
   28.22 +  have "P \<longlonglongrightarrow> Q"
   28.23    proof (rule metric_LIMSEQ_I)
   28.24      fix e::real assume "0 < e"
   28.25      have "\<exists>ni. \<forall>i\<in>d. \<forall>n\<ge>ni i. dist (p i n) (q i) < e"
    29.1 --- a/src/HOL/Probability/Infinite_Product_Measure.thy	Tue Dec 29 22:41:22 2015 +0100
    29.2 +++ b/src/HOL/Probability/Infinite_Product_Measure.thy	Tue Dec 29 23:04:53 2015 +0100
    29.3 @@ -187,7 +187,7 @@
    29.4  
    29.5  lemma measure_PiM_countable:
    29.6    fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets M"
    29.7 -  shows "(\<lambda>n. \<Prod>i\<le>n. measure M (E i)) ----> measure S (Pi UNIV E)"
    29.8 +  shows "(\<lambda>n. \<Prod>i\<le>n. measure M (E i)) \<longlonglongrightarrow> measure S (Pi UNIV E)"
    29.9  proof -
   29.10    let ?E = "\<lambda>n. emb UNIV {..n} (Pi\<^sub>E {.. n} E)"
   29.11    have "\<And>n. (\<Prod>i\<le>n. measure M (E i)) = measure S (?E n)"
    30.1 --- a/src/HOL/Probability/Interval_Integral.thy	Tue Dec 29 22:41:22 2015 +0100
    30.2 +++ b/src/HOL/Probability/Interval_Integral.thy	Tue Dec 29 23:04:53 2015 +0100
    30.3 @@ -62,16 +62,16 @@
    30.4    fixes a b :: ereal
    30.5    assumes "a < b"
    30.6    obtains X :: "nat \<Rightarrow> real" where 
    30.7 -    "incseq X" "\<And>i. a < X i" "\<And>i. X i < b" "X ----> b"
    30.8 +    "incseq X" "\<And>i. a < X i" "\<And>i. X i < b" "X \<longlonglongrightarrow> b"
    30.9  proof (cases b)
   30.10    case PInf
   30.11    with \<open>a < b\<close> have "a = -\<infinity> \<or> (\<exists>r. a = ereal r)"
   30.12      by (cases a) auto
   30.13 -  moreover have "(\<lambda>x. ereal (real (Suc x))) ----> \<infinity>"
   30.14 +  moreover have "(\<lambda>x. ereal (real (Suc x))) \<longlonglongrightarrow> \<infinity>"
   30.15        apply (subst LIMSEQ_Suc_iff)
   30.16        apply (simp add: Lim_PInfty)
   30.17        using nat_ceiling_le_eq by blast
   30.18 -  moreover have "\<And>r. (\<lambda>x. ereal (r + real (Suc x))) ----> \<infinity>"
   30.19 +  moreover have "\<And>r. (\<lambda>x. ereal (r + real (Suc x))) \<longlonglongrightarrow> \<infinity>"
   30.20      apply (subst LIMSEQ_Suc_iff)
   30.21      apply (subst Lim_PInfty)
   30.22      apply (metis add.commute diff_le_eq nat_ceiling_le_eq ereal_less_eq(3))
   30.23 @@ -89,7 +89,7 @@
   30.24      by (intro mult_strict_left_mono) auto
   30.25    with \<open>a < b\<close> a' have "\<And>i. a < ereal (b' - d / real (Suc (Suc i)))"
   30.26      by (cases a) (auto simp: real d_def field_simps)
   30.27 -  moreover have "(\<lambda>i. b' - d / Suc (Suc i)) ----> b'"
   30.28 +  moreover have "(\<lambda>i. b' - d / Suc (Suc i)) \<longlonglongrightarrow> b'"
   30.29      apply (subst filterlim_sequentially_Suc)
   30.30      apply (subst filterlim_sequentially_Suc)
   30.31      apply (rule tendsto_eq_intros)
   30.32 @@ -105,7 +105,7 @@
   30.33    fixes a b :: ereal
   30.34    assumes "a < b"
   30.35    obtains X :: "nat \<Rightarrow> real" where 
   30.36 -    "decseq X" "\<And>i. a < X i" "\<And>i. X i < b" "X ----> a"
   30.37 +    "decseq X" "\<And>i. a < X i" "\<And>i. X i < b" "X \<longlonglongrightarrow> a"
   30.38  proof -
   30.39    have "-b < -a" using \<open>a < b\<close> by simp
   30.40    from ereal_incseq_approx[OF this] guess X .
   30.41 @@ -123,7 +123,7 @@
   30.42    obtains u l :: "nat \<Rightarrow> real" where 
   30.43      "einterval a b = (\<Union>i. {l i .. u i})"
   30.44      "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
   30.45 -    "l ----> a" "u ----> b"
   30.46 +    "l \<longlonglongrightarrow> a" "u \<longlonglongrightarrow> b"
   30.47  proof -
   30.48    from dense[OF \<open>a < b\<close>] obtain c where "a < c" "c < b" by safe
   30.49    from ereal_incseq_approx[OF \<open>c < b\<close>] guess u . note u = this
   30.50 @@ -494,12 +494,12 @@
   30.51    fixes u l :: "nat \<Rightarrow> real"
   30.52    assumes  approx: "einterval a b = (\<Union>i. {l i .. u i})"
   30.53      "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
   30.54 -    "l ----> a" "u ----> b"
   30.55 +    "l \<longlonglongrightarrow> a" "u \<longlonglongrightarrow> b"
   30.56    fixes f :: "real \<Rightarrow> real"
   30.57    assumes f_integrable: "\<And>i. set_integrable lborel {l i..u i} f"
   30.58    assumes f_nonneg: "AE x in lborel. a < ereal x \<longrightarrow> ereal x < b \<longrightarrow> 0 \<le> f x"
   30.59    assumes f_measurable: "set_borel_measurable lborel (einterval a b) f"
   30.60 -  assumes lbint_lim: "(\<lambda>i. LBINT x=l i.. u i. f x) ----> C"
   30.61 +  assumes lbint_lim: "(\<lambda>i. LBINT x=l i.. u i. f x) \<longlonglongrightarrow> C"
   30.62    shows 
   30.63      "set_integrable lborel (einterval a b) f"
   30.64      "(LBINT x=a..b. f x) = C"
   30.65 @@ -517,7 +517,7 @@
   30.66        apply (metis approx(2) incseqD order_trans)
   30.67        done
   30.68    qed
   30.69 -  have 3: "AE x in lborel. (\<lambda>i. indicator {l i..u i} x *\<^sub>R f x) ----> indicator (einterval a b) x *\<^sub>R f x"
   30.70 +  have 3: "AE x in lborel. (\<lambda>i. indicator {l i..u i} x *\<^sub>R f x) \<longlonglongrightarrow> indicator (einterval a b) x *\<^sub>R f x"
   30.71    proof -
   30.72      { fix x i assume "l i \<le> x" "x \<le> u i"
   30.73        then have "eventually (\<lambda>i. l i \<le> x \<and> x \<le> u i) sequentially"
   30.74 @@ -530,7 +530,7 @@
   30.75      then show ?thesis
   30.76        unfolding approx(1) by (auto intro!: AE_I2 Lim_eventually split: split_indicator)
   30.77    qed
   30.78 -  have 4: "(\<lambda>i. \<integral> x. indicator {l i..u i} x *\<^sub>R f x \<partial>lborel) ----> C"
   30.79 +  have 4: "(\<lambda>i. \<integral> x. indicator {l i..u i} x *\<^sub>R f x \<partial>lborel) \<longlonglongrightarrow> C"
   30.80      using lbint_lim by (simp add: interval_integral_Icc approx less_imp_le)
   30.81    have 5: "set_borel_measurable lborel (einterval a b) f" by (rule assms)
   30.82    have "(LBINT x=a..b. f x) = lebesgue_integral lborel (\<lambda>x. indicator (einterval a b) x *\<^sub>R f x)"
   30.83 @@ -548,11 +548,11 @@
   30.84    assumes "a < b"
   30.85    assumes  approx: "einterval a b = (\<Union>i. {l i .. u i})"
   30.86      "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
   30.87 -    "l ----> a" "u ----> b"
   30.88 +    "l \<longlonglongrightarrow> a" "u \<longlonglongrightarrow> b"
   30.89    assumes f_integrable: "set_integrable lborel (einterval a b) f"
   30.90 -  shows "(\<lambda>i. LBINT x=l i.. u i. f x) ----> (LBINT x=a..b. f x)"
   30.91 +  shows "(\<lambda>i. LBINT x=l i.. u i. f x) \<longlonglongrightarrow> (LBINT x=a..b. f x)"
   30.92  proof -
   30.93 -  have "(\<lambda>i. LBINT x:{l i.. u i}. f x) ----> (LBINT x:einterval a b. f x)"
   30.94 +  have "(\<lambda>i. LBINT x:{l i.. u i}. f x) \<longlonglongrightarrow> (LBINT x:einterval a b. f x)"
   30.95    proof (rule integral_dominated_convergence)
   30.96      show "integrable lborel (\<lambda>x. norm (indicator (einterval a b) x *\<^sub>R f x))"
   30.97        by (rule integrable_norm) fact
   30.98 @@ -562,7 +562,7 @@
   30.99        by (rule set_borel_measurable_subset) (auto simp: approx)
  30.100      show "\<And>i. AE x in lborel. norm (indicator {l i..u i} x *\<^sub>R f x) \<le> norm (indicator (einterval a b) x *\<^sub>R f x)"
  30.101        by (intro AE_I2) (auto simp: approx split: split_indicator)
  30.102 -    show "AE x in lborel. (\<lambda>i. indicator {l i..u i} x *\<^sub>R f x) ----> indicator (einterval a b) x *\<^sub>R f x"
  30.103 +    show "AE x in lborel. (\<lambda>i. indicator {l i..u i} x *\<^sub>R f x) \<longlonglongrightarrow> indicator (einterval a b) x *\<^sub>R f x"
  30.104      proof (intro AE_I2 tendsto_intros Lim_eventually)
  30.105        fix x
  30.106        { fix i assume "l i \<le> x" "x \<le> u i" 
  30.107 @@ -570,7 +570,7 @@
  30.108          have "eventually (\<lambda>i. l i \<le> x \<and> x \<le> u i) sequentially"
  30.109            by (auto simp: eventually_sequentially decseq_def incseq_def intro: order_trans) }
  30.110        then show "eventually (\<lambda>xa. indicator {l xa..u xa} x = (indicator (einterval a b) x::real)) sequentially"
  30.111 -        using approx order_tendstoD(2)[OF \<open>l ----> a\<close>, of x] order_tendstoD(1)[OF \<open>u ----> b\<close>, of x]
  30.112 +        using approx order_tendstoD(2)[OF \<open>l \<longlonglongrightarrow> a\<close>, of x] order_tendstoD(1)[OF \<open>u \<longlonglongrightarrow> b\<close>, of x]
  30.113          by (auto split: split_indicator)
  30.114      qed
  30.115    qed
  30.116 @@ -649,7 +649,7 @@
  30.117    have 2: "set_borel_measurable lborel (einterval a b) f"
  30.118      by (auto simp del: real_scaleR_def intro!: set_borel_measurable_continuous 
  30.119               simp: continuous_on_eq_continuous_at einterval_iff f)
  30.120 -  have 3: "(\<lambda>i. LBINT x=l i..u i. f x) ----> B - A"
  30.121 +  have 3: "(\<lambda>i. LBINT x=l i..u i. f x) \<longlonglongrightarrow> B - A"
  30.122      apply (subst FTCi)
  30.123      apply (intro tendsto_intros)
  30.124      using B approx unfolding tendsto_at_iff_sequentially comp_def
  30.125 @@ -683,14 +683,14 @@
  30.126      by (auto simp add: less_imp_le min_def max_def
  30.127               intro!: f continuous_at_imp_continuous_on interval_integral_FTC_finite
  30.128               intro: has_vector_derivative_at_within)
  30.129 -  have "(\<lambda>i. LBINT x=l i..u i. f x) ----> B - A"
  30.130 +  have "(\<lambda>i. LBINT x=l i..u i. f x) \<longlonglongrightarrow> B - A"
  30.131      apply (subst FTCi)
  30.132      apply (intro tendsto_intros)
  30.133      using B approx unfolding tendsto_at_iff_sequentially comp_def
  30.134      apply (elim allE[of _ "\<lambda>i. ereal (u i)"], auto)
  30.135      using A approx unfolding tendsto_at_iff_sequentially comp_def
  30.136      by (elim allE[of _ "\<lambda>i. ereal (l i)"], auto)
  30.137 -  moreover have "(\<lambda>i. LBINT x=l i..u i. f x) ----> (LBINT x=a..b. f x)"
  30.138 +  moreover have "(\<lambda>i. LBINT x=l i..u i. f x) \<longlonglongrightarrow> (LBINT x=a..b. f x)"
  30.139      by (rule interval_integral_Icc_approx_integrable [OF \<open>a < b\<close> approx f_integrable])
  30.140    ultimately show ?thesis
  30.141      by (elim LIMSEQ_unique)
  30.142 @@ -863,12 +863,12 @@
  30.143      by (rule less_imp_le, rule le_less_trans, subst ereal_less_eq(3), assumption, auto)
  30.144    have "A \<le> B" and un: "einterval A B = (\<Union>i. {g(l i)<..<g(u i)})"
  30.145    proof - 
  30.146 -    have A2: "(\<lambda>i. g (l i)) ----> A"
  30.147 +    have A2: "(\<lambda>i. g (l i)) \<longlonglongrightarrow> A"
  30.148        using A apply (auto simp add: einterval_def tendsto_at_iff_sequentially comp_def)
  30.149        by (drule_tac x = "\<lambda>i. ereal (l i)" in spec, auto)
  30.150      hence A3: "\<And>i. g (l i) \<ge> A"
  30.151        by (intro decseq_le, auto simp add: decseq_def)
  30.152 -    have B2: "(\<lambda>i. g (u i)) ----> B"
  30.153 +    have B2: "(\<lambda>i. g (u i)) \<longlonglongrightarrow> B"
  30.154        using B apply (auto simp add: einterval_def tendsto_at_iff_sequentially comp_def)
  30.155        by (drule_tac x = "\<lambda>i. ereal (u i)" in spec, auto)
  30.156      hence B3: "\<And>i. g (u i) \<le> B"
  30.157 @@ -903,17 +903,17 @@
  30.158          apply (auto intro!: continuous_at_imp_continuous_on contf contg')
  30.159          done
  30.160    } note eq1 = this
  30.161 -  have "(\<lambda>i. LBINT x=l i..u i. g' x *\<^sub>R f (g x)) ----> (LBINT x=a..b. g' x *\<^sub>R f (g x))"
  30.162 +  have "(\<lambda>i. LBINT x=l i..u i. g' x *\<^sub>R f (g x)) \<longlonglongrightarrow> (LBINT x=a..b. g' x *\<^sub>R f (g x))"
  30.163      apply (rule interval_integral_Icc_approx_integrable [OF \<open>a < b\<close> approx])
  30.164      by (rule assms)
  30.165 -  hence 2: "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) ----> (LBINT x=a..b. g' x *\<^sub>R f (g x))"
  30.166 +  hence 2: "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) \<longlonglongrightarrow> (LBINT x=a..b. g' x *\<^sub>R f (g x))"
  30.167      by (simp add: eq1)
  30.168    have incseq: "incseq (\<lambda>i. {g (l i)<..<g (u i)})"
  30.169      apply (auto simp add: incseq_def)
  30.170      apply (rule order_le_less_trans)
  30.171      prefer 2 apply (assumption, auto)
  30.172      by (erule order_less_le_trans, rule g_nondec, auto)
  30.173 -  have "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) ----> (LBINT x = A..B. f x)"
  30.174 +  have "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) \<longlonglongrightarrow> (LBINT x = A..B. f x)"
  30.175      apply (subst interval_lebesgue_integral_le_eq, auto simp del: real_scaleR_def)
  30.176      apply (subst interval_lebesgue_integral_le_eq, rule \<open>A \<le> B\<close>)
  30.177      apply (subst un, rule set_integral_cont_up, auto simp del: real_scaleR_def)
  30.178 @@ -965,12 +965,12 @@
  30.179      by (rule less_imp_le, rule le_less_trans, subst ereal_less_eq(3), assumption, auto)
  30.180    have "A \<le> B" and un: "einterval A B = (\<Union>i. {g(l i)<..<g(u i)})"
  30.181    proof - 
  30.182 -    have A2: "(\<lambda>i. g (l i)) ----> A"
  30.183 +    have A2: "(\<lambda>i. g (l i)) \<longlonglongrightarrow> A"
  30.184        using A apply (auto simp add: einterval_def tendsto_at_iff_sequentially comp_def)
  30.185        by (drule_tac x = "\<lambda>i. ereal (l i)" in spec, auto)
  30.186      hence A3: "\<And>i. g (l i) \<ge> A"
  30.187        by (intro decseq_le, auto simp add: decseq_def)
  30.188 -    have B2: "(\<lambda>i. g (u i)) ----> B"
  30.189 +    have B2: "(\<lambda>i. g (u i)) \<longlonglongrightarrow> B"
  30.190        using B apply (auto simp add: einterval_def tendsto_at_iff_sequentially comp_def)
  30.191        by (drule_tac x = "\<lambda>i. ereal (u i)" in spec, auto)
  30.192      hence B3: "\<And>i. g (u i) \<le> B"
  30.193 @@ -1006,10 +1006,10 @@
  30.194         by (simp add: ac_simps)
  30.195    } note eq1 = this
  30.196    have "(\<lambda>i. LBINT x=l i..u i. f (g x) * g' x)
  30.197 -      ----> (LBINT x=a..b. f (g x) * g' x)"
  30.198 +      \<longlonglongrightarrow> (LBINT x=a..b. f (g x) * g' x)"
  30.199      apply (rule interval_integral_Icc_approx_integrable [OF \<open>a < b\<close> approx])
  30.200      by (rule assms)
  30.201 -  hence 2: "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) ----> (LBINT x=a..b. f (g x) * g' x)"
  30.202 +  hence 2: "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) \<longlonglongrightarrow> (LBINT x=a..b. f (g x) * g' x)"
  30.203      by (simp add: eq1)
  30.204    have incseq: "incseq (\<lambda>i. {g (l i)<..<g (u i)})"
  30.205      apply (auto simp add: incseq_def)
    31.1 --- a/src/HOL/Probability/Lebesgue_Measure.thy	Tue Dec 29 22:41:22 2015 +0100
    31.2 +++ b/src/HOL/Probability/Lebesgue_Measure.thy	Tue Dec 29 23:04:53 2015 +0100
    31.3 @@ -325,8 +325,8 @@
    31.4    show "((\<lambda>a. F b - F a) ---> emeasure ?F {a..b}) (at_left a)"
    31.5    proof (rule tendsto_at_left_sequentially)
    31.6      show "a - 1 < a" by simp
    31.7 -    fix X assume "\<And>n. X n < a" "incseq X" "X ----> a"
    31.8 -    with \<open>a \<le> b\<close> have "(\<lambda>n. emeasure ?F {X n<..b}) ----> emeasure ?F (\<Inter>n. {X n <..b})"
    31.9 +    fix X assume "\<And>n. X n < a" "incseq X" "X \<longlonglongrightarrow> a"
   31.10 +    with \<open>a \<le> b\<close> have "(\<lambda>n. emeasure ?F {X n<..b}) \<longlonglongrightarrow> emeasure ?F (\<Inter>n. {X n <..b})"
   31.11        apply (intro Lim_emeasure_decseq)
   31.12        apply (auto simp: decseq_def incseq_def emeasure_interval_measure_Ioc *)
   31.13        apply force
   31.14 @@ -336,13 +336,13 @@
   31.15      also have "(\<Inter>n. {X n <..b}) = {a..b}"
   31.16        using \<open>\<And>n. X n < a\<close>
   31.17        apply auto
   31.18 -      apply (rule LIMSEQ_le_const2[OF \<open>X ----> a\<close>])
   31.19 +      apply (rule LIMSEQ_le_const2[OF \<open>X \<longlonglongrightarrow> a\<close>])
   31.20        apply (auto intro: less_imp_le)
   31.21        apply (auto intro: less_le_trans)
   31.22        done
   31.23      also have "(\<lambda>n. emeasure ?F {X n<..b}) = (\<lambda>n. F b - F (X n))"
   31.24        using \<open>\<And>n. X n < a\<close> \<open>a \<le> b\<close> by (subst *) (auto intro: less_imp_le less_le_trans)
   31.25 -    finally show "(\<lambda>n. F b - F (X n)) ----> emeasure ?F {a..b}" .
   31.26 +    finally show "(\<lambda>n. F b - F (X n)) \<longlonglongrightarrow> emeasure ?F {a..b}" .
   31.27    qed
   31.28    show "((\<lambda>a. ereal (F b - F a)) ---> F b - F a) (at_left a)"
   31.29      using cont_F
   31.30 @@ -808,7 +808,7 @@
   31.31            by (intro setsum_mono2) auto
   31.32          from union(1) have *: "\<And>x i j. x \<in> F i \<Longrightarrow> x \<in> F j \<longleftrightarrow> j = i"
   31.33            by (auto simp add: disjoint_family_on_def)
   31.34 -        show "\<And>x. (\<lambda>k. ?f k x) ----> (if x \<in> UNION UNIV F \<inter> box a b then 1 else 0)"
   31.35 +        show "\<And>x. (\<lambda>k. ?f k x) \<longlonglongrightarrow> (if x \<in> UNION UNIV F \<inter> box a b then 1 else 0)"
   31.36            apply (auto simp: * setsum.If_cases Iio_Int_singleton)
   31.37            apply (rule_tac k="Suc xa" in LIMSEQ_offset)
   31.38            apply (simp add: tendsto_const)
   31.39 @@ -816,7 +816,7 @@
   31.40          have *: "emeasure lborel ((\<Union>x. F x) \<inter> box a b) \<le> emeasure lborel (box a b)"
   31.41            by (intro emeasure_mono) auto
   31.42  
   31.43 -        with union(1) show "(\<lambda>k. \<Sum>i<k. ?M (F i)) ----> ?M (\<Union>i. F i)"
   31.44 +        with union(1) show "(\<lambda>k. \<Sum>i<k. ?M (F i)) \<longlonglongrightarrow> ?M (\<Union>i. F i)"
   31.45            unfolding sums_def[symmetric] UN_extend_simps
   31.46            by (intro measure_UNION) (auto simp: disjoint_family_on_def emeasure_lborel_box_eq)
   31.47        qed
   31.48 @@ -836,12 +836,12 @@
   31.49      show "\<And>k x. ?f k x \<le> ?f (Suc k) x"
   31.50        by (auto simp: box_def)
   31.51      { fix x assume "x \<in> A"
   31.52 -      moreover have "(\<lambda>k. indicator (A \<inter> ?B k) x :: real) ----> indicator (\<Union>k::nat. A \<inter> ?B k) x"
   31.53 +      moreover have "(\<lambda>k. indicator (A \<inter> ?B k) x :: real) \<longlonglongrightarrow> indicator (\<Union>k::nat. A \<inter> ?B k) x"
   31.54          by (intro LIMSEQ_indicator_incseq) (auto simp: incseq_def box_def)
   31.55 -      ultimately show "(\<lambda>k. if x \<in> A \<inter> ?B k then 1 else 0::real) ----> 1"
   31.56 +      ultimately show "(\<lambda>k. if x \<in> A \<inter> ?B k then 1 else 0::real) \<longlonglongrightarrow> 1"
   31.57          by (simp add: indicator_def UN_box_eq_UNIV) }
   31.58  
   31.59 -    have "(\<lambda>n. emeasure lborel (A \<inter> ?B n)) ----> emeasure lborel (\<Union>n::nat. A \<inter> ?B n)"
   31.60 +    have "(\<lambda>n. emeasure lborel (A \<inter> ?B n)) \<longlonglongrightarrow> emeasure lborel (\<Union>n::nat. A \<inter> ?B n)"
   31.61        by (intro Lim_emeasure_incseq) (auto simp: incseq_def box_def)
   31.62      also have "(\<lambda>n. emeasure lborel (A \<inter> ?B n)) = (\<lambda>n. measure lborel (A \<inter> ?B n))"
   31.63      proof (intro ext emeasure_eq_ereal_measure)
   31.64 @@ -850,7 +850,7 @@
   31.65        then show "emeasure lborel (A \<inter> ?B n) \<noteq> \<infinity>"
   31.66          by auto
   31.67      qed
   31.68 -    finally show "(\<lambda>n. measure lborel (A \<inter> ?B n)) ----> measure lborel A"
   31.69 +    finally show "(\<lambda>n. measure lborel (A \<inter> ?B n)) \<longlonglongrightarrow> measure lborel A"
   31.70        using emeasure_eq_ereal_measure[of lborel A] finite
   31.71        by (simp add: UN_box_eq_UNIV)
   31.72    qed
   31.73 @@ -912,16 +912,16 @@
   31.74  
   31.75    have int_eq: "\<And>i. integral UNIV (U i) = p i" using U_int by (rule integral_unique)
   31.76  
   31.77 -  have *: "f integrable_on UNIV \<and> (\<lambda>k. integral UNIV (U k)) ----> integral UNIV f"
   31.78 +  have *: "f integrable_on UNIV \<and> (\<lambda>k. integral UNIV (U k)) \<longlonglongrightarrow> integral UNIV f"
   31.79    proof (rule monotone_convergence_increasing)
   31.80      show "\<forall>k. U k integrable_on UNIV" using U_int by auto
   31.81      show "\<forall>k. \<forall>x\<in>UNIV. U k x \<le> U (Suc k) x" using \<open>incseq U\<close> by (auto simp: incseq_def le_fun_def)
   31.82      then show "bounded {integral UNIV (U k) |k. True}"
   31.83        using bnd int_eq by (auto simp: bounded_real intro!: exI[of _ r])
   31.84 -    show "\<forall>x\<in>UNIV. (\<lambda>k. U k x) ----> f x"
   31.85 +    show "\<forall>x\<in>UNIV. (\<lambda>k. U k x) \<longlonglongrightarrow> f x"
   31.86        using seq by auto
   31.87    qed
   31.88 -  moreover have "(\<lambda>i. (\<integral>\<^sup>+x. U i x \<partial>lborel)) ----> (\<integral>\<^sup>+x. f x \<partial>lborel)"
   31.89 +  moreover have "(\<lambda>i. (\<integral>\<^sup>+x. U i x \<partial>lborel)) \<longlonglongrightarrow> (\<integral>\<^sup>+x. f x \<partial>lborel)"
   31.90      using seq U_le_f by (intro nn_integral_dominated_convergence[where w=f]) auto
   31.91    ultimately have "integral UNIV f = r"
   31.92      by (auto simp add: int_eq p seq intro: LIMSEQ_unique)
   31.93 @@ -1073,9 +1073,9 @@
   31.94                 simp del: times_ereal.simps)
   31.95      show "\<And>k. \<forall>x\<in>UNIV. norm (s k x) \<le> norm (2 * f x)"
   31.96        using lim by (auto simp add: abs_mult)
   31.97 -    show "\<forall>x\<in>UNIV. (\<lambda>k. s k x) ----> f x"
   31.98 +    show "\<forall>x\<in>UNIV. (\<lambda>k. s k x) \<longlonglongrightarrow> f x"
   31.99        using lim by auto
  31.100 -    show "(\<lambda>k. integral\<^sup>L lborel (s k)) ----> integral\<^sup>L lborel f"
  31.101 +    show "(\<lambda>k. integral\<^sup>L lborel (s k)) \<longlonglongrightarrow> integral\<^sup>L lborel f"
  31.102        using lim lim(1)[THEN borel_measurable_integrable]
  31.103        by (intro integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"]) auto
  31.104    qed
  31.105 @@ -1258,7 +1258,7 @@
  31.106      from reals_Archimedean2[of "x - a"] guess n ..
  31.107      then have "eventually (\<lambda>n. ?f n x = ?fR x) sequentially"
  31.108        by (auto intro!: eventually_sequentiallyI[where c=n] split: split_indicator)
  31.109 -    then show "(\<lambda>n. ?f n x) ----> ?fR x"
  31.110 +    then show "(\<lambda>n. ?f n x) \<longlonglongrightarrow> ?fR x"
  31.111        by (rule Lim_eventually)
  31.112    qed
  31.113    then have "integral\<^sup>N lborel ?fR = (\<integral>\<^sup>+ x. (SUP i::nat. ?f i x) \<partial>lborel)"
  31.114 @@ -1281,12 +1281,12 @@
  31.115          by (intro DERIV_nonneg_imp_nondecreasing[where f=F])
  31.116             (simp, metis add_increasing2 order_refl order_trans of_nat_0_le_iff)
  31.117      qed
  31.118 -    have "(\<lambda>x. F (a + real x)) ----> T"
  31.119 +    have "(\<lambda>x. F (a + real x)) \<longlonglongrightarrow> T"
  31.120        apply (rule filterlim_compose[OF lim filterlim_tendsto_add_at_top])
  31.121        apply (rule LIMSEQ_const_iff[THEN iffD2, OF refl])
  31.122        apply (rule filterlim_real_sequentially)
  31.123        done
  31.124 -    then show "(\<lambda>n. ereal (F (a + real n) - F a)) ----> ereal (T - F a)"
  31.125 +    then show "(\<lambda>n. ereal (F (a + real n) - F a)) \<longlonglongrightarrow> ereal (T - F a)"
  31.126        unfolding lim_ereal
  31.127        by (intro tendsto_diff) auto
  31.128    qed
    32.1 --- a/src/HOL/Probability/Measure_Space.thy	Tue Dec 29 22:41:22 2015 +0100
    32.2 +++ b/src/HOL/Probability/Measure_Space.thy	Tue Dec 29 23:04:53 2015 +0100
    32.3 @@ -63,7 +63,7 @@
    32.4  
    32.5  lemma LIMSEQ_binaryset:
    32.6    assumes f: "f {} = 0"
    32.7 -  shows  "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) ----> f A + f B"
    32.8 +  shows  "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) \<longlonglongrightarrow> f A + f B"
    32.9  proof -
   32.10    have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
   32.11      proof
   32.12 @@ -72,11 +72,11 @@
   32.13          by (induct n)  (auto simp add: binaryset_def f)
   32.14      qed
   32.15    moreover
   32.16 -  have "... ----> f A + f B" by (rule tendsto_const)
   32.17 +  have "... \<longlonglongrightarrow> f A + f B" by (rule tendsto_const)
   32.18    ultimately
   32.19 -  have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
   32.20 +  have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) \<longlonglongrightarrow> f A + f B"
   32.21      by metis
   32.22 -  hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) ----> f A + f B"
   32.23 +  hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) \<longlonglongrightarrow> f A + f B"
   32.24      by simp
   32.25    thus ?thesis by (rule LIMSEQ_offset [where k=2])
   32.26  qed
   32.27 @@ -281,7 +281,7 @@
   32.28  lemma (in ring_of_sets) countably_additive_iff_continuous_from_below:
   32.29    assumes f: "positive M f" "additive M f"
   32.30    shows "countably_additive M f \<longleftrightarrow>
   32.31 -    (\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i))"
   32.32 +    (\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i))"
   32.33    unfolding countably_additive_def
   32.34  proof safe
   32.35    assume count_sum: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> UNION UNIV A \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (UNION UNIV A)"
   32.36 @@ -290,20 +290,20 @@
   32.37    with count_sum[THEN spec, of "disjointed A"] A(3)
   32.38    have f_UN: "(\<Sum>i. f (disjointed A i)) = f (\<Union>i. A i)"
   32.39      by (auto simp: UN_disjointed_eq disjoint_family_disjointed)
   32.40 -  moreover have "(\<lambda>n. (\<Sum>i<n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))"
   32.41 +  moreover have "(\<lambda>n. (\<Sum>i<n. f (disjointed A i))) \<longlonglongrightarrow> (\<Sum>i. f (disjointed A i))"
   32.42      using f(1)[unfolded positive_def] dA
   32.43      by (auto intro!: summable_LIMSEQ summable_ereal_pos)
   32.44    from LIMSEQ_Suc[OF this]
   32.45 -  have "(\<lambda>n. (\<Sum>i\<le>n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))"
   32.46 +  have "(\<lambda>n. (\<Sum>i\<le>n. f (disjointed A i))) \<longlonglongrightarrow> (\<Sum>i. f (disjointed A i))"
   32.47      unfolding lessThan_Suc_atMost .
   32.48    moreover have "\<And>n. (\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"
   32.49      using disjointed_additive[OF f A(1,2)] .
   32.50 -  ultimately show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" by simp
   32.51 +  ultimately show "(\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i)" by simp
   32.52  next
   32.53 -  assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i)"
   32.54 +  assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i)"
   32.55    fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "disjoint_family A" "(\<Union>i. A i) \<in> M"
   32.56    have *: "(\<Union>n. (\<Union>i<n. A i)) = (\<Union>i. A i)" by auto
   32.57 -  have "(\<lambda>n. f (\<Union>i<n. A i)) ----> f (\<Union>i. A i)"
   32.58 +  have "(\<lambda>n. f (\<Union>i<n. A i)) \<longlonglongrightarrow> f (\<Union>i. A i)"
   32.59    proof (unfold *[symmetric], intro cont[rule_format])
   32.60      show "range (\<lambda>i. \<Union>i<i. A i) \<subseteq> M" "(\<Union>i. \<Union>i<i. A i) \<in> M"
   32.61        using A * by auto
   32.62 @@ -321,15 +321,15 @@
   32.63  
   32.64  lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous:
   32.65    assumes f: "positive M f" "additive M f"
   32.66 -  shows "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))
   32.67 -     \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0)"
   32.68 +  shows "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Inter>i. A i))
   32.69 +     \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0)"
   32.70  proof safe
   32.71 -  assume cont: "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))"
   32.72 +  assume cont: "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Inter>i. A i))"
   32.73    fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) = {}" "\<forall>i. f (A i) \<noteq> \<infinity>"
   32.74 -  with cont[THEN spec, of A] show "(\<lambda>i. f (A i)) ----> 0"
   32.75 +  with cont[THEN spec, of A] show "(\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
   32.76      using \<open>positive M f\<close>[unfolded positive_def] by auto
   32.77  next
   32.78 -  assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0"
   32.79 +  assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
   32.80    fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) \<in> M" "\<forall>i. f (A i) \<noteq> \<infinity>"
   32.81  
   32.82    have f_mono: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<subseteq> b \<Longrightarrow> f a \<le> f b"
   32.83 @@ -350,7 +350,7 @@
   32.84      then have "f (A i - (\<Inter>i. A i)) \<noteq> \<infinity>"
   32.85        using A by auto }
   32.86    note f_fin = this
   32.87 -  have "(\<lambda>i. f (A i - (\<Inter>i. A i))) ----> 0"
   32.88 +  have "(\<lambda>i. f (A i - (\<Inter>i. A i))) \<longlonglongrightarrow> 0"
   32.89    proof (intro cont[rule_format, OF _ decseq _ f_fin])
   32.90      show "range (\<lambda>i. A i - (\<Inter>i. A i)) \<subseteq> M" "(\<Inter>i. A i - (\<Inter>i. A i)) = {}"
   32.91        using A by auto
   32.92 @@ -372,14 +372,14 @@
   32.93    ultimately have "(INF n. f (A n)) = f (\<Inter>i. A i)"
   32.94      by simp
   32.95    with LIMSEQ_INF[OF decseq_fA]
   32.96 -  show "(\<lambda>i. f (A i)) ----> f (\<Inter>i. A i)" by simp
   32.97 +  show "(\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Inter>i. A i)" by simp
   32.98  qed
   32.99  
  32.100  lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below:
  32.101    assumes f: "positive M f" "additive M f" "\<forall>A\<in>M. f A \<noteq> \<infinity>"
  32.102 -  assumes cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<lambda>i. f (A i)) ----> 0"
  32.103 +  assumes cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
  32.104    assumes A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M"
  32.105 -  shows "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)"
  32.106 +  shows "(\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i)"
  32.107  proof -
  32.108    have "\<forall>A\<in>M. \<exists>x. f A = ereal x"
  32.109    proof
  32.110 @@ -387,7 +387,7 @@
  32.111        unfolding positive_def by (cases "f A") auto
  32.112    qed
  32.113    from bchoice[OF this] guess f' .. note f' = this[rule_format]
  32.114 -  from A have "(\<lambda>i. f ((\<Union>i. A i) - A i)) ----> 0"
  32.115 +  from A have "(\<lambda>i. f ((\<Union>i. A i) - A i)) \<longlonglongrightarrow> 0"
  32.116      by (intro cont[rule_format]) (auto simp: decseq_def incseq_def)
  32.117    moreover
  32.118    { fix i
  32.119 @@ -399,17 +399,17 @@
  32.120        using A by (subst (asm) (1 2 3) f') auto
  32.121      then have "f ((\<Union>i. A i) - A i) = ereal (f' (\<Union>i. A i) - f' (A i))"
  32.122        using A f' by auto }
  32.123 -  ultimately have "(\<lambda>i. f' (\<Union>i. A i) - f' (A i)) ----> 0"
  32.124 +  ultimately have "(\<lambda>i. f' (\<Union>i. A i) - f' (A i)) \<longlonglongrightarrow> 0"
  32.125      by (simp add: zero_ereal_def)
  32.126 -  then have "(\<lambda>i. f' (A i)) ----> f' (\<Union>i. A i)"
  32.127 +  then have "(\<lambda>i. f' (A i)) \<longlonglongrightarrow> f' (\<Union>i. A i)"
  32.128      by (rule Lim_transform2[OF tendsto_const])
  32.129 -  then show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)"
  32.130 +  then show "(\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i)"
  32.131      using A by (subst (1 2) f') auto
  32.132  qed
  32.133  
  32.134  lemma (in ring_of_sets) empty_continuous_imp_countably_additive:
  32.135    assumes f: "positive M f" "additive M f" and fin: "\<forall>A\<in>M. f A \<noteq> \<infinity>"
  32.136 -  assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0"
  32.137 +  assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
  32.138    shows "countably_additive M f"
  32.139    using countably_additive_iff_continuous_from_below[OF f]
  32.140    using empty_continuous_imp_continuous_from_below[OF f fin] cont
  32.141 @@ -503,7 +503,7 @@
  32.142  qed
  32.143  
  32.144  lemma Lim_emeasure_incseq:
  32.145 -  "range A \<subseteq> sets M \<Longrightarrow> incseq A \<Longrightarrow> (\<lambda>i. (emeasure M (A i))) ----> emeasure M (\<Union>i. A i)"
  32.146 +  "range A \<subseteq> sets M \<Longrightarrow> incseq A \<Longrightarrow> (\<lambda>i. (emeasure M (A i))) \<longlonglongrightarrow> emeasure M (\<Union>i. A i)"
  32.147    using emeasure_countably_additive
  32.148    by (auto simp add: sets.countably_additive_iff_continuous_from_below emeasure_positive
  32.149      emeasure_additive)
  32.150 @@ -604,7 +604,7 @@
  32.151  
  32.152  lemma Lim_emeasure_decseq:
  32.153    assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
  32.154 -  shows "(\<lambda>i. emeasure M (A i)) ----> emeasure M (\<Inter>i. A i)"
  32.155 +  shows "(\<lambda>i. emeasure M (A i)) \<longlonglongrightarrow> emeasure M (\<Inter>i. A i)"
  32.156    using LIMSEQ_INF[OF decseq_emeasure, OF A]
  32.157    using INF_emeasure_decseq[OF A fin] by simp
  32.158  
  32.159 @@ -1525,7 +1525,7 @@
  32.160  
  32.161  lemma Lim_measure_incseq:
  32.162    assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
  32.163 -  shows "(\<lambda>i. (measure M (A i))) ----> (measure M (\<Union>i. A i))"
  32.164 +  shows "(\<lambda>i. (measure M (A i))) \<longlonglongrightarrow> (measure M (\<Union>i. A i))"
  32.165  proof -
  32.166    have "ereal ((measure M (\<Union>i. A i))) = emeasure M (\<Union>i. A i)"
  32.167      using fin by (auto simp: emeasure_eq_ereal_measure)
  32.168 @@ -1537,7 +1537,7 @@
  32.169  
  32.170  lemma Lim_measure_decseq:
  32.171    assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
  32.172 -  shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)"
  32.173 +  shows "(\<lambda>n. measure M (A n)) \<longlonglongrightarrow> measure M (\<Inter>i. A i)"
  32.174  proof -
  32.175    have "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"
  32.176      using A by (auto intro!: emeasure_mono)
  32.177 @@ -1624,12 +1624,12 @@
  32.178  
  32.179  lemma (in finite_measure) finite_Lim_measure_incseq:
  32.180    assumes A: "range A \<subseteq> sets M" "incseq A"
  32.181 -  shows "(\<lambda>i. measure M (A i)) ----> measure M (\<Union>i. A i)"
  32.182 +  shows "(\<lambda>i. measure M (A i)) \<longlonglongrightarrow> measure M (\<Union>i. A i)"
  32.183    using Lim_measure_incseq[OF A] by simp
  32.184  
  32.185  lemma (in finite_measure) finite_Lim_measure_decseq:
  32.186    assumes A: "range A \<subseteq> sets M" "decseq A"
  32.187 -  shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)"
  32.188 +  shows "(\<lambda>n. measure M (A n)) \<longlonglongrightarrow> measure M (\<Inter>i. A i)"
  32.189    using Lim_measure_decseq[OF A] by simp
  32.190  
  32.191  lemma (in finite_measure) finite_measure_compl:
  32.192 @@ -1805,7 +1805,7 @@
  32.193      unfolding countably_additive_iff_continuous_from_below[OF positive additive]
  32.194    proof safe
  32.195      fix F :: "nat \<Rightarrow> 'a set" assume "incseq F"
  32.196 -    show "(\<lambda>i. ?M (F i)) ----> ?M (\<Union>i. F i)"
  32.197 +    show "(\<lambda>i. ?M (F i)) \<longlonglongrightarrow> ?M (\<Union>i. F i)"
  32.198      proof cases
  32.199        assume "\<exists>i. \<forall>j\<ge>i. F i = F j"
  32.200        then guess i .. note i = this
  32.201 @@ -1823,7 +1823,7 @@
  32.202  
  32.203        have "incseq (\<lambda>i. ?M (F i))"
  32.204          using \<open>incseq F\<close> unfolding incseq_def by (auto simp: card_mono dest: finite_subset)
  32.205 -      then have "(\<lambda>i. ?M (F i)) ----> (SUP n. ?M (F n))"
  32.206 +      then have "(\<lambda>i. ?M (F i)) \<longlonglongrightarrow> (SUP n. ?M (F n))"
  32.207          by (rule LIMSEQ_SUP)
  32.208  
  32.209        moreover have "(SUP n. ?M (F n)) = \<infinity>"
    33.1 --- a/src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy	Tue Dec 29 22:41:22 2015 +0100
    33.2 +++ b/src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy	Tue Dec 29 23:04:53 2015 +0100
    33.3 @@ -1450,10 +1450,10 @@
    33.4  
    33.5  lemma nn_integral_LIMSEQ:
    33.6    assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>n x. 0 \<le> f n x"
    33.7 -    and u: "\<And>x. (\<lambda>i. f i x) ----> u x"
    33.8 -  shows "(\<lambda>n. integral\<^sup>N M (f n)) ----> integral\<^sup>N M u"
    33.9 +    and u: "\<And>x. (\<lambda>i. f i x) \<longlonglongrightarrow> u x"
   33.10 +  shows "(\<lambda>n. integral\<^sup>N M (f n)) \<longlonglongrightarrow> integral\<^sup>N M u"
   33.11  proof -
   33.12 -  have "(\<lambda>n. integral\<^sup>N M (f n)) ----> (SUP n. integral\<^sup>N M (f n))"
   33.13 +  have "(\<lambda>n. integral\<^sup>N M (f n)) \<longlonglongrightarrow> (SUP n. integral\<^sup>N M (f n))"
   33.14      using f by (intro LIMSEQ_SUP[of "\<lambda>n. integral\<^sup>N M (f n)"] incseq_nn_integral)
   33.15    also have "(SUP n. integral\<^sup>N M (f n)) = integral\<^sup>N M (\<lambda>x. SUP n. f n x)"
   33.16      using f by (intro nn_integral_monotone_convergence_SUP[symmetric])
   33.17 @@ -1467,8 +1467,8 @@
   33.18         "\<And>i. u i \<in> borel_measurable M" "u' \<in> borel_measurable M" "w \<in> borel_measurable M"
   33.19      and bound: "\<And>j. AE x in M. 0 \<le> u j x" "\<And>j. AE x in M. u j x \<le> w x"
   33.20      and w: "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>"
   33.21 -    and u': "AE x in M. (\<lambda>i. u i x) ----> u' x"
   33.22 -  shows "(\<lambda>i. (\<integral>\<^sup>+x. u i x \<partial>M)) ----> (\<integral>\<^sup>+x. u' x \<partial>M)"
   33.23 +    and u': "AE x in M. (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
   33.24 +  shows "(\<lambda>i. (\<integral>\<^sup>+x. u i x \<partial>M)) \<longlonglongrightarrow> (\<integral>\<^sup>+x. u' x \<partial>M)"
   33.25  proof -
   33.26    have "limsup (\<lambda>n. integral\<^sup>N M (u n)) \<le> (\<integral>\<^sup>+ x. limsup (\<lambda>n. u n x) \<partial>M)"
   33.27      by (intro nn_integral_limsup[OF _ _ bound w]) auto
   33.28 @@ -1489,9 +1489,9 @@
   33.29    assumes "(\<integral>\<^sup>+ x. f 0 x \<partial>M) < \<infinity>" and nn: "\<And>x i. 0 \<le> f i x"
   33.30    shows "(\<integral>\<^sup>+ x. (INF i. f i x) \<partial>M) = (INF i. integral\<^sup>N M (f i))"
   33.31  proof (rule LIMSEQ_unique)
   33.32 -  show "(\<lambda>i. integral\<^sup>N M (f i)) ----> (INF i. integral\<^sup>N M (f i))"
   33.33 +  show "(\<lambda>i. integral\<^sup>N M (f i)) \<longlonglongrightarrow> (INF i. integral\<^sup>N M (f i))"
   33.34      using f by (intro LIMSEQ_INF) (auto intro!: nn_integral_mono simp: decseq_def le_fun_def)
   33.35 -  show "(\<lambda>i. integral\<^sup>N M (f i)) ----> \<integral>\<^sup>+ x. (INF i. f i x) \<partial>M"
   33.36 +  show "(\<lambda>i. integral\<^sup>N M (f i)) \<longlonglongrightarrow> \<integral>\<^sup>+ x. (INF i. f i x) \<partial>M"
   33.37    proof (rule nn_integral_dominated_convergence)
   33.38      show "(\<integral>\<^sup>+ x. f 0 x \<partial>M) < \<infinity>" "\<And>i. f i \<in> borel_measurable M" "f 0 \<in> borel_measurable M"
   33.39        by fact+
   33.40 @@ -1499,7 +1499,7 @@
   33.41        using nn by auto
   33.42      show "\<And>j. AE x in M. f j x \<le> f 0 x"
   33.43        using f by (auto simp: decseq_def le_fun_def)
   33.44 -    show "AE x in M. (\<lambda>i. f i x) ----> (INF i. f i x)"
   33.45 +    show "AE x in M. (\<lambda>i. f i x) \<longlonglongrightarrow> (INF i. f i x)"
   33.46        using f by (auto intro!: LIMSEQ_INF simp: decseq_def le_fun_def)
   33.47      show "(\<lambda>x. INF i. f i x) \<in> borel_measurable M"
   33.48        by auto
    34.1 --- a/src/HOL/Probability/Projective_Family.thy	Tue Dec 29 22:41:22 2015 +0100
    34.2 +++ b/src/HOL/Probability/Projective_Family.thy	Tue Dec 29 23:04:53 2015 +0100
    34.3 @@ -207,7 +207,7 @@
    34.4        with \<open>(\<Inter>i. A i) = {}\<close> * show False
    34.5          by (subst (asm) prod_emb_trans) (auto simp: A[abs_def])
    34.6      qed
    34.7 -    moreover have "(\<lambda>i. P (J i) (X i)) ----> (INF i. P (J i) (X i))"
    34.8 +    moreover have "(\<lambda>i. P (J i) (X i)) \<longlonglongrightarrow> (INF i. P (J i) (X i))"
    34.9      proof (intro LIMSEQ_INF antimonoI)
   34.10        fix x y :: nat assume "x \<le> y"
   34.11        have "P (J y \<union> J x) (emb (J y \<union> J x) (J y) (X y)) \<le> P (J y \<union> J x) (emb (J y \<union> J x) (J x) (X x))"
   34.12 @@ -217,7 +217,7 @@
   34.13        then show "P (J y) (X y) \<le> P (J x) (X x)"
   34.14          using * by (simp add: emeasure_P)
   34.15      qed
   34.16 -    ultimately show "(\<lambda>i. \<mu>G (A i)) ----> 0"
   34.17 +    ultimately show "(\<lambda>i. \<mu>G (A i)) \<longlonglongrightarrow> 0"
   34.18        by (auto simp: A[abs_def] mu_G_spec *)
   34.19    qed
   34.20    then obtain \<mu> where eq: "\<forall>s\<in>generator. \<mu> s = \<mu>G s"
    35.1 --- a/src/HOL/Probability/Projective_Limit.thy	Tue Dec 29 22:41:22 2015 +0100
    35.2 +++ b/src/HOL/Probability/Projective_Limit.thy	Tue Dec 29 23:04:53 2015 +0100
    35.3 @@ -53,12 +53,12 @@
    35.4    have "subseq (op + m)" by (simp add: subseq_def)
    35.5    have "\<forall>n. (f o (\<lambda>i. m + i)) n \<in> S" using assms by auto
    35.6    from seq_compactE[OF \<open>compact S\<close>[unfolded compact_eq_seq_compact_metric] this] guess l r .
    35.7 -  hence "l \<in> S" "subseq ((\<lambda>i. m + i) o r) \<and> (f \<circ> ((\<lambda>i. m + i) o r)) ----> l"
    35.8 +  hence "l \<in> S" "subseq ((\<lambda>i. m + i) o r) \<and> (f \<circ> ((\<lambda>i. m + i) o r)) \<longlonglongrightarrow> l"
    35.9      using subseq_o[OF \<open>subseq (op + m)\<close> \<open>subseq r\<close>] by (auto simp: o_def)
   35.10 -  thus "\<exists>l r. l \<in> S \<and> subseq r \<and> (f \<circ> r) ----> l" by blast
   35.11 +  thus "\<exists>l r. l \<in> S \<and> subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l" by blast
   35.12  qed
   35.13  
   35.14 -sublocale finmap_seqs_into_compact \<subseteq> subseqs "\<lambda>n s. (\<exists>l. (\<lambda>i. ((f o s) i)\<^sub>F n) ----> l)"
   35.15 +sublocale finmap_seqs_into_compact \<subseteq> subseqs "\<lambda>n s. (\<exists>l. (\<lambda>i. ((f o s) i)\<^sub>F n) \<longlonglongrightarrow> l)"
   35.16  proof
   35.17    fix n s
   35.18    assume "subseq s"
   35.19 @@ -72,24 +72,24 @@
   35.20        by auto
   35.21    qed
   35.22    from compactE'[OF compact_projset this] guess ls rs .
   35.23 -  thus "\<exists>r'. subseq r' \<and> (\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r')) i)\<^sub>F n) ----> l)" by (auto simp: o_def)
   35.24 +  thus "\<exists>r'. subseq r' \<and> (\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r')) i)\<^sub>F n) \<longlonglongrightarrow> l)" by (auto simp: o_def)
   35.25  qed
   35.26  
   35.27 -lemma (in finmap_seqs_into_compact) diagonal_tendsto: "\<exists>l. (\<lambda>i. (f (diagseq i))\<^sub>F n) ----> l"
   35.28 +lemma (in finmap_seqs_into_compact) diagonal_tendsto: "\<exists>l. (\<lambda>i. (f (diagseq i))\<^sub>F n) \<longlonglongrightarrow> l"
   35.29  proof -
   35.30 -  obtain l where "(\<lambda>i. ((f o (diagseq o op + (Suc n))) i)\<^sub>F n) ----> l"
   35.31 +  obtain l where "(\<lambda>i. ((f o (diagseq o op + (Suc n))) i)\<^sub>F n) \<longlonglongrightarrow> l"
   35.32    proof (atomize_elim, rule diagseq_holds)
   35.33      fix r s n
   35.34      assume "subseq r"
   35.35 -    assume "\<exists>l. (\<lambda>i. ((f \<circ> s) i)\<^sub>F n) ----> l"
   35.36 -    then obtain l where "((\<lambda>i. (f i)\<^sub>F n) o s) ----> l"
   35.37 +    assume "\<exists>l. (\<lambda>i. ((f \<circ> s) i)\<^sub>F n) \<longlonglongrightarrow> l"
   35.38 +    then obtain l where "((\<lambda>i. (f i)\<^sub>F n) o s) \<longlonglongrightarrow> l"
   35.39        by (auto simp: o_def)
   35.40 -    hence "((\<lambda>i. (f i)\<^sub>F n) o s o r) ----> l" using \<open>subseq r\<close>
   35.41 +    hence "((\<lambda>i. (f i)\<^sub>F n) o s o r) \<longlonglongrightarrow> l" using \<open>subseq r\<close>
   35.42        by (rule LIMSEQ_subseq_LIMSEQ)
   35.43 -    thus "\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r)) i)\<^sub>F n) ----> l" by (auto simp add: o_def)
   35.44 +    thus "\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r)) i)\<^sub>F n) \<longlonglongrightarrow> l" by (auto simp add: o_def)
   35.45    qed
   35.46 -  hence "(\<lambda>i. ((f (diagseq (i + Suc n))))\<^sub>F n) ----> l" by (simp add: ac_simps)
   35.47 -  hence "(\<lambda>i. (f (diagseq i))\<^sub>F n) ----> l" by (rule LIMSEQ_offset)
   35.48 +  hence "(\<lambda>i. ((f (diagseq (i + Suc n))))\<^sub>F n) \<longlonglongrightarrow> l" by (simp add: ac_simps)
   35.49 +  hence "(\<lambda>i. (f (diagseq i))\<^sub>F n) \<longlonglongrightarrow> l" by (rule LIMSEQ_offset)
   35.50    thus ?thesis ..
   35.51  qed
   35.52  
   35.53 @@ -362,10 +362,10 @@
   35.54        using \<open>j \<in> J (Suc n)\<close> \<open>j \<in> J (Suc m)\<close>
   35.55        unfolding j by (subst proj_fm, auto)+
   35.56    qed
   35.57 -  have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) ----> z"
   35.58 +  have "\<forall>t. \<exists>z. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) \<longlonglongrightarrow> z"
   35.59      using diagonal_tendsto ..
   35.60    then obtain z where z:
   35.61 -    "\<And>t. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) ----> z t"
   35.62 +    "\<And>t. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F t) \<longlonglongrightarrow> z t"
   35.63      unfolding choice_iff by blast
   35.64    {
   35.65      fix n :: nat assume "n \<ge> 1"
   35.66 @@ -377,7 +377,7 @@
   35.67        assume t: "t \<in> domain (finmap_of (Utn ` J n) z)"
   35.68        hence "t \<in> Utn ` J n" by simp
   35.69        then obtain j where j: "t = Utn j" "j \<in> J n" by auto
   35.70 -      have "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) ----> z t"
   35.71 +      have "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) \<longlonglongrightarrow> z t"
   35.72          apply (subst (2) tendsto_iff, subst eventually_sequentially)
   35.73        proof safe
   35.74          fix e :: real assume "0 < e"
   35.75 @@ -407,12 +407,12 @@
   35.76            finally show "dist ((fm n (y (Suc (diagseq na))))\<^sub>F t) (z t) < e" .
   35.77          qed
   35.78        qed
   35.79 -      hence "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) ----> (finmap_of (Utn ` J n) z)\<^sub>F t"
   35.80 +      hence "(\<lambda>i. (fm n (y (Suc (diagseq i))))\<^sub>F t) \<longlonglongrightarrow> (finmap_of (Utn ` J n) z)\<^sub>F t"
   35.81          by (simp add: tendsto_intros)
   35.82      } ultimately
   35.83 -    have "(\<lambda>i. fm n (y (Suc (diagseq i)))) ----> finmap_of (Utn ` J n) z"
   35.84 +    have "(\<lambda>i. fm n (y (Suc (diagseq i)))) \<longlonglongrightarrow> finmap_of (Utn ` J n) z"
   35.85        by (rule tendsto_finmap)
   35.86 -    hence "((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) ----> finmap_of (Utn ` J n) z"
   35.87 +    hence "((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) \<longlonglongrightarrow> finmap_of (Utn ` J n) z"
   35.88        by (intro lim_subseq) (simp add: subseq_def)
   35.89      moreover
   35.90      have "(\<forall>i. ((\<lambda>i. fm n (y (Suc (diagseq i)))) o (\<lambda>i. i + n)) i \<in> K' n)"
    36.1 --- a/src/HOL/Probability/Radon_Nikodym.thy	Tue Dec 29 22:41:22 2015 +0100
    36.2 +++ b/src/HOL/Probability/Radon_Nikodym.thy	Tue Dec 29 23:04:53 2015 +0100
    36.3 @@ -216,7 +216,7 @@
    36.4      have A: "incseq A" by (auto intro!: incseq_SucI)
    36.5      from finite_Lim_measure_incseq[OF _ A] \<open>range A \<subseteq> sets M\<close>
    36.6        M'.finite_Lim_measure_incseq[OF _ A]
    36.7 -    have convergent: "(\<lambda>i. ?d (A i)) ----> ?d (\<Union>i. A i)"
    36.8 +    have convergent: "(\<lambda>i. ?d (A i)) \<longlonglongrightarrow> ?d (\<Union>i. A i)"
    36.9        by (auto intro!: tendsto_diff simp: sets_eq)
   36.10      obtain n :: nat where "- ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto
   36.11      moreover from order_trans[OF decseq_le[OF decseq convergent] dA_less]
   36.12 @@ -261,7 +261,7 @@
   36.13      show "(\<Inter>i. A i) \<in> sets M" using \<open>\<And>n. A n \<in> sets M\<close> by auto
   36.14      have "decseq A" using A by (auto intro!: decseq_SucI)
   36.15      from A(1) finite_Lim_measure_decseq[OF _ this] N.finite_Lim_measure_decseq[OF _ this]
   36.16 -    have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (auto intro!: tendsto_diff simp: sets_eq)
   36.17 +    have "(\<lambda>i. ?d (A i)) \<longlonglongrightarrow> ?d (\<Inter>i. A i)" by (auto intro!: tendsto_diff simp: sets_eq)
   36.18      thus "?d (space M) \<le> ?d (\<Inter>i. A i)" using mono_dA[THEN monoD, of 0 _]
   36.19        by (rule_tac LIMSEQ_le_const) auto
   36.20    next
    37.1 --- a/src/HOL/Probability/Regularity.thy	Tue Dec 29 22:41:22 2015 +0100
    37.2 +++ b/src/HOL/Probability/Regularity.thy	Tue Dec 29 23:04:53 2015 +0100
    37.3 @@ -131,7 +131,7 @@
    37.4      have x: "space M = (\<Union>x\<in>X. cball x r)"
    37.5        by (auto simp add: sU) (metis dist_commute order_less_imp_le)
    37.6      let ?U = "\<Union>k. (\<Union>n\<in>{0..k}. cball (from_nat_into X n) r)"
    37.7 -    have "(\<lambda>k. emeasure M (\<Union>n\<in>{0..k}. cball (from_nat_into X n) r)) ----> M ?U"
    37.8 +    have "(\<lambda>k. emeasure M (\<Union>n\<in>{0..k}. cball (from_nat_into X n) r)) \<longlonglongrightarrow> M ?U"
    37.9        by (rule Lim_emeasure_incseq)
   37.10          (auto intro!: borel_closed bexI simp: closed_cball incseq_def Us sb)
   37.11      also have "?U = space M"
   37.12 @@ -140,13 +140,13 @@
   37.13        show "x \<in> ?U"
   37.14          using X(1) d by (auto intro!: exI[where x="to_nat_on X d"] simp: dist_commute Bex_def)
   37.15      qed (simp add: sU)
   37.16 -    finally have "(\<lambda>k. M (\<Union>n\<in>{0..k}. cball (from_nat_into X n) r)) ----> M (space M)" .
   37.17 +    finally have "(\<lambda>k. M (\<Union>n\<in>{0..k}. cball (from_nat_into X n) r)) \<longlonglongrightarrow> M (space M)" .
   37.18    } note M_space = this
   37.19    {
   37.20      fix e ::real and n :: nat assume "e > 0" "n > 0"
   37.21      hence "1/n > 0" "e * 2 powr - n > 0" by (auto)
   37.22      from M_space[OF \<open>1/n>0\<close>]
   37.23 -    have "(\<lambda>k. measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n))) ----> measure M (space M)"
   37.24 +    have "(\<lambda>k. measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n))) \<longlonglongrightarrow> measure M (space M)"
   37.25        unfolding emeasure_eq_measure by simp
   37.26      from metric_LIMSEQ_D[OF this \<open>0 < e * 2 powr -n\<close>]
   37.27      obtain k where "dist (measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1/real n))) (measure M (space M)) <
   37.28 @@ -351,9 +351,9 @@
   37.29      case (union D)
   37.30      then have "range D \<subseteq> sets M" by (auto simp: sb borel_eq_closed)
   37.31      with union have M[symmetric]: "(\<Sum>i. M (D i)) = M (\<Union>i. D i)" by (intro suminf_emeasure)
   37.32 -    also have "(\<lambda>n. \<Sum>i<n. M (D i)) ----> (\<Sum>i. M (D i))"
   37.33 +    also have "(\<lambda>n. \<Sum>i<n. M (D i)) \<longlonglongrightarrow> (\<Sum>i. M (D i))"
   37.34        by (intro summable_LIMSEQ summable_ereal_pos emeasure_nonneg)
   37.35 -    finally have measure_LIMSEQ: "(\<lambda>n. \<Sum>i<n. measure M (D i)) ----> measure M (\<Union>i. D i)"
   37.36 +    finally have measure_LIMSEQ: "(\<lambda>n. \<Sum>i<n. measure M (D i)) \<longlonglongrightarrow> measure M (\<Union>i. D i)"
   37.37        by (simp add: emeasure_eq_measure)
   37.38      have "(\<Union>i. D i) \<in> sets M" using \<open>range D \<subseteq> sets M\<close> by auto
   37.39  
    38.1 --- a/src/HOL/Probability/Set_Integral.thy	Tue Dec 29 22:41:22 2015 +0100
    38.2 +++ b/src/HOL/Probability/Set_Integral.thy	Tue Dec 29 23:04:53 2015 +0100
    38.3 @@ -327,7 +327,7 @@
    38.4    assumes "\<And>i. A i \<in> sets M" "mono A"
    38.5    assumes nneg: "\<And>x i. x \<in> A i \<Longrightarrow> 0 \<le> f x"
    38.6    and intgbl: "\<And>i::nat. set_integrable M (A i) f"
    38.7 -  and lim: "(\<lambda>i::nat. LINT x:A i|M. f x) ----> l"
    38.8 +  and lim: "(\<lambda>i::nat. LINT x:A i|M. f x) \<longlonglongrightarrow> l"
    38.9    shows "set_integrable M (\<Union>i. A i) f"
   38.10    apply (rule integrable_monotone_convergence[where f = "\<lambda>i::nat. \<lambda>x. indicator (A i) x *\<^sub>R f x" and x = l])
   38.11    apply (rule intgbl)
   38.12 @@ -336,7 +336,7 @@
   38.13    using \<open>mono A\<close> apply (auto simp: mono_def nneg split: split_indicator) []
   38.14  proof (rule AE_I2)
   38.15    { fix x assume "x \<in> space M"
   38.16 -    show "(\<lambda>i. indicator (A i) x *\<^sub>R f x) ----> indicator (\<Union>i. A i) x *\<^sub>R f x"
   38.17 +    show "(\<lambda>i. indicator (A i) x *\<^sub>R f x) \<longlonglongrightarrow> indicator (\<Union>i. A i) x *\<^sub>R f x"
   38.18      proof cases
   38.19        assume "\<exists>i. x \<in> A i"
   38.20        then guess i ..
   38.21 @@ -409,7 +409,7 @@
   38.22    fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
   38.23    assumes [measurable]: "\<And>i. A i \<in> sets M" and A: "incseq A"
   38.24    and intgbl: "set_integrable M (\<Union>i. A i) f"
   38.25 -  shows "(\<lambda>i. LINT x:(A i)|M. f x) ----> LINT x:(\<Union>i. A i)|M. f x"
   38.26 +  shows "(\<lambda>i. LINT x:(A i)|M. f x) \<longlonglongrightarrow> LINT x:(\<Union>i. A i)|M. f x"
   38.27  proof (intro integral_dominated_convergence[where w="\<lambda>x. indicator (\<Union>i. A i) x *\<^sub>R norm (f x)"])
   38.28    have int_A: "\<And>i. set_integrable M (A i) f"
   38.29      using intgbl by (rule set_integrable_subset) auto
   38.30 @@ -418,10 +418,10 @@
   38.31      using intgbl integrable_norm[OF intgbl] by auto
   38.32  
   38.33    { fix x i assume "x \<in> A i"
   38.34 -    with A have "(\<lambda>xa. indicator (A xa) x::real) ----> 1 \<longleftrightarrow> (\<lambda>xa. 1::real) ----> 1"
   38.35 +    with A have "(\<lambda>xa. indicator (A xa) x::real) \<longlonglongrightarrow> 1 \<longleftrightarrow> (\<lambda>xa. 1::real) \<longlonglongrightarrow> 1"
   38.36        by (intro filterlim_cong refl)
   38.37           (fastforce simp: eventually_sequentially incseq_def subset_eq intro!: exI[of _ i]) }
   38.38 -  then show "AE x in M. (\<lambda>i. indicator (A i) x *\<^sub>R f x) ----> indicator (\<Union>i. A i) x *\<^sub>R f x"
   38.39 +  then show "AE x in M. (\<lambda>i. indicator (A i) x *\<^sub>R f x) \<longlonglongrightarrow> indicator (\<Union>i. A i) x *\<^sub>R f x"
   38.40      by (intro AE_I2 tendsto_intros) (auto split: split_indicator)
   38.41  qed (auto split: split_indicator)
   38.42  
   38.43 @@ -430,7 +430,7 @@
   38.44    fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
   38.45    assumes [measurable]: "\<And>i. A i \<in> sets M" and A: "decseq A"
   38.46    and int0: "set_integrable M (A 0) f"
   38.47 -  shows "(\<lambda>i::nat. LINT x:(A i)|M. f x) ----> LINT x:(\<Inter>i. A i)|M. f x"
   38.48 +  shows "(\<lambda>i::nat. LINT x:(A i)|M. f x) \<longlonglongrightarrow> LINT x:(\<Inter>i. A i)|M. f x"
   38.49  proof (rule integral_dominated_convergence)
   38.50    have int_A: "\<And>i. set_integrable M (A i) f"
   38.51      using int0 by (rule set_integrable_subset) (insert A, auto simp: decseq_def)
   38.52 @@ -443,10 +443,10 @@
   38.53    show "\<And>i. AE x in M. norm (indicator (A i) x *\<^sub>R f x) \<le> indicator (A 0) x *\<^sub>R norm (f x)"
   38.54      using A by (auto split: split_indicator simp: decseq_def)
   38.55    { fix x i assume "x \<in> space M" "x \<notin> A i"
   38.56 -    with A have "(\<lambda>i. indicator (A i) x::real) ----> 0 \<longleftrightarrow> (\<lambda>i. 0::real) ----> 0"
   38.57 +    with A have "(\<lambda>i. indicator (A i) x::real) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<lambda>i. 0::real) \<longlonglongrightarrow> 0"
   38.58        by (intro filterlim_cong refl)
   38.59           (auto split: split_indicator simp: eventually_sequentially decseq_def intro!: exI[of _ i]) }
   38.60 -  then show "AE x in M. (\<lambda>i. indicator (A i) x *\<^sub>R f x) ----> indicator (\<Inter>i. A i) x *\<^sub>R f x"
   38.61 +  then show "AE x in M. (\<lambda>i. indicator (A i) x *\<^sub>R f x) \<longlonglongrightarrow> indicator (\<Inter>i. A i) x *\<^sub>R f x"
   38.62      by (intro AE_I2 tendsto_intros) (auto split: split_indicator)
   38.63  qed
   38.64  
    39.1 --- a/src/HOL/Real_Vector_Spaces.thy	Tue Dec 29 22:41:22 2015 +0100
    39.2 +++ b/src/HOL/Real_Vector_Spaces.thy	Tue Dec 29 23:04:53 2015 +0100
    39.3 @@ -1708,20 +1708,20 @@
    39.4  
    39.5  subsubsection \<open>Limits of Sequences\<close>
    39.6  
    39.7 -lemma lim_sequentially: "X ----> (L::'a::metric_space) \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
    39.8 +lemma lim_sequentially: "X \<longlonglongrightarrow> (L::'a::metric_space) \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
    39.9    unfolding tendsto_iff eventually_sequentially ..
   39.10  
   39.11  lemmas LIMSEQ_def = lim_sequentially  (*legacy binding*)
   39.12  
   39.13 -lemma LIMSEQ_iff_nz: "X ----> (L::'a::metric_space) = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
   39.14 +lemma LIMSEQ_iff_nz: "X \<longlonglongrightarrow> (L::'a::metric_space) = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
   39.15    unfolding lim_sequentially by (metis Suc_leD zero_less_Suc)
   39.16  
   39.17  lemma metric_LIMSEQ_I:
   39.18 -  "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> (L::'a::metric_space)"
   39.19 +  "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X \<longlonglongrightarrow> (L::'a::metric_space)"
   39.20  by (simp add: lim_sequentially)
   39.21  
   39.22  lemma metric_LIMSEQ_D:
   39.23 -  "\<lbrakk>X ----> (L::'a::metric_space); 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
   39.24 +  "\<lbrakk>X \<longlonglongrightarrow> (L::'a::metric_space); 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
   39.25  by (simp add: lim_sequentially)
   39.26  
   39.27  
   39.28 @@ -1840,7 +1840,7 @@
   39.29  done
   39.30  
   39.31  theorem LIMSEQ_imp_Cauchy:
   39.32 -  assumes X: "X ----> a" shows "Cauchy X"
   39.33 +  assumes X: "X \<longlonglongrightarrow> a" shows "Cauchy X"
   39.34  proof (rule metric_CauchyI)
   39.35    fix e::real assume "0 < e"
   39.36    hence "0 < e/2" by simp
   39.37 @@ -1890,7 +1890,7 @@
   39.38    assumes inc: "\<And>n. f n \<le> f (Suc n)"
   39.39        and bdd: "\<And>n. f n \<le> l"
   39.40        and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
   39.41 -  shows "f ----> l"
   39.42 +  shows "f \<longlonglongrightarrow> l"
   39.43  proof (rule increasing_tendsto)
   39.44    fix x assume "x < l"
   39.45    with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x"
   39.46 @@ -1937,7 +1937,7 @@
   39.47      thus "\<And>s. s \<in> S \<Longrightarrow>  s \<le> X N + 1"
   39.48        by (rule bound_isUb)
   39.49    qed
   39.50 -  have "X ----> Sup S"
   39.51 +  have "X \<longlonglongrightarrow> Sup S"
   39.52    proof (rule metric_LIMSEQ_I)
   39.53    fix r::real assume "0 < r"
   39.54    hence r: "0 < r/2" by simp
   39.55 @@ -1976,7 +1976,7 @@
   39.56  
   39.57  lemma tendsto_at_topI_sequentially:
   39.58    fixes f :: "real \<Rightarrow> 'b::first_countable_topology"
   39.59 -  assumes *: "\<And>X. filterlim X at_top sequentially \<Longrightarrow> (\<lambda>n. f (X n)) ----> y"
   39.60 +  assumes *: "\<And>X. filterlim X at_top sequentially \<Longrightarrow> (\<lambda>n. f (X n)) \<longlonglongrightarrow> y"
   39.61    shows "(f ---> y) at_top"
   39.62  proof -
   39.63    from nhds_countable[of y] guess A . note A = this
   39.64 @@ -2008,7 +2008,7 @@
   39.65  lemma tendsto_at_topI_sequentially_real:
   39.66    fixes f :: "real \<Rightarrow> real"
   39.67    assumes mono: "mono f"
   39.68 -  assumes limseq: "(\<lambda>n. f (real n)) ----> y"
   39.69 +  assumes limseq: "(\<lambda>n. f (real n)) \<longlonglongrightarrow> y"
   39.70    shows "(f ---> y) at_top"
   39.71  proof (rule tendstoI)
   39.72    fix e :: real assume "0 < e"
    40.1 --- a/src/HOL/Series.thy	Tue Dec 29 22:41:22 2015 +0100
    40.2 +++ b/src/HOL/Series.thy	Tue Dec 29 23:04:53 2015 +0100
    40.3 @@ -19,7 +19,7 @@
    40.4    sums :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool"
    40.5    (infixr "sums" 80)
    40.6  where
    40.7 -  "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i<n. f i) ----> s"
    40.8 +  "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> s"
    40.9  
   40.10  definition summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool" where
   40.11     "summable f \<longleftrightarrow> (\<exists>s. f sums s)"
   40.12 @@ -152,7 +152,7 @@
   40.13    by (simp add: summable_def sums_def suminf_def)
   40.14       (metis convergent_LIMSEQ_iff convergent_def lim_def)
   40.15  
   40.16 -lemma summable_LIMSEQ: "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i) ----> suminf f"
   40.17 +lemma summable_LIMSEQ: "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> suminf f"
   40.18    by (rule summable_sums [unfolded sums_def])
   40.19  
   40.20  lemma sums_unique: "f sums s \<Longrightarrow> s = suminf f"
   40.21 @@ -223,7 +223,7 @@
   40.22  lemma suminf_eq_zero_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> suminf f = 0 \<longleftrightarrow> (\<forall>n. f n = 0)"
   40.23  proof
   40.24    assume "summable f" "suminf f = 0" and pos: "\<forall>n. 0 \<le> f n"
   40.25 -  then have f: "(\<lambda>n. \<Sum>i<n. f i) ----> 0"
   40.26 +  then have f: "(\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> 0"
   40.27      using summable_LIMSEQ[of f] by simp
   40.28    then have "\<And>i. (\<Sum>n\<in>{i}. f n) \<le> 0"
   40.29    proof (rule LIMSEQ_le_const)
   40.30 @@ -268,13 +268,13 @@
   40.31    fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
   40.32    shows "(\<lambda>n. f (Suc n)) sums s \<longleftrightarrow> f sums (s + f 0)"
   40.33  proof -
   40.34 -  have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) ----> s + f 0"
   40.35 +  have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) \<longlonglongrightarrow> s + f 0"
   40.36      by (subst LIMSEQ_Suc_iff) (simp add: sums_def)
   40.37 -  also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) ----> s + f 0"
   40.38 +  also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
   40.39      by (simp add: ac_simps setsum.reindex image_iff lessThan_Suc_eq_insert_0)
   40.40    also have "\<dots> \<longleftrightarrow> (\<lambda>n. f (Suc n)) sums s"
   40.41    proof
   40.42 -    assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) ----> s + f 0"
   40.43 +    assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
   40.44      with tendsto_add[OF this tendsto_const, of "- f 0"]
   40.45      show "(\<lambda>i. f (Suc i)) sums s"
   40.46        by (simp add: sums_def)
   40.47 @@ -361,7 +361,7 @@
   40.48      by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF \<open>summable f\<close>])
   40.49  qed
   40.50  
   40.51 -lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f ----> 0"
   40.52 +lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f \<longlonglongrightarrow> 0"
   40.53    apply (drule summable_iff_convergent [THEN iffD1])
   40.54    apply (drule convergent_Cauchy)
   40.55    apply (simp only: Cauchy_iff LIMSEQ_iff, safe)
   40.56 @@ -503,11 +503,11 @@
   40.57    assume less_1: "norm c < 1"
   40.58    hence neq_1: "c \<noteq> 1" by auto
   40.59    hence neq_0: "c - 1 \<noteq> 0" by simp
   40.60 -  from less_1 have lim_0: "(\<lambda>n. c^n) ----> 0"
   40.61 +  from less_1 have lim_0: "(\<lambda>n. c^n) \<longlonglongrightarrow> 0"
   40.62      by (rule LIMSEQ_power_zero)
   40.63 -  hence "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) ----> 0 / (c - 1) - 1 / (c - 1)"
   40.64 +  hence "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) \<longlonglongrightarrow> 0 / (c - 1) - 1 / (c - 1)"
   40.65      using neq_0 by (intro tendsto_intros)
   40.66 -  hence "(\<lambda>n. (c ^ n - 1) / (c - 1)) ----> 1 / (1 - c)"
   40.67 +  hence "(\<lambda>n. (c ^ n - 1) / (c - 1)) \<longlonglongrightarrow> 1 / (1 - c)"
   40.68      by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
   40.69    thus "(\<lambda>n. c ^ n) sums (1 / (1 - c))"
   40.70      by (simp add: sums_def geometric_sum neq_1)
   40.71 @@ -522,7 +522,7 @@
   40.72  lemma summable_geometric_iff: "summable (\<lambda>n. c ^ n) \<longleftrightarrow> norm c < 1"
   40.73  proof
   40.74    assume "summable (\<lambda>n. c ^ n :: 'a :: real_normed_field)"
   40.75 -  hence "(\<lambda>n. norm c ^ n) ----> 0"
   40.76 +  hence "(\<lambda>n. norm c ^ n) \<longlonglongrightarrow> 0"
   40.77      by (simp add: norm_power [symmetric] tendsto_norm_zero_iff summable_LIMSEQ_zero)
   40.78    from order_tendstoD(2)[OF this zero_less_one] obtain n where "norm c ^ n < 1"
   40.79      by (auto simp: eventually_at_top_linorder)
   40.80 @@ -545,28 +545,28 @@
   40.81  subsection \<open>Telescoping\<close>
   40.82  
   40.83  lemma telescope_sums:
   40.84 -  assumes "f ----> (c :: 'a :: real_normed_vector)"
   40.85 +  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
   40.86    shows   "(\<lambda>n. f (Suc n) - f n) sums (c - f 0)"
   40.87    unfolding sums_def
   40.88  proof (subst LIMSEQ_Suc_iff [symmetric])
   40.89    have "(\<lambda>n. \<Sum>k<Suc n. f (Suc k) - f k) = (\<lambda>n. f (Suc n) - f 0)"
   40.90      by (simp add: lessThan_Suc_atMost atLeast0AtMost [symmetric] setsum_Suc_diff)
   40.91 -  also have "\<dots> ----> c - f 0" by (intro tendsto_diff LIMSEQ_Suc[OF assms] tendsto_const)
   40.92 -  finally show "(\<lambda>n. \<Sum>n<Suc n. f (Suc n) - f n) ----> c - f 0" .
   40.93 +  also have "\<dots> \<longlonglongrightarrow> c - f 0" by (intro tendsto_diff LIMSEQ_Suc[OF assms] tendsto_const)
   40.94 +  finally show "(\<lambda>n. \<Sum>n<Suc n. f (Suc n) - f n) \<longlonglongrightarrow> c - f 0" .
   40.95  qed
   40.96  
   40.97  lemma telescope_sums':
   40.98 -  assumes "f ----> (c :: 'a :: real_normed_vector)"
   40.99 +  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
  40.100    shows   "(\<lambda>n. f n - f (Suc n)) sums (f 0 - c)"
  40.101    using sums_minus[OF telescope_sums[OF assms]] by (simp add: algebra_simps)
  40.102  
  40.103  lemma telescope_summable:
  40.104 -  assumes "f ----> (c :: 'a :: real_normed_vector)"
  40.105 +  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
  40.106    shows   "summable (\<lambda>n. f (Suc n) - f n)"
  40.107    using telescope_sums[OF assms] by (simp add: sums_iff)
  40.108  
  40.109  lemma telescope_summable':
  40.110 -  assumes "f ----> (c :: 'a :: real_normed_vector)"
  40.111 +  assumes "f \<longlonglongrightarrow> (c :: 'a :: real_normed_vector)"
  40.112    shows   "summable (\<lambda>n. f n - f (Suc n))"
  40.113    using summable_minus[OF telescope_summable[OF assms]] by (simp add: algebra_simps)
  40.114  
  40.115 @@ -733,14 +733,14 @@
  40.116      unfolding real_norm_def
  40.117      by (simp only: abs_of_nonneg setsum_nonneg [rule_format])
  40.118  
  40.119 -  have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
  40.120 +  have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
  40.121      by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b])
  40.122 -  hence 1: "(\<lambda>n. setsum ?g (?S1 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
  40.123 +  hence 1: "(\<lambda>n. setsum ?g (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
  40.124      by (simp only: setsum_product setsum.Sigma [rule_format] finite_lessThan)
  40.125  
  40.126 -  have "(\<lambda>n. (\<Sum>k<n. norm (a k)) * (\<Sum>k<n. norm (b k))) ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
  40.127 +  have "(\<lambda>n. (\<Sum>k<n. norm (a k)) * (\<Sum>k<n. norm (b k))) \<longlonglongrightarrow> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
  40.128      using a b by (intro tendsto_mult summable_LIMSEQ)
  40.129 -  hence "(\<lambda>n. setsum ?f (?S1 n)) ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
  40.130 +  hence "(\<lambda>n. setsum ?f (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
  40.131      by (simp only: setsum_product setsum.Sigma [rule_format] finite_lessThan)
  40.132    hence "convergent (\<lambda>n. setsum ?f (?S1 n))"
  40.133      by (rule convergentI)
  40.134 @@ -774,11 +774,11 @@
  40.135      apply (rule order_trans [OF norm_setsum setsum_mono])
  40.136      apply (auto simp add: norm_mult_ineq)
  40.137      done
  40.138 -  hence 2: "(\<lambda>n. setsum ?g (?S1 n) - setsum ?g (?S2 n)) ----> 0"
  40.139 +  hence 2: "(\<lambda>n. setsum ?g (?S1 n) - setsum ?g (?S2 n)) \<longlonglongrightarrow> 0"
  40.140      unfolding tendsto_Zfun_iff diff_0_right
  40.141      by (simp only: setsum_diff finite_S1 S2_le_S1)
  40.142  
  40.143 -  with 1 have "(\<lambda>n. setsum ?g (?S2 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
  40.144 +  with 1 have "(\<lambda>n. setsum ?g (?S2 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
  40.145      by (rule Lim_transform2)
  40.146    thus ?thesis by (simp only: sums_def setsum_triangle_reindex)
  40.147  qed
  40.148 @@ -933,12 +933,12 @@
  40.149  
  40.150    have "incseq (\<lambda>n. \<Sum>i<n. (f \<circ> g) i)"
  40.151      by (rule incseq_SucI) (auto simp add: pos)
  40.152 -  then obtain  L where L: "(\<lambda> n. \<Sum>i<n. (f \<circ> g) i) ----> L"
  40.153 +  then obtain  L where L: "(\<lambda> n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> L"
  40.154      using smaller by(rule incseq_convergent)
  40.155    hence "(f \<circ> g) sums L" by (simp add: sums_def)
  40.156    thus "summable (f o g)" by (auto simp add: sums_iff)
  40.157  
  40.158 -  hence "(\<lambda>n. \<Sum>i<n. (f \<circ> g) i) ----> suminf (f \<circ> g)"
  40.159 +  hence "(\<lambda>n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> suminf (f \<circ> g)"
  40.160      by(rule summable_LIMSEQ)
  40.161    thus le: "suminf (f \<circ> g) \<le> suminf f"
  40.162      by(rule LIMSEQ_le_const2)(blast intro: smaller[rule_format])
  40.163 @@ -971,7 +971,7 @@
  40.164    shows   "(\<lambda>n. f (g n)) sums c \<longleftrightarrow> f sums c"
  40.165  unfolding sums_def
  40.166  proof
  40.167 -  assume lim: "(\<lambda>n. \<Sum>k<n. f k) ----> c"
  40.168 +  assume lim: "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c"
  40.169    have "(\<lambda>n. \<Sum>k<n. f (g k)) = (\<lambda>n. \<Sum>k<g n. f k)"
  40.170    proof
  40.171      fix n :: nat
  40.172 @@ -982,10 +982,10 @@
  40.173           (auto dest: subseq_strict_mono simp: strict_mono_less strict_mono_less_eq)
  40.174      finally show "(\<Sum>k<n. f (g k)) = (\<Sum>k<g n. f k)" .
  40.175    qed
  40.176 -  also from LIMSEQ_subseq_LIMSEQ[OF lim subseq] have "\<dots> ----> c" unfolding o_def .
  40.177 -  finally show "(\<lambda>n. \<Sum>k<n. f (g k)) ----> c" .
  40.178 +  also from LIMSEQ_subseq_LIMSEQ[OF lim subseq] have "\<dots> \<longlonglongrightarrow> c" unfolding o_def .
  40.179 +  finally show "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c" .
  40.180  next
  40.181 -  assume lim: "(\<lambda>n. \<Sum>k<n. f (g k)) ----> c"
  40.182 +  assume lim: "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c"
  40.183    def g_inv \<equiv> "\<lambda>n. LEAST m. g m \<ge> n"
  40.184    from filterlim_subseq[OF subseq] have g_inv_ex: "\<exists>m. g m \<ge> n" for n
  40.185      by (auto simp: filterlim_at_top eventually_at_top_linorder)
  40.186 @@ -1020,8 +1020,8 @@
  40.187    }
  40.188    hence "filterlim g_inv at_top sequentially"
  40.189      by (auto simp: filterlim_at_top eventually_at_top_linorder)
  40.190 -  from lim and this have "(\<lambda>n. \<Sum>k<g_inv n. f (g k)) ----> c" by (rule filterlim_compose)
  40.191 -  finally show "(\<lambda>n. \<Sum>k<n. f k) ----> c" .
  40.192 +  from lim and this have "(\<lambda>n. \<Sum>k<g_inv n. f (g k)) \<longlonglongrightarrow> c" by (rule filterlim_compose)
  40.193 +  finally show "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c" .
  40.194  qed
  40.195  
  40.196  lemma summable_mono_reindex:
    41.1 --- a/src/HOL/Topological_Spaces.thy	Tue Dec 29 22:41:22 2015 +0100
    41.2 +++ b/src/HOL/Topological_Spaces.thy	Tue Dec 29 23:04:53 2015 +0100
    41.3 @@ -779,16 +779,16 @@
    41.4  
    41.5  abbreviation (in topological_space)
    41.6    LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
    41.7 -    ("((_)/ ----> (_))" [60, 60] 60) where
    41.8 -  "X ----> L \<equiv> (X ---> L) sequentially"
    41.9 +    ("((_)/ \<longlonglongrightarrow> (_))" [60, 60] 60) where
   41.10 +  "X \<longlonglongrightarrow> L \<equiv> (X ---> L) sequentially"
   41.11  
   41.12  abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" where
   41.13    "lim X \<equiv> Lim sequentially X"
   41.14  
   41.15  definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
   41.16 -  "convergent X = (\<exists>L. X ----> L)"
   41.17 +  "convergent X = (\<exists>L. X \<longlonglongrightarrow> L)"
   41.18  
   41.19 -lemma lim_def: "lim X = (THE L. X ----> L)"
   41.20 +lemma lim_def: "lim X = (THE L. X \<longlonglongrightarrow> L)"
   41.21    unfolding Lim_def ..
   41.22  
   41.23  subsubsection \<open>Monotone sequences and subsequences\<close>
   41.24 @@ -996,81 +996,81 @@
   41.25  
   41.26  lemma LIMSEQ_const_iff:
   41.27    fixes k l :: "'a::t2_space"
   41.28 -  shows "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
   41.29 +  shows "(\<lambda>n. k) \<longlonglongrightarrow> l \<longleftrightarrow> k = l"
   41.30    using trivial_limit_sequentially by (rule tendsto_const_iff)
   41.31  
   41.32  lemma LIMSEQ_SUP:
   41.33 -  "incseq X \<Longrightarrow> X ----> (SUP i. X i :: 'a :: {complete_linorder, linorder_topology})"
   41.34 +  "incseq X \<Longrightarrow> X \<longlonglongrightarrow> (SUP i. X i :: 'a :: {complete_linorder, linorder_topology})"
   41.35    by (intro increasing_tendsto)
   41.36       (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans)
   41.37  
   41.38  lemma LIMSEQ_INF:
   41.39 -  "decseq X \<Longrightarrow> X ----> (INF i. X i :: 'a :: {complete_linorder, linorder_topology})"
   41.40 +  "decseq X \<Longrightarrow> X \<longlonglongrightarrow> (INF i. X i :: 'a :: {complete_linorder, linorder_topology})"
   41.41    by (intro decreasing_tendsto)
   41.42       (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans)
   41.43  
   41.44  lemma LIMSEQ_ignore_initial_segment:
   41.45 -  "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
   41.46 +  "f \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (n + k)) \<longlonglongrightarrow> a"
   41.47    unfolding tendsto_def
   41.48    by (subst eventually_sequentially_seg[where k=k])
   41.49  
   41.50  lemma LIMSEQ_offset:
   41.51 -  "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
   41.52 +  "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a \<Longrightarrow> f \<longlonglongrightarrow> a"
   41.53    unfolding tendsto_def
   41.54    by (subst (asm) eventually_sequentially_seg[where k=k])
   41.55  
   41.56 -lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
   41.57 +lemma LIMSEQ_Suc: "f \<longlonglongrightarrow> l \<Longrightarrow> (\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l"
   41.58  by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
   41.59  
   41.60 -lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
   41.61 +lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l \<Longrightarrow> f \<longlonglongrightarrow> l"
   41.62  by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
   41.63  
   41.64 -lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
   41.65 +lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l = f \<longlonglongrightarrow> l"
   41.66  by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
   41.67  
   41.68  lemma LIMSEQ_unique:
   41.69    fixes a b :: "'a::t2_space"
   41.70 -  shows "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
   41.71 +  shows "\<lbrakk>X \<longlonglongrightarrow> a; X \<longlonglongrightarrow> b\<rbrakk> \<Longrightarrow> a = b"
   41.72    using trivial_limit_sequentially by (rule tendsto_unique)
   41.73  
   41.74  lemma LIMSEQ_le_const:
   41.75 -  "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
   41.76 +  "\<lbrakk>X \<longlonglongrightarrow> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
   41.77    using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially)
   41.78  
   41.79  lemma LIMSEQ_le:
   41.80 -  "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::'a::linorder_topology)"
   41.81 +  "\<lbrakk>X \<longlonglongrightarrow> x; Y \<longlonglongrightarrow> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::'a::linorder_topology)"
   41.82    using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)
   41.83  
   41.84  lemma LIMSEQ_le_const2:
   41.85 -  "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
   41.86 +  "\<lbrakk>X \<longlonglongrightarrow> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
   41.87    by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) auto
   41.88  
   41.89 -lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
   41.90 +lemma convergentD: "convergent X ==> \<exists>L. (X \<longlonglongrightarrow> L)"
   41.91  by (simp add: convergent_def)
   41.92  
   41.93 -lemma convergentI: "(X ----> L) ==> convergent X"
   41.94 +lemma convergentI: "(X \<longlonglongrightarrow> L) ==> convergent X"
   41.95  by (auto simp add: convergent_def)
   41.96  
   41.97 -lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
   41.98 +lemma convergent_LIMSEQ_iff: "convergent X = (X \<longlonglongrightarrow> lim X)"
   41.99  by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
  41.100  
  41.101  lemma convergent_const: "convergent (\<lambda>n. c)"
  41.102    by (rule convergentI, rule tendsto_const)
  41.103  
  41.104  lemma monoseq_le:
  41.105 -  "monoseq a \<Longrightarrow> a ----> (x::'a::linorder_topology) \<Longrightarrow>
  41.106 +  "monoseq a \<Longrightarrow> a \<longlonglongrightarrow> (x::'a::linorder_topology) \<Longrightarrow>
  41.107      ((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or> ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
  41.108    by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)
  41.109  
  41.110  lemma LIMSEQ_subseq_LIMSEQ:
  41.111 -  "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
  41.112 +  "\<lbrakk> X \<longlonglongrightarrow> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) \<longlonglongrightarrow> L"
  41.113    unfolding comp_def by (rule filterlim_compose[of X, OF _ filterlim_subseq])
  41.114  
  41.115  lemma convergent_subseq_convergent:
  41.116    "\<lbrakk>convergent X; subseq f\<rbrakk> \<Longrightarrow> convergent (X o f)"
  41.117    unfolding convergent_def by (auto intro: LIMSEQ_subseq_LIMSEQ)
  41.118  
  41.119 -lemma limI: "X ----> L ==> lim X = L"
  41.120 +lemma limI: "X \<longlonglongrightarrow> L ==> lim X = L"
  41.121    by (rule tendsto_Lim) (rule trivial_limit_sequentially)
  41.122  
  41.123  lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> (x::'a::linorder_topology)) \<Longrightarrow> lim f \<le> x"
  41.124 @@ -1078,10 +1078,10 @@
  41.125  
  41.126  subsubsection\<open>Increasing and Decreasing Series\<close>
  41.127  
  41.128 -lemma incseq_le: "incseq X \<Longrightarrow> X ----> L \<Longrightarrow> X n \<le> (L::'a::linorder_topology)"
  41.129 +lemma incseq_le: "incseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> X n \<le> (L::'a::linorder_topology)"
  41.130    by (metis incseq_def LIMSEQ_le_const)
  41.131  
  41.132 -lemma decseq_le: "decseq X \<Longrightarrow> X ----> L \<Longrightarrow> (L::'a::linorder_topology) \<le> X n"
  41.133 +lemma decseq_le: "decseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> (L::'a::linorder_topology) \<le> X n"
  41.134    by (metis decseq_def LIMSEQ_le_const2)
  41.135  
  41.136  subsection \<open>First countable topologies\<close>
  41.137 @@ -1142,7 +1142,7 @@
  41.138  lemma (in first_countable_topology) countable_basis:
  41.139    obtains A :: "nat \<Rightarrow> 'a set" where
  41.140      "\<And>i. open (A i)" "\<And>i. x \<in> A i"
  41.141 -    "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F ----> x"
  41.142 +    "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F \<longlonglongrightarrow> x"
  41.143  proof atomize_elim
  41.144    obtain A :: "nat \<Rightarrow> 'a set" where A:
  41.145      "\<And>i. open (A i)"
  41.146 @@ -1154,25 +1154,25 @@
  41.147      with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially"
  41.148        by (auto elim: eventually_mono simp: subset_eq)
  41.149    }
  41.150 -  with A show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> (\<forall>F. (\<forall>n. F n \<in> A n) \<longrightarrow> F ----> x)"
  41.151 +  with A show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> (\<forall>F. (\<forall>n. F n \<in> A n) \<longrightarrow> F \<longlonglongrightarrow> x)"
  41.152      by (intro exI[of _ A]) (auto simp: tendsto_def)
  41.153  qed
  41.154  
  41.155  lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within:
  41.156 -  assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
  41.157 +  assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
  41.158    shows "eventually P (inf (nhds a) (principal s))"
  41.159  proof (rule ccontr)
  41.160    obtain A :: "nat \<Rightarrow> 'a set" where A:
  41.161      "\<And>i. open (A i)"
  41.162      "\<And>i. a \<in> A i"
  41.163 -    "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F ----> a"
  41.164 +    "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F \<longlonglongrightarrow> a"
  41.165      by (rule countable_basis) blast
  41.166    assume "\<not> ?thesis"
  41.167    with A have P: "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)"
  41.168      unfolding eventually_inf_principal eventually_nhds by (intro choice) fastforce
  41.169    then obtain F where F0: "\<forall>n. F n \<in> s" and F2: "\<forall>n. F n \<in> A n" and F3: "\<forall>n. \<not> P (F n)"
  41.170      by blast
  41.171 -  with A have "F ----> a" by auto
  41.172 +  with A have "F \<longlonglongrightarrow> a" by auto
  41.173    hence "eventually (\<lambda>n. P (F n)) sequentially"
  41.174      using assms F0 by simp
  41.175    thus "False" by (simp add: F3)
  41.176 @@ -1180,23 +1180,23 @@
  41.177  
  41.178  lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially:
  41.179    "eventually P (inf (nhds a) (principal s)) \<longleftrightarrow> 
  41.180 -    (\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
  41.181 +    (\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
  41.182  proof (safe intro!: sequentially_imp_eventually_nhds_within)
  41.183    assume "eventually P (inf (nhds a) (principal s))" 
  41.184    then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"
  41.185      by (auto simp: eventually_inf_principal eventually_nhds)
  41.186 -  moreover fix f assume "\<forall>n. f n \<in> s" "f ----> a"
  41.187 +  moreover fix f assume "\<forall>n. f n \<in> s" "f \<longlonglongrightarrow> a"
  41.188    ultimately show "eventually (\<lambda>n. P (f n)) sequentially"
  41.189      by (auto dest!: topological_tendstoD elim: eventually_mono)
  41.190  qed
  41.191  
  41.192  lemma (in first_countable_topology) eventually_nhds_iff_sequentially:
  41.193 -  "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
  41.194 +  "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
  41.195    using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
  41.196  
  41.197  lemma tendsto_at_iff_sequentially:
  41.198    fixes f :: "'a :: first_countable_topology \<Rightarrow> _"
  41.199 -  shows "(f ---> a) (at x within s) \<longleftrightarrow> (\<forall>X. (\<forall>i. X i \<in> s - {x}) \<longrightarrow> X ----> x \<longrightarrow> ((f \<circ> X) ----> a))"
  41.200 +  shows "(f ---> a) (at x within s) \<longleftrightarrow> (\<forall>X. (\<forall>i. X i \<in> s - {x}) \<longrightarrow> X \<longlonglongrightarrow> x \<longrightarrow> ((f \<circ> X) \<longlonglongrightarrow> a))"
  41.201    unfolding filterlim_def[of _ "nhds a"] le_filter_def eventually_filtermap at_within_def eventually_nhds_within_iff_sequentially comp_def
  41.202    by metis
  41.203  
  41.204 @@ -1269,39 +1269,39 @@
  41.205  subsubsection \<open>Relation of LIM and LIMSEQ\<close>
  41.206  
  41.207  lemma (in first_countable_topology) sequentially_imp_eventually_within:
  41.208 -  "(\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow>
  41.209 +  "(\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow>
  41.210      eventually P (at a within s)"
  41.211    unfolding at_within_def
  41.212    by (intro sequentially_imp_eventually_nhds_within) auto
  41.213  
  41.214  lemma (in first_countable_topology) sequentially_imp_eventually_at:
  41.215 -  "(\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> eventually P (at a)"
  41.216 +  "(\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> eventually P (at a)"
  41.217    using assms sequentially_imp_eventually_within [where s=UNIV] by simp
  41.218  
  41.219  lemma LIMSEQ_SEQ_conv1:
  41.220    fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
  41.221    assumes f: "f -- a --> l"
  41.222 -  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
  41.223 +  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l"
  41.224    using tendsto_compose_eventually [OF f, where F=sequentially] by simp
  41.225  
  41.226  lemma LIMSEQ_SEQ_conv2:
  41.227    fixes f :: "'a::first_countable_topology \<Rightarrow> 'b::topological_space"
  41.228 -  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
  41.229 +  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l"
  41.230    shows "f -- a --> l"
  41.231    using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at)
  41.232  
  41.233  lemma LIMSEQ_SEQ_conv:
  41.234 -  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::'a::first_countable_topology) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
  41.235 +  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> (a::'a::first_countable_topology) \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L) =
  41.236     (X -- a --> (L::'b::topological_space))"
  41.237    using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
  41.238  
  41.239  lemma sequentially_imp_eventually_at_left:
  41.240    fixes a :: "'a :: {linorder_topology, first_countable_topology}"
  41.241    assumes b[simp]: "b < a"
  41.242 -  assumes *: "\<And>f. (\<And>n. b < f n) \<Longrightarrow> (\<And>n. f n < a) \<Longrightarrow> incseq f \<Longrightarrow> f ----> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
  41.243 +  assumes *: "\<And>f. (\<And>n. b < f n) \<Longrightarrow> (\<And>n. f n < a) \<Longrightarrow> incseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
  41.244    shows "eventually P (at_left a)"
  41.245  proof (safe intro!: sequentially_imp_eventually_within)
  41.246 -  fix X assume X: "\<forall>n. X n \<in> {..< a} \<and> X n \<noteq> a" "X ----> a"
  41.247 +  fix X assume X: "\<forall>n. X n \<in> {..< a} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a"
  41.248    show "eventually (\<lambda>n. P (X n)) sequentially"
  41.249    proof (rule ccontr)
  41.250      assume neg: "\<not> eventually (\<lambda>n. P (X n)) sequentially"
  41.251 @@ -1319,8 +1319,8 @@
  41.252          by (auto dest!: not_eventuallyD)
  41.253      qed
  41.254      then guess s ..
  41.255 -    then have "\<And>n. b < X (s n)" "\<And>n. X (s n) < a" "incseq (\<lambda>n. X (s n))" "(\<lambda>n. X (s n)) ----> a" "\<And>n. \<not> P (X (s n))"
  41.256 -      using X by (auto simp: subseq_Suc_iff Suc_le_eq incseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X ----> a\<close>, unfolded comp_def])
  41.257 +    then have "\<And>n. b < X (s n)" "\<And>n. X (s n) < a" "incseq (\<lambda>n. X (s n))" "(\<lambda>n. X (s n)) \<longlonglongrightarrow> a" "\<And>n. \<not> P (X (s n))"
  41.258 +      using X by (auto simp: subseq_Suc_iff Suc_le_eq incseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def])
  41.259      from *[OF this(1,2,3,4)] this(5) show False by auto
  41.260    qed
  41.261  qed
  41.262 @@ -1328,7 +1328,7 @@
  41.263  lemma tendsto_at_left_sequentially:
  41.264    fixes a :: "_ :: {linorder_topology, first_countable_topology}"
  41.265    assumes "b < a"
  41.266 -  assumes *: "\<And>S. (\<And>n. S n < a) \<Longrightarrow> (\<And>n. b < S n) \<Longrightarrow> incseq S \<Longrightarrow> S ----> a \<Longrightarrow> (\<lambda>n. X (S n)) ----> L"
  41.267 +  assumes *: "\<And>S. (\<And>n. S n < a) \<Longrightarrow> (\<And>n. b < S n) \<Longrightarrow> incseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
  41.268    shows "(X ---> L) (at_left a)"
  41.269    using assms unfolding tendsto_def [where l=L]
  41.270    by (simp add: sequentially_imp_eventually_at_left)
  41.271 @@ -1336,10 +1336,10 @@
  41.272  lemma sequentially_imp_eventually_at_right:
  41.273    fixes a :: "'a :: {linorder_topology, first_countable_topology}"
  41.274    assumes b[simp]: "a < b"
  41.275 -  assumes *: "\<And>f. (\<And>n. a < f n) \<Longrightarrow> (\<And>n. f n < b) \<Longrightarrow> decseq f \<Longrightarrow> f ----> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
  41.276 +  assumes *: "\<And>f. (\<And>n. a < f n) \<Longrightarrow> (\<And>n. f n < b) \<Longrightarrow> decseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
  41.277    shows "eventually P (at_right a)"
  41.278  proof (safe intro!: sequentially_imp_eventually_within)
  41.279 -  fix X assume X: "\<forall>n. X n \<in> {a <..} \<and> X n \<noteq> a" "X ----> a"
  41.280 +  fix X assume X: "\<forall>n. X n \<in> {a <..} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a"
  41.281    show "eventually (\<lambda>n. P (X n)) sequentially"
  41.282    proof (rule ccontr)
  41.283      assume neg: "\<not> eventually (\<lambda>n. P (X n)) sequentially"
  41.284 @@ -1357,8 +1357,8 @@
  41.285          by (auto dest!: not_eventuallyD)
  41.286      qed
  41.287      then guess s ..
  41.288 -    then have "\<And>n. a < X (s n)" "\<And>n. X (s n) < b" "decseq (\<lambda>n. X (s n))" "(\<lambda>n. X (s n)) ----> a" "\<And>n. \<not> P (X (s n))"
  41.289 -      using X by (auto simp: subseq_Suc_iff Suc_le_eq decseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X ----> a\<close>, unfolded comp_def])
  41.290 +    then have "\<And>n. a < X (s n)" "\<And>n. X (s n) < b" "decseq (\<lambda>n. X (s n))" "(\<lambda>n. X (s n)) \<longlonglongrightarrow> a" "\<And>n. \<not> P (X (s n))"
  41.291 +      using X by (auto simp: subseq_Suc_iff Suc_le_eq decseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def])
  41.292      from *[OF this(1,2,3,4)] this(5) show False by auto
  41.293    qed
  41.294  qed
  41.295 @@ -1366,7 +1366,7 @@
  41.296  lemma tendsto_at_right_sequentially:
  41.297    fixes a :: "_ :: {linorder_topology, first_countable_topology}"
  41.298    assumes "a < b"
  41.299 -  assumes *: "\<And>S. (\<And>n. a < S n) \<Longrightarrow> (\<And>n. S n < b) \<Longrightarrow> decseq S \<Longrightarrow> S ----> a \<Longrightarrow> (\<lambda>n. X (S n)) ----> L"
  41.300 +  assumes *: "\<And>S. (\<And>n. a < S n) \<Longrightarrow> (\<And>n. S n < b) \<Longrightarrow> decseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
  41.301    shows "(X ---> L) (at_right a)"
  41.302    using assms unfolding tendsto_def [where l=L]
  41.303    by (simp add: sequentially_imp_eventually_at_right)
    42.1 --- a/src/HOL/Transcendental.thy	Tue Dec 29 22:41:22 2015 +0100
    42.2 +++ b/src/HOL/Transcendental.thy	Tue Dec 29 23:04:53 2015 +0100
    42.3 @@ -41,7 +41,7 @@
    42.4  
    42.5  lemma root_test_convergence:
    42.6    fixes f :: "nat \<Rightarrow> 'a::banach"
    42.7 -  assumes f: "(\<lambda>n. root n (norm (f n))) ----> x" \<comment> "could be weakened to lim sup"
    42.8 +  assumes f: "(\<lambda>n. root n (norm (f n))) \<longlonglongrightarrow> x" \<comment> "could be weakened to lim sup"
    42.9    assumes "x < 1"
   42.10    shows "summable f"
   42.11  proof -
   42.12 @@ -92,7 +92,7 @@
   42.13    shows "summable (\<lambda>n. norm (f n * z ^ n))"
   42.14  proof -
   42.15    from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
   42.16 -  from 1 have "(\<lambda>n. f n * x^n) ----> 0"
   42.17 +  from 1 have "(\<lambda>n. f n * x^n) \<longlonglongrightarrow> 0"
   42.18      by (rule summable_LIMSEQ_zero)
   42.19    hence "convergent (\<lambda>n. f n * x^n)"
   42.20      by (rule convergentI)
   42.21 @@ -148,7 +148,7 @@
   42.22  lemma powser_times_n_limit_0:
   42.23    fixes x :: "'a::{real_normed_div_algebra,banach}"
   42.24    assumes "norm x < 1"
   42.25 -    shows "(\<lambda>n. of_nat n * x ^ n) ----> 0"
   42.26 +    shows "(\<lambda>n. of_nat n * x ^ n) \<longlonglongrightarrow> 0"
   42.27  proof -
   42.28    have "norm x / (1 - norm x) \<ge> 0"
   42.29      using assms
   42.30 @@ -270,9 +270,9 @@
   42.31  
   42.32  lemma sums_alternating_upper_lower:
   42.33    fixes a :: "nat \<Rightarrow> real"
   42.34 -  assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
   42.35 -  shows "\<exists>l. ((\<forall>n. (\<Sum>i<2*n. (- 1)^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i<2*n. (- 1)^i*a i) ----> l) \<and>
   42.36 -             ((\<forall>n. l \<le> (\<Sum>i<2*n + 1. (- 1)^i*a i)) \<and> (\<lambda> n. \<Sum>i<2*n + 1. (- 1)^i*a i) ----> l)"
   42.37 +  assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a \<longlonglongrightarrow> 0"
   42.38 +  shows "\<exists>l. ((\<forall>n. (\<Sum>i<2*n. (- 1)^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i<2*n. (- 1)^i*a i) \<longlonglongrightarrow> l) \<and>
   42.39 +             ((\<forall>n. l \<le> (\<Sum>i<2*n + 1. (- 1)^i*a i)) \<and> (\<lambda> n. \<Sum>i<2*n + 1. (- 1)^i*a i) \<longlonglongrightarrow> l)"
   42.40    (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
   42.41  proof (rule nested_sequence_unique)
   42.42    have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
   42.43 @@ -294,11 +294,11 @@
   42.44      show "?f n \<le> ?g n" using fg_diff a_pos
   42.45        unfolding One_nat_def by auto
   42.46    qed
   42.47 -  show "(\<lambda>n. ?f n - ?g n) ----> 0" unfolding fg_diff
   42.48 +  show "(\<lambda>n. ?f n - ?g n) \<longlonglongrightarrow> 0" unfolding fg_diff
   42.49    proof (rule LIMSEQ_I)
   42.50      fix r :: real
   42.51      assume "0 < r"
   42.52 -    with \<open>a ----> 0\<close>[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
   42.53 +    with \<open>a \<longlonglongrightarrow> 0\<close>[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
   42.54        by auto
   42.55      hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
   42.56      thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
   42.57 @@ -307,14 +307,14 @@
   42.58  
   42.59  lemma summable_Leibniz':
   42.60    fixes a :: "nat \<Rightarrow> real"
   42.61 -  assumes a_zero: "a ----> 0"
   42.62 +  assumes a_zero: "a \<longlonglongrightarrow> 0"
   42.63      and a_pos: "\<And> n. 0 \<le> a n"
   42.64      and a_monotone: "\<And> n. a (Suc n) \<le> a n"
   42.65    shows summable: "summable (\<lambda> n. (-1)^n * a n)"
   42.66      and "\<And>n. (\<Sum>i<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
   42.67 -    and "(\<lambda>n. \<Sum>i<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
   42.68 +    and "(\<lambda>n. \<Sum>i<2*n. (-1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (-1)^i*a i)"
   42.69      and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i<2*n+1. (-1)^i*a i)"
   42.70 -    and "(\<lambda>n. \<Sum>i<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
   42.71 +    and "(\<lambda>n. \<Sum>i<2*n+1. (-1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (-1)^i*a i)"
   42.72  proof -
   42.73    let ?S = "\<lambda>n. (-1)^n * a n"
   42.74    let ?P = "\<lambda>n. \<Sum>i<n. ?S i"
   42.75 @@ -322,20 +322,20 @@
   42.76    let ?g = "\<lambda>n. ?P (2 * n + 1)"
   42.77    obtain l :: real
   42.78      where below_l: "\<forall> n. ?f n \<le> l"
   42.79 -      and "?f ----> l"
   42.80 +      and "?f \<longlonglongrightarrow> l"
   42.81        and above_l: "\<forall> n. l \<le> ?g n"
   42.82 -      and "?g ----> l"
   42.83 +      and "?g \<longlonglongrightarrow> l"
   42.84      using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
   42.85  
   42.86    let ?Sa = "\<lambda>m. \<Sum>n<m. ?S n"
   42.87 -  have "?Sa ----> l"
   42.88 +  have "?Sa \<longlonglongrightarrow> l"
   42.89    proof (rule LIMSEQ_I)
   42.90      fix r :: real
   42.91      assume "0 < r"
   42.92 -    with \<open>?f ----> l\<close>[THEN LIMSEQ_D]
   42.93 +    with \<open>?f \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D]
   42.94      obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
   42.95  
   42.96 -    from \<open>0 < r\<close> \<open>?g ----> l\<close>[THEN LIMSEQ_D]
   42.97 +    from \<open>0 < r\<close> \<open>?g \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D]
   42.98      obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
   42.99  
  42.100      {
  42.101 @@ -377,22 +377,22 @@
  42.102      unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
  42.103    show "?f n \<le> suminf ?S"
  42.104      unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
  42.105 -  show "?g ----> suminf ?S"
  42.106 -    using \<open>?g ----> l\<close> \<open>l = suminf ?S\<close> by auto
  42.107 -  show "?f ----> suminf ?S"
  42.108 -    using \<open>?f ----> l\<close> \<open>l = suminf ?S\<close> by auto
  42.109 +  show "?g \<longlonglongrightarrow> suminf ?S"
  42.110 +    using \<open>?g \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto
  42.111 +  show "?f \<longlonglongrightarrow> suminf ?S"
  42.112 +    using \<open>?f \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto
  42.113  qed
  42.114  
  42.115  theorem summable_Leibniz:
  42.116    fixes a :: "nat \<Rightarrow> real"
  42.117 -  assumes a_zero: "a ----> 0" and "monoseq a"
  42.118 +  assumes a_zero: "a \<longlonglongrightarrow> 0" and "monoseq a"
  42.119    shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
  42.120      and "0 < a 0 \<longrightarrow>
  42.121        (\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n. (- 1)^i * a i .. \<Sum>i<2*n+1. (- 1)^i * a i})" (is "?pos")
  42.122      and "a 0 < 0 \<longrightarrow>
  42.123        (\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n+1. (- 1)^i * a i .. \<Sum>i<2*n. (- 1)^i * a i})" (is "?neg")
  42.124 -    and "(\<lambda>n. \<Sum>i<2*n. (- 1)^i*a i) ----> (\<Sum>i. (- 1)^i*a i)" (is "?f")
  42.125 -    and "(\<lambda>n. \<Sum>i<2*n+1. (- 1)^i*a i) ----> (\<Sum>i. (- 1)^i*a i)" (is "?g")
  42.126 +    and "(\<lambda>n. \<Sum>i<2*n. (- 1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (- 1)^i*a i)" (is "?f")
  42.127 +    and "(\<lambda>n. \<Sum>i<2*n+1. (- 1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (- 1)^i*a i)" (is "?g")
  42.128  proof -
  42.129    have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
  42.130    proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
  42.131 @@ -404,13 +404,13 @@
  42.132        have "a (Suc n) \<le> a n"
  42.133          using ord[where n="Suc n" and m=n] by auto
  42.134      } note mono = this
  42.135 -    note leibniz = summable_Leibniz'[OF \<open>a ----> 0\<close> ge0]
  42.136 +    note leibniz = summable_Leibniz'[OF \<open>a \<longlonglongrightarrow> 0\<close> ge0]
  42.137      from leibniz[OF mono]
  42.138      show ?thesis using \<open>0 \<le> a 0\<close> by auto
  42.139    next
  42.140      let ?a = "\<lambda> n. - a n"
  42.141      case False
  42.142 -    with monoseq_le[OF \<open>monoseq a\<close> \<open>a ----> 0\<close>]
  42.143 +    with monoseq_le[OF \<open>monoseq a\<close> \<open>a \<longlonglongrightarrow> 0\<close>]
  42.144      have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
  42.145      hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
  42.146        by auto
  42.147 @@ -421,7 +421,7 @@
  42.148      } note monotone = this
  42.149      note leibniz =
  42.150        summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
  42.151 -        OF tendsto_minus[OF \<open>a ----> 0\<close>, unfolded minus_zero] monotone]
  42.152 +        OF tendsto_minus[OF \<open>a \<longlonglongrightarrow> 0\<close>, unfolded minus_zero] monotone]
  42.153      have "summable (\<lambda> n. (-1)^n * ?a n)"
  42.154        using leibniz(1) by auto
  42.155      then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
  42.156 @@ -724,7 +724,7 @@
  42.157    then obtain r::real where r: "norm x < norm r" "norm r < K" "r>0"
  42.158      using K False
  42.159      by (auto simp: field_simps abs_less_iff add_pos_pos intro: that [of "(norm x + K) / 2"])
  42.160 -  have "(\<lambda>n. of_nat n * (x / of_real r) ^ n) ----> 0"
  42.161 +  have "(\<lambda>n. of_nat n * (x / of_real r) ^ n) \<longlonglongrightarrow> 0"
  42.162      using r by (simp add: norm_divide powser_times_n_limit_0 [of "x / of_real r"])
  42.163    then obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> real_of_nat n * norm x ^ n < r ^ n"
  42.164      using r unfolding LIMSEQ_iff
  42.165 @@ -2646,7 +2646,7 @@
  42.166  
  42.167  lemma tendsto_exp_limit_sequentially:
  42.168    fixes x :: real
  42.169 -  shows "(\<lambda>n. (1 + x / n) ^ n) ----> exp x"
  42.170 +  shows "(\<lambda>n. (1 + x / n) ^ n) \<longlonglongrightarrow> exp x"
  42.171  proof (rule filterlim_mono_eventually)
  42.172    from reals_Archimedean2 [of "\<bar>x\<bar>"] obtain n :: nat where *: "real n > \<bar>x\<bar>" ..
  42.173    hence "eventually (\<lambda>n :: nat. 0 < 1 + x / real n) at_top"
  42.174 @@ -2658,7 +2658,7 @@
  42.175      done
  42.176    then show "eventually (\<lambda>n. (1 + x / n) powr n = (1 + x / n) ^ n) at_top"
  42.177      by (rule eventually_mono) (erule powr_realpow)
  42.178 -  show "(\<lambda>n. (1 + x / real n) powr real n) ----> exp x"
  42.179 +  show "(\<lambda>n. (1 + x / real n) powr real n) \<longlonglongrightarrow> exp x"
  42.180      by (rule filterlim_compose [OF tendsto_exp_limit_at_top filterlim_real_sequentially])
  42.181  qed auto
  42.182  
  42.183 @@ -4986,7 +4986,7 @@
  42.184  lemma zeroseq_arctan_series:
  42.185    fixes x :: real
  42.186    assumes "\<bar>x\<bar> \<le> 1"
  42.187 -  shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")
  42.188 +  shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) \<longlonglongrightarrow> 0" (is "?a \<longlonglongrightarrow> 0")
  42.189  proof (cases "x = 0")
  42.190    case True
  42.191    thus ?thesis
  42.192 @@ -4994,12 +4994,12 @@
  42.193  next
  42.194    case False
  42.195    have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
  42.196 -  show "?a ----> 0"
  42.197 +  show "?a \<longlonglongrightarrow> 0"
  42.198    proof (cases "\<bar>x\<bar> < 1")
  42.199      case True
  42.200      hence "norm x < 1" by auto
  42.201      from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF \<open>norm x < 1\<close>, THEN LIMSEQ_Suc]]
  42.202 -    have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) ----> 0"
  42.203 +    have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) \<longlonglongrightarrow> 0"
  42.204        unfolding inverse_eq_divide Suc_eq_plus1 by simp
  42.205      then show ?thesis using pos2 by (rule LIMSEQ_linear)
  42.206    next
  42.207 @@ -5252,15 +5252,15 @@
  42.208          by (rule LIM_less_bound)
  42.209        hence "?diff 1 n \<le> ?a 1 n" by auto
  42.210      }
  42.211 -    have "?a 1 ----> 0"
  42.212 +    have "?a 1 \<longlonglongrightarrow> 0"
  42.213        unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
  42.214        by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat simp del: of_nat_Suc)
  42.215 -    have "?diff 1 ----> 0"
  42.216 +    have "?diff 1 \<longlonglongrightarrow> 0"
  42.217      proof (rule LIMSEQ_I)
  42.218        fix r :: real
  42.219        assume "0 < r"
  42.220        obtain N :: nat where N_I: "\<And>n. N \<le> n \<Longrightarrow> ?a 1 n < r"
  42.221 -        using LIMSEQ_D[OF \<open>?a 1 ----> 0\<close> \<open>0 < r\<close>] by auto
  42.222 +        using LIMSEQ_D[OF \<open>?a 1 \<longlonglongrightarrow> 0\<close> \<open>0 < r\<close>] by auto
  42.223        {
  42.224          fix n
  42.225          assume "N \<le> n" from \<open>?diff 1 n \<le> ?a 1 n\<close> N_I[OF this]