author wenzelm Wed Dec 30 11:21:54 2015 +0100 (2015-12-30) changeset 61973 0c7e865fa7cb parent 61972 a70b89a3e02e child 61974 5b067c681989
more symbols;
```     1.1 --- a/NEWS	Tue Dec 29 23:50:44 2015 +0100
1.2 +++ b/NEWS	Wed Dec 30 11:21:54 2015 +0100
1.3 @@ -502,6 +502,7 @@
1.4    notation Preorder.equiv ("op ~~")
1.5      and Preorder.equiv ("(_/ ~~ _)" [51, 51] 50)
1.6
1.7 +  notation (in topological_space) tendsto (infixr "--->" 55)
1.8    notation (in topological_space) LIMSEQ ("((_)/ ----> (_))" [60, 60] 60)
1.9
1.10    notation NSLIMSEQ ("((_)/ ----NS> (_))" [60, 60] 60)
```
```     2.1 --- a/src/HOL/Complex.thy	Tue Dec 29 23:50:44 2015 +0100
2.2 +++ b/src/HOL/Complex.thy	Wed Dec 30 11:21:54 2015 +0100
2.3 @@ -382,14 +382,14 @@
2.4  lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]
2.5
2.6  lemma tendsto_Complex [tendsto_intros]:
2.7 -  "(f ---> a) F \<Longrightarrow> (g ---> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
2.8 +  "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) \<longlongrightarrow> Complex a b) F"
2.9    by (auto intro!: tendsto_intros)
2.10
2.11  lemma tendsto_complex_iff:
2.12 -  "(f ---> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) ---> Re x) F \<and> ((\<lambda>x. Im (f x)) ---> Im x) F)"
2.13 +  "(f \<longlongrightarrow> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F \<and> ((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F)"
2.14  proof safe
2.15 -  assume "((\<lambda>x. Re (f x)) ---> Re x) F" "((\<lambda>x. Im (f x)) ---> Im x) F"
2.16 -  from tendsto_Complex[OF this] show "(f ---> x) F"
2.17 +  assume "((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F" "((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F"
2.18 +  from tendsto_Complex[OF this] show "(f \<longlongrightarrow> x) F"
2.19      unfolding complex.collapse .
2.20  qed (auto intro: tendsto_intros)
2.21
2.22 @@ -530,7 +530,7 @@
2.23  lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]
2.24  lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]
2.25
2.26 -lemma lim_cnj: "((\<lambda>x. cnj(f x)) ---> cnj l) F \<longleftrightarrow> (f ---> l) F"
2.27 +lemma lim_cnj: "((\<lambda>x. cnj(f x)) \<longlongrightarrow> cnj l) F \<longleftrightarrow> (f \<longlongrightarrow> l) F"
2.28    by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)
2.29
2.30  lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
```
```     3.1 --- a/src/HOL/Deriv.thy	Tue Dec 29 23:50:44 2015 +0100
3.2 +++ b/src/HOL/Deriv.thy	Wed Dec 30 11:21:54 2015 +0100
3.3 @@ -20,7 +20,7 @@
3.4  where
3.5    "(f has_derivative f') F \<longleftrightarrow>
3.6      (bounded_linear f' \<and>
3.7 -     ((\<lambda>y. ((f y - f (Lim F (\<lambda>x. x))) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) ---> 0) F)"
3.8 +     ((\<lambda>y. ((f y - f (Lim F (\<lambda>x. x))) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) \<longlongrightarrow> 0) F)"
3.9
3.10  text \<open>
3.11    Usually the filter @{term F} is @{term "at x within s"}.  @{term "(f has_derivative D)
3.12 @@ -115,9 +115,9 @@
3.13  proof safe
3.14    let ?x = "Lim F (\<lambda>x. x)"
3.15    let ?D = "\<lambda>f f' y. ((f y - f ?x) - f' (y - ?x)) /\<^sub>R norm (y - ?x)"
3.16 -  have "((\<lambda>x. ?D f f' x + ?D g g' x) ---> (0 + 0)) F"
3.17 +  have "((\<lambda>x. ?D f f' x + ?D g g' x) \<longlongrightarrow> (0 + 0)) F"
3.18      using f g by (intro tendsto_add) (auto simp: has_derivative_def)
3.19 -  then show "(?D (\<lambda>x. f x + g x) (\<lambda>x. f' x + g' x) ---> 0) F"
3.20 +  then show "(?D (\<lambda>x. f x + g x) (\<lambda>x. f' x + g' x) \<longlongrightarrow> 0) F"
3.21      by (simp add: field_simps scaleR_add_right scaleR_diff_right)
3.22  qed (blast intro: bounded_linear_add f g has_derivative_bounded_linear)
3.23
3.24 @@ -138,12 +138,12 @@
3.25
3.26  lemma has_derivative_at_within:
3.27    "(f has_derivative f') (at x within s) \<longleftrightarrow>
3.28 -    (bounded_linear f' \<and> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s))"
3.29 +    (bounded_linear f' \<and> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within s))"
3.30    by (cases "at x within s = bot") (simp_all add: has_derivative_def Lim_ident_at)
3.31
3.32  lemma has_derivative_iff_norm:
3.33    "(f has_derivative f') (at x within s) \<longleftrightarrow>
3.34 -    (bounded_linear f' \<and> ((\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x)) ---> 0) (at x within s))"
3.35 +    (bounded_linear f' \<and> ((\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within s))"
3.36    using tendsto_norm_zero_iff[of _ "at x within s", where 'b="'b", symmetric]
3.37    by (simp add: has_derivative_at_within divide_inverse ac_simps)
3.38
3.39 @@ -164,20 +164,20 @@
3.40    done
3.41
3.42  lemma has_derivativeI:
3.43 -  "bounded_linear f' \<Longrightarrow> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s) \<Longrightarrow>
3.44 +  "bounded_linear f' \<Longrightarrow> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within s) \<Longrightarrow>
3.45    (f has_derivative f') (at x within s)"
3.46    by (simp add: has_derivative_at_within)
3.47
3.48  lemma has_derivativeI_sandwich:
3.49    assumes e: "0 < e" and bounded: "bounded_linear f'"
3.50      and sandwich: "(\<And>y. y \<in> s \<Longrightarrow> y \<noteq> x \<Longrightarrow> dist y x < e \<Longrightarrow> norm ((f y - f x) - f' (y - x)) / norm (y - x) \<le> H y)"
3.51 -    and "(H ---> 0) (at x within s)"
3.52 +    and "(H \<longlongrightarrow> 0) (at x within s)"
3.53    shows "(f has_derivative f') (at x within s)"
3.54    unfolding has_derivative_iff_norm
3.55  proof safe
3.56 -  show "((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) ---> 0) (at x within s)"
3.57 +  show "((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within s)"
3.58    proof (rule tendsto_sandwich[where f="\<lambda>x. 0"])
3.59 -    show "(H ---> 0) (at x within s)" by fact
3.60 +    show "(H \<longlongrightarrow> 0) (at x within s)" by fact
3.61      show "eventually (\<lambda>n. norm (f n - f x - f' (n - x)) / norm (n - x) \<le> H n) (at x within s)"
3.62        unfolding eventually_at using e sandwich by auto
3.63    qed (auto simp: le_divide_eq)
3.64 @@ -197,7 +197,7 @@
3.65  proof -
3.66    from f interpret F: bounded_linear f' by (rule has_derivative_bounded_linear)
3.67    note F.tendsto[tendsto_intros]
3.68 -  let ?L = "\<lambda>f. (f ---> 0) (at x within s)"
3.69 +  let ?L = "\<lambda>f. (f \<longlongrightarrow> 0) (at x within s)"
3.70    have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x))"
3.71      using f unfolding has_derivative_iff_norm by blast
3.72    then have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x) * norm (y - x))" (is ?m)
3.73 @@ -216,7 +216,7 @@
3.74
3.75  subsection \<open>Composition\<close>
3.76
3.77 -lemma tendsto_at_iff_tendsto_nhds_within: "f x = y \<Longrightarrow> (f ---> y) (at x within s) \<longleftrightarrow> (f ---> y) (inf (nhds x) (principal s))"
3.78 +lemma tendsto_at_iff_tendsto_nhds_within: "f x = y \<Longrightarrow> (f \<longlongrightarrow> y) (at x within s) \<longleftrightarrow> (f \<longlongrightarrow> y) (inf (nhds x) (principal s))"
3.79    unfolding tendsto_def eventually_inf_principal eventually_at_filter
3.80    by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)
3.81
3.82 @@ -231,7 +231,7 @@
3.83    from G.bounded obtain kG where kG: "\<And>x. norm (g' x) \<le> norm x * kG" by fast
3.84    note G.tendsto[tendsto_intros]
3.85
3.86 -  let ?L = "\<lambda>f. (f ---> 0) (at x within s)"
3.87 +  let ?L = "\<lambda>f. (f \<longlongrightarrow> 0) (at x within s)"
3.88    let ?D = "\<lambda>f f' x y. (f y - f x) - f' (y - x)"
3.89    let ?N = "\<lambda>f f' x y. norm (?D f f' x y) / norm (y - x)"
3.90    let ?gf = "\<lambda>x. g (f x)" and ?gf' = "\<lambda>x. g' (f' x)"
3.91 @@ -263,20 +263,20 @@
3.92    next
3.93      have [tendsto_intros]: "?L Nf"
3.94        using f unfolding has_derivative_iff_norm Nf_def ..
3.95 -    from f have "(f ---> f x) (at x within s)"
3.96 +    from f have "(f \<longlongrightarrow> f x) (at x within s)"
3.97        by (blast intro: has_derivative_continuous continuous_within[THEN iffD1])
3.98      then have f': "LIM x at x within s. f x :> inf (nhds (f x)) (principal (f`s))"
3.99        unfolding filterlim_def
3.100        by (simp add: eventually_filtermap eventually_at_filter le_principal)
3.101
3.102 -    have "((?N g  g' (f x)) ---> 0) (at (f x) within f`s)"
3.103 +    have "((?N g  g' (f x)) \<longlongrightarrow> 0) (at (f x) within f`s)"
3.104        using g unfolding has_derivative_iff_norm ..
3.105 -    then have g': "((?N g  g' (f x)) ---> 0) (inf (nhds (f x)) (principal (f`s)))"
3.106 +    then have g': "((?N g  g' (f x)) \<longlongrightarrow> 0) (inf (nhds (f x)) (principal (f`s)))"
3.107        by (rule tendsto_at_iff_tendsto_nhds_within[THEN iffD1, rotated]) simp
3.108
3.109      have [tendsto_intros]: "?L Ng"
3.110        unfolding Ng_def by (rule filterlim_compose[OF g' f'])
3.111 -    show "((\<lambda>y. Nf y * kG + Ng y * (Nf y + kF)) ---> 0) (at x within s)"
3.112 +    show "((\<lambda>y. Nf y * kG + Ng y * (Nf y + kF)) \<longlongrightarrow> 0) (at x within s)"
3.113        by (intro tendsto_eq_intros) auto
3.114    qed simp
3.115  qed
3.116 @@ -310,13 +310,13 @@
3.117          bounded_linear_compose [OF bounded_linear_right] bounded_linear_compose [OF bounded_linear_left]
3.118          has_derivative_bounded_linear [OF g] has_derivative_bounded_linear [OF f])
3.119    next
3.120 -    from g have "(g ---> g x) ?F"
3.121 +    from g have "(g \<longlongrightarrow> g x) ?F"
3.122        by (intro continuous_within[THEN iffD1] has_derivative_continuous)
3.123 -    moreover from f g have "(Nf ---> 0) ?F" "(Ng ---> 0) ?F"
3.124 +    moreover from f g have "(Nf \<longlongrightarrow> 0) ?F" "(Ng \<longlongrightarrow> 0) ?F"
3.125        by (simp_all add: has_derivative_iff_norm Ng_def Nf_def)
3.126 -    ultimately have "(?fun2 ---> norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F"
3.127 +    ultimately have "(?fun2 \<longlongrightarrow> norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F"
3.128        by (intro tendsto_intros) (simp_all add: LIM_zero_iff)
3.129 -    then show "(?fun2 ---> 0) ?F"
3.130 +    then show "(?fun2 \<longlongrightarrow> 0) ?F"
3.131        by simp
3.132    next
3.133      fix y::'d assume "y \<noteq> x"
3.134 @@ -377,7 +377,7 @@
3.135  next
3.136    show "0 < norm x" using x by simp
3.137  next
3.138 -  show "((\<lambda>y. norm (?inv y - ?inv x) * norm (?inv x)) ---> 0) (at x within s)"
3.139 +  show "((\<lambda>y. norm (?inv y - ?inv x) * norm (?inv x)) \<longlongrightarrow> 0) (at x within s)"
3.140      apply (rule tendsto_mult_left_zero)
3.141      apply (rule tendsto_norm_zero)
3.142      apply (rule LIM_zero)
3.143 @@ -1438,7 +1438,7 @@
3.144  lemma DERIV_pos_imp_increasing_at_bot:
3.145    fixes f :: "real => real"
3.146    assumes "\<And>x. x \<le> b \<Longrightarrow> (EX y. DERIV f x :> y & y > 0)"
3.147 -      and lim: "(f ---> flim) at_bot"
3.148 +      and lim: "(f \<longlongrightarrow> flim) at_bot"
3.149    shows "flim < f b"
3.150  proof -
3.151    have "flim \<le> f (b - 1)"
3.152 @@ -1455,7 +1455,7 @@
3.153  lemma DERIV_neg_imp_decreasing_at_top:
3.154    fixes f :: "real => real"
3.155    assumes der: "\<And>x. x \<ge> b \<Longrightarrow> (EX y. DERIV f x :> y & y < 0)"
3.156 -      and lim: "(f ---> flim) at_top"
3.157 +      and lim: "(f \<longlongrightarrow> flim) at_top"
3.158    shows "flim < f b"
3.159    apply (rule DERIV_pos_imp_increasing_at_bot [where f = "\<lambda>i. f (-i)" and b = "-b", simplified])
3.160    apply (metis DERIV_mirror der le_minus_iff neg_0_less_iff_less)
3.161 @@ -1592,7 +1592,7 @@
3.162
3.163  lemma isCont_If_ge:
3.164    fixes a :: "'a :: linorder_topology"
3.165 -  shows "continuous (at_left a) g \<Longrightarrow> (f ---> g a) (at_right a) \<Longrightarrow> isCont (\<lambda>x. if x \<le> a then g x else f x) a"
3.166 +  shows "continuous (at_left a) g \<Longrightarrow> (f \<longlongrightarrow> g a) (at_right a) \<Longrightarrow> isCont (\<lambda>x. if x \<le> a then g x else f x) a"
3.167    unfolding isCont_def continuous_within
3.168    apply (intro filterlim_split_at)
3.169    apply (subst filterlim_cong[OF refl refl, where g=g])
3.170 @@ -1603,15 +1603,15 @@
3.171
3.172  lemma lhopital_right_0:
3.173    fixes f0 g0 :: "real \<Rightarrow> real"
3.174 -  assumes f_0: "(f0 ---> 0) (at_right 0)"
3.175 -  assumes g_0: "(g0 ---> 0) (at_right 0)"
3.176 +  assumes f_0: "(f0 \<longlongrightarrow> 0) (at_right 0)"
3.177 +  assumes g_0: "(g0 \<longlongrightarrow> 0) (at_right 0)"
3.178    assumes ev:
3.179      "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
3.180      "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
3.181      "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
3.182      "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
3.183 -  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
3.184 -  shows "((\<lambda> x. f0 x / g0 x) ---> x) (at_right 0)"
3.185 +  assumes lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) (at_right 0)"
3.186 +  shows "((\<lambda> x. f0 x / g0 x) \<longlongrightarrow> x) (at_right 0)"
3.187  proof -
3.188    def f \<equiv> "\<lambda>x. if x \<le> 0 then 0 else f0 x"
3.189    then have "f 0 = 0" by simp
3.190 @@ -1674,15 +1674,15 @@
3.191    moreover
3.192    from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
3.193      by eventually_elim auto
3.194 -  then have "((\<lambda>x. norm (\<zeta> x)) ---> 0) (at_right 0)"
3.195 +  then have "((\<lambda>x. norm (\<zeta> x)) \<longlongrightarrow> 0) (at_right 0)"
3.196      by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"]) auto
3.197 -  then have "(\<zeta> ---> 0) (at_right 0)"
3.198 +  then have "(\<zeta> \<longlongrightarrow> 0) (at_right 0)"
3.199      by (rule tendsto_norm_zero_cancel)
3.200    with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
3.201      by (auto elim!: eventually_mono simp: filterlim_at)
3.202 -  from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) ---> x) (at_right 0)"
3.203 +  from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) \<longlongrightarrow> x) (at_right 0)"
3.204      by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
3.205 -  ultimately have "((\<lambda>t. f t / g t) ---> x) (at_right 0)" (is ?P)
3.206 +  ultimately have "((\<lambda>t. f t / g t) \<longlongrightarrow> x) (at_right 0)" (is ?P)
3.207      by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
3.208         (auto elim: eventually_mono)
3.209    also have "?P \<longleftrightarrow> ?thesis"
3.210 @@ -1691,35 +1691,35 @@
3.211  qed
3.212
3.213  lemma lhopital_right:
3.214 -  "((f::real \<Rightarrow> real) ---> 0) (at_right x) \<Longrightarrow> (g ---> 0) (at_right x) \<Longrightarrow>
3.215 +  "((f::real \<Rightarrow> real) \<longlongrightarrow> 0) (at_right x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at_right x) \<Longrightarrow>
3.216      eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
3.217      eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
3.218      eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
3.219      eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
3.220 -    ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
3.221 -  ((\<lambda> x. f x / g x) ---> y) (at_right x)"
3.222 +    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_right x) \<Longrightarrow>
3.223 +  ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_right x)"
3.224    unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
3.225    by (rule lhopital_right_0)
3.226
3.227  lemma lhopital_left:
3.228 -  "((f::real \<Rightarrow> real) ---> 0) (at_left x) \<Longrightarrow> (g ---> 0) (at_left x) \<Longrightarrow>
3.229 +  "((f::real \<Rightarrow> real) \<longlongrightarrow> 0) (at_left x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at_left x) \<Longrightarrow>
3.230      eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
3.231      eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
3.232      eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
3.233      eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
3.234 -    ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
3.235 -  ((\<lambda> x. f x / g x) ---> y) (at_left x)"
3.236 +    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_left x) \<Longrightarrow>
3.237 +  ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_left x)"
3.238    unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
3.239    by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
3.240
3.241  lemma lhopital:
3.242 -  "((f::real \<Rightarrow> real) ---> 0) (at x) \<Longrightarrow> (g ---> 0) (at x) \<Longrightarrow>
3.243 +  "((f::real \<Rightarrow> real) \<longlongrightarrow> 0) (at x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at x) \<Longrightarrow>
3.244      eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
3.245      eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
3.246      eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
3.247      eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
3.248 -    ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
3.249 -  ((\<lambda> x. f x / g x) ---> y) (at x)"
3.250 +    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at x) \<Longrightarrow>
3.251 +  ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at x)"
3.252    unfolding eventually_at_split filterlim_at_split
3.253    by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
3.254
3.255 @@ -1730,8 +1730,8 @@
3.256      "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
3.257      "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
3.258      "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
3.259 -  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
3.260 -  shows "((\<lambda> x. f x / g x) ---> x) (at_right 0)"
3.261 +  assumes lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) (at_right 0)"
3.262 +  shows "((\<lambda> x. f x / g x) \<longlongrightarrow> x) (at_right 0)"
3.263    unfolding tendsto_iff
3.264  proof safe
3.265    fix e :: real assume "0 < e"
3.266 @@ -1756,21 +1756,21 @@
3.267      using g_0 by (auto elim: eventually_mono simp: filterlim_at_top_dense)
3.268
3.269    moreover
3.270 -  have inv_g: "((\<lambda>x. inverse (g x)) ---> 0) (at_right 0)"
3.271 +  have inv_g: "((\<lambda>x. inverse (g x)) \<longlongrightarrow> 0) (at_right 0)"
3.272      using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]
3.273      by (rule filterlim_compose)
3.274 -  then have "((\<lambda>x. norm (1 - g a * inverse (g x))) ---> norm (1 - g a * 0)) (at_right 0)"
3.275 +  then have "((\<lambda>x. norm (1 - g a * inverse (g x))) \<longlongrightarrow> norm (1 - g a * 0)) (at_right 0)"
3.276      by (intro tendsto_intros)
3.277 -  then have "((\<lambda>x. norm (1 - g a / g x)) ---> 1) (at_right 0)"
3.278 +  then have "((\<lambda>x. norm (1 - g a / g x)) \<longlongrightarrow> 1) (at_right 0)"
3.279      by (simp add: inverse_eq_divide)
3.280    from this[unfolded tendsto_iff, rule_format, of 1]
3.281    have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
3.282      by (auto elim!: eventually_mono simp: dist_real_def)
3.283
3.284    moreover
3.285 -  from inv_g have "((\<lambda>t. norm ((f a - x * g a) * inverse (g t))) ---> norm ((f a - x * g a) * 0)) (at_right 0)"
3.286 +  from inv_g have "((\<lambda>t. norm ((f a - x * g a) * inverse (g t))) \<longlongrightarrow> norm ((f a - x * g a) * 0)) (at_right 0)"
3.287      by (intro tendsto_intros)
3.288 -  then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) ---> 0) (at_right 0)"
3.289 +  then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) \<longlongrightarrow> 0) (at_right 0)"
3.290      by (simp add: inverse_eq_divide)
3.291    from this[unfolded tendsto_iff, rule_format, of "e / 2"] \<open>0 < e\<close>
3.292    have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
3.293 @@ -1808,8 +1808,8 @@
3.294      eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
3.295      eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
3.296      eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
3.297 -    ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
3.298 -    ((\<lambda> x. f x / g x) ---> y) (at_right x)"
3.299 +    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_right x) \<Longrightarrow>
3.300 +    ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_right x)"
3.301    unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
3.302    by (rule lhopital_right_0_at_top)
3.303
3.304 @@ -1818,8 +1818,8 @@
3.305      eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
3.306      eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
3.307      eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
3.308 -    ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
3.309 -    ((\<lambda> x. f x / g x) ---> y) (at_left x)"
3.310 +    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_left x) \<Longrightarrow>
3.311 +    ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_left x)"
3.312    unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
3.313    by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
3.314
3.315 @@ -1828,8 +1828,8 @@
3.316      eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
3.317      eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
3.318      eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
3.319 -    ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
3.320 -    ((\<lambda> x. f x / g x) ---> y) (at x)"
3.321 +    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at x) \<Longrightarrow>
3.322 +    ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at x)"
3.323    unfolding eventually_at_split filterlim_at_split
3.324    by (auto intro!: lhopital_right_at_top[of g x g' f f'] lhopital_left_at_top[of g x g' f f'])
3.325
3.326 @@ -1839,8 +1839,8 @@
3.327    assumes g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
3.328    assumes Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
3.329    assumes Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
3.330 -  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) at_top"
3.331 -  shows "((\<lambda> x. f x / g x) ---> x) at_top"
3.332 +  assumes lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) at_top"
3.333 +  shows "((\<lambda> x. f x / g x) \<longlongrightarrow> x) at_top"
3.334    unfolding filterlim_at_top_to_right
3.335  proof (rule lhopital_right_0_at_top)
3.336    let ?F = "\<lambda>x. f (inverse x)"
3.337 @@ -1874,7 +1874,7 @@
3.338      using g' eventually_ge_at_top[where c="1::real"]
3.339      by eventually_elim auto
3.340
3.341 -  show "((\<lambda>x. ?D f' x / ?D g' x) ---> x) ?R"
3.342 +  show "((\<lambda>x. ?D f' x / ?D g' x) \<longlongrightarrow> x) ?R"
3.343      unfolding filterlim_at_right_to_top
3.344      apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
3.345      using eventually_ge_at_top[where c="1::real"]
```
```     4.1 --- a/src/HOL/Library/Extended_Real.thy	Tue Dec 29 23:50:44 2015 +0100
4.2 +++ b/src/HOL/Library/Extended_Real.thy	Wed Dec 30 11:21:54 2015 +0100
4.3 @@ -1749,11 +1749,11 @@
4.4    assumes f: "continuous_on s f" shows "continuous_on s (\<lambda>x. ereal (f x))"
4.5    by (rule continuous_on_compose2 [OF continuous_onI_mono[of ereal UNIV] f]) auto
4.6
4.7 -lemma tendsto_ereal[tendsto_intros, simp, intro]: "(f ---> x) F \<Longrightarrow> ((\<lambda>x. ereal (f x)) ---> ereal x) F"
4.8 +lemma tendsto_ereal[tendsto_intros, simp, intro]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. ereal (f x)) \<longlongrightarrow> ereal x) F"
4.9    using isCont_tendsto_compose[of x ereal f F] continuous_on_ereal[of UNIV "\<lambda>x. x"]
4.10    by (simp add: continuous_on_eq_continuous_at)
4.11
4.12 -lemma tendsto_uminus_ereal[tendsto_intros, simp, intro]: "(f ---> x) F \<Longrightarrow> ((\<lambda>x. - f x::ereal) ---> - x) F"
4.13 +lemma tendsto_uminus_ereal[tendsto_intros, simp, intro]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. - f x::ereal) \<longlongrightarrow> - x) F"
4.14    apply (rule tendsto_compose[where g=uminus])
4.15    apply (auto intro!: order_tendstoI simp: eventually_at_topological)
4.16    apply (rule_tac x="{..< -a}" in exI)
4.17 @@ -1770,7 +1770,7 @@
4.18    apply (metis PInfty_neq_ereal(2) ereal_less_eq(3) ereal_top le_cases order_trans)
4.19    done
4.20
4.21 -lemma ereal_Lim_uminus: "(f ---> f0) net \<longleftrightarrow> ((\<lambda>x. - f x::ereal) ---> - f0) net"
4.22 +lemma ereal_Lim_uminus: "(f \<longlongrightarrow> f0) net \<longleftrightarrow> ((\<lambda>x. - f x::ereal) \<longlongrightarrow> - f0) net"
4.23    using tendsto_uminus_ereal[of f f0 net] tendsto_uminus_ereal[of "\<lambda>x. - f x" "- f0" net]
4.24    by auto
4.25
4.26 @@ -1781,10 +1781,10 @@
4.27    by (cases a b c rule: ereal3_cases) (auto simp: field_simps)
4.28
4.29  lemma tendsto_cmult_ereal[tendsto_intros, simp, intro]:
4.30 -  assumes c: "\<bar>c\<bar> \<noteq> \<infinity>" and f: "(f ---> x) F" shows "((\<lambda>x. c * f x::ereal) ---> c * x) F"
4.31 +  assumes c: "\<bar>c\<bar> \<noteq> \<infinity>" and f: "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. c * f x::ereal) \<longlongrightarrow> c * x) F"
4.32  proof -
4.33    { fix c :: ereal assume "0 < c" "c < \<infinity>"
4.34 -    then have "((\<lambda>x. c * f x::ereal) ---> c * x) F"
4.35 +    then have "((\<lambda>x. c * f x::ereal) \<longlongrightarrow> c * x) F"
4.36        apply (intro tendsto_compose[OF _ f])
4.37        apply (auto intro!: order_tendstoI simp: eventually_at_topological)
4.38        apply (rule_tac x="{a/c <..}" in exI)
4.39 @@ -1801,7 +1801,7 @@
4.40      assume "- \<infinity> < c" "c < 0"
4.41      then have "0 < - c" "- c < \<infinity>"
4.42        by (auto simp: ereal_uminus_reorder ereal_less_uminus_reorder[of 0])
4.43 -    then have "((\<lambda>x. (- c) * f x) ---> (- c) * x) F"
4.44 +    then have "((\<lambda>x. (- c) * f x) \<longlongrightarrow> (- c) * x) F"
4.45        by (rule *)
4.46      from tendsto_uminus_ereal[OF this] show ?thesis
4.47        by simp
4.48 @@ -1809,7 +1809,7 @@
4.49  qed
4.50
4.51  lemma tendsto_cmult_ereal_not_0[tendsto_intros, simp, intro]:
4.52 -  assumes "x \<noteq> 0" and f: "(f ---> x) F" shows "((\<lambda>x. c * f x::ereal) ---> c * x) F"
4.53 +  assumes "x \<noteq> 0" and f: "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. c * f x::ereal) \<longlongrightarrow> c * x) F"
4.54  proof cases
4.55    assume "\<bar>c\<bar> = \<infinity>"
4.56    show ?thesis
4.57 @@ -1828,7 +1828,7 @@
4.58  qed (rule tendsto_cmult_ereal[OF _ f])
4.59
4.60  lemma tendsto_cadd_ereal[tendsto_intros, simp, intro]:
4.61 -  assumes c: "y \<noteq> - \<infinity>" "x \<noteq> - \<infinity>" and f: "(f ---> x) F" shows "((\<lambda>x. f x + y::ereal) ---> x + y) F"
4.62 +  assumes c: "y \<noteq> - \<infinity>" "x \<noteq> - \<infinity>" and f: "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. f x + y::ereal) \<longlongrightarrow> x + y) F"
4.63    apply (intro tendsto_compose[OF _ f])
4.64    apply (auto intro!: order_tendstoI simp: eventually_at_topological)
4.65    apply (rule_tac x="{a - y <..}" in exI)
4.66 @@ -1838,7 +1838,7 @@
4.67    done
4.68
4.69  lemma tendsto_add_left_ereal[tendsto_intros, simp, intro]:
4.70 -  assumes c: "\<bar>y\<bar> \<noteq> \<infinity>" and f: "(f ---> x) F" shows "((\<lambda>x. f x + y::ereal) ---> x + y) F"
4.71 +  assumes c: "\<bar>y\<bar> \<noteq> \<infinity>" and f: "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. f x + y::ereal) \<longlongrightarrow> x + y) F"
4.72    apply (intro tendsto_compose[OF _ f])
4.73    apply (auto intro!: order_tendstoI simp: eventually_at_topological)
4.74    apply (rule_tac x="{a - y <..}" in exI)
4.75 @@ -2337,10 +2337,10 @@
4.76
4.77  lemma eventually_finite:
4.78    fixes x :: ereal
4.79 -  assumes "\<bar>x\<bar> \<noteq> \<infinity>" "(f ---> x) F"
4.80 +  assumes "\<bar>x\<bar> \<noteq> \<infinity>" "(f \<longlongrightarrow> x) F"
4.81    shows "eventually (\<lambda>x. \<bar>f x\<bar> \<noteq> \<infinity>) F"
4.82  proof -
4.83 -  have "(f ---> ereal (real_of_ereal x)) F"
4.84 +  have "(f \<longlongrightarrow> ereal (real_of_ereal x)) F"
4.85      using assms by (cases x) auto
4.86    then have "eventually (\<lambda>x. f x \<in> ereal ` UNIV) F"
4.87      by (rule topological_tendstoD) (auto intro: open_ereal)
4.88 @@ -2436,8 +2436,8 @@
4.89  subsubsection \<open>Convergent sequences\<close>
4.90
4.91  lemma lim_real_of_ereal[simp]:
4.92 -  assumes lim: "(f ---> ereal x) net"
4.93 -  shows "((\<lambda>x. real_of_ereal (f x)) ---> x) net"
4.94 +  assumes lim: "(f \<longlongrightarrow> ereal x) net"
4.95 +  shows "((\<lambda>x. real_of_ereal (f x)) \<longlongrightarrow> x) net"
4.96  proof (intro topological_tendstoI)
4.97    fix S
4.98    assume "open S" and "x \<in> S"
4.99 @@ -2447,7 +2447,7 @@
4.100      by (auto intro: eventually_mono [OF lim[THEN topological_tendstoD, OF open_ereal, OF S]])
4.101  qed
4.102
4.103 -lemma lim_ereal[simp]: "((\<lambda>n. ereal (f n)) ---> ereal x) net \<longleftrightarrow> (f ---> x) net"
4.104 +lemma lim_ereal[simp]: "((\<lambda>n. ereal (f n)) \<longlongrightarrow> ereal x) net \<longleftrightarrow> (f \<longlongrightarrow> x) net"
4.105    by (auto dest!: lim_real_of_ereal)
4.106
4.107  lemma convergent_real_imp_convergent_ereal:
4.108 @@ -2460,7 +2460,7 @@
4.109    thus "lim (\<lambda>n. ereal (a n)) = ereal (lim a)" using lim L limI by metis
4.110  qed
4.111
4.112 -lemma tendsto_PInfty: "(f ---> \<infinity>) F \<longleftrightarrow> (\<forall>r. eventually (\<lambda>x. ereal r < f x) F)"
4.113 +lemma tendsto_PInfty: "(f \<longlongrightarrow> \<infinity>) F \<longleftrightarrow> (\<forall>r. eventually (\<lambda>x. ereal r < f x) F)"
4.114  proof -
4.115    {
4.116      fix l :: ereal
4.117 @@ -2473,10 +2473,10 @@
4.118  qed
4.119
4.120  lemma tendsto_PInfty_eq_at_top:
4.121 -  "((\<lambda>z. ereal (f z)) ---> \<infinity>) F \<longleftrightarrow> (LIM z F. f z :> at_top)"
4.122 +  "((\<lambda>z. ereal (f z)) \<longlongrightarrow> \<infinity>) F \<longleftrightarrow> (LIM z F. f z :> at_top)"
4.123    unfolding tendsto_PInfty filterlim_at_top_dense by simp
4.124
4.125 -lemma tendsto_MInfty: "(f ---> -\<infinity>) F \<longleftrightarrow> (\<forall>r. eventually (\<lambda>x. f x < ereal r) F)"
4.126 +lemma tendsto_MInfty: "(f \<longlongrightarrow> -\<infinity>) F \<longleftrightarrow> (\<forall>r. eventually (\<lambda>x. f x < ereal r) F)"
4.127    unfolding tendsto_def
4.128  proof safe
4.129    fix S :: "ereal set"
4.130 @@ -2558,8 +2558,8 @@
4.131  lemma tendsto_ereal_realD:
4.132    fixes f :: "'a \<Rightarrow> ereal"
4.133    assumes "x \<noteq> 0"
4.134 -    and tendsto: "((\<lambda>x. ereal (real_of_ereal (f x))) ---> x) net"
4.135 -  shows "(f ---> x) net"
4.136 +    and tendsto: "((\<lambda>x. ereal (real_of_ereal (f x))) \<longlongrightarrow> x) net"
4.137 +  shows "(f \<longlongrightarrow> x) net"
4.138  proof (intro topological_tendstoI)
4.139    fix S
4.140    assume S: "open S" "x \<in> S"
4.141 @@ -2572,8 +2572,8 @@
4.142
4.143  lemma tendsto_ereal_realI:
4.144    fixes f :: "'a \<Rightarrow> ereal"
4.145 -  assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f ---> x) net"
4.146 -  shows "((\<lambda>x. ereal (real_of_ereal (f x))) ---> x) net"
4.147 +  assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f \<longlongrightarrow> x) net"
4.148 +  shows "((\<lambda>x. ereal (real_of_ereal (f x))) \<longlongrightarrow> x) net"
4.149  proof (intro topological_tendstoI)
4.150    fix S
4.151    assume "open S" and "x \<in> S"
4.152 @@ -2592,15 +2592,15 @@
4.154    fixes x y :: ereal
4.155    assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and y: "\<bar>y\<bar> \<noteq> \<infinity>"
4.156 -  assumes f: "(f ---> x) F" and g: "(g ---> y) F"
4.157 -  shows "((\<lambda>x. f x + g x) ---> x + y) F"
4.158 +  assumes f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
4.159 +  shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
4.160  proof -
4.161    from x obtain r where x': "x = ereal r" by (cases x) auto
4.162 -  with f have "((\<lambda>i. real_of_ereal (f i)) ---> r) F" by simp
4.163 +  with f have "((\<lambda>i. real_of_ereal (f i)) \<longlongrightarrow> r) F" by simp
4.164    moreover
4.165    from y obtain p where y': "y = ereal p" by (cases y) auto
4.166 -  with g have "((\<lambda>i. real_of_ereal (g i)) ---> p) F" by simp
4.167 -  ultimately have "((\<lambda>i. real_of_ereal (f i) + real_of_ereal (g i)) ---> r + p) F"
4.168 +  with g have "((\<lambda>i. real_of_ereal (g i)) \<longlongrightarrow> p) F" by simp
4.169 +  ultimately have "((\<lambda>i. real_of_ereal (f i) + real_of_ereal (g i)) \<longlongrightarrow> r + p) F"
4.170      by (rule tendsto_add)
4.171    moreover
4.172    from eventually_finite[OF x f] eventually_finite[OF y g]
4.173 @@ -3408,7 +3408,7 @@
4.174  lemma Liminf_PInfty:
4.175    fixes f :: "'a \<Rightarrow> ereal"
4.176    assumes "\<not> trivial_limit net"
4.177 -  shows "(f ---> \<infinity>) net \<longleftrightarrow> Liminf net f = \<infinity>"
4.178 +  shows "(f \<longlongrightarrow> \<infinity>) net \<longleftrightarrow> Liminf net f = \<infinity>"
4.179    unfolding tendsto_iff_Liminf_eq_Limsup[OF assms]
4.180    using Liminf_le_Limsup[OF assms, of f]
4.181    by auto
4.182 @@ -3416,7 +3416,7 @@
4.183  lemma Limsup_MInfty:
4.184    fixes f :: "'a \<Rightarrow> ereal"
4.185    assumes "\<not> trivial_limit net"
4.186 -  shows "(f ---> -\<infinity>) net \<longleftrightarrow> Limsup net f = -\<infinity>"
4.187 +  shows "(f \<longlongrightarrow> -\<infinity>) net \<longleftrightarrow> Limsup net f = -\<infinity>"
4.188    unfolding tendsto_iff_Liminf_eq_Limsup[OF assms]
4.189    using Liminf_le_Limsup[OF assms, of f]
4.190    by auto
4.191 @@ -3582,23 +3582,23 @@
4.192    by (auto simp add: ereal_all_split ereal_ex_split)
4.193
4.194  lemma ereal_tendsto_simps1:
4.195 -  "((f \<circ> real_of_ereal) ---> y) (at_left (ereal x)) \<longleftrightarrow> (f ---> y) (at_left x)"
4.196 -  "((f \<circ> real_of_ereal) ---> y) (at_right (ereal x)) \<longleftrightarrow> (f ---> y) (at_right x)"
4.197 -  "((f \<circ> real_of_ereal) ---> y) (at_left (\<infinity>::ereal)) \<longleftrightarrow> (f ---> y) at_top"
4.198 -  "((f \<circ> real_of_ereal) ---> y) (at_right (-\<infinity>::ereal)) \<longleftrightarrow> (f ---> y) at_bot"
4.199 +  "((f \<circ> real_of_ereal) \<longlongrightarrow> y) (at_left (ereal x)) \<longleftrightarrow> (f \<longlongrightarrow> y) (at_left x)"
4.200 +  "((f \<circ> real_of_ereal) \<longlongrightarrow> y) (at_right (ereal x)) \<longleftrightarrow> (f \<longlongrightarrow> y) (at_right x)"
4.201 +  "((f \<circ> real_of_ereal) \<longlongrightarrow> y) (at_left (\<infinity>::ereal)) \<longleftrightarrow> (f \<longlongrightarrow> y) at_top"
4.202 +  "((f \<circ> real_of_ereal) \<longlongrightarrow> y) (at_right (-\<infinity>::ereal)) \<longleftrightarrow> (f \<longlongrightarrow> y) at_bot"
4.203    unfolding tendsto_compose_filtermap at_left_ereal at_right_ereal at_left_PInf at_right_MInf
4.204    by (auto simp: filtermap_filtermap filtermap_ident)
4.205
4.206  lemma ereal_tendsto_simps2:
4.207 -  "((ereal \<circ> f) ---> ereal a) F \<longleftrightarrow> (f ---> a) F"
4.208 -  "((ereal \<circ> f) ---> \<infinity>) F \<longleftrightarrow> (LIM x F. f x :> at_top)"
4.209 -  "((ereal \<circ> f) ---> -\<infinity>) F \<longleftrightarrow> (LIM x F. f x :> at_bot)"
4.210 +  "((ereal \<circ> f) \<longlongrightarrow> ereal a) F \<longleftrightarrow> (f \<longlongrightarrow> a) F"
4.211 +  "((ereal \<circ> f) \<longlongrightarrow> \<infinity>) F \<longleftrightarrow> (LIM x F. f x :> at_top)"
4.212 +  "((ereal \<circ> f) \<longlongrightarrow> -\<infinity>) F \<longleftrightarrow> (LIM x F. f x :> at_bot)"
4.213    unfolding tendsto_PInfty filterlim_at_top_dense tendsto_MInfty filterlim_at_bot_dense
4.214    using lim_ereal by (simp_all add: comp_def)
4.215
4.216  lemma inverse_infty_ereal_tendsto_0: "inverse -- \<infinity> --> (0::ereal)"
4.217  proof -
4.218 -  have **: "((\<lambda>x. ereal (inverse x)) ---> ereal 0) at_infinity"
4.219 +  have **: "((\<lambda>x. ereal (inverse x)) \<longlongrightarrow> ereal 0) at_infinity"
4.220      by (intro tendsto_intros tendsto_inverse_0)
4.221
4.222    show ?thesis
4.223 @@ -3619,7 +3619,7 @@
4.224               intro!: Lim_transform_eventually[OF _ **] t1_space_nhds)
4.225  qed (insert \<open>0 < x\<close>, auto intro!: inverse_infty_ereal_tendsto_0)
4.226
4.227 -lemma inverse_ereal_tendsto_at_right_0: "(inverse ---> \<infinity>) (at_right (0::ereal))"
4.228 +lemma inverse_ereal_tendsto_at_right_0: "(inverse \<longlongrightarrow> \<infinity>) (at_right (0::ereal))"
4.229    unfolding tendsto_compose_filtermap[symmetric] at_right_ereal zero_ereal_def
4.230    by (subst filterlim_cong[OF refl refl, where g="\<lambda>x. ereal (inverse x)"])
4.231       (auto simp: eventually_at_filter tendsto_PInfty_eq_at_top filterlim_inverse_at_top_right)
```
```     5.1 --- a/src/HOL/Library/Liminf_Limsup.thy	Tue Dec 29 23:50:44 2015 +0100
5.2 +++ b/src/HOL/Library/Liminf_Limsup.thy	Wed Dec 30 11:21:54 2015 +0100
5.3 @@ -192,7 +192,7 @@
5.4  lemma lim_imp_Liminf:
5.5    fixes f :: "'a \<Rightarrow> _ :: {complete_linorder,linorder_topology}"
5.6    assumes ntriv: "\<not> trivial_limit F"
5.7 -  assumes lim: "(f ---> f0) F"
5.8 +  assumes lim: "(f \<longlongrightarrow> f0) F"
5.9    shows "Liminf F f = f0"
5.10  proof (intro Liminf_eqI)
5.11    fix P assume P: "eventually P F"
5.12 @@ -231,7 +231,7 @@
5.13  lemma lim_imp_Limsup:
5.14    fixes f :: "'a \<Rightarrow> _ :: {complete_linorder,linorder_topology}"
5.15    assumes ntriv: "\<not> trivial_limit F"
5.16 -  assumes lim: "(f ---> f0) F"
5.17 +  assumes lim: "(f \<longlongrightarrow> f0) F"
5.18    shows "Limsup F f = f0"
5.19  proof (intro Limsup_eqI)
5.20    fix P assume P: "eventually P F"
5.21 @@ -271,7 +271,7 @@
5.22    fixes f0 :: "'a :: {complete_linorder,linorder_topology}"
5.23    assumes ntriv: "\<not> trivial_limit F"
5.24      and lim: "Liminf F f = f0" "Limsup F f = f0"
5.25 -  shows "(f ---> f0) F"
5.26 +  shows "(f \<longlongrightarrow> f0) F"
5.27  proof (rule order_tendstoI)
5.28    fix a assume "f0 < a"
5.29    with assms have "Limsup F f < a" by simp
5.30 @@ -290,7 +290,7 @@
5.31
5.32  lemma tendsto_iff_Liminf_eq_Limsup:
5.33    fixes f0 :: "'a :: {complete_linorder,linorder_topology}"
5.34 -  shows "\<not> trivial_limit F \<Longrightarrow> (f ---> f0) F \<longleftrightarrow> (Liminf F f = f0 \<and> Limsup F f = f0)"
5.35 +  shows "\<not> trivial_limit F \<Longrightarrow> (f \<longlongrightarrow> f0) F \<longleftrightarrow> (Liminf F f = f0 \<and> Limsup F f = f0)"
5.36    by (metis Liminf_eq_Limsup lim_imp_Limsup lim_imp_Liminf)
5.37
5.38  lemma liminf_subseq_mono:
```
```     6.1 --- a/src/HOL/Library/Product_Vector.thy	Tue Dec 29 23:50:44 2015 +0100
6.2 +++ b/src/HOL/Library/Product_Vector.thy	Wed Dec 30 11:21:54 2015 +0100
6.3 @@ -152,8 +152,8 @@
6.4  subsubsection \<open>Continuity of operations\<close>
6.5
6.6  lemma tendsto_fst [tendsto_intros]:
6.7 -  assumes "(f ---> a) F"
6.8 -  shows "((\<lambda>x. fst (f x)) ---> fst a) F"
6.9 +  assumes "(f \<longlongrightarrow> a) F"
6.10 +  shows "((\<lambda>x. fst (f x)) \<longlongrightarrow> fst a) F"
6.11  proof (rule topological_tendstoI)
6.12    fix S assume "open S" and "fst a \<in> S"
6.13    then have "open (fst -` S)" and "a \<in> fst -` S"
6.14 @@ -165,8 +165,8 @@
6.15  qed
6.16
6.17  lemma tendsto_snd [tendsto_intros]:
6.18 -  assumes "(f ---> a) F"
6.19 -  shows "((\<lambda>x. snd (f x)) ---> snd a) F"
6.20 +  assumes "(f \<longlongrightarrow> a) F"
6.21 +  shows "((\<lambda>x. snd (f x)) \<longlongrightarrow> snd a) F"
6.22  proof (rule topological_tendstoI)
6.23    fix S assume "open S" and "snd a \<in> S"
6.24    then have "open (snd -` S)" and "a \<in> snd -` S"
6.25 @@ -178,18 +178,18 @@
6.26  qed
6.27
6.28  lemma tendsto_Pair [tendsto_intros]:
6.29 -  assumes "(f ---> a) F" and "(g ---> b) F"
6.30 -  shows "((\<lambda>x. (f x, g x)) ---> (a, b)) F"
6.31 +  assumes "(f \<longlongrightarrow> a) F" and "(g \<longlongrightarrow> b) F"
6.32 +  shows "((\<lambda>x. (f x, g x)) \<longlongrightarrow> (a, b)) F"
6.33  proof (rule topological_tendstoI)
6.34    fix S assume "open S" and "(a, b) \<in> S"
6.35    then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
6.36      unfolding open_prod_def by fast
6.37    have "eventually (\<lambda>x. f x \<in> A) F"
6.38 -    using \<open>(f ---> a) F\<close> \<open>open A\<close> \<open>a \<in> A\<close>
6.39 +    using \<open>(f \<longlongrightarrow> a) F\<close> \<open>open A\<close> \<open>a \<in> A\<close>
6.40      by (rule topological_tendstoD)
6.41    moreover
6.42    have "eventually (\<lambda>x. g x \<in> B) F"
6.43 -    using \<open>(g ---> b) F\<close> \<open>open B\<close> \<open>b \<in> B\<close>
6.44 +    using \<open>(g \<longlongrightarrow> b) F\<close> \<open>open B\<close> \<open>b \<in> B\<close>
6.45      by (rule topological_tendstoD)
6.46    ultimately
6.47    show "eventually (\<lambda>x. (f x, g x) \<in> S) F"
6.48 @@ -491,7 +491,7 @@
6.49    let ?Rg = "\<lambda>y. g y - g x - g' (y - x)"
6.50    let ?R = "\<lambda>y. ((f y, g y) - (f x, g x) - (f' (y - x), g' (y - x)))"
6.51
6.52 -  show "((\<lambda>y. norm (?Rf y) / norm (y - x) + norm (?Rg y) / norm (y - x)) ---> 0) (at x within s)"
6.53 +  show "((\<lambda>y. norm (?Rf y) / norm (y - x) + norm (?Rg y) / norm (y - x)) \<longlongrightarrow> 0) (at x within s)"
6.54      using f g by (intro tendsto_add_zero) (auto simp: has_derivative_iff_norm)
6.55
6.56    fix y :: 'a assume "y \<noteq> x"
```
```     7.1 --- a/src/HOL/Limits.thy	Tue Dec 29 23:50:44 2015 +0100
7.2 +++ b/src/HOL/Limits.thy	Wed Dec 30 11:21:54 2015 +0100
7.3 @@ -43,7 +43,7 @@
7.4    by (rule filterlim_mono[OF _ at_top_le_at_infinity order_refl])
7.5
7.6  lemma lim_infinity_imp_sequentially:
7.7 -  "(f ---> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) ---> l) sequentially"
7.8 +  "(f \<longlongrightarrow> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) \<longlongrightarrow> l) sequentially"
7.9  by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)
7.10
7.11
7.12 @@ -485,19 +485,19 @@
7.13  lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
7.14  lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
7.15
7.16 -lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
7.17 +lemma tendsto_Zfun_iff: "(f \<longlongrightarrow> a) F = Zfun (\<lambda>x. f x - a) F"
7.18    by (simp only: tendsto_iff Zfun_def dist_norm)
7.19
7.20 -lemma tendsto_0_le: "\<lbrakk>(f ---> 0) F; eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F\<rbrakk>
7.21 -                     \<Longrightarrow> (g ---> 0) F"
7.22 +lemma tendsto_0_le: "\<lbrakk>(f \<longlongrightarrow> 0) F; eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F\<rbrakk>
7.23 +                     \<Longrightarrow> (g \<longlongrightarrow> 0) F"
7.24    by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)
7.25
7.26  subsubsection \<open>Distance and norms\<close>
7.27
7.28  lemma tendsto_dist [tendsto_intros]:
7.29    fixes l m :: "'a :: metric_space"
7.30 -  assumes f: "(f ---> l) F" and g: "(g ---> m) F"
7.31 -  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
7.32 +  assumes f: "(f \<longlongrightarrow> l) F" and g: "(g \<longlongrightarrow> m) F"
7.33 +  shows "((\<lambda>x. dist (f x) (g x)) \<longlongrightarrow> dist l m) F"
7.34  proof (rule tendstoI)
7.35    fix e :: real assume "0 < e"
7.36    hence e2: "0 < e/2" by simp
7.37 @@ -526,7 +526,7 @@
7.38    unfolding continuous_on_def by (auto intro: tendsto_dist)
7.39
7.40  lemma tendsto_norm [tendsto_intros]:
7.41 -  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
7.42 +  "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> norm a) F"
7.43    unfolding norm_conv_dist by (intro tendsto_intros)
7.44
7.45  lemma continuous_norm [continuous_intros]:
7.46 @@ -538,19 +538,19 @@
7.47    unfolding continuous_on_def by (auto intro: tendsto_norm)
7.48
7.49  lemma tendsto_norm_zero:
7.50 -  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
7.51 +  "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F"
7.52    by (drule tendsto_norm, simp)
7.53
7.54  lemma tendsto_norm_zero_cancel:
7.55 -  "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
7.56 +  "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
7.57    unfolding tendsto_iff dist_norm by simp
7.58
7.59  lemma tendsto_norm_zero_iff:
7.60 -  "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
7.61 +  "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
7.62    unfolding tendsto_iff dist_norm by simp
7.63
7.64  lemma tendsto_rabs [tendsto_intros]:
7.65 -  "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
7.66 +  "(f \<longlongrightarrow> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> \<bar>l\<bar>) F"
7.67    by (fold real_norm_def, rule tendsto_norm)
7.68
7.69  lemma continuous_rabs [continuous_intros]:
7.70 @@ -562,22 +562,22 @@
7.71    unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
7.72
7.73  lemma tendsto_rabs_zero:
7.74 -  "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
7.75 +  "(f \<longlongrightarrow> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F"
7.76    by (fold real_norm_def, rule tendsto_norm_zero)
7.77
7.78  lemma tendsto_rabs_zero_cancel:
7.79 -  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
7.80 +  "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
7.81    by (fold real_norm_def, rule tendsto_norm_zero_cancel)
7.82
7.83  lemma tendsto_rabs_zero_iff:
7.84 -  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
7.85 +  "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
7.86    by (fold real_norm_def, rule tendsto_norm_zero_iff)
7.87
7.88  subsubsection \<open>Addition and subtraction\<close>
7.89
7.90  lemma tendsto_add [tendsto_intros]:
7.91    fixes a b :: "'a::real_normed_vector"
7.92 -  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
7.93 +  shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> a + b) F"
7.95
7.96  lemma continuous_add [continuous_intros]:
7.97 @@ -592,12 +592,12 @@
7.98
7.100    fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
7.101 -  shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
7.102 +  shows "\<lbrakk>(f \<longlongrightarrow> 0) F; (g \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> 0) F"
7.103    by (drule (1) tendsto_add, simp)
7.104
7.105  lemma tendsto_minus [tendsto_intros]:
7.106    fixes a :: "'a::real_normed_vector"
7.107 -  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
7.108 +  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> - a) F"
7.109    by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
7.110
7.111  lemma continuous_minus [continuous_intros]:
7.112 @@ -612,17 +612,17 @@
7.113
7.114  lemma tendsto_minus_cancel:
7.115    fixes a :: "'a::real_normed_vector"
7.116 -  shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
7.117 +  shows "((\<lambda>x. - f x) \<longlongrightarrow> - a) F \<Longrightarrow> (f \<longlongrightarrow> a) F"
7.118    by (drule tendsto_minus, simp)
7.119
7.120  lemma tendsto_minus_cancel_left:
7.121 -    "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
7.122 +    "(f \<longlongrightarrow> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> y) F"
7.123    using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
7.124    by auto
7.125
7.126  lemma tendsto_diff [tendsto_intros]:
7.127    fixes a b :: "'a::real_normed_vector"
7.128 -  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
7.129 +  shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> a - b) F"
7.130    using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)
7.131
7.132  lemma continuous_diff [continuous_intros]:
7.133 @@ -640,8 +640,8 @@
7.134
7.135  lemma tendsto_setsum [tendsto_intros]:
7.136    fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
7.137 -  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
7.138 -  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
7.139 +  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> a i) F"
7.140 +  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) \<longlongrightarrow> (\<Sum>i\<in>S. a i)) F"
7.141  proof (cases "finite S")
7.142    assume "finite S" thus ?thesis using assms
7.143      by (induct, simp, simp add: tendsto_add)
7.144 @@ -666,7 +666,7 @@
7.145    by (auto simp: linearI distrib_left)
7.146
7.147  lemma (in bounded_linear) tendsto:
7.148 -  "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
7.149 +  "(g \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
7.150    by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
7.151
7.152  lemma (in bounded_linear) continuous:
7.153 @@ -678,11 +678,11 @@
7.154    using tendsto[of g] by (auto simp: continuous_on_def)
7.155
7.156  lemma (in bounded_linear) tendsto_zero:
7.157 -  "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
7.158 +  "(g \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> 0) F"
7.159    by (drule tendsto, simp only: zero)
7.160
7.161  lemma (in bounded_bilinear) tendsto:
7.162 -  "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
7.163 +  "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) \<longlongrightarrow> a ** b) F"
7.164    by (simp only: tendsto_Zfun_iff prod_diff_prod
7.165                   Zfun_add Zfun Zfun_left Zfun_right)
7.166
7.167 @@ -695,17 +695,17 @@
7.168    using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
7.169
7.170  lemma (in bounded_bilinear) tendsto_zero:
7.171 -  assumes f: "(f ---> 0) F"
7.172 -  assumes g: "(g ---> 0) F"
7.173 -  shows "((\<lambda>x. f x ** g x) ---> 0) F"
7.174 +  assumes f: "(f \<longlongrightarrow> 0) F"
7.175 +  assumes g: "(g \<longlongrightarrow> 0) F"
7.176 +  shows "((\<lambda>x. f x ** g x) \<longlongrightarrow> 0) F"
7.177    using tendsto [OF f g] by (simp add: zero_left)
7.178
7.179  lemma (in bounded_bilinear) tendsto_left_zero:
7.180 -  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
7.181 +  "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) \<longlongrightarrow> 0) F"
7.182    by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
7.183
7.184  lemma (in bounded_bilinear) tendsto_right_zero:
7.185 -  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
7.186 +  "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) \<longlongrightarrow> 0) F"
7.187    by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
7.188
7.189  lemmas tendsto_of_real [tendsto_intros] =
7.190 @@ -719,12 +719,12 @@
7.191
7.192  lemma tendsto_mult_left:
7.193    fixes c::"'a::real_normed_algebra"
7.194 -  shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) ---> c * l) F"
7.195 +  shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) \<longlongrightarrow> c * l) F"
7.196  by (rule tendsto_mult [OF tendsto_const])
7.197
7.198  lemma tendsto_mult_right:
7.199    fixes c::"'a::real_normed_algebra"
7.200 -  shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) ---> l * c) F"
7.201 +  shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) \<longlongrightarrow> l * c) F"
7.202  by (rule tendsto_mult [OF _ tendsto_const])
7.203
7.204  lemmas continuous_of_real [continuous_intros] =
7.205 @@ -756,7 +756,7 @@
7.206
7.207  lemma tendsto_power [tendsto_intros]:
7.208    fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
7.209 -  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
7.210 +  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) \<longlongrightarrow> a ^ n) F"
7.211    by (induct n) (simp_all add: tendsto_mult)
7.212
7.213  lemma continuous_power [continuous_intros]:
7.214 @@ -771,8 +771,8 @@
7.215
7.216  lemma tendsto_setprod [tendsto_intros]:
7.217    fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
7.218 -  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
7.219 -  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
7.220 +  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> L i) F"
7.221 +  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) \<longlongrightarrow> (\<Prod>i\<in>S. L i)) F"
7.222  proof (cases "finite S")
7.223    assume "finite S" thus ?thesis using assms
7.224      by (induct, simp, simp add: tendsto_mult)
7.225 @@ -789,11 +789,11 @@
7.226    unfolding continuous_on_def by (auto intro: tendsto_setprod)
7.227
7.228  lemma tendsto_of_real_iff:
7.229 -  "((\<lambda>x. of_real (f x) :: 'a :: real_normed_div_algebra) ---> of_real c) F \<longleftrightarrow> (f ---> c) F"
7.230 +  "((\<lambda>x. of_real (f x) :: 'a :: real_normed_div_algebra) \<longlongrightarrow> of_real c) F \<longleftrightarrow> (f \<longlongrightarrow> c) F"
7.231    unfolding tendsto_iff by simp
7.232
7.234 -  "((\<lambda>x. c + f x :: 'a :: real_normed_vector) ---> c + d) F \<longleftrightarrow> (f ---> d) F"
7.235 +  "((\<lambda>x. c + f x :: 'a :: real_normed_vector) \<longlongrightarrow> c + d) F \<longleftrightarrow> (f \<longlongrightarrow> d) F"
7.236    using tendsto_add[OF tendsto_const[of c], of f d]
7.237          tendsto_add[OF tendsto_const[of "-c"], of "\<lambda>x. c + f x" "c + d"] by auto
7.238
7.239 @@ -842,7 +842,7 @@
7.240
7.241  lemma Bfun_inverse:
7.242    fixes a :: "'a::real_normed_div_algebra"
7.243 -  assumes f: "(f ---> a) F"
7.244 +  assumes f: "(f \<longlongrightarrow> a) F"
7.245    assumes a: "a \<noteq> 0"
7.246    shows "Bfun (\<lambda>x. inverse (f x)) F"
7.247  proof -
7.248 @@ -877,9 +877,9 @@
7.249
7.250  lemma tendsto_inverse [tendsto_intros]:
7.251    fixes a :: "'a::real_normed_div_algebra"
7.252 -  assumes f: "(f ---> a) F"
7.253 +  assumes f: "(f \<longlongrightarrow> a) F"
7.254    assumes a: "a \<noteq> 0"
7.255 -  shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
7.256 +  shows "((\<lambda>x. inverse (f x)) \<longlongrightarrow> inverse a) F"
7.257  proof -
7.258    from a have "0 < norm a" by simp
7.259    with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
7.260 @@ -923,8 +923,8 @@
7.261
7.262  lemma tendsto_divide [tendsto_intros]:
7.263    fixes a b :: "'a::real_normed_field"
7.264 -  shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
7.265 -    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
7.266 +  shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F; b \<noteq> 0\<rbrakk>
7.267 +    \<Longrightarrow> ((\<lambda>x. f x / g x) \<longlongrightarrow> a / b) F"
7.268    by (simp add: tendsto_mult tendsto_inverse divide_inverse)
7.269
7.270  lemma continuous_divide:
7.271 @@ -953,7 +953,7 @@
7.272
7.273  lemma tendsto_sgn [tendsto_intros]:
7.274    fixes l :: "'a::real_normed_vector"
7.275 -  shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
7.276 +  shows "\<lbrakk>(f \<longlongrightarrow> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) \<longlongrightarrow> sgn l) F"
7.277    unfolding sgn_div_norm by (simp add: tendsto_intros)
7.278
7.279  lemma continuous_sgn:
7.280 @@ -1001,7 +1001,7 @@
7.281
7.282  lemma not_tendsto_and_filterlim_at_infinity:
7.283    assumes "F \<noteq> bot"
7.284 -  assumes "(f ---> (c :: 'a :: real_normed_vector)) F"
7.285 +  assumes "(f \<longlongrightarrow> (c :: 'a :: real_normed_vector)) F"
7.286    assumes "filterlim f at_infinity F"
7.287    shows   False
7.288  proof -
7.289 @@ -1142,7 +1142,7 @@
7.290
7.291  lemma tendsto_inverse_0:
7.292    fixes x :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
7.293 -  shows "(inverse ---> (0::'a)) at_infinity"
7.294 +  shows "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
7.295    unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
7.296  proof safe
7.297    fix r :: real assume "0 < r"
7.298 @@ -1157,12 +1157,12 @@
7.299  qed
7.300
7.302 -  assumes "(f ---> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
7.303 +  assumes "(f \<longlongrightarrow> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
7.304    assumes "filterlim g at_infinity F"
7.305    shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"
7.306  proof (subst filterlim_at_infinity[OF order_refl], safe)
7.307    fix r :: real assume r: "r > 0"
7.308 -  from assms(1) have "((\<lambda>x. norm (f x)) ---> norm c) F" by (rule tendsto_norm)
7.309 +  from assms(1) have "((\<lambda>x. norm (f x)) \<longlongrightarrow> norm c) F" by (rule tendsto_norm)
7.310    hence "eventually (\<lambda>x. norm (f x) < norm c + 1) F" by (rule order_tendstoD) simp_all
7.311    moreover from r have "r + norm c + 1 > 0" by (intro add_pos_nonneg) simp_all
7.312    with assms(2) have "eventually (\<lambda>x. norm (g x) \<ge> r + norm c + 1) F"
7.313 @@ -1178,7 +1178,7 @@
7.314
7.316    assumes "filterlim f at_infinity F"
7.317 -  assumes "(g ---> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
7.318 +  assumes "(g \<longlongrightarrow> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
7.319    shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"
7.320    by (subst add.commute) (rule tendsto_add_filterlim_at_infinity assms)+
7.321
7.322 @@ -1188,7 +1188,7 @@
7.323       (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
7.324
7.325  lemma filterlim_inverse_at_top:
7.326 -  "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
7.327 +  "(f \<longlongrightarrow> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
7.328    by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
7.329       (simp add: filterlim_def eventually_filtermap eventually_mono at_within_def le_principal)
7.330
7.331 @@ -1197,7 +1197,7 @@
7.332    by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
7.333
7.334  lemma filterlim_inverse_at_bot:
7.335 -  "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
7.336 +  "(f \<longlongrightarrow> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
7.337    unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
7.338    by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
7.339
7.340 @@ -1258,11 +1258,11 @@
7.341  qed
7.342
7.343  lemma tendsto_mult_filterlim_at_infinity:
7.344 -  assumes "F \<noteq> bot" "(f ---> (c :: 'a :: real_normed_field)) F" "c \<noteq> 0"
7.345 +  assumes "F \<noteq> bot" "(f \<longlongrightarrow> (c :: 'a :: real_normed_field)) F" "c \<noteq> 0"
7.346    assumes "filterlim g at_infinity F"
7.347    shows   "filterlim (\<lambda>x. f x * g x) at_infinity F"
7.348  proof -
7.349 -  have "((\<lambda>x. inverse (f x) * inverse (g x)) ---> inverse c * 0) F"
7.350 +  have "((\<lambda>x. inverse (f x) * inverse (g x)) \<longlongrightarrow> inverse c * 0) F"
7.351      by (intro tendsto_mult tendsto_inverse assms filterlim_compose[OF tendsto_inverse_0])
7.352    hence "filterlim (\<lambda>x. inverse (f x) * inverse (g x)) (at (inverse c * 0)) F"
7.353      unfolding filterlim_at using assms
7.354 @@ -1270,7 +1270,7 @@
7.355    thus ?thesis by (subst filterlim_inverse_at_iff[symmetric]) simp_all
7.356  qed
7.357
7.358 -lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
7.359 +lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) \<longlongrightarrow> 0) F"
7.360   by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
7.361
7.362  lemma mult_nat_left_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. c * x :: nat) at_top sequentially"
7.363 @@ -1283,7 +1283,7 @@
7.364    fixes x :: "'a :: {real_normed_field,field}"
7.365    shows "(at (0::'a)) = filtermap inverse at_infinity"
7.366  proof (rule antisym)
7.367 -  have "(inverse ---> (0::'a)) at_infinity"
7.368 +  have "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
7.369      by (fact tendsto_inverse_0)
7.370    then show "filtermap inverse at_infinity \<le> at (0::'a)"
7.371      apply (simp add: le_principal eventually_filtermap eventually_at_infinity filterlim_def at_within_def)
7.372 @@ -1299,12 +1299,12 @@
7.373
7.374  lemma lim_at_infinity_0:
7.375    fixes l :: "'a :: {real_normed_field,field}"
7.376 -  shows "(f ---> l) at_infinity \<longleftrightarrow> ((f o inverse) ---> l) (at (0::'a))"
7.377 +  shows "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> ((f o inverse) \<longlongrightarrow> l) (at (0::'a))"
7.378  by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)
7.379
7.380  lemma lim_zero_infinity:
7.381    fixes l :: "'a :: {real_normed_field,field}"
7.382 -  shows "((\<lambda>x. f(1 / x)) ---> l) (at (0::'a)) \<Longrightarrow> (f ---> l) at_infinity"
7.383 +  shows "((\<lambda>x. f(1 / x)) \<longlongrightarrow> l) (at (0::'a)) \<Longrightarrow> (f \<longlongrightarrow> l) at_infinity"
7.384  by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)
7.385
7.386
7.387 @@ -1316,7 +1316,7 @@
7.388  \<close>
7.389
7.390  lemma filterlim_tendsto_pos_mult_at_top:
7.391 -  assumes f: "(f ---> c) F" and c: "0 < c"
7.392 +  assumes f: "(f \<longlongrightarrow> c) F" and c: "0 < c"
7.393    assumes g: "LIM x F. g x :> at_top"
7.394    shows "LIM x F. (f x * g x :: real) :> at_top"
7.395    unfolding filterlim_at_top_gt[where c=0]
7.396 @@ -1359,13 +1359,13 @@
7.397  qed
7.398
7.399  lemma filterlim_tendsto_pos_mult_at_bot:
7.400 -  assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
7.401 +  assumes "(f \<longlongrightarrow> c) F" "0 < (c::real)" "filterlim g at_bot F"
7.402    shows "LIM x F. f x * g x :> at_bot"
7.403    using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
7.404    unfolding filterlim_uminus_at_bot by simp
7.405
7.406  lemma filterlim_tendsto_neg_mult_at_bot:
7.407 -  assumes c: "(f ---> c) F" "(c::real) < 0" and g: "filterlim g at_top F"
7.408 +  assumes c: "(f \<longlongrightarrow> c) F" "(c::real) < 0" and g: "filterlim g at_top F"
7.409    shows "LIM x F. f x * g x :> at_bot"
7.410    using c filterlim_tendsto_pos_mult_at_top[of "\<lambda>x. - f x" "- c" F, OF _ _ g]
7.411    unfolding filterlim_uminus_at_bot tendsto_minus_cancel_left by simp
7.412 @@ -1390,7 +1390,7 @@
7.413    using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_bot)
7.414
7.416 -  assumes f: "(f ---> c) F"
7.417 +  assumes f: "(f \<longlongrightarrow> c) F"
7.418    assumes g: "LIM x F. g x :> at_top"
7.419    shows "LIM x F. (f x + g x :: real) :> at_top"
7.420    unfolding filterlim_at_top_gt[where c=0]
7.421 @@ -1406,8 +1406,8 @@
7.422
7.423  lemma LIM_at_top_divide:
7.424    fixes f g :: "'a \<Rightarrow> real"
7.425 -  assumes f: "(f ---> a) F" "0 < a"
7.426 -  assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
7.427 +  assumes f: "(f \<longlongrightarrow> a) F" "0 < a"
7.428 +  assumes g: "(g \<longlongrightarrow> 0) F" "eventually (\<lambda>x. 0 < g x) F"
7.429    shows "LIM x F. f x / g x :> at_top"
7.430    unfolding divide_inverse
7.431    by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
7.432 @@ -1429,9 +1429,9 @@
7.433
7.434  lemma tendsto_divide_0:
7.435    fixes f :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
7.436 -  assumes f: "(f ---> c) F"
7.437 +  assumes f: "(f \<longlongrightarrow> c) F"
7.438    assumes g: "LIM x F. g x :> at_infinity"
7.439 -  shows "((\<lambda>x. f x / g x) ---> 0) F"
7.440 +  shows "((\<lambda>x. f x / g x) \<longlongrightarrow> 0) F"
7.441    using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
7.442
7.443  lemma linear_plus_1_le_power:
7.444 @@ -1510,16 +1510,16 @@
7.445
7.446  lemma Lim_transform:
7.447    fixes a b :: "'a::real_normed_vector"
7.448 -  shows "\<lbrakk>(g ---> a) F; ((\<lambda>x. f x - g x) ---> 0) F\<rbrakk> \<Longrightarrow> (f ---> a) F"
7.449 +  shows "\<lbrakk>(g \<longlongrightarrow> a) F; ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> a) F"
7.450    using tendsto_add [of g a F "\<lambda>x. f x - g x" 0] by simp
7.451
7.452  lemma Lim_transform2:
7.453    fixes a b :: "'a::real_normed_vector"
7.454 -  shows "\<lbrakk>(f ---> a) F; ((\<lambda>x. f x - g x) ---> 0) F\<rbrakk> \<Longrightarrow> (g ---> a) F"
7.455 +  shows "\<lbrakk>(f \<longlongrightarrow> a) F; ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> (g \<longlongrightarrow> a) F"
7.456    by (erule Lim_transform) (simp add: tendsto_minus_cancel)
7.457
7.458  lemma Lim_transform_eventually:
7.459 -  "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
7.460 +  "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> (g \<longlongrightarrow> l) net"
7.461    apply (rule topological_tendstoI)
7.462    apply (drule (2) topological_tendstoD)
7.463    apply (erule (1) eventually_elim2, simp)
7.464 @@ -1528,19 +1528,19 @@
7.465  lemma Lim_transform_within:
7.466    assumes "0 < d"
7.467      and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
7.468 -    and "(f ---> l) (at x within S)"
7.469 -  shows "(g ---> l) (at x within S)"
7.470 +    and "(f \<longlongrightarrow> l) (at x within S)"
7.471 +  shows "(g \<longlongrightarrow> l) (at x within S)"
7.472  proof (rule Lim_transform_eventually)
7.473    show "eventually (\<lambda>x. f x = g x) (at x within S)"
7.474      using assms(1,2) by (auto simp: dist_nz eventually_at)
7.475 -  show "(f ---> l) (at x within S)" by fact
7.476 +  show "(f \<longlongrightarrow> l) (at x within S)" by fact
7.477  qed
7.478
7.479  lemma Lim_transform_at:
7.480    assumes "0 < d"
7.481      and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
7.482 -    and "(f ---> l) (at x)"
7.483 -  shows "(g ---> l) (at x)"
7.484 +    and "(f \<longlongrightarrow> l) (at x)"
7.485 +  shows "(g \<longlongrightarrow> l) (at x)"
7.486    using _ assms(3)
7.487  proof (rule Lim_transform_eventually)
7.488    show "eventually (\<lambda>x. f x = g x) (at x)"
7.489 @@ -1554,10 +1554,10 @@
7.490    fixes a b :: "'a::t1_space"
7.491    assumes "a \<noteq> b"
7.492      and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
7.493 -    and "(f ---> l) (at a within S)"
7.494 -  shows "(g ---> l) (at a within S)"
7.495 +    and "(f \<longlongrightarrow> l) (at a within S)"
7.496 +  shows "(g \<longlongrightarrow> l) (at a within S)"
7.497  proof (rule Lim_transform_eventually)
7.498 -  show "(f ---> l) (at a within S)" by fact
7.499 +  show "(f \<longlongrightarrow> l) (at a within S)" by fact
7.500    show "eventually (\<lambda>x. f x = g x) (at a within S)"
7.501      unfolding eventually_at_topological
7.502      by (rule exI [where x="- {b}"], simp add: open_Compl assms)
7.503 @@ -1567,8 +1567,8 @@
7.504    fixes a b :: "'a::t1_space"
7.505    assumes ab: "a\<noteq>b"
7.506      and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
7.507 -    and fl: "(f ---> l) (at a)"
7.508 -  shows "(g ---> l) (at a)"
7.509 +    and fl: "(f \<longlongrightarrow> l) (at a)"
7.510 +  shows "(g \<longlongrightarrow> l) (at a)"
7.511    using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp
7.512
7.513  text\<open>Alternatively, within an open set.\<close>
7.514 @@ -1576,13 +1576,13 @@
7.515  lemma Lim_transform_within_open:
7.516    assumes "open S" and "a \<in> S"
7.517      and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
7.518 -    and "(f ---> l) (at a)"
7.519 -  shows "(g ---> l) (at a)"
7.520 +    and "(f \<longlongrightarrow> l) (at a)"
7.521 +  shows "(g \<longlongrightarrow> l) (at a)"
7.522  proof (rule Lim_transform_eventually)
7.523    show "eventually (\<lambda>x. f x = g x) (at a)"
7.524      unfolding eventually_at_topological
7.525      using assms(1,2,3) by auto
7.526 -  show "(f ---> l) (at a)" by fact
7.527 +  show "(f \<longlongrightarrow> l) (at a)" by fact
7.528  qed
7.529
7.530  text\<open>A congruence rule allowing us to transform limits assuming not at point.\<close>
7.531 @@ -1594,14 +1594,14 @@
7.532      and "x = y"
7.533      and "S = T"
7.534      and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
7.535 -  shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
7.536 +  shows "(f \<longlongrightarrow> x) (at a within S) \<longleftrightarrow> (g \<longlongrightarrow> y) (at b within T)"
7.537    unfolding tendsto_def eventually_at_topological
7.538    using assms by simp
7.539
7.540  lemma Lim_cong_at(*[cong add]*):
7.541    assumes "a = b" "x = y"
7.542      and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
7.543 -  shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
7.544 +  shows "((\<lambda>x. f x) \<longlongrightarrow> x) (at a) \<longleftrightarrow> ((g \<longlongrightarrow> y) (at a))"
7.545    unfolding tendsto_def eventually_at_topological
7.546    using assms by simp
7.547  text\<open>An unbounded sequence's inverse tends to 0\<close>
7.548 @@ -1636,7 +1636,7 @@
7.549    using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
7.550    by auto
7.551
7.552 -lemma lim_1_over_n: "((\<lambda>n. 1 / of_nat n) ---> (0::'a::real_normed_field)) sequentially"
7.553 +lemma lim_1_over_n: "((\<lambda>n. 1 / of_nat n) \<longlongrightarrow> (0::'a::real_normed_field)) sequentially"
7.554  proof (subst lim_sequentially, intro allI impI exI)
7.555    fix e :: real assume e: "e > 0"
7.556    fix n :: nat assume n: "n \<ge> nat \<lceil>inverse e + 1\<rceil>"
7.557 @@ -1646,7 +1646,7 @@
7.558      by (simp add: divide_simps mult.commute norm_conv_dist[symmetric] norm_divide)
7.559  qed
7.560
7.561 -lemma lim_inverse_n: "((\<lambda>n. inverse(of_nat n)) ---> (0::'a::real_normed_field)) sequentially"
7.562 +lemma lim_inverse_n: "((\<lambda>n. inverse(of_nat n)) \<longlongrightarrow> (0::'a::real_normed_field)) sequentially"
7.563    using lim_1_over_n by (simp add: inverse_eq_divide)
7.564
7.565  lemma LIMSEQ_Suc_n_over_n: "(\<lambda>n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
7.566 @@ -1979,17 +1979,17 @@
7.567
7.568  lemma LIM_zero:
7.569    fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
7.570 -  shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. f x - l) ---> 0) F"
7.571 +  shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. f x - l) \<longlongrightarrow> 0) F"
7.572  unfolding tendsto_iff dist_norm by simp
7.573
7.574  lemma LIM_zero_cancel:
7.575    fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
7.576 -  shows "((\<lambda>x. f x - l) ---> 0) F \<Longrightarrow> (f ---> l) F"
7.577 +  shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> l) F"
7.578  unfolding tendsto_iff dist_norm by simp
7.579
7.580  lemma LIM_zero_iff:
7.581    fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
7.582 -  shows "((\<lambda>x. f x - l) ---> 0) F = (f ---> l) F"
7.583 +  shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> l) F"
7.584  unfolding tendsto_iff dist_norm by simp
7.585
7.586  lemma LIM_imp_LIM:
7.587 @@ -2152,7 +2152,7 @@
7.588    assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
7.589    shows "0 \<le> f x"
7.590  proof (rule tendsto_le_const)
7.591 -  show "(f ---> f x) (at_left x)"
7.592 +  show "(f \<longlongrightarrow> f x) (at_left x)"
7.593      using \<open>isCont f x\<close> by (simp add: filterlim_at_split isCont_def)
7.594    show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
7.595      using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])
```
```     8.1 --- a/src/HOL/Multivariate_Analysis/Bounded_Linear_Function.thy	Tue Dec 29 23:50:44 2015 +0100
8.2 +++ b/src/HOL/Multivariate_Analysis/Bounded_Linear_Function.thy	Wed Dec 30 11:21:54 2015 +0100
8.3 @@ -355,8 +355,8 @@
8.4  lemma tendsto_componentwise1:
8.5    fixes a::"'a::euclidean_space \<Rightarrow>\<^sub>L 'b::real_normed_vector"
8.6      and b::"'c \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
8.7 -  assumes "(\<And>j. j \<in> Basis \<Longrightarrow> ((\<lambda>n. b n j) ---> a j) F)"
8.8 -  shows "(b ---> a) F"
8.9 +  assumes "(\<And>j. j \<in> Basis \<Longrightarrow> ((\<lambda>n. b n j) \<longlongrightarrow> a j) F)"
8.10 +  shows "(b \<longlongrightarrow> a) F"
8.11  proof -
8.12    have "\<And>j. j \<in> Basis \<Longrightarrow> Zfun (\<lambda>x. norm (b x j - a j)) F"
8.13      using assms unfolding tendsto_Zfun_iff Zfun_norm_iff .
8.14 @@ -408,8 +408,8 @@
8.15    done
8.16
8.17  lemma tendsto_blinfun_of_matrix:
8.18 -  assumes "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> ((\<lambda>n. b n i j) ---> a i j) F"
8.19 -  shows "((\<lambda>n. blinfun_of_matrix (b n)) ---> blinfun_of_matrix a) F"
8.20 +  assumes "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> ((\<lambda>n. b n i j) \<longlongrightarrow> a i j) F"
8.21 +  shows "((\<lambda>n. blinfun_of_matrix (b n)) \<longlongrightarrow> blinfun_of_matrix a) F"
8.22  proof -
8.23    have "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> Zfun (\<lambda>x. norm (b x i j - a i j)) F"
8.24      using assms unfolding tendsto_Zfun_iff Zfun_norm_iff .
8.25 @@ -423,7 +423,7 @@
8.26  lemma tendsto_componentwise:
8.27    fixes a::"'a::euclidean_space \<Rightarrow>\<^sub>L 'b::euclidean_space"
8.28      and b::"'c \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
8.29 -  shows "(\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> ((\<lambda>n. b n j \<bullet> i) ---> a j \<bullet> i) F) \<Longrightarrow> (b ---> a) F"
8.30 +  shows "(\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> ((\<lambda>n. b n j \<bullet> i) \<longlongrightarrow> a j \<bullet> i) F) \<Longrightarrow> (b \<longlongrightarrow> a) F"
8.31    apply (subst blinfun_of_matrix_works[of a, symmetric])
8.32    apply (subst blinfun_of_matrix_works[of "b x" for x, symmetric, abs_def])
8.33    by (rule tendsto_blinfun_of_matrix)
8.34 @@ -495,7 +495,7 @@
8.35        by (metis (lifting) bounded_subset f' image_subsetI s')
8.36      then obtain l2 r2
8.37        where r2: "subseq r2"
8.38 -      and lr2: "((\<lambda>i. f (r1 (r2 i)) k) ---> l2) sequentially"
8.39 +      and lr2: "((\<lambda>i. f (r1 (r2 i)) k) \<longlongrightarrow> l2) sequentially"
8.40        using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) k"]
8.41        by (auto simp: o_def)
8.42      def r \<equiv> "r1 \<circ> r2"
8.43 @@ -585,9 +585,9 @@
8.44      ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
8.45        using eventually_elim2 by force
8.46    }
8.47 -  then have *: "((f \<circ> r) ---> l) sequentially"
8.48 +  then have *: "((f \<circ> r) \<longlongrightarrow> l) sequentially"
8.49      unfolding o_def tendsto_iff by simp
8.50 -  with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
8.51 +  with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
8.52      by auto
8.53  qed
8.54
```
```     9.1 --- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy	Tue Dec 29 23:50:44 2015 +0100
9.2 +++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy	Wed Dec 30 11:21:54 2015 +0100
9.3 @@ -842,7 +842,7 @@
9.4      have "bounded (range (\<lambda>i. f (r1 i) \$ k))"
9.5        by (metis (lifting) bounded_subset image_subsetI f' s')
9.6      then obtain l2 r2 where r2: "subseq r2"
9.7 -      and lr2: "((\<lambda>i. f (r1 (r2 i)) \$ k) ---> l2) sequentially"
9.8 +      and lr2: "((\<lambda>i. f (r1 (r2 i)) \$ k) \<longlongrightarrow> l2) sequentially"
9.9        using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \$ k"] by (auto simp: o_def)
9.10      def r \<equiv> "r1 \<circ> r2"
9.11      have r: "subseq r"
9.12 @@ -888,8 +888,8 @@
9.13      ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
9.14        by (rule eventually_mono)
9.15    }
9.16 -  hence "((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
9.17 -  with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
9.18 +  hence "((f \<circ> r) \<longlongrightarrow> l) sequentially" unfolding o_def tendsto_iff by simp
9.19 +  with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto
9.20  qed
9.21
9.22  lemma interval_cart:
9.23 @@ -1014,19 +1014,19 @@
9.24
9.25  lemma Lim_component_le_cart:
9.26    fixes f :: "'a \<Rightarrow> real^'n"
9.27 -  assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x \$i \<le> b) net"
9.28 +  assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x \$i \<le> b) net"
9.29    shows "l\$i \<le> b"
9.30    by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
9.31
9.32  lemma Lim_component_ge_cart:
9.33    fixes f :: "'a \<Rightarrow> real^'n"
9.34 -  assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)\$i) net"
9.35 +  assumes "(f \<longlongrightarrow> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)\$i) net"
9.36    shows "b \<le> l\$i"
9.37    by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
9.38
9.39  lemma Lim_component_eq_cart:
9.40    fixes f :: "'a \<Rightarrow> real^'n"
9.41 -  assumes net: "(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)\$i = b) net"
9.42 +  assumes net: "(f \<longlongrightarrow> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)\$i = b) net"
9.43    shows "l\$i = b"
9.44    using ev[unfolded order_eq_iff eventually_conj_iff] and
9.45      Lim_component_ge_cart[OF net, of b i] and
```
```    10.1 --- a/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy	Tue Dec 29 23:50:44 2015 +0100
10.2 +++ b/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy	Wed Dec 30 11:21:54 2015 +0100
10.3 @@ -2664,7 +2664,7 @@
10.4      then have "\<exists>d>0. \<forall>y\<in>s. y \<noteq> x \<and> cmod (y-x) < d \<longrightarrow> cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
10.5        using d1 by blast
10.6    }
10.7 -  then have *: "((\<lambda>y. (contour_integral (linepath x y) f - f x * (y - x)) /\<^sub>R cmod (y - x)) ---> 0) (at x within s)"
10.8 +  then have *: "((\<lambda>y. (contour_integral (linepath x y) f - f x * (y - x)) /\<^sub>R cmod (y - x)) \<longlongrightarrow> 0) (at x within s)"
10.9      by (simp add: Lim_within dist_norm inverse_eq_divide)
10.10    show ?thesis
10.11      apply (simp add: has_field_derivative_def has_derivative_within bounded_linear_mult_right)
10.12 @@ -5237,7 +5237,7 @@
10.13        and noleB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
10.14        and \<gamma>: "valid_path \<gamma>"
10.15        and [simp]: "~ (trivial_limit F)"
10.16 -  shows "l contour_integrable_on \<gamma>" "((\<lambda>n. contour_integral \<gamma> (f n)) ---> contour_integral \<gamma> l) F"
10.17 +  shows "l contour_integrable_on \<gamma>" "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
10.18  proof -
10.19    have "0 \<le> B" by (meson noleB [of 0] atLeastAtMost_iff norm_ge_zero order_refl order_trans zero_le_one)
10.20    { fix e::real
10.21 @@ -5292,7 +5292,7 @@
10.22        apply (blast intro: *)+
10.23        done
10.24    }
10.25 -  then show "((\<lambda>n. contour_integral \<gamma> (f n)) ---> contour_integral \<gamma> l) F"
10.26 +  then show "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
10.27      by (rule tendstoI)
10.28  qed
10.29
10.30 @@ -5300,7 +5300,7 @@
10.31    assumes ev_fint: "eventually (\<lambda>n::'a. (f n) contour_integrable_on (circlepath z r)) F"
10.32        and ev_no: "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>n. \<forall>x \<in> path_image (circlepath z r). norm(f n x - l x) < e) F"
10.33        and [simp]: "~ (trivial_limit F)" "0 < r"
10.34 -  shows "l contour_integrable_on (circlepath z r)" "((\<lambda>n. contour_integral (circlepath z r) (f n)) ---> contour_integral (circlepath z r) l) F"
10.35 +  shows "l contour_integrable_on (circlepath z r)" "((\<lambda>n. contour_integral (circlepath z r) (f n)) \<longlongrightarrow> contour_integral (circlepath z r) l) F"
10.36  by (auto simp: vector_derivative_circlepath norm_mult intro: contour_integral_uniform_limit assms)
10.37
10.38
10.39 @@ -6156,7 +6156,7 @@
10.40  text\<open> These weak Liouville versions don't even need the derivative formula.\<close>
10.41
10.42  lemma Liouville_weak_0:
10.43 -  assumes holf: "f holomorphic_on UNIV" and inf: "(f ---> 0) at_infinity"
10.44 +  assumes holf: "f holomorphic_on UNIV" and inf: "(f \<longlongrightarrow> 0) at_infinity"
10.45      shows "f z = 0"
10.46  proof (rule ccontr)
10.47    assume fz: "f z \<noteq> 0"
10.48 @@ -6182,7 +6182,7 @@
10.49  qed
10.50
10.51  proposition Liouville_weak:
10.52 -  assumes "f holomorphic_on UNIV" and "(f ---> l) at_infinity"
10.53 +  assumes "f holomorphic_on UNIV" and "(f \<longlongrightarrow> l) at_infinity"
10.54      shows "f z = l"
10.55    using Liouville_weak_0 [of "\<lambda>z. f z - l"]
10.56    by (simp add: assms holomorphic_on_const holomorphic_on_diff LIM_zero)
10.57 @@ -6195,7 +6195,7 @@
10.58    { assume f: "\<And>z. f z \<noteq> 0"
10.59      have 1: "(\<lambda>x. 1 / f x) holomorphic_on UNIV"
10.60        by (simp add: holomorphic_on_divide holomorphic_on_const assms f)
10.61 -    have 2: "((\<lambda>x. 1 / f x) ---> 0) at_infinity"
10.62 +    have 2: "((\<lambda>x. 1 / f x) \<longlongrightarrow> 0) at_infinity"
10.63        apply (rule tendstoI [OF eventually_mono])
10.64        apply (rule_tac B="2/e" in unbounded)
10.65        apply (simp add: dist_norm norm_divide divide_simps mult_ac)
10.66 @@ -6274,16 +6274,16 @@
10.67        done
10.68      have g_cint: "(\<lambda>u. g u/(u - w)) contour_integrable_on circlepath z r"
10.69        by (rule contour_integral_uniform_limit_circlepath [OF ev_int ev_less F \<open>0 < r\<close>])
10.70 -    have cif_tends_cig: "((\<lambda>n. contour_integral(circlepath z r) (\<lambda>u. f n u / (u - w))) ---> contour_integral(circlepath z r) (\<lambda>u. g u/(u - w))) F"
10.71 +    have cif_tends_cig: "((\<lambda>n. contour_integral(circlepath z r) (\<lambda>u. f n u / (u - w))) \<longlongrightarrow> contour_integral(circlepath z r) (\<lambda>u. g u/(u - w))) F"
10.72        by (rule contour_integral_uniform_limit_circlepath [OF ev_int ev_less F \<open>0 < r\<close>])
10.73 -    have f_tends_cig: "((\<lambda>n. 2 * of_real pi * ii * f n w) ---> contour_integral (circlepath z r) (\<lambda>u. g u / (u - w))) F"
10.74 +    have f_tends_cig: "((\<lambda>n. 2 * of_real pi * ii * f n w) \<longlongrightarrow> contour_integral (circlepath z r) (\<lambda>u. g u / (u - w))) F"
10.75        apply (rule Lim_transform_eventually [where f = "\<lambda>n. contour_integral (circlepath z r) (\<lambda>u. f n u/(u - w))"])
10.76        apply (rule eventually_mono [OF cont])
10.77        apply (rule contour_integral_unique [OF Cauchy_integral_circlepath])
10.78        using w
10.79        apply (auto simp: norm_minus_commute dist_norm cif_tends_cig)
10.80        done
10.81 -    have "((\<lambda>n. 2 * of_real pi * \<i> * f n w) ---> 2 * of_real pi * \<i> * g w) F"
10.82 +    have "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> 2 * of_real pi * \<i> * g w) F"
10.83        apply (rule tendsto_mult_left [OF tendstoI])
10.84        apply (rule eventually_mono [OF lim], assumption)
10.85        using w
10.86 @@ -6314,7 +6314,7 @@
10.87        and F:  "~ trivial_limit F" and "0 < r"
10.88    obtains g' where
10.89        "continuous_on (cball z r) g"
10.90 -      "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) ---> g' w) F"
10.91 +      "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
10.92  proof -
10.93    let ?conint = "contour_integral (circlepath z r)"
10.94    have g: "continuous_on (cball z r) g" "g holomorphic_on ball z r"
10.95 @@ -6325,7 +6325,7 @@
10.96      by (fastforce simp add: holomorphic_on_open)
10.97    then have derg: "\<And>x. x \<in> ball z r \<Longrightarrow> deriv g x = g' x"
10.98      by (simp add: DERIV_imp_deriv)
10.99 -  have tends_f'n_g': "((\<lambda>n. f' n w) ---> g' w) F" if w: "w \<in> ball z r" for w
10.100 +  have tends_f'n_g': "((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" if w: "w \<in> ball z r" for w
10.101    proof -
10.102      have eq_f': "?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2) = (f' n w - g' w) * (2 * of_real pi * \<i>)"
10.103               if cont_fn: "continuous_on (cball z r) (f n)"
10.104 @@ -6359,17 +6359,17 @@
10.105        apply (force simp add: dist_norm dle intro: less_le_trans)
10.106        done
10.107      have "((\<lambda>n. contour_integral (circlepath z r) (\<lambda>x. f n x / (x - w)\<^sup>2))
10.108 -             ---> contour_integral (circlepath z r) ((\<lambda>x. g x / (x - w)\<^sup>2))) F"
10.109 +             \<longlongrightarrow> contour_integral (circlepath z r) ((\<lambda>x. g x / (x - w)\<^sup>2))) F"
10.110        by (rule Cauchy_Integral_Thm.contour_integral_uniform_limit_circlepath [OF 1 2 F \<open>0 < r\<close>])
10.111 -    then have tendsto_0: "((\<lambda>n. 1 / (2 * of_real pi * \<i>) * (?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2))) ---> 0) F"
10.112 +    then have tendsto_0: "((\<lambda>n. 1 / (2 * of_real pi * \<i>) * (?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2))) \<longlongrightarrow> 0) F"
10.113        using Lim_null by (force intro!: tendsto_mult_right_zero)
10.114 -    have "((\<lambda>n. f' n w - g' w) ---> 0) F"
10.115 +    have "((\<lambda>n. f' n w - g' w) \<longlongrightarrow> 0) F"
10.116        apply (rule Lim_transform_eventually [OF _ tendsto_0])
10.117        apply (force simp add: divide_simps intro: eq_f' eventually_mono [OF cont])
10.118        done
10.119      then show ?thesis using Lim_null by blast
10.120    qed
10.121 -  obtain g' where "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) ---> g' w) F"
10.122 +  obtain g' where "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
10.123        by (blast intro: tends_f'n_g' g' )
10.124    then show ?thesis using g
10.125      using that by blast
```
```    11.1 --- a/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy	Tue Dec 29 23:50:44 2015 +0100
11.2 +++ b/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy	Wed Dec 30 11:21:54 2015 +0100
11.3 @@ -42,28 +42,28 @@
11.4
11.5  lemma tendsto_Re_upper:
11.6    assumes "~ (trivial_limit F)"
11.7 -          "(f ---> l) F"
11.8 +          "(f \<longlongrightarrow> l) F"
11.9            "eventually (\<lambda>x. Re(f x) \<le> b) F"
11.10      shows  "Re(l) \<le> b"
11.11    by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Re)
11.12
11.13  lemma tendsto_Re_lower:
11.14    assumes "~ (trivial_limit F)"
11.15 -          "(f ---> l) F"
11.16 +          "(f \<longlongrightarrow> l) F"
11.17            "eventually (\<lambda>x. b \<le> Re(f x)) F"
11.18      shows  "b \<le> Re(l)"
11.19    by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Re)
11.20
11.21  lemma tendsto_Im_upper:
11.22    assumes "~ (trivial_limit F)"
11.23 -          "(f ---> l) F"
11.24 +          "(f \<longlongrightarrow> l) F"
11.25            "eventually (\<lambda>x. Im(f x) \<le> b) F"
11.26      shows  "Im(l) \<le> b"
11.27    by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Im)
11.28
11.29  lemma tendsto_Im_lower:
11.30    assumes "~ (trivial_limit F)"
11.31 -          "(f ---> l) F"
11.32 +          "(f \<longlongrightarrow> l) F"
11.33            "eventually (\<lambda>x. b \<le> Im(f x)) F"
11.34      shows  "b \<le> Im(l)"
11.35    by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Im)
11.36 @@ -237,7 +237,7 @@
11.37
11.38  lemma real_lim:
11.39    fixes l::complex
11.40 -  assumes "(f ---> l) F" and "~(trivial_limit F)" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>"
11.41 +  assumes "(f \<longlongrightarrow> l) F" and "~(trivial_limit F)" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>"
11.42    shows  "l \<in> \<real>"
11.43  proof (rule Lim_in_closed_set[OF closed_complex_Reals _ assms(2,1)])
11.44    show "eventually (\<lambda>x. f x \<in> \<real>) F"
11.45 @@ -246,7 +246,7 @@
11.46
11.47  lemma real_lim_sequentially:
11.48    fixes l::complex
11.49 -  shows "(f ---> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
11.50 +  shows "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
11.51  by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially)
11.52
11.53  lemma real_series:
11.54 @@ -256,7 +256,7 @@
11.55  by (metis real_lim_sequentially setsum_in_Reals)
11.56
11.57  lemma Lim_null_comparison_Re:
11.58 -  assumes "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F" "(g ---> 0) F" shows "(f ---> 0) F"
11.59 +  assumes "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F" "(g \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F"
11.60    by (rule Lim_null_comparison[OF assms(1)] tendsto_eq_intros assms(2))+ simp
11.61
11.62  subsection\<open>Holomorphic functions\<close>
11.63 @@ -812,11 +812,11 @@
11.64    assumes cvs: "convex s"
11.65        and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
11.66        and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s \<longrightarrow> norm (f' n x - g' x) \<le> e"
11.67 -      and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) ---> l) sequentially"
11.68 -    shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) ---> g x) sequentially \<and>
11.69 +      and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
11.70 +    shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and>
11.71                         (g has_field_derivative (g' x)) (at x within s)"
11.72  proof -
11.73 -  from assms obtain x l where x: "x \<in> s" and tf: "((\<lambda>n. f n x) ---> l) sequentially"
11.74 +  from assms obtain x l where x: "x \<in> s" and tf: "((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
11.75      by blast
11.76    { fix e::real assume e: "e > 0"
11.77      then obtain N where N: "\<forall>n\<ge>N. \<forall>x. x \<in> s \<longrightarrow> cmod (f' n x - g' x) \<le> e"
```
```    12.1 --- a/src/HOL/Multivariate_Analysis/Complex_Transcendental.thy	Tue Dec 29 23:50:44 2015 +0100
12.2 +++ b/src/HOL/Multivariate_Analysis/Complex_Transcendental.thy	Wed Dec 30 11:21:54 2015 +0100
12.3 @@ -1564,7 +1564,7 @@
12.4  lemma lim_Ln_over_power:
12.5    fixes s::complex
12.6    assumes "0 < Re s"
12.7 -    shows "((\<lambda>n. Ln n / (n powr s)) ---> 0) sequentially"
12.8 +    shows "((\<lambda>n. Ln n / (n powr s)) \<longlongrightarrow> 0) sequentially"
12.9  proof (simp add: lim_sequentially dist_norm, clarify)
12.10    fix e::real
12.11    assume e: "0 < e"
12.12 @@ -1603,7 +1603,7 @@
12.13      done
12.14  qed
12.15
12.16 -lemma lim_Ln_over_n: "((\<lambda>n. Ln(of_nat n) / of_nat n) ---> 0) sequentially"
12.17 +lemma lim_Ln_over_n: "((\<lambda>n. Ln(of_nat n) / of_nat n) \<longlongrightarrow> 0) sequentially"
12.18    using lim_Ln_over_power [of 1]
12.19    by simp
12.20
12.21 @@ -1616,14 +1616,14 @@
12.22  lemma lim_ln_over_power:
12.23    fixes s :: real
12.24    assumes "0 < s"
12.25 -    shows "((\<lambda>n. ln n / (n powr s)) ---> 0) sequentially"
12.26 +    shows "((\<lambda>n. ln n / (n powr s)) \<longlongrightarrow> 0) sequentially"
12.27    using lim_Ln_over_power [of "of_real s", THEN filterlim_sequentially_Suc [THEN iffD2]] assms
12.28    apply (subst filterlim_sequentially_Suc [symmetric])
12.29    apply (simp add: lim_sequentially dist_norm
12.30            Ln_Reals_eq norm_powr_real_powr norm_divide)
12.31    done
12.32
12.33 -lemma lim_ln_over_n: "((\<lambda>n. ln(real_of_nat n) / of_nat n) ---> 0) sequentially"
12.34 +lemma lim_ln_over_n: "((\<lambda>n. ln(real_of_nat n) / of_nat n) \<longlongrightarrow> 0) sequentially"
12.35    using lim_ln_over_power [of 1, THEN filterlim_sequentially_Suc [THEN iffD2]]
12.36    apply (subst filterlim_sequentially_Suc [symmetric])
12.37    apply (simp add: lim_sequentially dist_norm)
12.38 @@ -1631,7 +1631,7 @@
12.39
12.40  lemma lim_1_over_complex_power:
12.41    assumes "0 < Re s"
12.42 -    shows "((\<lambda>n. 1 / (of_nat n powr s)) ---> 0) sequentially"
12.43 +    shows "((\<lambda>n. 1 / (of_nat n powr s)) \<longlongrightarrow> 0) sequentially"
12.44  proof -
12.45    have "\<forall>n>0. 3 \<le> n \<longrightarrow> 1 \<le> ln (real_of_nat n)"
12.46      using ln3_gt_1
12.47 @@ -1648,14 +1648,14 @@
12.48  lemma lim_1_over_real_power:
12.49    fixes s :: real
12.50    assumes "0 < s"
12.51 -    shows "((\<lambda>n. 1 / (of_nat n powr s)) ---> 0) sequentially"
12.52 +    shows "((\<lambda>n. 1 / (of_nat n powr s)) \<longlongrightarrow> 0) sequentially"
12.53    using lim_1_over_complex_power [of "of_real s", THEN filterlim_sequentially_Suc [THEN iffD2]] assms
12.54    apply (subst filterlim_sequentially_Suc [symmetric])
12.55    apply (simp add: lim_sequentially dist_norm)
12.56    apply (simp add: Ln_Reals_eq norm_powr_real_powr norm_divide)
12.57    done
12.58
12.59 -lemma lim_1_over_Ln: "((\<lambda>n. 1 / Ln(of_nat n)) ---> 0) sequentially"
12.60 +lemma lim_1_over_Ln: "((\<lambda>n. 1 / Ln(of_nat n)) \<longlongrightarrow> 0) sequentially"
12.61  proof (clarsimp simp add: lim_sequentially dist_norm norm_divide divide_simps)
12.62    fix r::real
12.63    assume "0 < r"
12.64 @@ -1680,7 +1680,7 @@
12.65      done
12.66  qed
12.67
12.68 -lemma lim_1_over_ln: "((\<lambda>n. 1 / ln(real_of_nat n)) ---> 0) sequentially"
12.69 +lemma lim_1_over_ln: "((\<lambda>n. 1 / ln(real_of_nat n)) \<longlongrightarrow> 0) sequentially"
12.70    using lim_1_over_Ln [THEN filterlim_sequentially_Suc [THEN iffD2]] assms
12.71    apply (subst filterlim_sequentially_Suc [symmetric])
12.72    apply (simp add: lim_sequentially dist_norm)
```
```    13.1 --- a/src/HOL/Multivariate_Analysis/Derivative.thy	Tue Dec 29 23:50:44 2015 +0100
13.2 +++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Wed Dec 30 11:21:54 2015 +0100
13.3 @@ -70,12 +70,12 @@
13.4  text \<open>These are the only cases we'll care about, probably.\<close>
13.5
13.6  lemma has_derivative_within: "(f has_derivative f') (at x within s) \<longleftrightarrow>
13.7 -    bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)"
13.8 +    bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) \<longlongrightarrow> 0) (at x within s)"
13.9    unfolding has_derivative_def Lim
13.10    by (auto simp add: netlimit_within field_simps)
13.11
13.12  lemma has_derivative_at: "(f has_derivative f') (at x) \<longleftrightarrow>
13.13 -    bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)"
13.14 +    bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) \<longlongrightarrow> 0) (at x)"
13.15    using has_derivative_within [of f f' x UNIV]
13.16    by simp
13.17
13.18 @@ -111,14 +111,14 @@
13.19    fixes f :: "real \<Rightarrow> real"
13.20      and y :: "real"
13.21    shows "(f has_derivative (op * y)) (at x within ({x <..} \<inter> I)) \<longleftrightarrow>
13.22 -    ((\<lambda>t. (f x - f t) / (x - t)) ---> y) (at x within ({x <..} \<inter> I))"
13.23 +    ((\<lambda>t. (f x - f t) / (x - t)) \<longlongrightarrow> y) (at x within ({x <..} \<inter> I))"
13.24  proof -
13.25 -  have "((\<lambda>t. (f t - (f x + y * (t - x))) / \<bar>t - x\<bar>) ---> 0) (at x within ({x<..} \<inter> I)) \<longleftrightarrow>
13.26 -    ((\<lambda>t. (f t - f x) / (t - x) - y) ---> 0) (at x within ({x<..} \<inter> I))"
13.27 +  have "((\<lambda>t. (f t - (f x + y * (t - x))) / \<bar>t - x\<bar>) \<longlongrightarrow> 0) (at x within ({x<..} \<inter> I)) \<longleftrightarrow>
13.28 +    ((\<lambda>t. (f t - f x) / (t - x) - y) \<longlongrightarrow> 0) (at x within ({x<..} \<inter> I))"
13.29      by (intro Lim_cong_within) (auto simp add: diff_divide_distrib add_divide_distrib)
13.30 -  also have "\<dots> \<longleftrightarrow> ((\<lambda>t. (f t - f x) / (t - x)) ---> y) (at x within ({x<..} \<inter> I))"
13.31 +  also have "\<dots> \<longleftrightarrow> ((\<lambda>t. (f t - f x) / (t - x)) \<longlongrightarrow> y) (at x within ({x<..} \<inter> I))"
13.32      by (simp add: Lim_null[symmetric])
13.33 -  also have "\<dots> \<longleftrightarrow> ((\<lambda>t. (f x - f t) / (x - t)) ---> y) (at x within ({x<..} \<inter> I))"
13.34 +  also have "\<dots> \<longleftrightarrow> ((\<lambda>t. (f x - f t) / (x - t)) \<longlongrightarrow> y) (at x within ({x<..} \<inter> I))"
13.35      by (intro Lim_cong_within) (simp_all add: field_simps)
13.36    finally show ?thesis
13.37      by (simp add: bounded_linear_mult_right has_derivative_within)
13.38 @@ -127,7 +127,7 @@
13.39  subsubsection \<open>Caratheodory characterization\<close>
13.40
13.41  lemma DERIV_within_iff:
13.42 -  "(f has_field_derivative D) (at a within s) \<longleftrightarrow> ((\<lambda>z. (f z - f a) / (z - a)) ---> D) (at a within s)"
13.43 +  "(f has_field_derivative D) (at a within s) \<longleftrightarrow> ((\<lambda>z. (f z - f a) / (z - a)) \<longlongrightarrow> D) (at a within s)"
13.44  proof -
13.45    have 1: "\<And>w y. ~(w = a) ==> y / (w - a) - D = (y - (w - a)*D)/(w - a)"
13.46      by (metis divide_diff_eq_iff eq_iff_diff_eq_0 mult.commute)
13.47 @@ -816,8 +816,8 @@
13.48            using \<open>a < x2\<close>
13.49            by (auto simp: trivial_limit_within islimpt_in_closure)
13.50          moreover
13.51 -        have "((\<lambda>x1. (\<phi> x1 - \<phi> a) + e * (x1 - a) + e) ---> (\<phi> x2 - \<phi> a) + e * (x2 - a) + e) (at x2 within {a <..<x2})"
13.52 -          "((\<lambda>x1. norm (f x1 - f a)) ---> norm (f x2 - f a)) (at x2 within {a <..<x2})"
13.53 +        have "((\<lambda>x1. (\<phi> x1 - \<phi> a) + e * (x1 - a) + e) \<longlongrightarrow> (\<phi> x2 - \<phi> a) + e * (x2 - a) + e) (at x2 within {a <..<x2})"
13.54 +          "((\<lambda>x1. norm (f x1 - f a)) \<longlongrightarrow> norm (f x2 - f a)) (at x2 within {a <..<x2})"
13.55            using a
13.56            by (auto intro!: tendsto_eq_intros f_tendsto phi_tendsto
13.57              intro: tendsto_within_subset[where S="{a .. b}"])
13.58 @@ -1899,13 +1899,13 @@
13.59      and "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
13.60      and "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
13.61      and "x0 \<in> s"
13.62 -    and "((\<lambda>n. f n x0) ---> l) sequentially"
13.63 -  shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially \<and> (g has_derivative g'(x)) (at x within s)"
13.64 +    and "((\<lambda>n. f n x0) \<longlongrightarrow> l) sequentially"
13.65 +  shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and> (g has_derivative g'(x)) (at x within s)"
13.66  proof -
13.67    have lem1: "\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s.
13.68        norm ((f m x - f n x) - (f m y - f n y)) \<le> e * norm (x - y)"
13.69      using assms(1,2,3) by (rule has_derivative_sequence_lipschitz)
13.70 -  have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially"
13.71 +  have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially"
13.72      apply (rule bchoice)
13.73      unfolding convergent_eq_cauchy
13.74    proof
13.75 @@ -1970,7 +1970,7 @@
13.76      proof rule+
13.77        fix n x y
13.78        assume as: "N \<le> n" "x \<in> s" "y \<in> s"
13.79 -      have "((\<lambda>m. norm (f n x - f n y - (f m x - f m y))) ---> norm (f n x - f n y - (g x - g y))) sequentially"
13.80 +      have "((\<lambda>m. norm (f n x - f n y - (f m x - f m y))) \<longlongrightarrow> norm (f n x - f n y - (g x - g y))) sequentially"
13.81          by (intro tendsto_intros g[rule_format] as)
13.82        moreover have "eventually (\<lambda>m. norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)) sequentially"
13.83          unfolding eventually_sequentially
13.84 @@ -1988,14 +1988,14 @@
13.85          by (rule tendsto_ge_const[OF trivial_limit_sequentially])
13.86      qed
13.87    qed
13.88 -  have "\<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially \<and> (g has_derivative g' x) (at x within s)"
13.89 +  have "\<forall>x\<in>s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and> (g has_derivative g' x) (at x within s)"
13.90      unfolding has_derivative_within_alt2
13.91    proof (intro ballI conjI)
13.92      fix x
13.93      assume "x \<in> s"
13.94 -    then show "((\<lambda>n. f n x) ---> g x) sequentially"
13.95 +    then show "((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially"
13.96        by (simp add: g)
13.97 -    have lem3: "\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially"
13.98 +    have lem3: "\<forall>u. ((\<lambda>n. f' n x u) \<longlongrightarrow> g' x u) sequentially"
13.99        unfolding filterlim_def le_nhds_metric_le eventually_filtermap dist_norm
13.100      proof (intro allI impI)
13.101        fix u
13.102 @@ -2554,7 +2554,7 @@
13.103    from c have "c \<in> interior A \<inter> closure {c<..}" by auto
13.104    also have "\<dots> \<subseteq> closure (interior A \<inter> {c<..})" by (intro open_inter_closure_subset) auto
13.105    finally have "at c within ?A' \<noteq> bot" by (subst at_within_eq_bot_iff) auto
13.106 -  moreover from deriv have "((\<lambda>y. (f y - f c) / (y - c)) ---> f') (at c within ?A')"
13.107 +  moreover from deriv have "((\<lambda>y. (f y - f c) / (y - c)) \<longlongrightarrow> f') (at c within ?A')"
13.108      unfolding DERIV_within_iff using interior_subset[of A] by (blast intro: tendsto_mono at_le)
13.109    moreover from eventually_at_right_real[OF xc]
13.110      have "eventually (\<lambda>y. (f y - f c) / (y - c) \<le> (f x - f c) / (x - c)) (at_right c)"
13.111 @@ -2576,7 +2576,7 @@
13.112    from c have "c \<in> interior A \<inter> closure {..<c}" by auto
13.113    also have "\<dots> \<subseteq> closure (interior A \<inter> {..<c})" by (intro open_inter_closure_subset) auto
13.114    finally have "at c within ?A' \<noteq> bot" by (subst at_within_eq_bot_iff) auto
13.115 -  moreover from deriv have "((\<lambda>y. (f y - f c) / (y - c)) ---> f') (at c within ?A')"
13.116 +  moreover from deriv have "((\<lambda>y. (f y - f c) / (y - c)) \<longlongrightarrow> f') (at c within ?A')"
13.117      unfolding DERIV_within_iff using interior_subset[of A] by (blast intro: tendsto_mono at_le)
13.118    moreover from eventually_at_left_real[OF xc]
13.119      have "eventually (\<lambda>y. (f y - f c) / (y - c) \<ge> (f x - f c) / (x - c)) (at_left c)"
```
```    14.1 --- a/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy	Tue Dec 29 23:50:44 2015 +0100
14.2 +++ b/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy	Wed Dec 30 11:21:54 2015 +0100
14.3 @@ -324,7 +324,7 @@
14.4      by (auto simp: filter_eq_iff eventually_at_filter le_less)
14.5    moreover have "0 < x \<Longrightarrow> at x within {0 ..} = at x"
14.6      using at_within_interior[of x "{0 ..}"] by (simp add: interior_Ici[of "- \<infinity>"])
14.7 -  ultimately show "(inverse ---> inverse x) (at x within {0..})"
14.8 +  ultimately show "(inverse \<longlongrightarrow> inverse x) (at x within {0..})"
14.9      by (auto simp: le_less inverse_ereal_tendsto_at_right_0 inverse_ereal_tendsto_pos)
14.10  qed
14.11
```
```    15.1 --- a/src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy	Tue Dec 29 23:50:44 2015 +0100
15.2 +++ b/src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy	Wed Dec 30 11:21:54 2015 +0100
15.3 @@ -196,8 +196,8 @@
15.4  qed
15.5
15.6  lemma tendsto_vec_nth [tendsto_intros]:
15.7 -  assumes "((\<lambda>x. f x) ---> a) net"
15.8 -  shows "((\<lambda>x. f x \$ i) ---> a \$ i) net"
15.9 +  assumes "((\<lambda>x. f x) \<longlongrightarrow> a) net"
15.10 +  shows "((\<lambda>x. f x \$ i) \<longlongrightarrow> a \$ i) net"
15.11  proof (rule topological_tendstoI)
15.12    fix S assume "open S" "a \$ i \<in> S"
15.13    then have "open ((\<lambda>y. y \$ i) -` S)" "a \<in> ((\<lambda>y. y \$ i) -` S)"
15.14 @@ -212,8 +212,8 @@
15.15    unfolding isCont_def by (rule tendsto_vec_nth)
15.16
15.17  lemma vec_tendstoI:
15.18 -  assumes "\<And>i. ((\<lambda>x. f x \$ i) ---> a \$ i) net"
15.19 -  shows "((\<lambda>x. f x) ---> a) net"
15.20 +  assumes "\<And>i. ((\<lambda>x. f x \$ i) \<longlongrightarrow> a \$ i) net"
15.21 +  shows "((\<lambda>x. f x) \<longlongrightarrow> a) net"
15.22  proof (rule topological_tendstoI)
15.23    fix S assume "open S" and "a \<in> S"
15.24    then obtain A where A: "\<And>i. open (A i)" "\<And>i. a \$ i \<in> A i"
15.25 @@ -228,8 +228,8 @@
15.26  qed
15.27
15.28  lemma tendsto_vec_lambda [tendsto_intros]:
15.29 -  assumes "\<And>i. ((\<lambda>x. f x i) ---> a i) net"
15.30 -  shows "((\<lambda>x. \<chi> i. f x i) ---> (\<chi> i. a i)) net"
15.31 +  assumes "\<And>i. ((\<lambda>x. f x i) \<longlongrightarrow> a i) net"
15.32 +  shows "((\<lambda>x. \<chi> i. f x i) \<longlongrightarrow> (\<chi> i. a i)) net"
15.33    using assms by (simp add: vec_tendstoI)
15.34
15.35  lemma open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x \$ i) ` S)"
```
```    16.1 --- a/src/HOL/Multivariate_Analysis/Integration.thy	Tue Dec 29 23:50:44 2015 +0100
16.2 +++ b/src/HOL/Multivariate_Analysis/Integration.thy	Wed Dec 30 11:21:54 2015 +0100
16.3 @@ -9665,9 +9665,9 @@
16.4    fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
16.5    assumes "\<forall>k. (f k) integrable_on cbox a b"
16.6      and "\<forall>k. \<forall>x\<in>cbox a b.(f k x) \<le> f (Suc k) x"
16.7 -    and "\<forall>x\<in>cbox a b. ((\<lambda>k. f k x) ---> g x) sequentially"
16.8 +    and "\<forall>x\<in>cbox a b. ((\<lambda>k. f k x) \<longlongrightarrow> g x) sequentially"
16.9      and "bounded {integral (cbox a b) (f k) | k . k \<in> UNIV}"
16.10 -  shows "g integrable_on cbox a b \<and> ((\<lambda>k. integral (cbox a b) (f k)) ---> integral (cbox a b) g) sequentially"
16.11 +  shows "g integrable_on cbox a b \<and> ((\<lambda>k. integral (cbox a b) (f k)) \<longlongrightarrow> integral (cbox a b) g) sequentially"
16.12  proof (cases "content (cbox a b) = 0")
16.13    case True
16.14    show ?thesis
16.15 @@ -9692,7 +9692,7 @@
16.16        apply auto
16.17        done
16.18    qed
16.19 -  have "\<exists>i. ((\<lambda>k. integral (cbox a b) (f k)) ---> i) sequentially"
16.20 +  have "\<exists>i. ((\<lambda>k. integral (cbox a b) (f k)) \<longlongrightarrow> i) sequentially"
16.21      apply (rule bounded_increasing_convergent)
16.22      defer
16.23      apply rule
16.24 @@ -9935,15 +9935,15 @@
16.25    fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
16.26    assumes "\<forall>k. (f k) integrable_on s"
16.27      and "\<forall>k. \<forall>x\<in>s. (f k x) \<le> (f (Suc k) x)"
16.28 -    and "\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially"
16.29 +    and "\<forall>x\<in>s. ((\<lambda>k. f k x) \<longlongrightarrow> g x) sequentially"
16.30      and "bounded {integral s (f k)| k. True}"
16.31 -  shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
16.32 -proof -
16.33 -  have lem: "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
16.34 +  shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) \<longlongrightarrow> integral s g) sequentially"
16.35 +proof -
16.36 +  have lem: "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) \<longlongrightarrow> integral s g) sequentially"
16.37      if "\<forall>k. \<forall>x\<in>s. 0 \<le> f k x"
16.38      and "\<forall>k. (f k) integrable_on s"
16.39      and "\<forall>k. \<forall>x\<in>s. f k x \<le> f (Suc k) x"
16.40 -    and "\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially"
16.41 +    and "\<forall>x\<in>s. ((\<lambda>k. f k x) \<longlongrightarrow> g x) sequentially"
16.42      and "bounded {integral s (f k)| k. True}"
16.43      for f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real" and g s
16.44    proof -
16.45 @@ -9962,7 +9962,7 @@
16.46        done
16.47      note fg=this[rule_format]
16.48
16.49 -    have "\<exists>i. ((\<lambda>k. integral s (f k)) ---> i) sequentially"
16.50 +    have "\<exists>i. ((\<lambda>k. integral s (f k)) \<longlongrightarrow> i) sequentially"
16.51        apply (rule bounded_increasing_convergent)
16.52        apply (rule that(5))
16.53        apply rule
16.54 @@ -10004,7 +10004,7 @@
16.55        apply (rule int)
16.56        done
16.57      have "\<And>a b. (\<lambda>x. if x \<in> s then g x else 0) integrable_on cbox a b \<and>
16.58 -      ((\<lambda>k. integral (cbox a b) (\<lambda>x. if x \<in> s then f k x else 0)) --->
16.59 +      ((\<lambda>k. integral (cbox a b) (\<lambda>x. if x \<in> s then f k x else 0)) \<longlongrightarrow>
16.60        integral (cbox a b) (\<lambda>x. if x \<in> s then g x else 0)) sequentially"
16.61      proof (rule monotone_convergence_interval, safe, goal_cases)
16.62        case 1
16.63 @@ -10140,7 +10140,7 @@
16.64      apply auto
16.65      done
16.66    note * = this[rule_format]
16.67 -  have "(\<lambda>x. g x - f 0 x) integrable_on s \<and> ((\<lambda>k. integral s (\<lambda>x. f (Suc k) x - f 0 x)) --->
16.68 +  have "(\<lambda>x. g x - f 0 x) integrable_on s \<and> ((\<lambda>k. integral s (\<lambda>x. f (Suc k) x - f 0 x)) \<longlongrightarrow>
16.69      integral s (\<lambda>x. g x - f 0 x)) sequentially"
16.70      apply (rule lem)
16.71      apply safe
16.72 @@ -10225,9 +10225,9 @@
16.73    fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
16.74    assumes "\<forall>k. (f k) integrable_on s"
16.75      and "\<forall>k. \<forall>x\<in>s. f (Suc k) x \<le> f k x"
16.76 -    and "\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially"
16.77 +    and "\<forall>x\<in>s. ((\<lambda>k. f k x) \<longlongrightarrow> g x) sequentially"
16.78      and "bounded {integral s (f k)| k. True}"
16.79 -  shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
16.80 +  shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) \<longlongrightarrow> integral s g) sequentially"
16.81  proof -
16.82    note assm = assms[rule_format]
16.83    have *: "{integral s (\<lambda>x. - f k x) |k. True} = op *\<^sub>R (- 1) ` {integral s (f k)| k. True}"
16.84 @@ -10240,7 +10240,7 @@
16.85      apply auto
16.86      done
16.87    have "(\<lambda>x. - g x) integrable_on s \<and>
16.88 -    ((\<lambda>k. integral s (\<lambda>x. - f k x)) ---> integral s (\<lambda>x. - g x)) sequentially"
16.89 +    ((\<lambda>k. integral s (\<lambda>x. - f k x)) \<longlongrightarrow> integral s (\<lambda>x. - g x)) sequentially"
16.90      apply (rule monotone_convergence_increasing)
16.91      apply safe
16.92      apply (rule integrable_neg)
16.93 @@ -11842,7 +11842,7 @@
16.94    obtains I J where
16.95      "\<And>n. (f n has_integral I n) (cbox a b)"
16.96      "(g has_integral J) (cbox a b)"
16.97 -    "(I ---> J) F"
16.98 +    "(I \<longlongrightarrow> J) F"
16.99  proof -
16.100    have fi[simp]: "f n integrable_on (cbox a b)" for n
16.101      by (auto intro!: integrable_continuous assms)
16.102 @@ -11858,7 +11858,7 @@
16.103
16.104    moreover
16.105
16.106 -  have "(I ---> J) F"
16.107 +  have "(I \<longlongrightarrow> J) F"
16.108    proof cases
16.109      assume "content (cbox a b) = 0"
16.110      hence "I = (\<lambda>_. 0)" "J = 0"
16.111 @@ -11906,9 +11906,9 @@
16.112    fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
16.113    assumes "\<And>k. (f k) integrable_on s" "h integrable_on s"
16.114      and "\<And>k. \<forall>x \<in> s. norm (f k x) \<le> h x"
16.115 -    and "\<forall>x \<in> s. ((\<lambda>k. f k x) ---> g x) sequentially"
16.116 +    and "\<forall>x \<in> s. ((\<lambda>k. f k x) \<longlongrightarrow> g x) sequentially"
16.117    shows "g integrable_on s"
16.118 -    and "((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
16.119 +    and "((\<lambda>k. integral s (f k)) \<longlongrightarrow> integral s g) sequentially"
16.120  proof -
16.121    have bdd_below[simp]: "\<And>x P. x \<in> s \<Longrightarrow> bdd_below {f n x |n. P n}"
16.122    proof (safe intro!: bdd_belowI)
16.123 @@ -11922,7 +11922,7 @@
16.124    qed
16.125
16.126    have "\<And>m. (\<lambda>x. Inf {f j x |j. m \<le> j}) integrable_on s \<and>
16.127 -    ((\<lambda>k. integral s (\<lambda>x. Inf {f j x |j. j \<in> {m..m + k}})) --->
16.128 +    ((\<lambda>k. integral s (\<lambda>x. Inf {f j x |j. j \<in> {m..m + k}})) \<longlongrightarrow>
16.129      integral s (\<lambda>x. Inf {f j x |j. m \<le> j}))sequentially"
16.130    proof (rule monotone_convergence_decreasing, safe)
16.131      fix m :: nat
16.132 @@ -11962,7 +11962,7 @@
16.133      show "Inf {f j x |j. j \<in> {m..m + Suc k}} \<le> Inf {f j x |j. j \<in> {m..m + k}}"
16.134        by (rule cInf_superset_mono) auto
16.135      let ?S = "{f j x| j. m \<le> j}"
16.136 -    show "((\<lambda>k. Inf {f j x |j. j \<in> {m..m + k}}) ---> Inf ?S) sequentially"
16.137 +    show "((\<lambda>k. Inf {f j x |j. j \<in> {m..m + k}}) \<longlongrightarrow> Inf ?S) sequentially"
16.138      proof (rule LIMSEQ_I, goal_cases)
16.139        case r: (1 r)
16.140
16.141 @@ -11993,7 +11993,7 @@
16.142    note dec1 = conjunctD2[OF this]
16.143
16.144    have "\<And>m. (\<lambda>x. Sup {f j x |j. m \<le> j}) integrable_on s \<and>
16.145 -    ((\<lambda>k. integral s (\<lambda>x. Sup {f j x |j. j \<in> {m..m + k}})) --->
16.146 +    ((\<lambda>k. integral s (\<lambda>x. Sup {f j x |j. j \<in> {m..m + k}})) \<longlongrightarrow>
16.147      integral s (\<lambda>x. Sup {f j x |j. m \<le> j})) sequentially"
16.148    proof (rule monotone_convergence_increasing,safe)
16.149      fix m :: nat
16.150 @@ -12033,7 +12033,7 @@
16.151      show "Sup {f j x |j. j \<in> {m..m + Suc k}} \<ge> Sup {f j x |j. j \<in> {m..m + k}}"
16.152        by (rule cSup_subset_mono) auto
16.153      let ?S = "{f j x| j. m \<le> j}"
16.154 -    show "((\<lambda>k. Sup {f j x |j. j \<in> {m..m + k}}) ---> Sup ?S) sequentially"
16.155 +    show "((\<lambda>k. Sup {f j x |j. j \<in> {m..m + k}}) \<longlongrightarrow> Sup ?S) sequentially"
16.156      proof (rule LIMSEQ_I, goal_cases)
16.157        case r: (1 r)
16.158        have "\<exists>y\<in>?S. Sup ?S - r < y"
16.159 @@ -12062,7 +12062,7 @@
16.160    note inc1 = conjunctD2[OF this]
16.161
16.162    have "g integrable_on s \<and>
16.163 -    ((\<lambda>k. integral s (\<lambda>x. Inf {f j x |j. k \<le> j})) ---> integral s g) sequentially"
16.164 +    ((\<lambda>k. integral s (\<lambda>x. Inf {f j x |j. k \<le> j})) \<longlongrightarrow> integral s g) sequentially"
16.165      apply (rule monotone_convergence_increasing,safe)
16.166      apply fact
16.167    proof -
16.168 @@ -12110,7 +12110,7 @@
16.169    note inc2 = conjunctD2[OF this]
16.170
16.171    have "g integrable_on s \<and>
16.172 -    ((\<lambda>k. integral s (\<lambda>x. Sup {f j x |j. k \<le> j})) ---> integral s g) sequentially"
16.173 +    ((\<lambda>k. integral s (\<lambda>x. Sup {f j x |j. k \<le> j})) \<longlongrightarrow> integral s g) sequentially"
16.174      apply (rule monotone_convergence_decreasing,safe)
16.175      apply fact
16.176    proof -
16.177 @@ -12135,7 +12135,7 @@
16.178
16.179      show "Sup {f j x |j. k \<le> j} \<ge> Sup {f j x |j. Suc k \<le> j}"
16.180        by (rule cSup_subset_mono) (auto simp: \<open>x\<in>s\<close>)
16.181 -    show "((\<lambda>k. Sup {f j x |j. k \<le> j}) ---> g x) sequentially"
16.182 +    show "((\<lambda>k. Sup {f j x |j. k \<le> j}) \<longlongrightarrow> g x) sequentially"
16.183      proof (rule LIMSEQ_I, goal_cases)
16.184        case r: (1 r)
16.185        then have "0<r/2"
16.186 @@ -12157,7 +12157,7 @@
16.187    note dec2 = conjunctD2[OF this]
16.188
16.189    show "g integrable_on s" by fact
16.190 -  show "((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
16.191 +  show "((\<lambda>k. integral s (f k)) \<longlongrightarrow> integral s g) sequentially"
16.192    proof (rule LIMSEQ_I, goal_cases)
16.193      case r: (1 r)
16.194      from LIMSEQ_D [OF inc2(2) r] guess N1 .. note N1=this[unfolded real_norm_def]
```
```    17.1 --- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Tue Dec 29 23:50:44 2015 +0100
17.2 +++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Wed Dec 30 11:21:54 2015 +0100
17.3 @@ -2881,8 +2881,8 @@
17.4  qed
17.5
17.6  lemma tendsto_infnorm [tendsto_intros]:
17.7 -  assumes "(f ---> a) F"
17.8 -  shows "((\<lambda>x. infnorm (f x)) ---> infnorm a) F"
17.9 +  assumes "(f \<longlongrightarrow> a) F"
17.10 +  shows "((\<lambda>x. infnorm (f x)) \<longlongrightarrow> infnorm a) F"
17.11  proof (rule tendsto_compose [OF LIM_I assms])
17.12    fix r :: real
17.13    assume "r > 0"
```
```    18.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Tue Dec 29 23:50:44 2015 +0100
18.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Wed Dec 30 11:21:54 2015 +0100
18.3 @@ -48,7 +48,7 @@
18.4
18.5  lemma Lim_within_open:
18.6    fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
18.7 -  shows "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"
18.8 +  shows "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f \<longlongrightarrow> l)(at a within S) \<longleftrightarrow> (f \<longlongrightarrow> l)(at a)"
18.9    by (fact tendsto_within_open)
18.10
18.11  lemma continuous_on_union:
18.12 @@ -2293,48 +2293,48 @@
18.13  subsection \<open>Limits\<close>
18.14
18.15  lemma Lim:
18.16 -  "(f ---> l) net \<longleftrightarrow>
18.17 +  "(f \<longlongrightarrow> l) net \<longleftrightarrow>
18.18          trivial_limit net \<or>
18.19          (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
18.20    unfolding tendsto_iff trivial_limit_eq by auto
18.21
18.22  text\<open>Show that they yield usual definitions in the various cases.\<close>
18.23
18.24 -lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
18.25 +lemma Lim_within_le: "(f \<longlongrightarrow> l)(at a within S) \<longleftrightarrow>
18.26      (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> d \<longrightarrow> dist (f x) l < e)"
18.27    by (auto simp add: tendsto_iff eventually_at_le dist_nz)
18.28
18.29 -lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
18.30 +lemma Lim_within: "(f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow>
18.31      (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a  < d \<longrightarrow> dist (f x) l < e)"
18.32    by (auto simp add: tendsto_iff eventually_at dist_nz)
18.33
18.34 -lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
18.35 +lemma Lim_at: "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow>
18.36      (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d  \<longrightarrow> dist (f x) l < e)"
18.37    by (auto simp add: tendsto_iff eventually_at2)
18.38
18.39  lemma Lim_at_infinity:
18.40 -  "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"
18.41 +  "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"
18.42    by (auto simp add: tendsto_iff eventually_at_infinity)
18.43
18.44 -lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
18.45 +lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f \<longlongrightarrow> l) net"
18.46    by (rule topological_tendstoI, auto elim: eventually_mono)
18.47
18.48  text\<open>The expected monotonicity property.\<close>
18.49
18.50  lemma Lim_Un:
18.51 -  assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"
18.52 -  shows "(f ---> l) (at x within (S \<union> T))"
18.53 +  assumes "(f \<longlongrightarrow> l) (at x within S)" "(f \<longlongrightarrow> l) (at x within T)"
18.54 +  shows "(f \<longlongrightarrow> l) (at x within (S \<union> T))"
18.55    using assms unfolding at_within_union by (rule filterlim_sup)
18.56
18.57  lemma Lim_Un_univ:
18.58 -  "(f ---> l) (at x within S) \<Longrightarrow> (f ---> l) (at x within T) \<Longrightarrow>
18.59 -    S \<union> T = UNIV \<Longrightarrow> (f ---> l) (at x)"
18.60 +  "(f \<longlongrightarrow> l) (at x within S) \<Longrightarrow> (f \<longlongrightarrow> l) (at x within T) \<Longrightarrow>
18.61 +    S \<union> T = UNIV \<Longrightarrow> (f \<longlongrightarrow> l) (at x)"
18.62    by (metis Lim_Un)
18.63
18.64  text\<open>Interrelations between restricted and unrestricted limits.\<close>
18.65
18.66  lemma Lim_at_imp_Lim_at_within:
18.67 -  "(f ---> l) (at x) \<Longrightarrow> (f ---> l) (at x within S)"
18.68 +  "(f \<longlongrightarrow> l) (at x) \<Longrightarrow> (f \<longlongrightarrow> l) (at x within S)"
18.69    by (metis order_refl filterlim_mono subset_UNIV at_le)
18.70
18.71  lemma eventually_within_interior:
18.72 @@ -2366,7 +2366,7 @@
18.73  lemma Lim_within_LIMSEQ:
18.74    fixes a :: "'a::first_countable_topology"
18.75    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"
18.76 -  shows "(X ---> L) (at a within T)"
18.77 +  shows "(X \<longlongrightarrow> L) (at a within T)"
18.78    using assms unfolding tendsto_def [where l=L]
18.79    by (simp add: sequentially_imp_eventually_within)
18.80
18.81 @@ -2375,7 +2375,7 @@
18.82      'b::{linorder_topology, conditionally_complete_linorder}"
18.83    assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
18.84      and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
18.85 -  shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
18.86 +  shows "(f \<longlongrightarrow> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
18.87  proof (cases "{x<..} \<inter> I = {}")
18.88    case True
18.89    then show ?thesis by simp
18.90 @@ -2411,7 +2411,7 @@
18.91
18.92  lemma islimpt_sequential:
18.93    fixes x :: "'a::first_countable_topology"
18.94 -  shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"
18.95 +  shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f \<longlongrightarrow> x) sequentially)"
18.96      (is "?lhs = ?rhs")
18.97  proof
18.98    assume ?lhs
18.99 @@ -2456,13 +2456,13 @@
18.100
18.101  lemma Lim_null:
18.102    fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
18.103 -  shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"
18.104 +  shows "(f \<longlongrightarrow> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) \<longlongrightarrow> 0) net"
18.105    by (simp add: Lim dist_norm)
18.106
18.107  lemma Lim_null_comparison:
18.108    fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
18.109 -  assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
18.110 -  shows "(f ---> 0) net"
18.111 +  assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g \<longlongrightarrow> 0) net"
18.112 +  shows "(f \<longlongrightarrow> 0) net"
18.113    using assms(2)
18.114  proof (rule metric_tendsto_imp_tendsto)
18.115    show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
18.116 @@ -2473,8 +2473,8 @@
18.117    fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
18.118      and g :: "'a \<Rightarrow> 'c::real_normed_vector"
18.119    assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"
18.120 -    and "(g ---> 0) net"
18.121 -  shows "(f ---> 0) net"
18.122 +    and "(g \<longlongrightarrow> 0) net"
18.123 +  shows "(f \<longlongrightarrow> 0) net"
18.124    using assms(1) tendsto_norm_zero [OF assms(2)]
18.125    by (rule Lim_null_comparison)
18.126
18.127 @@ -2483,7 +2483,7 @@
18.128  lemma Lim_in_closed_set:
18.129    assumes "closed S"
18.130      and "eventually (\<lambda>x. f(x) \<in> S) net"
18.131 -    and "\<not> trivial_limit net" "(f ---> l) net"
18.132 +    and "\<not> trivial_limit net" "(f \<longlongrightarrow> l) net"
18.133    shows "l \<in> S"
18.134  proof (rule ccontr)
18.135    assume "l \<notin> S"
18.136 @@ -2501,21 +2501,21 @@
18.137
18.138  lemma Lim_dist_ubound:
18.139    assumes "\<not>(trivial_limit net)"
18.140 -    and "(f ---> l) net"
18.141 +    and "(f \<longlongrightarrow> l) net"
18.142      and "eventually (\<lambda>x. dist a (f x) \<le> e) net"
18.143    shows "dist a l \<le> e"
18.144    using assms by (fast intro: tendsto_le tendsto_intros)
18.145
18.146  lemma Lim_norm_ubound:
18.147    fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
18.148 -  assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"
18.149 +  assumes "\<not>(trivial_limit net)" "(f \<longlongrightarrow> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"
18.150    shows "norm(l) \<le> e"
18.151    using assms by (fast intro: tendsto_le tendsto_intros)
18.152
18.153  lemma Lim_norm_lbound:
18.154    fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
18.155    assumes "\<not> trivial_limit net"
18.156 -    and "(f ---> l) net"
18.157 +    and "(f \<longlongrightarrow> l) net"
18.158      and "eventually (\<lambda>x. e \<le> norm (f x)) net"
18.159    shows "e \<le> norm l"
18.160    using assms by (fast intro: tendsto_le tendsto_intros)
18.161 @@ -2523,25 +2523,25 @@
18.162  text\<open>Limit under bilinear function\<close>
18.163
18.164  lemma Lim_bilinear:
18.165 -  assumes "(f ---> l) net"
18.166 -    and "(g ---> m) net"
18.167 +  assumes "(f \<longlongrightarrow> l) net"
18.168 +    and "(g \<longlongrightarrow> m) net"
18.169      and "bounded_bilinear h"
18.170 -  shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
18.171 -  using \<open>bounded_bilinear h\<close> \<open>(f ---> l) net\<close> \<open>(g ---> m) net\<close>
18.172 +  shows "((\<lambda>x. h (f x) (g x)) \<longlongrightarrow> (h l m)) net"
18.173 +  using \<open>bounded_bilinear h\<close> \<open>(f \<longlongrightarrow> l) net\<close> \<open>(g \<longlongrightarrow> m) net\<close>
18.174    by (rule bounded_bilinear.tendsto)
18.175
18.176  text\<open>These are special for limits out of the same vector space.\<close>
18.177
18.178 -lemma Lim_within_id: "(id ---> a) (at a within s)"
18.179 +lemma Lim_within_id: "(id \<longlongrightarrow> a) (at a within s)"
18.180    unfolding id_def by (rule tendsto_ident_at)
18.181
18.182 -lemma Lim_at_id: "(id ---> a) (at a)"
18.183 +lemma Lim_at_id: "(id \<longlongrightarrow> a) (at a)"
18.184    unfolding id_def by (rule tendsto_ident_at)
18.185
18.186  lemma Lim_at_zero:
18.187    fixes a :: "'a::real_normed_vector"
18.188      and l :: "'b::topological_space"
18.189 -  shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)"
18.190 +  shows "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) \<longlongrightarrow> l) (at 0)"
18.191    using LIM_offset_zero LIM_offset_zero_cancel ..
18.192
18.193  text\<open>It's also sometimes useful to extract the limit point from the filter.\<close>
18.194 @@ -2558,7 +2558,7 @@
18.195    using netlimit_within [of a UNIV] by simp
18.196
18.197  lemma lim_within_interior:
18.198 -  "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
18.199 +  "x \<in> interior S \<Longrightarrow> (f \<longlongrightarrow> l) (at x within S) \<longleftrightarrow> (f \<longlongrightarrow> l) (at x)"
18.200    by (metis at_within_interior)
18.201
18.202  lemma netlimit_within_interior:
18.203 @@ -2587,7 +2587,7 @@
18.204
18.205  lemma closure_sequential:
18.206    fixes l :: "'a::first_countable_topology"
18.207 -  shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)"
18.208 +  shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially)"
18.209    (is "?lhs = ?rhs")
18.210  proof
18.211    assume "?lhs"
18.212 @@ -2610,7 +2610,7 @@
18.213
18.214  lemma closed_sequential_limits:
18.215    fixes S :: "'a::first_countable_topology set"
18.216 -  shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
18.217 +  shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially \<longrightarrow> l \<in> S)"
18.218  by (metis closure_sequential closure_subset_eq subset_iff)
18.219
18.220  lemma closure_approachable:
18.221 @@ -2795,8 +2795,8 @@
18.222  qed
18.223
18.224  lemma tendsto_infdist [tendsto_intros]:
18.225 -  assumes f: "(f ---> l) F"
18.226 -  shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"
18.227 +  assumes f: "(f \<longlongrightarrow> l) F"
18.228 +  shows "((\<lambda>x. infdist (f x) A) \<longlongrightarrow> infdist l A) F"
18.229  proof (rule tendstoI)
18.230    fix e ::real
18.231    assume "e > 0"
18.232 @@ -2820,13 +2820,13 @@
18.233    using assms by (rule eventually_sequentially_seg [THEN iffD2])
18.234
18.235  lemma seq_offset_neg: (* TODO: move to Topological_Spaces.thy *)
18.236 -  "(f ---> l) sequentially \<Longrightarrow> ((\<lambda>i. f(i - k)) ---> l) sequentially"
18.237 +  "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> ((\<lambda>i. f(i - k)) \<longlongrightarrow> l) sequentially"
18.238    apply (erule filterlim_compose)
18.239    apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially)
18.240    apply arith
18.241    done
18.242
18.243 -lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
18.244 +lemma seq_harmonic: "((\<lambda>n. inverse (real n)) \<longlongrightarrow> 0) sequentially"
18.245    using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc) (* TODO: move to Limits.thy *)
18.246
18.247  subsection \<open>More properties of closed balls\<close>
18.248 @@ -3224,7 +3224,7 @@
18.249    {
18.250      fix y
18.251      assume "y \<in> closure S"
18.252 -    then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"
18.253 +    then obtain f where f: "\<forall>n. f n \<in> S"  "(f \<longlongrightarrow> y) sequentially"
18.254        unfolding closure_sequential by auto
18.255      have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
18.256      then have "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
18.257 @@ -3567,7 +3567,7 @@
18.258  lemma sequence_infinite_lemma:
18.259    fixes f :: "nat \<Rightarrow> 'a::t1_space"
18.260    assumes "\<forall>n. f n \<noteq> l"
18.261 -    and "(f ---> l) sequentially"
18.262 +    and "(f \<longlongrightarrow> l) sequentially"
18.263    shows "infinite (range f)"
18.264  proof
18.265    assume "finite (range f)"
18.266 @@ -3664,7 +3664,7 @@
18.267
18.268  lemma sequence_unique_limpt:
18.269    fixes f :: "nat \<Rightarrow> 'a::t2_space"
18.270 -  assumes "(f ---> l) sequentially"
18.271 +  assumes "(f \<longlongrightarrow> l) sequentially"
18.272      and "l' islimpt (range f)"
18.273    shows "l' = l"
18.274  proof (rule ccontr)
18.275 @@ -3701,7 +3701,7 @@
18.276  proof -
18.277    {
18.278      fix x l
18.279 -    assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
18.280 +    assume as: "\<forall>n::nat. x n \<in> s" "(x \<longlongrightarrow> l) sequentially"
18.281      then have "l \<in> s"
18.282      proof (cases "\<forall>n. x n \<noteq> l")
18.283        case False
18.284 @@ -3974,16 +3974,16 @@
18.285
18.286  definition seq_compact :: "'a::topological_space set \<Rightarrow> bool"
18.287    where "seq_compact S \<longleftrightarrow>
18.288 -    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially))"
18.289 +    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially))"
18.290
18.291  lemma seq_compactI:
18.292 -  assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
18.293 +  assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
18.294    shows "seq_compact S"
18.295    unfolding seq_compact_def using assms by fast
18.296
18.297  lemma seq_compactE:
18.298    assumes "seq_compact S" "\<forall>n. f n \<in> S"
18.299 -  obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
18.300 +  obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) \<longlongrightarrow> l) sequentially"
18.301    using assms unfolding seq_compact_def by fast
18.302
18.303  lemma closed_sequentially: (* TODO: move upwards *)
18.304 @@ -4184,7 +4184,7 @@
18.305    {
18.306      fix f :: "nat \<Rightarrow> 'a"
18.307      assume f: "\<forall>n. f n \<in> s"
18.308 -    have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
18.309 +    have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
18.310      proof (cases "finite (range f)")
18.311        case True
18.312        obtain l where "infinite {n. f n = f l}"
18.313 @@ -4200,9 +4200,9 @@
18.314        with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)"
18.315          by auto
18.316        then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..
18.317 -      from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
18.318 +      from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
18.319          using acc_point_range_imp_convergent_subsequence[of l f] by auto
18.320 -      with \<open>l \<in> s\<close> show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..
18.321 +      with \<open>l \<in> s\<close> show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" ..
18.322      qed
18.323    }
18.324    then show ?thesis
18.325 @@ -4261,7 +4261,7 @@
18.326      qed simp
18.327      then obtain x where "\<forall>n::nat. x n \<in> s" and x:"\<And>n m. m < n \<Longrightarrow> \<not> (dist (x m) (x n) < e)"
18.328        by blast
18.329 -    then obtain l r where "l \<in> s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially"
18.330 +    then obtain l r where "l \<in> s" and r:"subseq r" and "((x \<circ> r) \<longlongrightarrow> l) sequentially"
18.331        using assms by (metis seq_compact_def)
18.332      from this(3) have "Cauchy (x \<circ> r)"
18.333        using LIMSEQ_imp_Cauchy by auto
18.334 @@ -4358,7 +4358,7 @@
18.335
18.336  class heine_borel = metric_space +
18.337    assumes bounded_imp_convergent_subsequence:
18.338 -    "bounded (range f) \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
18.339 +    "bounded (range f) \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
18.340
18.341  lemma bounded_closed_imp_seq_compact:
18.342    fixes s::"'a::heine_borel set"
18.343 @@ -4370,13 +4370,13 @@
18.344    assume f: "\<forall>n. f n \<in> s"
18.345    with \<open>bounded s\<close> have "bounded (range f)"
18.346      by (auto intro: bounded_subset)
18.347 -  obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
18.348 +  obtain l r where r: "subseq r" and l: "((f \<circ> r) \<longlongrightarrow> l) sequentially"
18.349      using bounded_imp_convergent_subsequence [OF \<open>bounded (range f)\<close>] by auto
18.350    from f have fr: "\<forall>n. (f \<circ> r) n \<in> s"
18.351      by simp
18.352    have "l \<in> s" using \<open>closed s\<close> fr l
18.353      by (rule closed_sequentially)
18.354 -  show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
18.355 +  show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
18.356      using \<open>l \<in> s\<close> r l by blast
18.357  qed
18.358
18.359 @@ -4451,7 +4451,7 @@
18.360        by simp
18.361      have "bounded (range (\<lambda>i. f (r1 i) \<bullet> k))"
18.362        by (metis (lifting) bounded_subset f' image_subsetI s')
18.363 -    then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"
18.364 +    then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) \<longlongrightarrow> l2) sequentially"
18.365        using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"]
18.366        by (auto simp: o_def)
18.367      def r \<equiv> "r1 \<circ> r2"
18.368 @@ -4509,9 +4509,9 @@
18.369      ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
18.370        by (rule eventually_mono)
18.371    }
18.372 -  then have *: "((f \<circ> r) ---> l) sequentially"
18.373 +  then have *: "((f \<circ> r) \<longlongrightarrow> l) sequentially"
18.374      unfolding o_def tendsto_iff by simp
18.375 -  with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
18.376 +  with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
18.377      by auto
18.378  qed
18.379
18.380 @@ -4535,16 +4535,16 @@
18.381      using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast
18.382    from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"
18.383      by (auto simp add: image_comp intro: bounded_snd bounded_subset)
18.384 -  obtain l2 r2 where r2: "subseq r2" and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
18.385 +  obtain l2 r2 where r2: "subseq r2" and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) \<longlongrightarrow> l2) sequentially"
18.386      using bounded_imp_convergent_subsequence [OF s2]
18.387      unfolding o_def by fast
18.388 -  have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
18.389 +  have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) \<longlongrightarrow> l1) sequentially"
18.390      using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .
18.391 -  have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
18.392 +  have l: "((f \<circ> (r1 \<circ> r2)) \<longlongrightarrow> (l1, l2)) sequentially"
18.393      using tendsto_Pair [OF l1' l2] unfolding o_def by simp
18.394    have r: "subseq (r1 \<circ> r2)"
18.395      using r1 r2 unfolding subseq_def by simp
18.396 -  show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
18.397 +  show "\<exists>l r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
18.398      using l r by fast
18.399  qed
18.400
18.401 @@ -4601,7 +4601,7 @@
18.402        }
18.403        then have "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast
18.404      }
18.405 -    then have "\<exists>l\<in>s. (f ---> l) sequentially" using \<open>l\<in>s\<close>
18.406 +    then have "\<exists>l\<in>s. (f \<longlongrightarrow> l) sequentially" using \<open>l\<in>s\<close>
18.407        unfolding lim_sequentially by auto
18.408    }
18.409    then show ?thesis unfolding complete_def by auto
18.410 @@ -4777,7 +4777,7 @@
18.411      by (rule compact_imp_complete)
18.412    moreover have "\<forall>n. f n \<in> closure (range f)"
18.413      using closure_subset [of "range f"] by auto
18.414 -  ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
18.415 +  ultimately have "\<exists>l\<in>closure (range f). (f \<longlongrightarrow> l) sequentially"
18.416      using \<open>Cauchy f\<close> unfolding complete_def by auto
18.417    then show "convergent f"
18.418      unfolding convergent_def by auto
18.419 @@ -4840,12 +4840,12 @@
18.420
18.421  lemma convergent_eq_cauchy:
18.422    fixes s :: "nat \<Rightarrow> 'a::complete_space"
18.423 -  shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"
18.424 +  shows "(\<exists>l. (s \<longlongrightarrow> l) sequentially) \<longleftrightarrow> Cauchy s"
18.425    unfolding Cauchy_convergent_iff convergent_def ..
18.426
18.427  lemma convergent_imp_bounded:
18.428    fixes s :: "nat \<Rightarrow> 'a::metric_space"
18.429 -  shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"
18.430 +  shows "(s \<longlongrightarrow> l) sequentially \<Longrightarrow> bounded (range s)"
18.431    by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)
18.432
18.433  lemma compact_cball[simp]:
18.434 @@ -4940,7 +4940,7 @@
18.435    }
18.436    then have "Cauchy t"
18.437      unfolding cauchy_def by auto
18.438 -  then obtain l where l:"(t ---> l) sequentially"
18.439 +  then obtain l where l:"(t \<longlongrightarrow> l) sequentially"
18.440      using complete_UNIV unfolding complete_def by auto
18.441    {
18.442      fix n :: nat
18.443 @@ -5035,9 +5035,9 @@
18.444      apply (erule_tac x=e in allE)
18.445      apply auto
18.446      done
18.447 -  then obtain l where l: "\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially"
18.448 +  then obtain l where l: "\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) \<longlongrightarrow> l x) sequentially"
18.449      unfolding convergent_eq_cauchy[symmetric]
18.450 -    using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"]
18.451 +    using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) \<longlongrightarrow> l) sequentially"]
18.452      by auto
18.453    {
18.454      fix e :: real
18.455 @@ -5230,7 +5230,7 @@
18.456    "continuous (at x) f \<Longrightarrow> continuous (at x within s) f"
18.457    unfolding continuous_within continuous_at using Lim_at_imp_Lim_at_within by auto
18.458
18.459 -lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"
18.460 +lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f \<longlongrightarrow> l) net"
18.461    by simp
18.462
18.463  lemmas continuous_on = continuous_on_def \<comment> "legacy theorem name"
18.464 @@ -5253,8 +5253,8 @@
18.465  lemma continuous_within_sequentially:
18.466    fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
18.467    shows "continuous (at a within s) f \<longleftrightarrow>
18.468 -    (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
18.469 -         \<longrightarrow> ((f \<circ> x) ---> f a) sequentially)"
18.470 +    (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x \<longlongrightarrow> a) sequentially
18.471 +         \<longrightarrow> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
18.472    (is "?lhs = ?rhs")
18.473  proof
18.474    assume ?lhs
18.475 @@ -5286,14 +5286,14 @@
18.476  lemma continuous_at_sequentially:
18.477    fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
18.478    shows "continuous (at a) f \<longleftrightarrow>
18.479 -    (\<forall>x. (x ---> a) sequentially --> ((f \<circ> x) ---> f a) sequentially)"
18.480 +    (\<forall>x. (x \<longlongrightarrow> a) sequentially --> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
18.481    using continuous_within_sequentially[of a UNIV f] by simp
18.482
18.483  lemma continuous_on_sequentially:
18.484    fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
18.485    shows "continuous_on s f \<longleftrightarrow>
18.486 -    (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
18.487 -      --> ((f \<circ> x) ---> f a) sequentially)"
18.488 +    (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x \<longlongrightarrow> a) sequentially
18.489 +      --> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
18.490    (is "?lhs = ?rhs")
18.491  proof
18.492    assume ?rhs
18.493 @@ -5311,15 +5311,15 @@
18.494
18.495  lemma uniformly_continuous_on_sequentially:
18.496    "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
18.497 -                    ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially
18.498 -                    \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")
18.499 +                    ((\<lambda>n. dist (x n) (y n)) \<longlongrightarrow> 0) sequentially
18.500 +                    \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) \<longlongrightarrow> 0) sequentially)" (is "?lhs = ?rhs")
18.501  proof
18.502    assume ?lhs
18.503    {
18.504      fix x y
18.505      assume x: "\<forall>n. x n \<in> s"
18.506        and y: "\<forall>n. y n \<in> s"
18.507 -      and xy: "((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"
18.508 +      and xy: "((\<lambda>n. dist (x n) (y n)) \<longlongrightarrow> 0) sequentially"
18.509      {
18.510        fix e :: real
18.511        assume "e > 0"
18.512 @@ -5340,7 +5340,7 @@
18.513        then have "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"
18.514          by auto
18.515      }
18.516 -    then have "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially"
18.517 +    then have "((\<lambda>n. dist (f(x n)) (f(y n))) \<longlongrightarrow> 0) sequentially"
18.518        unfolding lim_sequentially and dist_real_def by auto
18.519    }
18.520    then show ?rhs by auto
18.521 @@ -5390,14 +5390,14 @@
18.522
18.523  lemma continuous_on_tendsto_compose:
18.524    assumes f_cont: "continuous_on s f"
18.525 -  assumes g: "(g ---> l) F"
18.526 +  assumes g: "(g \<longlongrightarrow> l) F"
18.527    assumes l: "l \<in> s"
18.528    assumes ev: "\<forall>\<^sub>F x in F. g x \<in> s"
18.529 -  shows "((\<lambda>x. f (g x)) ---> f l) F"
18.530 +  shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f l) F"
18.531  proof -
18.532 -  from f_cont have f: "(f ---> f l) (at l within s)"
18.533 +  from f_cont have f: "(f \<longlongrightarrow> f l) (at l within s)"
18.534      by (auto simp: l continuous_on)
18.535 -  have i: "((\<lambda>x. if g x = l then f l else f (g x)) ---> f l) F"
18.536 +  have i: "((\<lambda>x. if g x = l then f l else f (g x)) \<longlongrightarrow> f l) F"
18.537      by (rule filterlim_If)
18.538        (auto intro!: filterlim_compose[OF f] eventually_conj tendsto_mono[OF _ g]
18.539          simp: filterlim_at eventually_inf_principal eventually_mono[OF ev])
18.540 @@ -5422,7 +5422,7 @@
18.541      using assms(3) by auto
18.542    have "f x = g x"
18.543      using assms(1,2,3) by auto
18.544 -  then show "(f ---> g x) (at x within s)"
18.545 +  then show "(f \<longlongrightarrow> g x) (at x within s)"
18.546      using assms(4) unfolding continuous_within by simp
18.547  qed
18.548
18.549 @@ -5822,7 +5822,7 @@
18.550  proof -
18.551    obtain U where "open U" and "f x \<in> U" and "a \<notin> U"
18.552      using t1_space [OF \<open>f x \<noteq> a\<close>] by fast
18.553 -  have "(f ---> f x) (at x within s)"
18.554 +  have "(f \<longlongrightarrow> f x) (at x within s)"
18.555      using assms(1) by (simp add: continuous_within)
18.556    then have "eventually (\<lambda>y. f y \<in> U) (at x within s)"
18.557      using \<open>open U\<close> and \<open>f x \<in> U\<close>
18.558 @@ -6015,10 +6015,10 @@
18.559
18.560  lemma linear_lim_0:
18.561    assumes "bounded_linear f"
18.562 -  shows "(f ---> 0) (at (0))"
18.563 +  shows "(f \<longlongrightarrow> 0) (at (0))"
18.564  proof -
18.565    interpret f: bounded_linear f by fact
18.566 -  have "(f ---> f 0) (at 0)"
18.567 +  have "(f \<longlongrightarrow> f 0) (at 0)"
18.568      using tendsto_ident_at by (rule f.tendsto)
18.569    then show ?thesis unfolding f.zero .
18.570  qed
18.571 @@ -6267,7 +6267,7 @@
18.572    {
18.573      fix x
18.574      assume "x\<in>s"
18.575 -    have "(dist a ---> dist a x) (at x within s)"
18.576 +    have "(dist a \<longlongrightarrow> dist a x) (at x within s)"
18.577        by (intro tendsto_dist tendsto_const tendsto_ident_at)
18.578    }
18.579    then show "continuous_on s (dist a)"
18.580 @@ -6570,12 +6570,12 @@
18.581    let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
18.582    {
18.583      fix x l
18.584 -    assume as: "\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"
18.585 +    assume as: "\<forall>n. x n \<in> ?S"  "(x \<longlongrightarrow> l) sequentially"
18.586      from as(1) obtain f where f: "\<forall>n. x n = fst (f n) + snd (f n)"  "\<forall>n. fst (f n) \<in> s"  "\<forall>n. snd (f n) \<in> t"
18.587        using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
18.588 -    obtain l' r where "l'\<in>s" and r: "subseq r" and lr: "(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
18.589 +    obtain l' r where "l'\<in>s" and r: "subseq r" and lr: "(((\<lambda>n. fst (f n)) \<circ> r) \<longlongrightarrow> l') sequentially"
18.590        using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
18.591 -    have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
18.592 +    have "((\<lambda>n. snd (f (r n))) \<longlongrightarrow> l - l') sequentially"
18.593        using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)
18.594        unfolding o_def
18.595        by auto
18.596 @@ -6906,7 +6906,7 @@
18.597
18.598  lemma Lim_component_le:
18.599    fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
18.600 -  assumes "(f ---> l) net"
18.601 +  assumes "(f \<longlongrightarrow> l) net"
18.602      and "\<not> (trivial_limit net)"
18.603      and "eventually (\<lambda>x. f(x)\<bullet>i \<le> b) net"
18.604    shows "l\<bullet>i \<le> b"
18.605 @@ -6914,7 +6914,7 @@
18.606
18.607  lemma Lim_component_ge:
18.608    fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
18.609 -  assumes "(f ---> l) net"
18.610 +  assumes "(f \<longlongrightarrow> l) net"
18.611      and "\<not> (trivial_limit net)"
18.612      and "eventually (\<lambda>x. b \<le> (f x)\<bullet>i) net"
18.613    shows "b \<le> l\<bullet>i"
18.614 @@ -6922,7 +6922,7 @@
18.615
18.616  lemma Lim_component_eq:
18.617    fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
18.618 -  assumes net: "(f ---> l) net" "\<not> trivial_limit net"
18.619 +  assumes net: "(f \<longlongrightarrow> l) net" "\<not> trivial_limit net"
18.620      and ev:"eventually (\<lambda>x. f(x)\<bullet>i = b) net"
18.621    shows "l\<bullet>i = b"
18.622    using ev[unfolded order_eq_iff eventually_conj_iff]
18.623 @@ -6939,13 +6939,13 @@
18.624    by (auto elim!: eventually_rev_mp)
18.625
18.626  lemma Lim_within_union:
18.627 - "(f ---> l) (at x within (s \<union> t)) \<longleftrightarrow>
18.628 -  (f ---> l) (at x within s) \<and> (f ---> l) (at x within t)"
18.629 + "(f \<longlongrightarrow> l) (at x within (s \<union> t)) \<longleftrightarrow>
18.630 +  (f \<longlongrightarrow> l) (at x within s) \<and> (f \<longlongrightarrow> l) (at x within t)"
18.631    unfolding tendsto_def
18.632    by (auto simp add: eventually_within_Un)
18.633
18.634  lemma Lim_topological:
18.635 -  "(f ---> l) net \<longleftrightarrow>
18.636 +  "(f \<longlongrightarrow> l) net \<longleftrightarrow>
18.637      trivial_limit net \<or> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
18.638    unfolding tendsto_def trivial_limit_eq by auto
18.639
18.640 @@ -7266,7 +7266,7 @@
18.641      }
18.642      moreover
18.643      {
18.644 -      assume "\<not> (f ---> x) sequentially"
18.645 +      assume "\<not> (f \<longlongrightarrow> x) sequentially"
18.646        {
18.647          fix e :: real
18.648          assume "e > 0"
18.649 @@ -7281,9 +7281,9 @@
18.650            by (auto intro!: that le_less_trans [OF _ N])
18.651          then have "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto
18.652        }
18.653 -      then have "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
18.654 +      then have "((\<lambda>n. inverse (real n + 1)) \<longlongrightarrow> 0) sequentially"
18.655          unfolding lim_sequentially by(auto simp add: dist_norm)
18.656 -      then have "(f ---> x) sequentially"
18.657 +      then have "(f \<longlongrightarrow> x) sequentially"
18.658          unfolding f_def
18.659          using tendsto_add[OF tendsto_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
18.660          using tendsto_scaleR [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"]
18.661 @@ -7815,11 +7815,11 @@
18.662      then have "f \<circ> x = g"
18.663        unfolding fun_eq_iff
18.664        by auto
18.665 -    then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
18.666 +    then obtain l where "l\<in>s" and l:"(x \<longlongrightarrow> l) sequentially"
18.667        using cs[unfolded complete_def, THEN spec[where x="x"]]
18.668        using cauchy_isometric[OF \<open>0 < e\<close> s f normf] and cfg and x(1)
18.669        by auto
18.670 -    then have "\<exists>l\<in>f ` s. (g ---> l) sequentially"
18.671 +    then have "\<exists>l\<in>f ` s. (g \<longlongrightarrow> l) sequentially"
18.672        using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
18.673        unfolding \<open>f \<circ> x = g\<close>
18.674        by auto
18.675 @@ -8186,7 +8186,7 @@
18.676    }
18.677    then have "Cauchy z"
18.678      unfolding cauchy_def by auto
18.679 -  then obtain x where "x\<in>s" and x:"(z ---> x) sequentially"
18.680 +  then obtain x where "x\<in>s" and x:"(z \<longlongrightarrow> x) sequentially"
18.681      using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto
18.682
18.683    def e \<equiv> "dist (f x) x"
```
```    19.1 --- a/src/HOL/Multivariate_Analysis/Uniform_Limit.thy	Tue Dec 29 23:50:44 2015 +0100
19.2 +++ b/src/HOL/Multivariate_Analysis/Uniform_Limit.thy	Wed Dec 30 11:21:54 2015 +0100
19.3 @@ -54,11 +54,11 @@
19.4    by (fastforce dest: spec[where x = "e / 2" for e])
19.5
19.6  lemma swap_uniform_limit:
19.7 -  assumes f: "\<forall>\<^sub>F n in F. (f n ---> g n) (at x within S)"
19.8 -  assumes g: "(g ---> l) F"
19.9 +  assumes f: "\<forall>\<^sub>F n in F. (f n \<longlongrightarrow> g n) (at x within S)"
19.10 +  assumes g: "(g \<longlongrightarrow> l) F"
19.11    assumes uc: "uniform_limit S f h F"
19.12    assumes "\<not>trivial_limit F"
19.13 -  shows "(h ---> l) (at x within S)"
19.14 +  shows "(h \<longlongrightarrow> l) (at x within S)"
19.15  proof (rule tendstoI)
19.16    fix e :: real
19.17    def e' \<equiv> "e/3"
19.18 @@ -101,7 +101,7 @@
19.19    tendsto_uniform_limitI:
19.20    assumes "uniform_limit S f l F"
19.21    assumes "x \<in> S"
19.22 -  shows "((\<lambda>y. f y x) ---> l x) F"
19.23 +  shows "((\<lambda>y. f y x) \<longlongrightarrow> l x) F"
19.24    using assms
19.25    by (auto intro!: tendstoI simp: eventually_mono dest!: uniform_limitD)
19.26
19.27 @@ -113,10 +113,10 @@
19.28    unfolding continuous_on_def
19.29  proof safe
19.30    fix x assume "x \<in> A"
19.31 -  then have "\<forall>\<^sub>F n in F. (f n ---> f n x) (at x within A)" "((\<lambda>n. f n x) ---> l x) F"
19.32 +  then have "\<forall>\<^sub>F n in F. (f n \<longlongrightarrow> f n x) (at x within A)" "((\<lambda>n. f n x) \<longlongrightarrow> l x) F"
19.33      using c ul
19.34      by (auto simp: continuous_on_def eventually_mono tendsto_uniform_limitI)
19.35 -  then show "(l ---> l x) (at x within A)"
19.36 +  then show "(l \<longlongrightarrow> l x) (at x within A)"
19.37      by (rule swap_uniform_limit) fact+
19.38  qed
19.39
```
```    20.1 --- a/src/HOL/NthRoot.thy	Tue Dec 29 23:50:44 2015 +0100
20.2 +++ b/src/HOL/NthRoot.thy	Wed Dec 30 11:21:54 2015 +0100
20.3 @@ -268,7 +268,7 @@
20.4  qed (simp add: root_def[abs_def])
20.5
20.6  lemma tendsto_real_root[tendsto_intros]:
20.7 -  "(f ---> x) F \<Longrightarrow> ((\<lambda>x. root n (f x)) ---> root n x) F"
20.8 +  "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. root n (f x)) \<longlongrightarrow> root n x) F"
20.9    using isCont_tendsto_compose[OF isCont_real_root, of f x F] .
20.10
20.11  lemma continuous_real_root[continuous_intros]:
20.12 @@ -457,7 +457,7 @@
20.13  unfolding sqrt_def by (rule isCont_real_root)
20.14
20.15  lemma tendsto_real_sqrt[tendsto_intros]:
20.16 -  "(f ---> x) F \<Longrightarrow> ((\<lambda>x. sqrt (f x)) ---> sqrt x) F"
20.17 +  "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. sqrt (f x)) \<longlongrightarrow> sqrt x) F"
20.18    unfolding sqrt_def by (rule tendsto_real_root)
20.19
20.20  lemma continuous_real_sqrt[continuous_intros]:
```
```    21.1 --- a/src/HOL/Probability/Bochner_Integration.thy	Tue Dec 29 23:50:44 2015 +0100
21.2 +++ b/src/HOL/Probability/Bochner_Integration.thy	Wed Dec 30 11:21:54 2015 +0100
21.3 @@ -1524,11 +1524,11 @@
21.4    fixes s :: "real \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}" and w :: "'a \<Rightarrow> real"
21.5      and f :: "'a \<Rightarrow> 'b" and M
21.6    assumes "f \<in> borel_measurable M" "\<And>t. s t \<in> borel_measurable M" "integrable M w"
21.7 -  assumes lim: "AE x in M. ((\<lambda>i. s i x) ---> f x) at_top"
21.8 +  assumes lim: "AE x in M. ((\<lambda>i. s i x) \<longlongrightarrow> f x) at_top"
21.9    assumes bound: "\<forall>\<^sub>F i in at_top. AE x in M. norm (s i x) \<le> w x"
21.10  begin
21.11
21.12 -lemma integral_dominated_convergence_at_top: "((\<lambda>t. integral\<^sup>L M (s t)) ---> integral\<^sup>L M f) at_top"
21.13 +lemma integral_dominated_convergence_at_top: "((\<lambda>t. integral\<^sup>L M (s t)) \<longlongrightarrow> integral\<^sup>L M f) at_top"
21.14  proof (rule tendsto_at_topI_sequentially)
21.15    fix X :: "nat \<Rightarrow> real" assume X: "filterlim X at_top sequentially"
21.16    from filterlim_iff[THEN iffD1, OF this, rule_format, OF bound]
21.17 @@ -1542,7 +1542,7 @@
21.18      show "AE x in M. (\<lambda>n. s (X (n + N)) x) \<longlonglongrightarrow> f x"
21.19        using lim
21.20      proof eventually_elim
21.21 -      fix x assume "((\<lambda>i. s i x) ---> f x) at_top"
21.22 +      fix x assume "((\<lambda>i. s i x) \<longlongrightarrow> f x) at_top"
21.23        then show "(\<lambda>n. s (X (n + N)) x) \<longlonglongrightarrow> f x"
21.24          by (intro LIMSEQ_ignore_initial_segment filterlim_compose[OF _ X])
21.25      qed
21.26 @@ -1560,7 +1560,7 @@
21.27      show "AE x in M. (\<lambda>i. s (N + real i) x) \<longlonglongrightarrow> f x"
21.28        using lim
21.29      proof eventually_elim
21.30 -      fix x assume "((\<lambda>i. s i x) ---> f x) at_top"
21.31 +      fix x assume "((\<lambda>i. s i x) \<longlongrightarrow> f x) at_top"
21.32        then show "(\<lambda>n. s (N + n) x) \<longlonglongrightarrow> f x"
21.33          by (rule filterlim_compose)
21.34             (auto intro!: filterlim_tendsto_add_at_top filterlim_real_sequentially)
21.35 @@ -2459,7 +2459,7 @@
21.36  lemma tendsto_integral_at_top:
21.37    fixes f :: "real \<Rightarrow> 'a::{banach, second_countable_topology}"
21.38    assumes [measurable_cong]: "sets M = sets borel" and f[measurable]: "integrable M f"
21.39 -  shows "((\<lambda>y. \<integral> x. indicator {.. y} x *\<^sub>R f x \<partial>M) ---> \<integral> x. f x \<partial>M) at_top"
21.40 +  shows "((\<lambda>y. \<integral> x. indicator {.. y} x *\<^sub>R f x \<partial>M) \<longlongrightarrow> \<integral> x. f x \<partial>M) at_top"
21.41  proof (rule tendsto_at_topI_sequentially)
21.42    fix X :: "nat \<Rightarrow> real" assume "filterlim X at_top sequentially"
21.43    show "(\<lambda>n. \<integral>x. indicator {..X n} x *\<^sub>R f x \<partial>M) \<longlonglongrightarrow> integral\<^sup>L M f"
21.44 @@ -2486,7 +2486,7 @@
21.45    assumes nonneg: "AE x in M. 0 \<le> f x"
21.46    assumes borel: "f \<in> borel_measurable borel"
21.47    assumes int: "\<And>y. integrable M (\<lambda>x. f x * indicator {.. y} x)"
21.48 -  assumes conv: "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> x) at_top"
21.49 +  assumes conv: "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) \<longlongrightarrow> x) at_top"
21.50    shows has_bochner_integral_monotone_convergence_at_top: "has_bochner_integral M f x"
21.51      and integrable_monotone_convergence_at_top: "integrable M f"
21.52      and integral_monotone_convergence_at_top:"integral\<^sup>L M f = x"
```
```    22.1 --- a/src/HOL/Probability/Distributions.thy	Tue Dec 29 23:50:44 2015 +0100
22.2 +++ b/src/HOL/Probability/Distributions.thy	Wed Dec 30 11:21:54 2015 +0100
22.3 @@ -123,7 +123,7 @@
22.4      apply (metis nat_ceiling_le_eq)
22.5      done
22.6
22.7 -  have "((\<lambda>x::real. (1 - (\<Sum>n\<le>k. (x ^ n / exp x) / (fact n))) * fact k) --->
22.8 +  have "((\<lambda>x::real. (1 - (\<Sum>n\<le>k. (x ^ n / exp x) / (fact n))) * fact k) \<longlongrightarrow>
22.9          (1 - (\<Sum>n\<le>k. 0 / (fact n))) * fact k) at_top"
22.10      by (intro tendsto_intros tendsto_power_div_exp_0) simp
22.11    then show "?X \<longlonglongrightarrow> real_of_nat (fact k)"
22.12 @@ -865,20 +865,20 @@
22.13    proof (intro nn_integral_cong ereal_right_mult_cong)
22.14      fix s :: real show "?pI (\<lambda>x. ?ff x s) = ereal (1 / (2 * (1 + s\<^sup>2)))"
22.15      proof (subst nn_integral_FTC_atLeast)
22.16 -      have "((\<lambda>a. - (exp (- (a\<^sup>2 * (1 + s\<^sup>2))) / (2 + 2 * s\<^sup>2))) ---> (- (0 / (2 + 2 * s\<^sup>2)))) at_top"
22.17 +      have "((\<lambda>a. - (exp (- (a\<^sup>2 * (1 + s\<^sup>2))) / (2 + 2 * s\<^sup>2))) \<longlongrightarrow> (- (0 / (2 + 2 * s\<^sup>2)))) at_top"
22.18          apply (intro tendsto_intros filterlim_compose[OF exp_at_bot] filterlim_compose[OF filterlim_uminus_at_bot_at_top])
22.19          apply (subst mult.commute)
22.20          apply (auto intro!: filterlim_tendsto_pos_mult_at_top
22.21                              filterlim_at_top_mult_at_top[OF filterlim_ident filterlim_ident]
22.23          done
22.24 -      then show "((\<lambda>a. - (exp (- a\<^sup>2 - s\<^sup>2 * a\<^sup>2) / (2 + 2 * s\<^sup>2))) ---> 0) at_top"
22.25 +      then show "((\<lambda>a. - (exp (- a\<^sup>2 - s\<^sup>2 * a\<^sup>2) / (2 + 2 * s\<^sup>2))) \<longlongrightarrow> 0) at_top"
22.26          by (simp add: field_simps)
22.27      qed (auto intro!: derivative_eq_intros simp: field_simps add_nonneg_eq_0_iff)
22.28    qed rule
22.29    also have "... = ereal (pi / 4)"
22.30    proof (subst nn_integral_FTC_atLeast)
22.31 -    show "((\<lambda>a. arctan a / 2) ---> (pi / 2) / 2 ) at_top"
22.32 +    show "((\<lambda>a. arctan a / 2) \<longlongrightarrow> (pi / 2) / 2 ) at_top"
22.33        by (intro tendsto_intros) (simp_all add: tendsto_arctan_at_top)
22.34    qed (auto intro!: derivative_eq_intros simp: add_nonneg_eq_0_iff field_simps power2_eq_square)
22.35    finally have "?pI ?gauss^2 = pi / 4"
22.36 @@ -904,12 +904,12 @@
22.37         (auto simp: ac_simps times_ereal.simps(1)[symmetric] ereal_indicator simp del: times_ereal.simps)
22.38    also have "\<dots> = ereal (0 - (- exp (- 0\<^sup>2) / 2))"
22.39    proof (rule nn_integral_FTC_atLeast)
22.40 -    have "((\<lambda>x::real. - exp (- x\<^sup>2) / 2) ---> - 0 / 2) at_top"
22.41 +    have "((\<lambda>x::real. - exp (- x\<^sup>2) / 2) \<longlongrightarrow> - 0 / 2) at_top"
22.42        by (intro tendsto_divide tendsto_minus filterlim_compose[OF exp_at_bot]
22.43                     filterlim_compose[OF filterlim_uminus_at_bot_at_top]
22.44                     filterlim_pow_at_top filterlim_ident)
22.45           auto
22.46 -    then show "((\<lambda>a::real. - exp (- a\<^sup>2) / 2) ---> 0) at_top"
22.47 +    then show "((\<lambda>a::real. - exp (- a\<^sup>2) / 2) \<longlongrightarrow> 0) at_top"
22.48        by simp
22.49    qed (auto intro!: derivative_eq_intros)
22.50    also have "\<dots> = ereal (1 / 2)"
22.51 @@ -934,13 +934,13 @@
22.52      have "2 \<noteq> (0::real)"
22.53        by linarith
22.54      let ?f = "\<lambda>b. \<integral>x. indicator {0..} x *\<^sub>R ?M (k + 2) x * indicator {..b} x \<partial>lborel"
22.55 -    have "((\<lambda>b. (k + 1) / 2 * (\<integral>x. indicator {..b} x *\<^sub>R (indicator {0..} x *\<^sub>R ?M k x) \<partial>lborel) - ?M (k + 1) b / 2) --->
22.56 +    have "((\<lambda>b. (k + 1) / 2 * (\<integral>x. indicator {..b} x *\<^sub>R (indicator {0..} x *\<^sub>R ?M k x) \<partial>lborel) - ?M (k + 1) b / 2) \<longlongrightarrow>
22.57          (k + 1) / 2 * (\<integral>x. indicator {0..} x *\<^sub>R ?M k x \<partial>lborel) - 0 / 2) at_top" (is ?tendsto)
22.58      proof (intro tendsto_intros \<open>2 \<noteq> 0\<close> tendsto_integral_at_top sets_lborel Mk[THEN integrable.intros])
22.59 -      show "(?M (k + 1) ---> 0) at_top"
22.60 +      show "(?M (k + 1) \<longlongrightarrow> 0) at_top"
22.61        proof cases
22.62          assume "even k"
22.63 -        have "((\<lambda>x. ((x\<^sup>2)^(k div 2 + 1) / exp (x\<^sup>2)) * (1 / x) :: real) ---> 0 * 0) at_top"
22.64 +        have "((\<lambda>x. ((x\<^sup>2)^(k div 2 + 1) / exp (x\<^sup>2)) * (1 / x) :: real) \<longlongrightarrow> 0 * 0) at_top"
22.65            by (intro tendsto_intros tendsto_divide_0[OF tendsto_const] filterlim_compose[OF tendsto_power_div_exp_0]
22.66                     filterlim_at_top_imp_at_infinity filterlim_ident filterlim_pow_at_top filterlim_ident)
22.67               auto
22.68 @@ -949,7 +949,7 @@
22.69          finally show ?thesis by simp
22.70        next
22.71          assume "odd k"
22.72 -        have "((\<lambda>x. ((x\<^sup>2)^((k - 1) div 2 + 1) / exp (x\<^sup>2)) :: real) ---> 0) at_top"
22.73 +        have "((\<lambda>x. ((x\<^sup>2)^((k - 1) div 2 + 1) / exp (x\<^sup>2)) :: real) \<longlongrightarrow> 0) at_top"
22.74            by (intro filterlim_compose[OF tendsto_power_div_exp_0] filterlim_at_top_imp_at_infinity
22.75                      filterlim_ident filterlim_pow_at_top)
22.76               auto
22.77 @@ -958,7 +958,7 @@
22.78          finally show ?thesis by simp
22.79        qed
22.80      qed
22.81 -    also have "?tendsto \<longleftrightarrow> ((?f ---> (k + 1) / 2 * (\<integral>x. indicator {0..} x *\<^sub>R ?M k x \<partial>lborel) - 0 / 2) at_top)"
22.82 +    also have "?tendsto \<longleftrightarrow> ((?f \<longlongrightarrow> (k + 1) / 2 * (\<integral>x. indicator {0..} x *\<^sub>R ?M k x \<partial>lborel) - 0 / 2) at_top)"
22.83      proof (intro filterlim_cong refl eventually_at_top_linorder[THEN iffD2] exI[of _ 0] allI impI)
22.84        fix b :: real assume b: "0 \<le> b"
22.85        have "Suc k * (\<integral>x. indicator {0..b} x *\<^sub>R ?M k x \<partial>lborel) = (\<integral>x. indicator {0..b} x *\<^sub>R (exp (- x\<^sup>2) * ((Suc k) * x ^ k)) \<partial>lborel)"
22.86 @@ -977,7 +977,7 @@
22.87        then show "(k + 1) / 2 * (\<integral>x. indicator {..b} x *\<^sub>R (indicator {0..} x *\<^sub>R ?M k x)\<partial>lborel) - exp (- b\<^sup>2) * b ^ (k + 1) / 2 = ?f b"
22.88          by (simp add: field_simps atLeastAtMost_def indicator_inter_arith)
22.89      qed
22.90 -    finally have int_M_at_top: "((?f ---> (k + 1) / 2 * (\<integral>x. indicator {0..} x *\<^sub>R ?M k x \<partial>lborel)) at_top)"
22.91 +    finally have int_M_at_top: "((?f \<longlongrightarrow> (k + 1) / 2 * (\<integral>x. indicator {0..} x *\<^sub>R ?M k x \<partial>lborel)) at_top)"
22.92        by simp
22.93
22.94      have "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R ?M (k + 2) x) ((k + 1) / 2 * I)"
22.95 @@ -988,7 +988,7 @@
22.96          by rule (simp split: split_indicator)
22.97        show "integrable lborel (\<lambda>x. indicator {0..} x *\<^sub>R (?M (k + 2) x) * indicator {..y} x::real)"
22.98          unfolding * by (rule borel_integrable_compact) (auto intro!: continuous_intros)
22.99 -      show "((?f ---> (k + 1) / 2 * I) at_top)"
22.100 +      show "((?f \<longlongrightarrow> (k + 1) / 2 * I) at_top)"
22.101          using int_M_at_top has_bochner_integral_integral_eq[OF Mk] by simp
22.102      qed (auto split: split_indicator) }
22.103    note step = this
```
```    23.1 --- a/src/HOL/Probability/Fin_Map.thy	Tue Dec 29 23:50:44 2015 +0100
23.2 +++ b/src/HOL/Probability/Fin_Map.thy	Wed Dec 30 11:21:54 2015 +0100
23.3 @@ -187,7 +187,7 @@
23.4    using open_restricted_space[of "\<lambda>x. \<not> P x"]
23.5    unfolding closed_def by (rule back_subst) auto
23.6
23.7 -lemma tendsto_proj: "((\<lambda>x. x) ---> a) F \<Longrightarrow> ((\<lambda>x. (x)\<^sub>F i) ---> (a)\<^sub>F i) F"
23.8 +lemma tendsto_proj: "((\<lambda>x. x) \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. (x)\<^sub>F i) \<longlongrightarrow> (a)\<^sub>F i) F"
23.9    unfolding tendsto_def
23.10  proof safe
23.11    fix S::"'b set"
```
```    24.1 --- a/src/HOL/Probability/Interval_Integral.thy	Tue Dec 29 23:50:44 2015 +0100
24.2 +++ b/src/HOL/Probability/Interval_Integral.thy	Wed Dec 30 11:21:54 2015 +0100
24.3 @@ -622,8 +622,8 @@
24.4    assumes F: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> DERIV F x :> f x"
24.5    assumes f: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f x"
24.6    assumes f_nonneg: "AE x in lborel. a < ereal x \<longrightarrow> ereal x < b \<longrightarrow> 0 \<le> f x"
24.7 -  assumes A: "((F \<circ> real_of_ereal) ---> A) (at_right a)"
24.8 -  assumes B: "((F \<circ> real_of_ereal) ---> B) (at_left b)"
24.9 +  assumes A: "((F \<circ> real_of_ereal) \<longlongrightarrow> A) (at_right a)"
24.10 +  assumes B: "((F \<circ> real_of_ereal) \<longlongrightarrow> B) (at_left b)"
24.11    shows
24.12      "set_integrable lborel (einterval a b) f"
24.13      "(LBINT x=a..b. f x) = B - A"
24.14 @@ -669,8 +669,8 @@
24.15    assumes F: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> (F has_vector_derivative f x) (at x)"
24.16    assumes f: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f x"
24.17    assumes f_integrable: "set_integrable lborel (einterval a b) f"
24.18 -  assumes A: "((F \<circ> real_of_ereal) ---> A) (at_right a)"
24.19 -  assumes B: "((F \<circ> real_of_ereal) ---> B) (at_left b)"
24.20 +  assumes A: "((F \<circ> real_of_ereal) \<longlongrightarrow> A) (at_right a)"
24.21 +  assumes B: "((F \<circ> real_of_ereal) \<longlongrightarrow> B) (at_left b)"
24.22    shows "(LBINT x=a..b. f x) = B - A"
24.23  proof -
24.24    from einterval_Icc_approximation[OF \<open>a < b\<close>] guess u l . note approx = this
24.25 @@ -834,8 +834,8 @@
24.26    and contf: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f (g x)"
24.27    and contg': "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont g' x"
24.28    and g'_nonneg: "\<And>x. a \<le> ereal x \<Longrightarrow> ereal x \<le> b \<Longrightarrow> 0 \<le> g' x"
24.29 -  and A: "((ereal \<circ> g \<circ> real_of_ereal) ---> A) (at_right a)"
24.30 -  and B: "((ereal \<circ> g \<circ> real_of_ereal) ---> B) (at_left b)"
24.31 +  and A: "((ereal \<circ> g \<circ> real_of_ereal) \<longlongrightarrow> A) (at_right a)"
24.32 +  and B: "((ereal \<circ> g \<circ> real_of_ereal) \<longlongrightarrow> B) (at_left b)"
24.33    and integrable: "set_integrable lborel (einterval a b) (\<lambda>x. g' x *\<^sub>R f (g x))"
24.34    and integrable2: "set_integrable lborel (einterval A B) (\<lambda>x. f x)"
24.35    shows "(LBINT x=A..B. f x) = (LBINT x=a..b. g' x *\<^sub>R f (g x))"
24.36 @@ -935,8 +935,8 @@
24.37    and contg': "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont g' x"
24.38    and f_nonneg: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> 0 \<le> f (g x)" (* TODO: make this AE? *)
24.39    and g'_nonneg: "\<And>x. a \<le> ereal x \<Longrightarrow> ereal x \<le> b \<Longrightarrow> 0 \<le> g' x"
24.40 -  and A: "((ereal \<circ> g \<circ> real_of_ereal) ---> A) (at_right a)"
24.41 -  and B: "((ereal \<circ> g \<circ> real_of_ereal) ---> B) (at_left b)"
24.42 +  and A: "((ereal \<circ> g \<circ> real_of_ereal) \<longlongrightarrow> A) (at_right a)"
24.43 +  and B: "((ereal \<circ> g \<circ> real_of_ereal) \<longlongrightarrow> B) (at_left b)"
24.44    and integrable_fg: "set_integrable lborel (einterval a b) (\<lambda>x. f (g x) * g' x)"
24.45    shows
24.46      "set_integrable lborel (einterval A B) f"
```
```    25.1 --- a/src/HOL/Probability/Lebesgue_Integral_Substitution.thy	Tue Dec 29 23:50:44 2015 +0100
25.2 +++ b/src/HOL/Probability/Lebesgue_Integral_Substitution.thy	Wed Dec 30 11:21:54 2015 +0100
25.3 @@ -148,7 +148,7 @@
25.4    shows "D \<ge> 0"
25.5  proof (rule tendsto_le_const)
25.6    let ?A' = "(\<lambda>y. y - x) ` interior A"
25.7 -  from deriv show "((\<lambda>h. (f (x + h) - f x) / h) ---> D) (at 0)"
25.8 +  from deriv show "((\<lambda>h. (f (x + h) - f x) / h) \<longlongrightarrow> D) (at 0)"
25.9        by (simp add: field_has_derivative_at has_field_derivative_def)
25.10    from mono have mono': "mono_on f (interior A)" by (rule mono_on_subset) (rule interior_subset)
25.11
```
```    26.1 --- a/src/HOL/Probability/Lebesgue_Measure.thy	Tue Dec 29 23:50:44 2015 +0100
26.2 +++ b/src/HOL/Probability/Lebesgue_Measure.thy	Wed Dec 30 11:21:54 2015 +0100
26.3 @@ -322,7 +322,7 @@
26.4    note * = this
26.5
26.6    let ?F = "interval_measure F"
26.7 -  show "((\<lambda>a. F b - F a) ---> emeasure ?F {a..b}) (at_left a)"
26.8 +  show "((\<lambda>a. F b - F a) \<longlongrightarrow> emeasure ?F {a..b}) (at_left a)"
26.9    proof (rule tendsto_at_left_sequentially)
26.10      show "a - 1 < a" by simp
26.11      fix X assume "\<And>n. X n < a" "incseq X" "X \<longlonglongrightarrow> a"
26.12 @@ -344,7 +344,7 @@
26.13        using \<open>\<And>n. X n < a\<close> \<open>a \<le> b\<close> by (subst *) (auto intro: less_imp_le less_le_trans)
26.14      finally show "(\<lambda>n. F b - F (X n)) \<longlonglongrightarrow> emeasure ?F {a..b}" .
26.15    qed
26.16 -  show "((\<lambda>a. ereal (F b - F a)) ---> F b - F a) (at_left a)"
26.17 +  show "((\<lambda>a. ereal (F b - F a)) \<longlongrightarrow> F b - F a) (at_left a)"
26.18      using cont_F
26.19      by (intro lim_ereal[THEN iffD2] tendsto_intros )
26.20         (auto simp: continuous_on_def intro: tendsto_within_subset)
26.21 @@ -1244,7 +1244,7 @@
26.22    assumes f_borel: "f \<in> borel_measurable borel"
26.23    assumes f: "\<And>x. a \<le> x \<Longrightarrow> DERIV F x :> f x"
26.24    assumes nonneg: "\<And>x. a \<le> x \<Longrightarrow> 0 \<le> f x"
26.25 -  assumes lim: "(F ---> T) at_top"
26.26 +  assumes lim: "(F \<longlongrightarrow> T) at_top"
26.27    shows "(\<integral>\<^sup>+x. ereal (f x) * indicator {a ..} x \<partial>lborel) = T - F a"
26.28  proof -
26.29    let ?f = "\<lambda>(i::nat) (x::real). ereal (f x) * indicator {a..a + real i} x"
```
```    27.1 --- a/src/HOL/Probability/Projective_Limit.thy	Tue Dec 29 23:50:44 2015 +0100
27.2 +++ b/src/HOL/Probability/Projective_Limit.thy	Wed Dec 30 11:21:54 2015 +0100
27.3 @@ -48,7 +48,7 @@
27.4  lemma compactE':
27.5    fixes S :: "'a :: metric_space set"
27.6    assumes "compact S" "\<forall>n\<ge>m. f n \<in> S"
27.7 -  obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
27.8 +  obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) \<longlongrightarrow> l) sequentially"
27.9  proof atomize_elim
27.10    have "subseq (op + m)" by (simp add: subseq_def)
27.11    have "\<forall>n. (f o (\<lambda>i. m + i)) n \<in> S" using assms by auto
```
```    28.1 --- a/src/HOL/Real_Vector_Spaces.thy	Tue Dec 29 23:50:44 2015 +0100
28.2 +++ b/src/HOL/Real_Vector_Spaces.thy	Wed Dec 30 11:21:54 2015 +0100
28.3 @@ -1651,13 +1651,13 @@
28.4  qed (auto intro!: exI[of _ "{y. dist x y < e}" for e] open_ball simp: dist_commute)
28.5
28.6  lemma (in metric_space) tendsto_iff:
28.7 -  "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
28.8 +  "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
28.9    unfolding nhds_metric filterlim_INF filterlim_principal by auto
28.10
28.11 -lemma (in metric_space) tendstoI: "(\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F) \<Longrightarrow> (f ---> l) F"
28.12 +lemma (in metric_space) tendstoI: "(\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F) \<Longrightarrow> (f \<longlongrightarrow> l) F"
28.13    by (auto simp: tendsto_iff)
28.14
28.15 -lemma (in metric_space) tendstoD: "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
28.16 +lemma (in metric_space) tendstoD: "(f \<longlongrightarrow> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
28.17    by (auto simp: tendsto_iff)
28.18
28.19  lemma (in metric_space) eventually_nhds_metric:
28.20 @@ -1688,9 +1688,9 @@
28.21
28.22  lemma metric_tendsto_imp_tendsto:
28.23    fixes a :: "'a :: metric_space" and b :: "'b :: metric_space"
28.24 -  assumes f: "(f ---> a) F"
28.25 +  assumes f: "(f \<longlongrightarrow> a) F"
28.26    assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
28.27 -  shows "(g ---> b) F"
28.28 +  shows "(g \<longlongrightarrow> b) F"
28.29  proof (rule tendstoI)
28.30    fix e :: real assume "0 < e"
28.31    with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
28.32 @@ -1977,7 +1977,7 @@
28.33  lemma tendsto_at_topI_sequentially:
28.34    fixes f :: "real \<Rightarrow> 'b::first_countable_topology"
28.35    assumes *: "\<And>X. filterlim X at_top sequentially \<Longrightarrow> (\<lambda>n. f (X n)) \<longlonglongrightarrow> y"
28.36 -  shows "(f ---> y) at_top"
28.37 +  shows "(f \<longlongrightarrow> y) at_top"
28.38  proof -
28.39    from nhds_countable[of y] guess A . note A = this
28.40
28.41 @@ -2009,7 +2009,7 @@
28.42    fixes f :: "real \<Rightarrow> real"
28.43    assumes mono: "mono f"
28.44    assumes limseq: "(\<lambda>n. f (real n)) \<longlonglongrightarrow> y"
28.45 -  shows "(f ---> y) at_top"
28.46 +  shows "(f \<longlongrightarrow> y) at_top"
28.47  proof (rule tendstoI)
28.48    fix e :: real assume "0 < e"
28.49    with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"
```
```    29.1 --- a/src/HOL/Topological_Spaces.thy	Tue Dec 29 23:50:44 2015 +0100
29.2 +++ b/src/HOL/Topological_Spaces.thy	Wed Dec 30 11:21:54 2015 +0100
29.3 @@ -480,13 +480,13 @@
29.4  subsubsection \<open>Tendsto\<close>
29.5
29.6  abbreviation (in topological_space)
29.7 -  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
29.8 -  "(f ---> l) F \<equiv> filterlim f (nhds l) F"
29.9 +  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "\<longlongrightarrow>" 55) where
29.10 +  "(f \<longlongrightarrow> l) F \<equiv> filterlim f (nhds l) F"
29.11
29.12  definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a" where
29.13 -  "Lim A f = (THE l. (f ---> l) A)"
29.14 +  "Lim A f = (THE l. (f \<longlongrightarrow> l) A)"
29.15
29.16 -lemma tendsto_eq_rhs: "(f ---> x) F \<Longrightarrow> x = y \<Longrightarrow> (f ---> y) F"
29.17 +lemma tendsto_eq_rhs: "(f \<longlongrightarrow> x) F \<Longrightarrow> x = y \<Longrightarrow> (f \<longlongrightarrow> y) F"
29.18    by simp
29.19
29.20  named_theorems tendsto_intros "introduction rules for tendsto"
29.21 @@ -498,55 +498,55 @@
29.22  \<close>
29.23
29.24  lemma (in topological_space) tendsto_def:
29.25 -   "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
29.26 +   "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
29.27     unfolding nhds_def filterlim_INF filterlim_principal by auto
29.28
29.29  lemma tendsto_cong:
29.30    assumes "eventually (\<lambda>x. f x = g x) F"
29.31 -  shows   "(f ---> c) F \<longleftrightarrow> (g ---> c) F"
29.32 +  shows   "(f \<longlongrightarrow> c) F \<longleftrightarrow> (g \<longlongrightarrow> c) F"
29.33    by (rule filterlim_cong[OF refl refl assms])
29.34
29.35
29.36 -lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
29.37 +lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f \<longlongrightarrow> l) F' \<Longrightarrow> (f \<longlongrightarrow> l) F"
29.38    unfolding tendsto_def le_filter_def by fast
29.39
29.40 -lemma tendsto_within_subset: "(f ---> l) (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (at x within T)"
29.41 +lemma tendsto_within_subset: "(f \<longlongrightarrow> l) (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f \<longlongrightarrow> l) (at x within T)"
29.42    by (blast intro: tendsto_mono at_le)
29.43
29.44  lemma filterlim_at:
29.45 -  "(LIM x F. f x :> at b within s) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> s \<and> f x \<noteq> b) F \<and> (f ---> b) F)"
29.46 +  "(LIM x F. f x :> at b within s) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> s \<and> f x \<noteq> b) F \<and> (f \<longlongrightarrow> b) F)"
29.47    by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute)
29.48
29.49  lemma (in topological_space) topological_tendstoI:
29.50 -  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f ---> l) F"
29.51 +  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f \<longlongrightarrow> l) F"
29.52    unfolding tendsto_def by auto
29.53
29.54  lemma (in topological_space) topological_tendstoD:
29.55 -  "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
29.56 +  "(f \<longlongrightarrow> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
29.57    unfolding tendsto_def by auto
29.58
29.59  lemma (in order_topology) order_tendsto_iff:
29.60 -  "(f ---> x) F \<longleftrightarrow> (\<forall>l<x. eventually (\<lambda>x. l < f x) F) \<and> (\<forall>u>x. eventually (\<lambda>x. f x < u) F)"
29.61 +  "(f \<longlongrightarrow> x) F \<longleftrightarrow> (\<forall>l<x. eventually (\<lambda>x. l < f x) F) \<and> (\<forall>u>x. eventually (\<lambda>x. f x < u) F)"
29.62    unfolding nhds_order filterlim_inf filterlim_INF filterlim_principal by auto
29.63
29.64  lemma (in order_topology) order_tendstoI:
29.65    "(\<And>a. a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F) \<Longrightarrow> (\<And>a. y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F) \<Longrightarrow>
29.66 -    (f ---> y) F"
29.67 +    (f \<longlongrightarrow> y) F"
29.68    unfolding order_tendsto_iff by auto
29.69
29.70  lemma (in order_topology) order_tendstoD:
29.71 -  assumes "(f ---> y) F"
29.72 +  assumes "(f \<longlongrightarrow> y) F"
29.73    shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
29.74      and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
29.75    using assms unfolding order_tendsto_iff by auto
29.76
29.77 -lemma tendsto_bot [simp]: "(f ---> a) bot"
29.78 +lemma tendsto_bot [simp]: "(f \<longlongrightarrow> a) bot"
29.79    unfolding tendsto_def by simp
29.80
29.81  lemma (in linorder_topology) tendsto_max:
29.82 -  assumes X: "(X ---> x) net"
29.83 -  assumes Y: "(Y ---> y) net"
29.84 -  shows "((\<lambda>x. max (X x) (Y x)) ---> max x y) net"
29.85 +  assumes X: "(X \<longlongrightarrow> x) net"
29.86 +  assumes Y: "(Y \<longlongrightarrow> y) net"
29.87 +  shows "((\<lambda>x. max (X x) (Y x)) \<longlongrightarrow> max x y) net"
29.88  proof (rule order_tendstoI)
29.89    fix a assume "a < max x y"
29.90    then show "eventually (\<lambda>x. a < max (X x) (Y x)) net"
29.91 @@ -560,9 +560,9 @@
29.92  qed
29.93
29.94  lemma (in linorder_topology) tendsto_min:
29.95 -  assumes X: "(X ---> x) net"
29.96 -  assumes Y: "(Y ---> y) net"
29.97 -  shows "((\<lambda>x. min (X x) (Y x)) ---> min x y) net"
29.98 +  assumes X: "(X \<longlongrightarrow> x) net"
29.99 +  assumes Y: "(Y \<longlongrightarrow> y) net"
29.100 +  shows "((\<lambda>x. min (X x) (Y x)) \<longlongrightarrow> min x y) net"
29.101  proof (rule order_tendstoI)
29.102    fix a assume "a < min x y"
29.103    then show "eventually (\<lambda>x. a < min (X x) (Y x)) net"
29.104 @@ -575,24 +575,24 @@
29.105      by (auto simp: min_less_iff_disj elim: eventually_mono)
29.106  qed
29.107
29.108 -lemma tendsto_ident_at [tendsto_intros, simp, intro]: "((\<lambda>x. x) ---> a) (at a within s)"
29.109 +lemma tendsto_ident_at [tendsto_intros, simp, intro]: "((\<lambda>x. x) \<longlongrightarrow> a) (at a within s)"
29.110    unfolding tendsto_def eventually_at_topological by auto
29.111
29.112 -lemma (in topological_space) tendsto_const [tendsto_intros, simp, intro]: "((\<lambda>x. k) ---> k) F"
29.113 +lemma (in topological_space) tendsto_const [tendsto_intros, simp, intro]: "((\<lambda>x. k) \<longlongrightarrow> k) F"
29.114    by (simp add: tendsto_def)
29.115
29.116  lemma (in t2_space) tendsto_unique:
29.117 -  assumes "F \<noteq> bot" and "(f ---> a) F" and "(f ---> b) F"
29.118 +  assumes "F \<noteq> bot" and "(f \<longlongrightarrow> a) F" and "(f \<longlongrightarrow> b) F"
29.119    shows "a = b"
29.120  proof (rule ccontr)
29.121    assume "a \<noteq> b"
29.122    obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
29.123      using hausdorff [OF \<open>a \<noteq> b\<close>] by fast
29.124    have "eventually (\<lambda>x. f x \<in> U) F"
29.125 -    using \<open>(f ---> a) F\<close> \<open>open U\<close> \<open>a \<in> U\<close> by (rule topological_tendstoD)
29.126 +    using \<open>(f \<longlongrightarrow> a) F\<close> \<open>open U\<close> \<open>a \<in> U\<close> by (rule topological_tendstoD)
29.127    moreover
29.128    have "eventually (\<lambda>x. f x \<in> V) F"
29.129 -    using \<open>(f ---> b) F\<close> \<open>open V\<close> \<open>b \<in> V\<close> by (rule topological_tendstoD)
29.130 +    using \<open>(f \<longlongrightarrow> b) F\<close> \<open>open V\<close> \<open>b \<in> V\<close> by (rule topological_tendstoD)
29.131    ultimately
29.132    have "eventually (\<lambda>x. False) F"
29.133    proof eventually_elim
29.134 @@ -605,28 +605,28 @@
29.135  qed
29.136
29.137  lemma (in t2_space) tendsto_const_iff:
29.138 -  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a :: 'a) ---> b) F \<longleftrightarrow> a = b"
29.139 +  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a :: 'a) \<longlongrightarrow> b) F \<longleftrightarrow> a = b"
29.140    by (auto intro!: tendsto_unique [OF assms tendsto_const])
29.141
29.142  lemma increasing_tendsto:
29.143    fixes f :: "_ \<Rightarrow> 'a::order_topology"
29.144    assumes bdd: "eventually (\<lambda>n. f n \<le> l) F"
29.145        and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F"
29.146 -  shows "(f ---> l) F"
29.147 +  shows "(f \<longlongrightarrow> l) F"
29.148    using assms by (intro order_tendstoI) (auto elim!: eventually_mono)
29.149
29.150  lemma decreasing_tendsto:
29.151    fixes f :: "_ \<Rightarrow> 'a::order_topology"
29.152    assumes bdd: "eventually (\<lambda>n. l \<le> f n) F"
29.153        and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F"
29.154 -  shows "(f ---> l) F"
29.155 +  shows "(f \<longlongrightarrow> l) F"
29.156    using assms by (intro order_tendstoI) (auto elim!: eventually_mono)
29.157
29.158  lemma tendsto_sandwich:
29.159    fixes f g h :: "'a \<Rightarrow> 'b::order_topology"
29.160    assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
29.161 -  assumes lim: "(f ---> c) net" "(h ---> c) net"
29.162 -  shows "(g ---> c) net"
29.163 +  assumes lim: "(f \<longlongrightarrow> c) net" "(h \<longlongrightarrow> c) net"
29.164 +  shows "(g \<longlongrightarrow> c) net"
29.165  proof (rule order_tendstoI)
29.166    fix a show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net"
29.167      using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2)
29.168 @@ -638,7 +638,7 @@
29.169  lemma limit_frequently_eq:
29.170    assumes "F \<noteq> bot"
29.171    assumes "frequently (\<lambda>x. f x = c) F"
29.172 -  assumes "(f ---> d) F"
29.173 +  assumes "(f \<longlongrightarrow> d) F"
29.174    shows   "d = (c :: 'a :: t1_space)"
29.175  proof (rule ccontr)
29.176    assume "d \<noteq> c"
29.177 @@ -649,7 +649,7 @@
29.178  qed
29.179
29.180  lemma tendsto_imp_eventually_ne:
29.181 -  assumes "F \<noteq> bot" "(f ---> c) F" "c \<noteq> (c' :: 'a :: t1_space)"
29.182 +  assumes "F \<noteq> bot" "(f \<longlongrightarrow> c) F" "c \<noteq> (c' :: 'a :: t1_space)"
29.183    shows   "eventually (\<lambda>z. f z \<noteq> c') F"
29.184  proof (rule ccontr)
29.185    assume "\<not>eventually (\<lambda>z. f z \<noteq> c') F"
29.186 @@ -660,7 +660,7 @@
29.187  lemma tendsto_le:
29.188    fixes f g :: "'a \<Rightarrow> 'b::linorder_topology"
29.189    assumes F: "\<not> trivial_limit F"
29.190 -  assumes x: "(f ---> x) F" and y: "(g ---> y) F"
29.191 +  assumes x: "(f \<longlongrightarrow> x) F" and y: "(g \<longlongrightarrow> y) F"
29.192    assumes ev: "eventually (\<lambda>x. g x \<le> f x) F"
29.193    shows "y \<le> x"
29.194  proof (rule ccontr)
29.195 @@ -678,14 +678,14 @@
29.196  lemma tendsto_le_const:
29.197    fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
29.198    assumes F: "\<not> trivial_limit F"
29.199 -  assumes x: "(f ---> x) F" and a: "eventually (\<lambda>i. a \<le> f i) F"
29.200 +  assumes x: "(f \<longlongrightarrow> x) F" and a: "eventually (\<lambda>i. a \<le> f i) F"
29.201    shows "a \<le> x"
29.202    using F x tendsto_const a by (rule tendsto_le)
29.203
29.204  lemma tendsto_ge_const:
29.205    fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
29.206    assumes F: "\<not> trivial_limit F"
29.207 -  assumes x: "(f ---> x) F" and a: "eventually (\<lambda>i. a \<ge> f i) F"
29.208 +  assumes x: "(f \<longlongrightarrow> x) F" and a: "eventually (\<lambda>i. a \<ge> f i) F"
29.209    shows "a \<ge> x"
29.210    by (rule tendsto_le [OF F tendsto_const x a])
29.211
29.212 @@ -695,7 +695,7 @@
29.213  subsubsection \<open>Rules about @{const Lim}\<close>
29.214
29.215  lemma tendsto_Lim:
29.216 -  "\<not>(trivial_limit net) \<Longrightarrow> (f ---> l) net \<Longrightarrow> Lim net f = l"
29.217 +  "\<not>(trivial_limit net) \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> Lim net f = l"
29.218    unfolding Lim_def using tendsto_unique[of net f] by auto
29.219
29.220  lemma Lim_ident_at: "\<not> trivial_limit (at x within s) \<Longrightarrow> Lim (at x within s) (\<lambda>x. x) = x"
29.221 @@ -771,7 +771,7 @@
29.222  qed
29.223
29.224  lemma tendsto_at_within_iff_tendsto_nhds:
29.225 -  "(g ---> g l) (at l within S) \<longleftrightarrow> (g ---> g l) (inf (nhds l) (principal S))"
29.226 +  "(g \<longlongrightarrow> g l) (at l within S) \<longleftrightarrow> (g \<longlongrightarrow> g l) (inf (nhds l) (principal S))"
29.227    unfolding tendsto_def eventually_at_filter eventually_inf_principal
29.228    by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)
29.229
29.230 @@ -780,7 +780,7 @@
29.231  abbreviation (in topological_space)
29.232    LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
29.233      ("((_)/ \<longlonglongrightarrow> (_))" [60, 60] 60) where
29.234 -  "X \<longlonglongrightarrow> L \<equiv> (X ---> L) sequentially"
29.235 +  "X \<longlonglongrightarrow> L \<equiv> (X \<longlongrightarrow> L) sequentially"
29.236
29.237  abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" where
29.238    "lim X \<equiv> Lim sequentially X"
29.239 @@ -1196,7 +1196,7 @@
29.240
29.241  lemma tendsto_at_iff_sequentially:
29.242    fixes f :: "'a :: first_countable_topology \<Rightarrow> _"
29.243 -  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))"
29.244 +  shows "(f \<longlongrightarrow> 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))"
29.245    unfolding filterlim_def[of _ "nhds a"] le_filter_def eventually_filtermap at_within_def eventually_nhds_within_iff_sequentially comp_def
29.246    by metis
29.247
29.248 @@ -1205,9 +1205,9 @@
29.249  abbreviation
29.250    LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
29.251          ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
29.252 -  "f -- a --> L \<equiv> (f ---> L) (at a)"
29.253 +  "f -- a --> L \<equiv> (f \<longlongrightarrow> L) (at a)"
29.254
29.255 -lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f ---> l) (at a within S) \<longleftrightarrow> (f -- a --> l)"
29.256 +lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow> (f -- a --> l)"
29.257    unfolding tendsto_def by (simp add: at_within_open[where S=S])
29.258
29.259  lemma LIM_const_not_eq[tendsto_intros]:
29.260 @@ -1241,19 +1241,19 @@
29.261    by simp
29.262
29.263  lemma tendsto_at_iff_tendsto_nhds:
29.264 -  "g -- l --> g l \<longleftrightarrow> (g ---> g l) (nhds l)"
29.265 +  "g -- l --> g l \<longleftrightarrow> (g \<longlongrightarrow> g l) (nhds l)"
29.266    unfolding tendsto_def eventually_at_filter
29.267    by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)
29.268
29.269  lemma tendsto_compose:
29.270 -  "g -- l --> g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
29.271 +  "g -- l --> g l \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
29.272    unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
29.273
29.274  lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
29.275    unfolding o_def by (rule tendsto_compose)
29.276
29.277  lemma tendsto_compose_eventually:
29.278 -  "g -- l --> m \<Longrightarrow> (f ---> l) F \<Longrightarrow> eventually (\<lambda>x. f x \<noteq> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> m) F"
29.279 +  "g -- l --> m \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> eventually (\<lambda>x. f x \<noteq> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> m) F"
29.280    by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)
29.281
29.282  lemma LIM_compose_eventually:
29.283 @@ -1263,7 +1263,7 @@
29.284    shows "(\<lambda>x. g (f x)) -- a --> c"
29.285    using g f inj by (rule tendsto_compose_eventually)
29.286
29.287 -lemma tendsto_compose_filtermap: "((g \<circ> f) ---> T) F \<longleftrightarrow> (g ---> T) (filtermap f F)"
29.288 +lemma tendsto_compose_filtermap: "((g \<circ> f) \<longlongrightarrow> T) F \<longleftrightarrow> (g \<longlongrightarrow> T) (filtermap f F)"
29.289    by (simp add: filterlim_def filtermap_filtermap comp_def)
29.290
29.291  subsubsection \<open>Relation of LIM and LIMSEQ\<close>
29.292 @@ -1329,7 +1329,7 @@
29.293    fixes a :: "_ :: {linorder_topology, first_countable_topology}"
29.294    assumes "b < a"
29.295    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"
29.296 -  shows "(X ---> L) (at_left a)"
29.297 +  shows "(X \<longlongrightarrow> L) (at_left a)"
29.298    using assms unfolding tendsto_def [where l=L]
29.299    by (simp add: sequentially_imp_eventually_at_left)
29.300
29.301 @@ -1367,7 +1367,7 @@
29.302    fixes a :: "_ :: {linorder_topology, first_countable_topology}"
29.303    assumes "a < b"
29.304    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"
29.305 -  shows "(X ---> L) (at_right a)"
29.306 +  shows "(X \<longlongrightarrow> L) (at_right a)"
29.307    using assms unfolding tendsto_def [where l=L]
29.308    by (simp add: sequentially_imp_eventually_at_right)
29.309
29.310 @@ -1376,7 +1376,7 @@
29.311  subsubsection \<open>Continuity on a set\<close>
29.312
29.313  definition continuous_on :: "'a set \<Rightarrow> ('a :: topological_space \<Rightarrow> 'b :: topological_space) \<Rightarrow> bool" where
29.314 -  "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
29.315 +  "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
29.316
29.317  lemma continuous_on_cong [cong]:
29.318    "s = t \<Longrightarrow> (\<And>x. x \<in> t \<Longrightarrow> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g"
29.319 @@ -1559,7 +1559,7 @@
29.320  subsubsection \<open>Continuity at a point\<close>
29.321
29.322  definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where
29.323 -  "continuous F f \<longleftrightarrow> (f ---> f (Lim F (\<lambda>x. x))) F"
29.324 +  "continuous F f \<longleftrightarrow> (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F"
29.325
29.326  lemma continuous_bot[continuous_intros, simp]: "continuous bot f"
29.327    unfolding continuous_def by auto
29.328 @@ -1567,7 +1567,7 @@
29.329  lemma continuous_trivial_limit: "trivial_limit net \<Longrightarrow> continuous net f"
29.330    by simp
29.331
29.332 -lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f x) (at x within s)"
29.333 +lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f \<longlongrightarrow> f x) (at x within s)"
29.334    by (cases "trivial_limit (at x within s)") (auto simp add: Lim_ident_at continuous_def)
29.335
29.336  lemma continuous_within_topological:
29.337 @@ -1619,7 +1619,7 @@
29.338  lemma isCont_o[continuous_intros]: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (g \<circ> f) a"
29.339    unfolding o_def by (rule isCont_o2)
29.340
29.341 -lemma isCont_tendsto_compose: "isCont g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
29.342 +lemma isCont_tendsto_compose: "isCont g l \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
29.343    unfolding isCont_def by (rule tendsto_compose)
29.344
29.345  lemma continuous_within_compose3:
29.346 @@ -2271,7 +2271,7 @@
29.347    assumes S: "S \<noteq> {}" "bdd_above S"
29.348    shows "f (Sup S) = (SUP s:S. f s)"
29.349  proof (rule antisym)
29.350 -  have f: "(f ---> f (Sup S)) (at_left (Sup S))"
29.351 +  have f: "(f \<longlongrightarrow> f (Sup S)) (at_left (Sup S))"
29.352      using cont unfolding continuous_within .
29.353
29.354    show "f (Sup S) \<le> (SUP s:S. f s)"
29.355 @@ -2308,7 +2308,7 @@
29.356    assumes S: "S \<noteq> {}" "bdd_above S"
29.357    shows "f (Sup S) = (INF s:S. f s)"
29.358  proof (rule antisym)
29.359 -  have f: "(f ---> f (Sup S)) (at_left (Sup S))"
29.360 +  have f: "(f \<longlongrightarrow> f (Sup S)) (at_left (Sup S))"
29.361      using cont unfolding continuous_within .
29.362
29.363    show "(INF s:S. f s) \<le> f (Sup S)"
29.364 @@ -2345,7 +2345,7 @@
29.365    assumes S: "S \<noteq> {}" "bdd_below S"
29.366    shows "f (Inf S) = (INF s:S. f s)"
29.367  proof (rule antisym)
29.368 -  have f: "(f ---> f (Inf S)) (at_right (Inf S))"
29.369 +  have f: "(f \<longlongrightarrow> f (Inf S)) (at_right (Inf S))"
29.370      using cont unfolding continuous_within .
29.371
29.372    show "(INF s:S. f s) \<le> f (Inf S)"
29.373 @@ -2382,7 +2382,7 @@
29.374    assumes S: "S \<noteq> {}" "bdd_below S"
29.375    shows "f (Inf S) = (SUP s:S. f s)"
29.376  proof (rule antisym)
29.377 -  have f: "(f ---> f (Inf S)) (at_right (Inf S))"
29.378 +  have f: "(f \<longlongrightarrow> f (Inf S)) (at_right (Inf S))"
29.379      using cont unfolding continuous_within .
29.380
29.381    show "f (Inf S) \<le> (SUP s:S. f s)"
```
```    30.1 --- a/src/HOL/Transcendental.thy	Tue Dec 29 23:50:44 2015 +0100
30.2 +++ b/src/HOL/Transcendental.thy	Wed Dec 30 11:21:54 2015 +0100
30.3 @@ -181,7 +181,7 @@
30.4
30.5  corollary lim_n_over_pown:
30.6    fixes x :: "'a::{real_normed_field,banach}"
30.7 -  shows "1 < norm x \<Longrightarrow> ((\<lambda>n. of_nat n / x^n) ---> 0) sequentially"
30.8 +  shows "1 < norm x \<Longrightarrow> ((\<lambda>n. of_nat n / x^n) \<longlongrightarrow> 0) sequentially"
30.9  using powser_times_n_limit_0 [of "inverse x"]
30.10  by (simp add: norm_divide divide_simps)
30.11
30.12 @@ -811,7 +811,7 @@
30.13    fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
30.14    assumes s: "0 < s"
30.15        and sm: "\<And>x. norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
30.16 -    shows "(f ---> a 0) (at 0)"
30.17 +    shows "(f \<longlongrightarrow> a 0) (at 0)"
30.18  proof -
30.19    have "summable (\<lambda>n. a n * (of_real s / 2) ^ n)"
30.20      apply (rule sums_summable [where l = "f (of_real s / 2)", OF sm])
30.21 @@ -825,7 +825,7 @@
30.22      done
30.23    then have "isCont (\<lambda>x. \<Sum>n. a n * x ^ n) 0"
30.24      by (blast intro: DERIV_continuous)
30.25 -  then have "((\<lambda>x. \<Sum>n. a n * x ^ n) ---> a 0) (at 0)"
30.26 +  then have "((\<lambda>x. \<Sum>n. a n * x ^ n) \<longlongrightarrow> a 0) (at 0)"
30.27      by (simp add: continuous_within powser_zero)
30.28    then show ?thesis
30.29      apply (rule Lim_transform)
30.30 @@ -840,9 +840,9 @@
30.31    fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
30.32    assumes s: "0 < s"
30.33        and sm: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
30.34 -    shows "(f ---> a 0) (at 0)"
30.35 +    shows "(f \<longlongrightarrow> a 0) (at 0)"
30.36  proof -
30.37 -  have *: "((\<lambda>x. if x = 0 then a 0 else f x) ---> a 0) (at 0)"
30.38 +  have *: "((\<lambda>x. if x = 0 then a 0 else f x) \<longlongrightarrow> a 0) (at 0)"
30.39      apply (rule powser_limit_0 [OF s])
30.40      apply (case_tac "x=0")
30.41      apply (auto simp add: powser_sums_zero sm)
30.42 @@ -1208,7 +1208,7 @@
30.43
30.44  lemma tendsto_exp [tendsto_intros]:
30.45    fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
30.46 -  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
30.47 +  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) \<longlongrightarrow> exp a) F"
30.48    by (rule isCont_tendsto_compose [OF isCont_exp])
30.49
30.50  lemma continuous_exp [continuous_intros]:
30.51 @@ -1597,7 +1597,7 @@
30.52
30.53  lemma tendsto_ln [tendsto_intros]:
30.54    fixes a::real shows
30.55 -  "(f ---> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
30.56 +  "(f \<longlongrightarrow> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) \<longlongrightarrow> ln a) F"
30.57    by (rule isCont_tendsto_compose [OF isCont_ln])
30.58
30.59  lemma continuous_ln:
30.60 @@ -2015,7 +2015,7 @@
30.61    qed
30.62  qed
30.63
30.64 -lemma exp_at_bot: "(exp ---> (0::real)) at_bot"
30.65 +lemma exp_at_bot: "(exp \<longlongrightarrow> (0::real)) at_bot"
30.66    unfolding tendsto_Zfun_iff
30.67  proof (rule ZfunI, simp add: eventually_at_bot_dense)
30.68    fix r :: real assume "0 < r"
30.69 @@ -2036,7 +2036,7 @@
30.70
30.71  lemma lim_exp_minus_1:
30.72    fixes x :: "'a::{real_normed_field,banach}"
30.73 -  shows "((\<lambda>z::'a. (exp(z) - 1) / z) ---> 1) (at 0)"
30.74 +  shows "((\<lambda>z::'a. (exp(z) - 1) / z) \<longlongrightarrow> 1) (at 0)"
30.75  proof -
30.76    have "((\<lambda>z::'a. exp(z) - 1) has_field_derivative 1) (at 0)"
30.77      by (intro derivative_eq_intros | simp)+
30.78 @@ -2059,10 +2059,10 @@
30.79    by (intro filtermap_fun_inverse[of ln] exp_at_top ln_at_top)
30.80       (auto simp: eventually_at_top_dense)
30.81
30.82 -lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) ---> (0::real)) at_top"
30.83 +lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) \<longlongrightarrow> (0::real)) at_top"
30.84  proof (induct k)
30.85    case 0
30.86 -  show "((\<lambda>x. x ^ 0 / exp x) ---> (0::real)) at_top"
30.87 +  show "((\<lambda>x. x ^ 0 / exp x) \<longlongrightarrow> (0::real)) at_top"
30.88      by (simp add: inverse_eq_divide[symmetric])
30.89         (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
30.90                at_top_le_at_infinity order_refl)
30.91 @@ -2077,7 +2077,7 @@
30.92      show "eventually (\<lambda>x. exp x \<noteq> 0) at_top"
30.93        by auto
30.94      from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"]
30.95 -    show "((\<lambda>x. real (Suc k) * x ^ k / exp x) ---> 0) at_top"
30.96 +    show "((\<lambda>x. real (Suc k) * x ^ k / exp x) \<longlongrightarrow> 0) at_top"
30.97        by simp
30.98    qed (rule exp_at_top)
30.99  qed
30.100 @@ -2089,7 +2089,7 @@
30.101
30.102
30.103  lemma tendsto_log [tendsto_intros]:
30.104 -  "\<lbrakk>(f ---> a) F; (g ---> b) F; 0 < a; a \<noteq> 1; 0 < b\<rbrakk> \<Longrightarrow> ((\<lambda>x. log (f x) (g x)) ---> log a b) F"
30.105 +  "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F; 0 < a; a \<noteq> 1; 0 < b\<rbrakk> \<Longrightarrow> ((\<lambda>x. log (f x) (g x)) \<longlongrightarrow> log a b) F"
30.106    unfolding log_def by (intro tendsto_intros) auto
30.107
30.108  lemma continuous_log:
30.109 @@ -2505,11 +2505,11 @@
30.110
30.111  lemma tendsto_powr [tendsto_intros]:
30.112    fixes a::real
30.113 -  assumes f: "(f ---> a) F" and g: "(g ---> b) F" and a: "a \<noteq> 0"
30.114 -  shows "((\<lambda>x. f x powr g x) ---> a powr b) F"
30.115 +  assumes f: "(f \<longlongrightarrow> a) F" and g: "(g \<longlongrightarrow> b) F" and a: "a \<noteq> 0"
30.116 +  shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
30.117    unfolding powr_def
30.118  proof (rule filterlim_If)
30.119 -  from f show "((\<lambda>x. 0) ---> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))"
30.120 +  from f show "((\<lambda>x. 0) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))"
30.121      by simp (auto simp: filterlim_iff eventually_inf_principal elim: eventually_mono dest: t1_space_nhds)
30.122  qed (insert f g a, auto intro!: tendsto_intros intro: tendsto_mono inf_le1)
30.123
30.124 @@ -2539,22 +2539,22 @@
30.125
30.126  lemma tendsto_powr2:
30.127    fixes a::real
30.128 -  assumes f: "(f ---> a) F" and g: "(g ---> b) F" and f_nonneg: "\<forall>\<^sub>F x in F. 0 \<le> f x" and b: "0 < b"
30.129 -  shows "((\<lambda>x. f x powr g x) ---> a powr b) F"
30.130 +  assumes f: "(f \<longlongrightarrow> a) F" and g: "(g \<longlongrightarrow> b) F" and f_nonneg: "\<forall>\<^sub>F x in F. 0 \<le> f x" and b: "0 < b"
30.131 +  shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
30.132    unfolding powr_def
30.133  proof (rule filterlim_If)
30.134 -  from f show "((\<lambda>x. 0) ---> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))"
30.135 +  from f show "((\<lambda>x. 0) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))"
30.136      by simp (auto simp: filterlim_iff eventually_inf_principal elim: eventually_mono dest: t1_space_nhds)
30.137  next
30.138    { assume "a = 0"
30.139      with f f_nonneg have "LIM x inf F (principal {x. f x \<noteq> 0}). f x :> at_right 0"
30.140        by (auto simp add: filterlim_at eventually_inf_principal le_less
30.141                 elim: eventually_mono intro: tendsto_mono inf_le1)
30.142 -    then have "((\<lambda>x. exp (g x * ln (f x))) ---> 0) (inf F (principal {x. f x \<noteq> 0}))"
30.143 +    then have "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> 0) (inf F (principal {x. f x \<noteq> 0}))"
30.144        by (auto intro!: filterlim_compose[OF exp_at_bot] filterlim_compose[OF ln_at_0]
30.145                         filterlim_tendsto_pos_mult_at_bot[OF _ \<open>0 < b\<close>]
30.146                 intro: tendsto_mono inf_le1 g) }
30.147 -  then show "((\<lambda>x. exp (g x * ln (f x))) ---> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x \<noteq> 0}))"
30.148 +  then show "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x \<noteq> 0}))"
30.149      using f g by (auto intro!: tendsto_intros intro: tendsto_mono inf_le1)
30.150  qed
30.151
30.152 @@ -2596,19 +2596,19 @@
30.153  declare has_real_derivative_powr[THEN DERIV_chain2, derivative_intros]
30.154
30.155  lemma tendsto_zero_powrI:
30.156 -  assumes "(f ---> (0::real)) F" "(g ---> b) F" "\<forall>\<^sub>F x in F. 0 \<le> f x" "0 < b"
30.157 -  shows "((\<lambda>x. f x powr g x) ---> 0) F"
30.158 +  assumes "(f \<longlongrightarrow> (0::real)) F" "(g \<longlongrightarrow> b) F" "\<forall>\<^sub>F x in F. 0 \<le> f x" "0 < b"
30.159 +  shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> 0) F"
30.160    using tendsto_powr2[OF assms] by simp
30.161
30.162  lemma tendsto_neg_powr:
30.163    assumes "s < 0"
30.164      and f: "LIM x F. f x :> at_top"
30.165 -  shows "((\<lambda>x. f x powr s) ---> (0::real)) F"
30.166 +  shows "((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F"
30.167  proof -
30.168 -  have "((\<lambda>x. exp (s * ln (f x))) ---> (0::real)) F" (is "?X")
30.169 +  have "((\<lambda>x. exp (s * ln (f x))) \<longlongrightarrow> (0::real)) F" (is "?X")
30.170      by (auto intro!: filterlim_compose[OF exp_at_bot] filterlim_compose[OF ln_at_top]
30.171                       filterlim_tendsto_neg_mult_at_bot assms)
30.172 -  also have "?X \<longleftrightarrow> ((\<lambda>x. f x powr s) ---> (0::real)) F"
30.173 +  also have "?X \<longleftrightarrow> ((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F"
30.174      using f filterlim_at_top_dense[of f F]
30.175      by (intro filterlim_cong[OF refl refl]) (auto simp: neq_iff powr_def elim: eventually_mono)
30.176    finally show ?thesis .
30.177 @@ -2616,14 +2616,14 @@
30.178
30.179  lemma tendsto_exp_limit_at_right:
30.180    fixes x :: real
30.181 -  shows "((\<lambda>y. (1 + x * y) powr (1 / y)) ---> exp x) (at_right 0)"
30.182 +  shows "((\<lambda>y. (1 + x * y) powr (1 / y)) \<longlongrightarrow> exp x) (at_right 0)"
30.183  proof cases
30.184    assume "x \<noteq> 0"
30.185    have "((\<lambda>y. ln (1 + x * y)::real) has_real_derivative 1 * x) (at 0)"
30.186      by (auto intro!: derivative_eq_intros)
30.187 -  then have "((\<lambda>y. ln (1 + x * y) / y) ---> x) (at 0)"
30.188 +  then have "((\<lambda>y. ln (1 + x * y) / y) \<longlongrightarrow> x) (at 0)"
30.189      by (auto simp add: has_field_derivative_def field_has_derivative_at)
30.190 -  then have *: "((\<lambda>y. exp (ln (1 + x * y) / y)) ---> exp x) (at 0)"
30.191 +  then have *: "((\<lambda>y. exp (ln (1 + x * y) / y)) \<longlongrightarrow> exp x) (at 0)"
30.192      by (rule tendsto_intros)
30.193    then show ?thesis
30.194    proof (rule filterlim_mono_eventually)
30.195 @@ -2638,7 +2638,7 @@
30.196
30.197  lemma tendsto_exp_limit_at_top:
30.198    fixes x :: real
30.199 -  shows "((\<lambda>y. (1 + x / y) powr y) ---> exp x) at_top"
30.200 +  shows "((\<lambda>y. (1 + x / y) powr y) \<longlongrightarrow> exp x) at_top"
30.201    apply (subst filterlim_at_top_to_right)
30.202    apply (simp add: inverse_eq_divide)
30.203    apply (rule tendsto_exp_limit_at_right)
30.204 @@ -2819,12 +2819,12 @@
30.205
30.206  lemma tendsto_sin [tendsto_intros]:
30.207    fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
30.208 -  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F"
30.209 +  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) \<longlongrightarrow> sin a) F"
30.210    by (rule isCont_tendsto_compose [OF isCont_sin])
30.211
30.212  lemma tendsto_cos [tendsto_intros]:
30.213    fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
30.214 -  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F"
30.215 +  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) \<longlongrightarrow> cos a) F"
30.216    by (rule isCont_tendsto_compose [OF isCont_cos])
30.217
30.218  lemma continuous_sin [continuous_intros]:
30.219 @@ -4049,7 +4049,7 @@
30.220
30.221  lemma tendsto_tan [tendsto_intros]:
30.222    fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
30.223 -  shows "\<lbrakk>(f ---> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) ---> tan a) F"
30.224 +  shows "\<lbrakk>(f \<longlongrightarrow> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) \<longlongrightarrow> tan a) F"
30.225    by (rule isCont_tendsto_compose [OF isCont_tan])
30.226
30.227  lemma continuous_tan:
30.228 @@ -4318,7 +4318,7 @@
30.229
30.230  lemma tendsto_cot [tendsto_intros]:
30.231    fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
30.232 -  shows "\<lbrakk>(f ---> a) F; sin a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. cot (f x)) ---> cot a) F"
30.233 +  shows "\<lbrakk>(f \<longlongrightarrow> a) F; sin a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. cot (f x)) \<longlongrightarrow> cot a) F"
30.234    by (rule isCont_tendsto_compose [OF isCont_cot])
30.235
30.236  lemma continuous_cot:
30.237 @@ -4631,7 +4631,7 @@
30.238    apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
30.239    done
30.240
30.241 -lemma tendsto_arctan [tendsto_intros]: "(f ---> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) ---> arctan x) F"
30.242 +lemma tendsto_arctan [tendsto_intros]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) \<longlongrightarrow> arctan x) F"
30.243    by (rule isCont_tendsto_compose [OF isCont_arctan])
30.244
30.245  lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))"
30.246 @@ -4690,7 +4690,7 @@
30.247       (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
30.248             intro!: tan_monotone exI[of _ "pi/2"])
30.249
30.250 -lemma tendsto_arctan_at_top: "(arctan ---> (pi/2)) at_top"
30.251 +lemma tendsto_arctan_at_top: "(arctan \<longlongrightarrow> (pi/2)) at_top"
30.252  proof (rule tendstoI)
30.253    fix e :: real
30.254    assume "0 < e"
30.255 @@ -4712,7 +4712,7 @@
30.256    qed
30.257  qed
30.258
30.259 -lemma tendsto_arctan_at_bot: "(arctan ---> - (pi/2)) at_bot"
30.260 +lemma tendsto_arctan_at_bot: "(arctan \<longlongrightarrow> - (pi/2)) at_bot"
30.261    unfolding filterlim_at_bot_mirror arctan_minus
30.262    by (intro tendsto_minus tendsto_arctan_at_top)
30.263
```