# Theory Limits

Up to index of Isabelle/HOL

theory Limits
imports RealVector
`(*  Title       : Limits.thy    Author      : Brian Huffman*)header {* Filters and Limits *}theory Limitsimports RealVectorbeginsubsection {* Filters *}text {*  This definition also allows non-proper filters.*}locale is_filter =  fixes F :: "('a => bool) => bool"  assumes True: "F (λx. True)"  assumes conj: "F (λx. P x) ==> F (λx. Q x) ==> F (λx. P x ∧ Q x)"  assumes mono: "∀x. P x --> Q x ==> F (λx. P x) ==> F (λx. Q x)"typedef 'a filter = "{F :: ('a => bool) => bool. is_filter F}"proof  show "(λx. True) ∈ ?filter" by (auto intro: is_filter.intro)qedlemma is_filter_Rep_filter: "is_filter (Rep_filter F)"  using Rep_filter [of F] by simplemma Abs_filter_inverse':  assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"  using assms by (simp add: Abs_filter_inverse)subsection {* Eventually *}definition eventually :: "('a => bool) => 'a filter => bool"  where "eventually P F <-> Rep_filter F P"lemma eventually_Abs_filter:  assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"  unfolding eventually_def using assms by (simp add: Abs_filter_inverse)lemma filter_eq_iff:  shows "F = F' <-> (∀P. eventually P F = eventually P F')"  unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..lemma eventually_True [simp]: "eventually (λx. True) F"  unfolding eventually_def  by (rule is_filter.True [OF is_filter_Rep_filter])lemma always_eventually: "∀x. P x ==> eventually P F"proof -  assume "∀x. P x" hence "P = (λx. True)" by (simp add: ext)  thus "eventually P F" by simpqedlemma eventually_mono:  "(∀x. P x --> Q x) ==> eventually P F ==> eventually Q F"  unfolding eventually_def  by (rule is_filter.mono [OF is_filter_Rep_filter])lemma eventually_conj:  assumes P: "eventually (λx. P x) F"  assumes Q: "eventually (λx. Q x) F"  shows "eventually (λx. P x ∧ Q x) F"  using assms unfolding eventually_def  by (rule is_filter.conj [OF is_filter_Rep_filter])lemma eventually_Ball_finite:  assumes "finite A" and "∀y∈A. eventually (λx. P x y) net"  shows "eventually (λx. ∀y∈A. P x y) net"using assms by (induct set: finite, simp, simp add: eventually_conj)lemma eventually_all_finite:  fixes P :: "'a => 'b::finite => bool"  assumes "!!y. eventually (λx. P x y) net"  shows "eventually (λx. ∀y. P x y) net"using eventually_Ball_finite [of UNIV P] assms by simplemma eventually_mp:  assumes "eventually (λx. P x --> Q x) F"  assumes "eventually (λx. P x) F"  shows "eventually (λx. Q x) F"proof (rule eventually_mono)  show "∀x. (P x --> Q x) ∧ P x --> Q x" by simp  show "eventually (λx. (P x --> Q x) ∧ P x) F"    using assms by (rule eventually_conj)qedlemma eventually_rev_mp:  assumes "eventually (λx. P x) F"  assumes "eventually (λx. P x --> Q x) F"  shows "eventually (λx. Q x) F"using assms(2) assms(1) by (rule eventually_mp)lemma eventually_conj_iff:  "eventually (λx. P x ∧ Q x) F <-> eventually P F ∧ eventually Q F"  by (auto intro: eventually_conj elim: eventually_rev_mp)lemma eventually_elim1:  assumes "eventually (λi. P i) F"  assumes "!!i. P i ==> Q i"  shows "eventually (λi. Q i) F"  using assms by (auto elim!: eventually_rev_mp)lemma eventually_elim2:  assumes "eventually (λi. P i) F"  assumes "eventually (λi. Q i) F"  assumes "!!i. P i ==> Q i ==> R i"  shows "eventually (λi. R i) F"  using assms by (auto elim!: eventually_rev_mp)lemma eventually_subst:  assumes "eventually (λn. P n = Q n) F"  shows "eventually P F = eventually Q F" (is "?L = ?R")proof -  from assms have "eventually (λx. P x --> Q x) F"      and "eventually (λx. Q x --> P x) F"    by (auto elim: eventually_elim1)  then show ?thesis by (auto elim: eventually_elim2)qedML {*  fun eventually_elim_tac ctxt thms thm =    let      val thy = Proof_Context.theory_of ctxt      val mp_thms = thms RL [@{thm eventually_rev_mp}]      val raw_elim_thm =        (@{thm allI} RS @{thm always_eventually})        |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms        |> fold (fn _ => fn thm => @{thm impI} RS thm) thms      val cases_prop = prop_of (raw_elim_thm RS thm)      val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])])    in      CASES cases (rtac raw_elim_thm 1) thm    end*}method_setup eventually_elim = {*  Scan.succeed (fn ctxt => METHOD_CASES (eventually_elim_tac ctxt))*} "elimination of eventually quantifiers"subsection {* Finer-than relation *}text {* @{term "F ≤ F'"} means that filter @{term F} is finer thanfilter @{term F'}. *}instantiation filter :: (type) complete_latticebegindefinition le_filter_def:  "F ≤ F' <-> (∀P. eventually P F' --> eventually P F)"definition  "(F :: 'a filter) < F' <-> F ≤ F' ∧ ¬ F' ≤ F"definition  "top = Abs_filter (λP. ∀x. P x)"definition  "bot = Abs_filter (λP. True)"definition  "sup F F' = Abs_filter (λP. eventually P F ∧ eventually P F')"definition  "inf F F' = Abs_filter      (λP. ∃Q R. eventually Q F ∧ eventually R F' ∧ (∀x. Q x ∧ R x --> P x))"definition  "Sup S = Abs_filter (λP. ∀F∈S. eventually P F)"definition  "Inf S = Sup {F::'a filter. ∀F'∈S. F ≤ F'}"lemma eventually_top [simp]: "eventually P top <-> (∀x. P x)"  unfolding top_filter_def  by (rule eventually_Abs_filter, rule is_filter.intro, auto)lemma eventually_bot [simp]: "eventually P bot"  unfolding bot_filter_def  by (subst eventually_Abs_filter, rule is_filter.intro, auto)lemma eventually_sup:  "eventually P (sup F F') <-> eventually P F ∧ eventually P F'"  unfolding sup_filter_def  by (rule eventually_Abs_filter, rule is_filter.intro)     (auto elim!: eventually_rev_mp)lemma eventually_inf:  "eventually P (inf F F') <->   (∃Q R. eventually Q F ∧ eventually R F' ∧ (∀x. Q x ∧ R x --> P x))"  unfolding inf_filter_def  apply (rule eventually_Abs_filter, rule is_filter.intro)  apply (fast intro: eventually_True)  apply clarify  apply (intro exI conjI)  apply (erule (1) eventually_conj)  apply (erule (1) eventually_conj)  apply simp  apply auto  donelemma eventually_Sup:  "eventually P (Sup S) <-> (∀F∈S. eventually P F)"  unfolding Sup_filter_def  apply (rule eventually_Abs_filter, rule is_filter.intro)  apply (auto intro: eventually_conj elim!: eventually_rev_mp)  doneinstance proof  fix F F' F'' :: "'a filter" and S :: "'a filter set"  { show "F < F' <-> F ≤ F' ∧ ¬ F' ≤ F"    by (rule less_filter_def) }  { show "F ≤ F"    unfolding le_filter_def by simp }  { assume "F ≤ F'" and "F' ≤ F''" thus "F ≤ F''"    unfolding le_filter_def by simp }  { assume "F ≤ F'" and "F' ≤ F" thus "F = F'"    unfolding le_filter_def filter_eq_iff by fast }  { show "F ≤ top"    unfolding le_filter_def eventually_top by (simp add: always_eventually) }  { show "bot ≤ F"    unfolding le_filter_def by simp }  { show "F ≤ sup F F'" and "F' ≤ sup F F'"    unfolding le_filter_def eventually_sup by simp_all }  { assume "F ≤ F''" and "F' ≤ F''" thus "sup F F' ≤ F''"    unfolding le_filter_def eventually_sup by simp }  { show "inf F F' ≤ F" and "inf F F' ≤ F'"    unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }  { assume "F ≤ F'" and "F ≤ F''" thus "F ≤ inf F' F''"    unfolding le_filter_def eventually_inf    by (auto elim!: eventually_mono intro: eventually_conj) }  { assume "F ∈ S" thus "F ≤ Sup S"    unfolding le_filter_def eventually_Sup by simp }  { assume "!!F. F ∈ S ==> F ≤ F'" thus "Sup S ≤ F'"    unfolding le_filter_def eventually_Sup by simp }  { assume "F'' ∈ S" thus "Inf S ≤ F''"    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }  { assume "!!F'. F' ∈ S ==> F ≤ F'" thus "F ≤ Inf S"    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }qedendlemma filter_leD:  "F ≤ F' ==> eventually P F' ==> eventually P F"  unfolding le_filter_def by simplemma filter_leI:  "(!!P. eventually P F' ==> eventually P F) ==> F ≤ F'"  unfolding le_filter_def by simplemma eventually_False:  "eventually (λx. False) F <-> F = bot"  unfolding filter_eq_iff by (auto elim: eventually_rev_mp)abbreviation (input) trivial_limit :: "'a filter => bool"  where "trivial_limit F ≡ F = bot"lemma trivial_limit_def: "trivial_limit F <-> eventually (λx. False) F"  by (rule eventually_False [symmetric])subsection {* Map function for filters *}definition filtermap :: "('a => 'b) => 'a filter => 'b filter"  where "filtermap f F = Abs_filter (λP. eventually (λx. P (f x)) F)"lemma eventually_filtermap:  "eventually P (filtermap f F) = eventually (λx. P (f x)) F"  unfolding filtermap_def  apply (rule eventually_Abs_filter)  apply (rule is_filter.intro)  apply (auto elim!: eventually_rev_mp)  donelemma filtermap_ident: "filtermap (λx. x) F = F"  by (simp add: filter_eq_iff eventually_filtermap)lemma filtermap_filtermap:  "filtermap f (filtermap g F) = filtermap (λx. f (g x)) F"  by (simp add: filter_eq_iff eventually_filtermap)lemma filtermap_mono: "F ≤ F' ==> filtermap f F ≤ filtermap f F'"  unfolding le_filter_def eventually_filtermap by simplemma filtermap_bot [simp]: "filtermap f bot = bot"  by (simp add: filter_eq_iff eventually_filtermap)lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"  by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)subsection {* Order filters *}definition at_top :: "('a::order) filter"  where "at_top = Abs_filter (λP. ∃k. ∀n≥k. P n)"lemma eventually_at_top_linorder: "eventually P at_top <-> (∃N::'a::linorder. ∀n≥N. P n)"  unfolding at_top_defproof (rule eventually_Abs_filter, rule is_filter.intro)  fix P Q :: "'a => bool"  assume "∃i. ∀n≥i. P n" and "∃j. ∀n≥j. Q n"  then obtain i j where "∀n≥i. P n" and "∀n≥j. Q n" by auto  then have "∀n≥max i j. P n ∧ Q n" by simp  then show "∃k. ∀n≥k. P n ∧ Q n" ..qed autolemma eventually_ge_at_top:  "eventually (λx. (c::_::linorder) ≤ x) at_top"  unfolding eventually_at_top_linorder by autolemma eventually_at_top_dense: "eventually P at_top <-> (∃N::'a::dense_linorder. ∀n>N. P n)"  unfolding eventually_at_top_linorderproof safe  fix N assume "∀n≥N. P n" then show "∃N. ∀n>N. P n" by (auto intro!: exI[of _ N])next  fix N assume "∀n>N. P n"  moreover from gt_ex[of N] guess y ..  ultimately show "∃N. ∀n≥N. P n" by (auto intro!: exI[of _ y])qedlemma eventually_gt_at_top:  "eventually (λx. (c::_::dense_linorder) < x) at_top"  unfolding eventually_at_top_dense by autodefinition at_bot :: "('a::order) filter"  where "at_bot = Abs_filter (λP. ∃k. ∀n≤k. P n)"lemma eventually_at_bot_linorder:  fixes P :: "'a::linorder => bool" shows "eventually P at_bot <-> (∃N. ∀n≤N. P n)"  unfolding at_bot_defproof (rule eventually_Abs_filter, rule is_filter.intro)  fix P Q :: "'a => bool"  assume "∃i. ∀n≤i. P n" and "∃j. ∀n≤j. Q n"  then obtain i j where "∀n≤i. P n" and "∀n≤j. Q n" by auto  then have "∀n≤min i j. P n ∧ Q n" by simp  then show "∃k. ∀n≤k. P n ∧ Q n" ..qed autolemma eventually_le_at_bot:  "eventually (λx. x ≤ (c::_::linorder)) at_bot"  unfolding eventually_at_bot_linorder by autolemma eventually_at_bot_dense:  fixes P :: "'a::dense_linorder => bool" shows "eventually P at_bot <-> (∃N. ∀n<N. P n)"  unfolding eventually_at_bot_linorderproof safe  fix N assume "∀n≤N. P n" then show "∃N. ∀n<N. P n" by (auto intro!: exI[of _ N])next  fix N assume "∀n<N. P n"   moreover from lt_ex[of N] guess y ..  ultimately show "∃N. ∀n≤N. P n" by (auto intro!: exI[of _ y])qedlemma eventually_gt_at_bot:  "eventually (λx. x < (c::_::dense_linorder)) at_bot"  unfolding eventually_at_bot_dense by autosubsection {* Sequentially *}abbreviation sequentially :: "nat filter"  where "sequentially == at_top"lemma sequentially_def: "sequentially = Abs_filter (λP. ∃k. ∀n≥k. P n)"  unfolding at_top_def by simplemma eventually_sequentially:  "eventually P sequentially <-> (∃N. ∀n≥N. P n)"  by (rule eventually_at_top_linorder)lemma sequentially_bot [simp, intro]: "sequentially ≠ bot"  unfolding filter_eq_iff eventually_sequentially by autolemmas trivial_limit_sequentially = sequentially_botlemma eventually_False_sequentially [simp]:  "¬ eventually (λn. False) sequentially"  by (simp add: eventually_False)lemma le_sequentially:  "F ≤ sequentially <-> (∀N. eventually (λn. N ≤ n) F)"  unfolding le_filter_def eventually_sequentially  by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)lemma eventually_sequentiallyI:  assumes "!!x. c ≤ x ==> P x"  shows "eventually P sequentially"using assms by (auto simp: eventually_sequentially)subsection {* Standard filters *}definition within :: "'a filter => 'a set => 'a filter" (infixr "within" 70)  where "F within S = Abs_filter (λP. eventually (λx. x ∈ S --> P x) F)"definition (in topological_space) nhds :: "'a => 'a filter"  where "nhds a = Abs_filter (λP. ∃S. open S ∧ a ∈ S ∧ (∀x∈S. P x))"definition (in topological_space) at :: "'a => 'a filter"  where "at a = nhds a within - {a}"abbreviation at_right :: "'a::{topological_space, order} => 'a filter" where  "at_right x ≡ at x within {x <..}"abbreviation at_left :: "'a::{topological_space, order} => 'a filter" where  "at_left x ≡ at x within {..< x}"definition at_infinity :: "'a::real_normed_vector filter" where  "at_infinity = Abs_filter (λP. ∃r. ∀x. r ≤ norm x --> P x)"lemma eventually_within:  "eventually P (F within S) = eventually (λx. x ∈ S --> P x) F"  unfolding within_def  by (rule eventually_Abs_filter, rule is_filter.intro)     (auto elim!: eventually_rev_mp)lemma within_UNIV [simp]: "F within UNIV = F"  unfolding filter_eq_iff eventually_within by simplemma within_empty [simp]: "F within {} = bot"  unfolding filter_eq_iff eventually_within by simplemma within_within_eq: "(F within S) within T = F within (S ∩ T)"  by (auto simp: filter_eq_iff eventually_within elim: eventually_elim1)lemma at_within_eq: "at x within T = nhds x within (T - {x})"  unfolding at_def within_within_eq by (simp add: ac_simps Diff_eq)lemma within_le: "F within S ≤ F"  unfolding le_filter_def eventually_within by (auto elim: eventually_elim1)lemma le_withinI: "F ≤ F' ==> eventually (λx. x ∈ S) F ==> F ≤ F' within S"  unfolding le_filter_def eventually_within by (auto elim: eventually_elim2)lemma le_within_iff: "eventually (λx. x ∈ S) F ==> F ≤ F' within S <-> F ≤ F'"  by (blast intro: within_le le_withinI order_trans)lemma eventually_nhds:  "eventually P (nhds a) <-> (∃S. open S ∧ a ∈ S ∧ (∀x∈S. P x))"unfolding nhds_defproof (rule eventually_Abs_filter, rule is_filter.intro)  have "open UNIV ∧ a ∈ UNIV ∧ (∀x∈UNIV. True)" by simp  thus "∃S. open S ∧ a ∈ S ∧ (∀x∈S. True)" ..next  fix P Q  assume "∃S. open S ∧ a ∈ S ∧ (∀x∈S. P x)"     and "∃T. open T ∧ a ∈ T ∧ (∀x∈T. Q x)"  then obtain S T where    "open S ∧ a ∈ S ∧ (∀x∈S. P x)"    "open T ∧ a ∈ T ∧ (∀x∈T. Q x)" by auto  hence "open (S ∩ T) ∧ a ∈ S ∩ T ∧ (∀x∈(S ∩ T). P x ∧ Q x)"    by (simp add: open_Int)  thus "∃S. open S ∧ a ∈ S ∧ (∀x∈S. P x ∧ Q x)" ..qed autolemma eventually_nhds_metric:  "eventually P (nhds a) <-> (∃d>0. ∀x. dist x a < d --> P x)"unfolding eventually_nhds open_distapply safeapply fastapply (rule_tac x="{x. dist x a < d}" in exI, simp)apply clarsimpapply (rule_tac x="d - dist x a" in exI, clarsimp)apply (simp only: less_diff_eq)apply (erule le_less_trans [OF dist_triangle])donelemma nhds_neq_bot [simp]: "nhds a ≠ bot"  unfolding trivial_limit_def eventually_nhds by simplemma eventually_at_topological:  "eventually P (at a) <-> (∃S. open S ∧ a ∈ S ∧ (∀x∈S. x ≠ a --> P x))"unfolding at_def eventually_within eventually_nhds by simplemma eventually_at:  fixes a :: "'a::metric_space"  shows "eventually P (at a) <-> (∃d>0. ∀x. x ≠ a ∧ dist x a < d --> P x)"unfolding at_def eventually_within eventually_nhds_metric by autolemma eventually_within_less: (* COPY FROM Topo/eventually_within *)  "eventually P (at a within S) <-> (∃d>0. ∀x∈S. 0 < dist x a ∧ dist x a < d --> P x)"  unfolding eventually_within eventually_at dist_nz by autolemma eventually_within_le: (* COPY FROM Topo/eventually_within_le *)  "eventually P (at a within S) <-> (∃d>0. ∀x∈S. 0 < dist x a ∧ dist x a <= d --> P x)"  unfolding eventually_within_less by auto (metis dense order_le_less_trans)lemma at_eq_bot_iff: "at a = bot <-> open {a}"  unfolding trivial_limit_def eventually_at_topological  by (safe, case_tac "S = {a}", simp, fast, fast)lemma at_neq_bot [simp]: "at (a::'a::perfect_space) ≠ bot"  by (simp add: at_eq_bot_iff not_open_singleton)lemma trivial_limit_at_left_real [simp]: (* maybe generalize type *)  "¬ trivial_limit (at_left (x::real))"  unfolding trivial_limit_def eventually_within_le  apply clarsimp  apply (rule_tac x="x - d/2" in bexI)  apply (auto simp: dist_real_def)  donelemma trivial_limit_at_right_real [simp]: (* maybe generalize type *)  "¬ trivial_limit (at_right (x::real))"  unfolding trivial_limit_def eventually_within_le  apply clarsimp  apply (rule_tac x="x + d/2" in bexI)  apply (auto simp: dist_real_def)  donelemma eventually_at_infinity:  "eventually P at_infinity <-> (∃b. ∀x. b ≤ norm x --> P x)"unfolding at_infinity_defproof (rule eventually_Abs_filter, rule is_filter.intro)  fix P Q :: "'a => bool"  assume "∃r. ∀x. r ≤ norm x --> P x" and "∃s. ∀x. s ≤ norm x --> Q x"  then obtain r s where    "∀x. r ≤ norm x --> P x" and "∀x. s ≤ norm x --> Q x" by auto  then have "∀x. max r s ≤ norm x --> P x ∧ Q x" by simp  then show "∃r. ∀x. r ≤ norm x --> P x ∧ Q x" ..qed autolemma at_infinity_eq_at_top_bot:  "(at_infinity :: real filter) = sup at_top at_bot"  unfolding sup_filter_def at_infinity_def eventually_at_top_linorder eventually_at_bot_linorderproof (intro arg_cong[where f=Abs_filter] ext iffI)  fix P :: "real => bool" assume "∃r. ∀x. r ≤ norm x --> P x"  then guess r ..  then have "(∀x≥r. P x) ∧ (∀x≤-r. P x)" by auto  then show "(∃r. ∀x≥r. P x) ∧ (∃r. ∀x≤r. P x)" by autonext  fix P :: "real => bool" assume "(∃r. ∀x≥r. P x) ∧ (∃r. ∀x≤r. P x)"  then obtain p q where "∀x≥p. P x" "∀x≤q. P x" by auto  then show "∃r. ∀x. r ≤ norm x --> P x"    by (intro exI[of _ "max p (-q)"])       (auto simp: abs_real_def)qedlemma at_top_le_at_infinity:  "at_top ≤ (at_infinity :: real filter)"  unfolding at_infinity_eq_at_top_bot by simplemma at_bot_le_at_infinity:  "at_bot ≤ (at_infinity :: real filter)"  unfolding at_infinity_eq_at_top_bot by simpsubsection {* Boundedness *}definition Bfun :: "('a => 'b::real_normed_vector) => 'a filter => bool"  where "Bfun f F = (∃K>0. eventually (λx. norm (f x) ≤ K) F)"lemma BfunI:  assumes K: "eventually (λx. norm (f x) ≤ K) F" shows "Bfun f F"unfolding Bfun_defproof (intro exI conjI allI)  show "0 < max K 1" by simpnext  show "eventually (λx. norm (f x) ≤ max K 1) F"    using K by (rule eventually_elim1, simp)qedlemma BfunE:  assumes "Bfun f F"  obtains B where "0 < B" and "eventually (λx. norm (f x) ≤ B) F"using assms unfolding Bfun_def by fastsubsection {* Convergence to Zero *}definition Zfun :: "('a => 'b::real_normed_vector) => 'a filter => bool"  where "Zfun f F = (∀r>0. eventually (λx. norm (f x) < r) F)"lemma ZfunI:  "(!!r. 0 < r ==> eventually (λx. norm (f x) < r) F) ==> Zfun f F"  unfolding Zfun_def by simplemma ZfunD:  "[|Zfun f F; 0 < r|] ==> eventually (λx. norm (f x) < r) F"  unfolding Zfun_def by simplemma Zfun_ssubst:  "eventually (λx. f x = g x) F ==> Zfun g F ==> Zfun f F"  unfolding Zfun_def by (auto elim!: eventually_rev_mp)lemma Zfun_zero: "Zfun (λx. 0) F"  unfolding Zfun_def by simplemma Zfun_norm_iff: "Zfun (λx. norm (f x)) F = Zfun (λx. f x) F"  unfolding Zfun_def by simplemma Zfun_imp_Zfun:  assumes f: "Zfun f F"  assumes g: "eventually (λx. norm (g x) ≤ norm (f x) * K) F"  shows "Zfun (λx. g x) F"proof (cases)  assume K: "0 < K"  show ?thesis  proof (rule ZfunI)    fix r::real assume "0 < r"    hence "0 < r / K"      using K by (rule divide_pos_pos)    then have "eventually (λx. norm (f x) < r / K) F"      using ZfunD [OF f] by fast    with g show "eventually (λx. norm (g x) < r) F"    proof eventually_elim      case (elim x)      hence "norm (f x) * K < r"        by (simp add: pos_less_divide_eq K)      thus ?case        by (simp add: order_le_less_trans [OF elim(1)])    qed  qednext  assume "¬ 0 < K"  hence K: "K ≤ 0" by (simp only: not_less)  show ?thesis  proof (rule ZfunI)    fix r :: real    assume "0 < r"    from g show "eventually (λx. norm (g x) < r) F"    proof eventually_elim      case (elim x)      also have "norm (f x) * K ≤ norm (f x) * 0"        using K norm_ge_zero by (rule mult_left_mono)      finally show ?case        using `0 < r` by simp    qed  qedqedlemma Zfun_le: "[|Zfun g F; ∀x. norm (f x) ≤ norm (g x)|] ==> Zfun f F"  by (erule_tac K="1" in Zfun_imp_Zfun, simp)lemma Zfun_add:  assumes f: "Zfun f F" and g: "Zfun g F"  shows "Zfun (λx. f x + g x) F"proof (rule ZfunI)  fix r::real assume "0 < r"  hence r: "0 < r / 2" by simp  have "eventually (λx. norm (f x) < r/2) F"    using f r by (rule ZfunD)  moreover  have "eventually (λx. norm (g x) < r/2) F"    using g r by (rule ZfunD)  ultimately  show "eventually (λx. norm (f x + g x) < r) F"  proof eventually_elim    case (elim x)    have "norm (f x + g x) ≤ norm (f x) + norm (g x)"      by (rule norm_triangle_ineq)    also have "… < r/2 + r/2"      using elim by (rule add_strict_mono)    finally show ?case      by simp  qedqedlemma Zfun_minus: "Zfun f F ==> Zfun (λx. - f x) F"  unfolding Zfun_def by simplemma Zfun_diff: "[|Zfun f F; Zfun g F|] ==> Zfun (λx. f x - g x) F"  by (simp only: diff_minus Zfun_add Zfun_minus)lemma (in bounded_linear) Zfun:  assumes g: "Zfun g F"  shows "Zfun (λx. f (g x)) F"proof -  obtain K where "!!x. norm (f x) ≤ norm x * K"    using bounded by fast  then have "eventually (λx. norm (f (g x)) ≤ norm (g x) * K) F"    by simp  with g show ?thesis    by (rule Zfun_imp_Zfun)qedlemma (in bounded_bilinear) Zfun:  assumes f: "Zfun f F"  assumes g: "Zfun g F"  shows "Zfun (λx. f x ** g x) F"proof (rule ZfunI)  fix r::real assume r: "0 < r"  obtain K where K: "0 < K"    and norm_le: "!!x y. norm (x ** y) ≤ norm x * norm y * K"    using pos_bounded by fast  from K have K': "0 < inverse K"    by (rule positive_imp_inverse_positive)  have "eventually (λx. norm (f x) < r) F"    using f r by (rule ZfunD)  moreover  have "eventually (λx. norm (g x) < inverse K) F"    using g K' by (rule ZfunD)  ultimately  show "eventually (λx. norm (f x ** g x) < r) F"  proof eventually_elim    case (elim x)    have "norm (f x ** g x) ≤ norm (f x) * norm (g x) * K"      by (rule norm_le)    also have "norm (f x) * norm (g x) * K < r * inverse K * K"      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)    also from K have "r * inverse K * K = r"      by simp    finally show ?case .  qedqedlemma (in bounded_bilinear) Zfun_left:  "Zfun f F ==> Zfun (λx. f x ** a) F"  by (rule bounded_linear_left [THEN bounded_linear.Zfun])lemma (in bounded_bilinear) Zfun_right:  "Zfun f F ==> Zfun (λx. a ** f x) F"  by (rule bounded_linear_right [THEN bounded_linear.Zfun])lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]subsection {* Limits *}definition filterlim :: "('a => 'b) => 'b filter => 'a filter => bool" where  "filterlim f F2 F1 <-> filtermap f F1 ≤ F2"syntax  "_LIM" :: "pttrns => 'a => 'b => 'a => bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)translations  "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"lemma filterlim_iff:  "(LIM x F1. f x :> F2) <-> (∀P. eventually P F2 --> eventually (λx. P (f x)) F1)"  unfolding filterlim_def le_filter_def eventually_filtermap ..lemma filterlim_compose:  "filterlim g F3 F2 ==> filterlim f F2 F1 ==> filterlim (λx. g (f x)) F3 F1"  unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)lemma filterlim_mono:  "filterlim f F2 F1 ==> F2 ≤ F2' ==> F1' ≤ F1 ==> filterlim f F2' F1'"  unfolding filterlim_def by (metis filtermap_mono order_trans)lemma filterlim_ident: "LIM x F. x :> F"  by (simp add: filterlim_def filtermap_ident)lemma filterlim_cong:  "F1 = F1' ==> F2 = F2' ==> eventually (λx. f x = g x) F2 ==> filterlim f F1 F2 = filterlim g F1' F2'"  by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)lemma filterlim_within:  "(LIM x F1. f x :> F2 within S) <-> (eventually (λx. f x ∈ S) F1 ∧ (LIM x F1. f x :> F2))"  unfolding filterlim_defproof safe  assume "filtermap f F1 ≤ F2 within S" then show "eventually (λx. f x ∈ S) F1"    by (auto simp: le_filter_def eventually_filtermap eventually_within elim!: allE[of _ "λx. x ∈ S"])qed (auto intro: within_le order_trans simp: le_within_iff eventually_filtermap)lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (λx. f (g x)) F1 F2"  unfolding filterlim_def filtermap_filtermap ..lemma filterlim_sup:  "filterlim f F F1 ==> filterlim f F F2 ==> filterlim f F (sup F1 F2)"  unfolding filterlim_def filtermap_sup by autolemma filterlim_Suc: "filterlim Suc sequentially sequentially"  by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)abbreviation (in topological_space)  tendsto :: "('b => 'a) => 'a => 'b filter => bool" (infixr "--->" 55) where  "(f ---> l) F ≡ filterlim f (nhds l) F"ML {*structure Tendsto_Intros = Named_Thms(  val name = @{binding tendsto_intros}  val description = "introduction rules for tendsto")*}setup Tendsto_Intros.setuplemma tendsto_def: "(f ---> l) F <-> (∀S. open S --> l ∈ S --> eventually (λx. f x ∈ S) F)"  unfolding filterlim_defproof safe  fix S assume "open S" "l ∈ S" "filtermap f F ≤ nhds l"  then show "eventually (λx. f x ∈ S) F"    unfolding eventually_nhds eventually_filtermap le_filter_def    by (auto elim!: allE[of _ "λx. x ∈ S"] eventually_rev_mp)qed (auto elim!: eventually_rev_mp simp: eventually_nhds eventually_filtermap le_filter_def)lemma filterlim_at:  "(LIM x F. f x :> at b) <-> (eventually (λx. f x ≠ b) F ∧ (f ---> b) F)"  by (simp add: at_def filterlim_within)lemma tendsto_mono: "F ≤ F' ==> (f ---> l) F' ==> (f ---> l) F"  unfolding tendsto_def le_filter_def by fastlemma topological_tendstoI:  "(!!S. open S ==> l ∈ S ==> eventually (λx. f x ∈ S) F)    ==> (f ---> l) F"  unfolding tendsto_def by autolemma topological_tendstoD:  "(f ---> l) F ==> open S ==> l ∈ S ==> eventually (λx. f x ∈ S) F"  unfolding tendsto_def by autolemma tendstoI:  assumes "!!e. 0 < e ==> eventually (λx. dist (f x) l < e) F"  shows "(f ---> l) F"  apply (rule topological_tendstoI)  apply (simp add: open_dist)  apply (drule (1) bspec, clarify)  apply (drule assms)  apply (erule eventually_elim1, simp)  donelemma tendstoD:  "(f ---> l) F ==> 0 < e ==> eventually (λx. dist (f x) l < e) F"  apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)  apply (clarsimp simp add: open_dist)  apply (rule_tac x="e - dist x l" in exI, clarsimp)  apply (simp only: less_diff_eq)  apply (erule le_less_trans [OF dist_triangle])  apply simp  apply simp  donelemma tendsto_iff:  "(f ---> l) F <-> (∀e>0. eventually (λx. dist (f x) l < e) F)"  using tendstoI tendstoD by fastlemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (λx. f x - a) F"  by (simp only: tendsto_iff Zfun_def dist_norm)lemma tendsto_bot [simp]: "(f ---> a) bot"  unfolding tendsto_def by simplemma tendsto_ident_at [tendsto_intros]: "((λx. x) ---> a) (at a)"  unfolding tendsto_def eventually_at_topological by autolemma tendsto_ident_at_within [tendsto_intros]:  "((λx. x) ---> a) (at a within S)"  unfolding tendsto_def eventually_within eventually_at_topological by autolemma tendsto_const [tendsto_intros]: "((λx. k) ---> k) F"  by (simp add: tendsto_def)lemma tendsto_unique:  fixes f :: "'a => 'b::t2_space"  assumes "¬ trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"  shows "a = b"proof (rule ccontr)  assume "a ≠ b"  obtain U V where "open U" "open V" "a ∈ U" "b ∈ V" "U ∩ V = {}"    using hausdorff [OF `a ≠ b`] by fast  have "eventually (λx. f x ∈ U) F"    using `(f ---> a) F` `open U` `a ∈ U` by (rule topological_tendstoD)  moreover  have "eventually (λx. f x ∈ V) F"    using `(f ---> b) F` `open V` `b ∈ V` by (rule topological_tendstoD)  ultimately  have "eventually (λx. False) F"  proof eventually_elim    case (elim x)    hence "f x ∈ U ∩ V" by simp    with `U ∩ V = {}` show ?case by simp  qed  with `¬ trivial_limit F` show "False"    by (simp add: trivial_limit_def)qedlemma tendsto_const_iff:  fixes a b :: "'a::t2_space"  assumes "¬ trivial_limit F" shows "((λx. a) ---> b) F <-> a = b"  by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])lemma tendsto_at_iff_tendsto_nhds:  "(g ---> g l) (at l) <-> (g ---> g l) (nhds l)"  unfolding tendsto_def at_def eventually_within  by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)lemma tendsto_compose:  "(g ---> g l) (at l) ==> (f ---> l) F ==> ((λx. g (f x)) ---> g l) F"  unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])lemma tendsto_compose_eventually:  "(g ---> m) (at l) ==> (f ---> l) F ==> eventually (λx. f x ≠ l) F ==> ((λx. g (f x)) ---> m) F"  by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)lemma metric_tendsto_imp_tendsto:  assumes f: "(f ---> a) F"  assumes le: "eventually (λx. dist (g x) b ≤ dist (f x) a) F"  shows "(g ---> b) F"proof (rule tendstoI)  fix e :: real assume "0 < e"  with f have "eventually (λx. dist (f x) a < e) F" by (rule tendstoD)  with le show "eventually (λx. dist (g x) b < e) F"    using le_less_trans by (rule eventually_elim2)qedsubsubsection {* Distance and norms *}lemma tendsto_dist [tendsto_intros]:  assumes f: "(f ---> l) F" and g: "(g ---> m) F"  shows "((λx. dist (f x) (g x)) ---> dist l m) F"proof (rule tendstoI)  fix e :: real assume "0 < e"  hence e2: "0 < e/2" by simp  from tendstoD [OF f e2] tendstoD [OF g e2]  show "eventually (λx. dist (dist (f x) (g x)) (dist l m) < e) F"  proof (eventually_elim)    case (elim x)    then show "dist (dist (f x) (g x)) (dist l m) < e"      unfolding dist_real_def      using dist_triangle2 [of "f x" "g x" "l"]      using dist_triangle2 [of "g x" "l" "m"]      using dist_triangle3 [of "l" "m" "f x"]      using dist_triangle [of "f x" "m" "g x"]      by arith  qedqedlemma norm_conv_dist: "norm x = dist x 0"  unfolding dist_norm by simplemma tendsto_norm [tendsto_intros]:  "(f ---> a) F ==> ((λx. norm (f x)) ---> norm a) F"  unfolding norm_conv_dist by (intro tendsto_intros)lemma tendsto_norm_zero:  "(f ---> 0) F ==> ((λx. norm (f x)) ---> 0) F"  by (drule tendsto_norm, simp)lemma tendsto_norm_zero_cancel:  "((λx. norm (f x)) ---> 0) F ==> (f ---> 0) F"  unfolding tendsto_iff dist_norm by simplemma tendsto_norm_zero_iff:  "((λx. norm (f x)) ---> 0) F <-> (f ---> 0) F"  unfolding tendsto_iff dist_norm by simplemma tendsto_rabs [tendsto_intros]:  "(f ---> (l::real)) F ==> ((λx. ¦f x¦) ---> ¦l¦) F"  by (fold real_norm_def, rule tendsto_norm)lemma tendsto_rabs_zero:  "(f ---> (0::real)) F ==> ((λx. ¦f x¦) ---> 0) F"  by (fold real_norm_def, rule tendsto_norm_zero)lemma tendsto_rabs_zero_cancel:  "((λx. ¦f x¦) ---> (0::real)) F ==> (f ---> 0) F"  by (fold real_norm_def, rule tendsto_norm_zero_cancel)lemma tendsto_rabs_zero_iff:  "((λx. ¦f x¦) ---> (0::real)) F <-> (f ---> 0) F"  by (fold real_norm_def, rule tendsto_norm_zero_iff)subsubsection {* Addition and subtraction *}lemma tendsto_add [tendsto_intros]:  fixes a b :: "'a::real_normed_vector"  shows "[|(f ---> a) F; (g ---> b) F|] ==> ((λx. f x + g x) ---> a + b) F"  by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)lemma tendsto_add_zero:  fixes f g :: "'a::type => 'b::real_normed_vector"  shows "[|(f ---> 0) F; (g ---> 0) F|] ==> ((λx. f x + g x) ---> 0) F"  by (drule (1) tendsto_add, simp)lemma tendsto_minus [tendsto_intros]:  fixes a :: "'a::real_normed_vector"  shows "(f ---> a) F ==> ((λx. - f x) ---> - a) F"  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)lemma tendsto_minus_cancel:  fixes a :: "'a::real_normed_vector"  shows "((λx. - f x) ---> - a) F ==> (f ---> a) F"  by (drule tendsto_minus, simp)lemma tendsto_minus_cancel_left:    "(f ---> - (y::_::real_normed_vector)) F <-> ((λx. - f x) ---> y) F"  using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]  by autolemma tendsto_diff [tendsto_intros]:  fixes a b :: "'a::real_normed_vector"  shows "[|(f ---> a) F; (g ---> b) F|] ==> ((λx. f x - g x) ---> a - b) F"  by (simp add: diff_minus tendsto_add tendsto_minus)lemma tendsto_setsum [tendsto_intros]:  fixes f :: "'a => 'b => 'c::real_normed_vector"  assumes "!!i. i ∈ S ==> (f i ---> a i) F"  shows "((λx. ∑i∈S. f i x) ---> (∑i∈S. a i)) F"proof (cases "finite S")  assume "finite S" thus ?thesis using assms    by (induct, simp add: tendsto_const, simp add: tendsto_add)next  assume "¬ finite S" thus ?thesis    by (simp add: tendsto_const)qedlemma real_tendsto_sandwich:  fixes f g h :: "'a => real"  assumes ev: "eventually (λn. f n ≤ g n) net" "eventually (λn. g n ≤ h n) net"  assumes lim: "(f ---> c) net" "(h ---> c) net"  shows "(g ---> c) net"proof -  have "((λn. g n - f n) ---> 0) net"  proof (rule metric_tendsto_imp_tendsto)    show "eventually (λn. dist (g n - f n) 0 ≤ dist (h n - f n) 0) net"      using ev by (rule eventually_elim2) (simp add: dist_real_def)    show "((λn. h n - f n) ---> 0) net"      using tendsto_diff[OF lim(2,1)] by simp  qed  from tendsto_add[OF this lim(1)] show ?thesis by simpqedsubsubsection {* Linear operators and multiplication *}lemma (in bounded_linear) tendsto:  "(g ---> a) F ==> ((λx. f (g x)) ---> f a) F"  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)lemma (in bounded_linear) tendsto_zero:  "(g ---> 0) F ==> ((λx. f (g x)) ---> 0) F"  by (drule tendsto, simp only: zero)lemma (in bounded_bilinear) tendsto:  "[|(f ---> a) F; (g ---> b) F|] ==> ((λx. f x ** g x) ---> a ** b) F"  by (simp only: tendsto_Zfun_iff prod_diff_prod                 Zfun_add Zfun Zfun_left Zfun_right)lemma (in bounded_bilinear) tendsto_zero:  assumes f: "(f ---> 0) F"  assumes g: "(g ---> 0) F"  shows "((λx. f x ** g x) ---> 0) F"  using tendsto [OF f g] by (simp add: zero_left)lemma (in bounded_bilinear) tendsto_left_zero:  "(f ---> 0) F ==> ((λx. f x ** c) ---> 0) F"  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])lemma (in bounded_bilinear) tendsto_right_zero:  "(f ---> 0) F ==> ((λx. c ** f x) ---> 0) F"  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])lemmas tendsto_of_real [tendsto_intros] =  bounded_linear.tendsto [OF bounded_linear_of_real]lemmas tendsto_scaleR [tendsto_intros] =  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]lemmas tendsto_mult [tendsto_intros] =  bounded_bilinear.tendsto [OF bounded_bilinear_mult]lemmas tendsto_mult_zero =  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]lemmas tendsto_mult_left_zero =  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]lemmas tendsto_mult_right_zero =  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]lemma tendsto_power [tendsto_intros]:  fixes f :: "'a => 'b::{power,real_normed_algebra}"  shows "(f ---> a) F ==> ((λx. f x ^ n) ---> a ^ n) F"  by (induct n) (simp_all add: tendsto_const tendsto_mult)lemma tendsto_setprod [tendsto_intros]:  fixes f :: "'a => 'b => 'c::{real_normed_algebra,comm_ring_1}"  assumes "!!i. i ∈ S ==> (f i ---> L i) F"  shows "((λx. ∏i∈S. f i x) ---> (∏i∈S. L i)) F"proof (cases "finite S")  assume "finite S" thus ?thesis using assms    by (induct, simp add: tendsto_const, simp add: tendsto_mult)next  assume "¬ finite S" thus ?thesis    by (simp add: tendsto_const)qedlemma tendsto_le_const:  fixes f :: "_ => real"   assumes F: "¬ trivial_limit F"  assumes x: "(f ---> x) F" and a: "eventually (λx. a ≤ f x) F"  shows "a ≤ x"proof (rule ccontr)  assume "¬ a ≤ x"  with x have "eventually (λx. f x < a) F"    by (auto simp add: tendsto_def elim!: allE[of _ "{..< a}"])  with a have "eventually (λx. False) F"    by eventually_elim auto  with F show False    by (simp add: eventually_False)qedlemma tendsto_le:  fixes f g :: "_ => real"   assumes F: "¬ trivial_limit F"  assumes x: "(f ---> x) F" and y: "(g ---> y) F"  assumes ev: "eventually (λx. g x ≤ f x) F"  shows "y ≤ x"  using tendsto_le_const[OF F tendsto_diff[OF x y], of 0] ev  by (simp add: sign_simps)subsubsection {* Inverse and division *}lemma (in bounded_bilinear) Zfun_prod_Bfun:  assumes f: "Zfun f F"  assumes g: "Bfun g F"  shows "Zfun (λx. f x ** g x) F"proof -  obtain K where K: "0 ≤ K"    and norm_le: "!!x y. norm (x ** y) ≤ norm x * norm y * K"    using nonneg_bounded by fast  obtain B where B: "0 < B"    and norm_g: "eventually (λx. norm (g x) ≤ B) F"    using g by (rule BfunE)  have "eventually (λx. norm (f x ** g x) ≤ norm (f x) * (B * K)) F"  using norm_g proof eventually_elim    case (elim x)    have "norm (f x ** g x) ≤ norm (f x) * norm (g x) * K"      by (rule norm_le)    also have "… ≤ norm (f x) * B * K"      by (intro mult_mono' order_refl norm_g norm_ge_zero                mult_nonneg_nonneg K elim)    also have "… = norm (f x) * (B * K)"      by (rule mult_assoc)    finally show "norm (f x ** g x) ≤ norm (f x) * (B * K)" .  qed  with f show ?thesis    by (rule Zfun_imp_Zfun)qedlemma (in bounded_bilinear) flip:  "bounded_bilinear (λx y. y ** x)"  apply default  apply (rule add_right)  apply (rule add_left)  apply (rule scaleR_right)  apply (rule scaleR_left)  apply (subst mult_commute)  using bounded by fastlemma (in bounded_bilinear) Bfun_prod_Zfun:  assumes f: "Bfun f F"  assumes g: "Zfun g F"  shows "Zfun (λx. f x ** g x) F"  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)lemma Bfun_inverse_lemma:  fixes x :: "'a::real_normed_div_algebra"  shows "[|r ≤ norm x; 0 < r|] ==> norm (inverse x) ≤ inverse r"  apply (subst nonzero_norm_inverse, clarsimp)  apply (erule (1) le_imp_inverse_le)  donelemma Bfun_inverse:  fixes a :: "'a::real_normed_div_algebra"  assumes f: "(f ---> a) F"  assumes a: "a ≠ 0"  shows "Bfun (λx. inverse (f x)) F"proof -  from a have "0 < norm a" by simp  hence "∃r>0. r < norm a" by (rule dense)  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast  have "eventually (λx. dist (f x) a < r) F"    using tendstoD [OF f r1] by fast  hence "eventually (λx. norm (inverse (f x)) ≤ inverse (norm a - r)) F"  proof eventually_elim    case (elim x)    hence 1: "norm (f x - a) < r"      by (simp add: dist_norm)    hence 2: "f x ≠ 0" using r2 by auto    hence "norm (inverse (f x)) = inverse (norm (f x))"      by (rule nonzero_norm_inverse)    also have "… ≤ inverse (norm a - r)"    proof (rule le_imp_inverse_le)      show "0 < norm a - r" using r2 by simp    next      have "norm a - norm (f x) ≤ norm (a - f x)"        by (rule norm_triangle_ineq2)      also have "… = norm (f x - a)"        by (rule norm_minus_commute)      also have "… < r" using 1 .      finally show "norm a - r ≤ norm (f x)" by simp    qed    finally show "norm (inverse (f x)) ≤ inverse (norm a - r)" .  qed  thus ?thesis by (rule BfunI)qedlemma tendsto_inverse [tendsto_intros]:  fixes a :: "'a::real_normed_div_algebra"  assumes f: "(f ---> a) F"  assumes a: "a ≠ 0"  shows "((λx. inverse (f x)) ---> inverse a) F"proof -  from a have "0 < norm a" by simp  with f have "eventually (λx. dist (f x) a < norm a) F"    by (rule tendstoD)  then have "eventually (λx. f x ≠ 0) F"    unfolding dist_norm by (auto elim!: eventually_elim1)  with a have "eventually (λx. inverse (f x) - inverse a =    - (inverse (f x) * (f x - a) * inverse a)) F"    by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)  moreover have "Zfun (λx. - (inverse (f x) * (f x - a) * inverse a)) F"    by (intro Zfun_minus Zfun_mult_left      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])  ultimately show ?thesis    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)qedlemma tendsto_divide [tendsto_intros]:  fixes a b :: "'a::real_normed_field"  shows "[|(f ---> a) F; (g ---> b) F; b ≠ 0|]    ==> ((λx. f x / g x) ---> a / b) F"  by (simp add: tendsto_mult tendsto_inverse divide_inverse)lemma tendsto_sgn [tendsto_intros]:  fixes l :: "'a::real_normed_vector"  shows "[|(f ---> l) F; l ≠ 0|] ==> ((λx. sgn (f x)) ---> sgn l) F"  unfolding sgn_div_norm by (simp add: tendsto_intros)subsection {* Limits to @{const at_top} and @{const at_bot} *}lemma filterlim_at_top:  fixes f :: "'a => ('b::linorder)"  shows "(LIM x F. f x :> at_top) <-> (∀Z. eventually (λx. Z ≤ f x) F)"  by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)lemma filterlim_at_top_dense:  fixes f :: "'a => ('b::dense_linorder)"  shows "(LIM x F. f x :> at_top) <-> (∀Z. eventually (λx. Z < f x) F)"  by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le            filterlim_at_top[of f F] filterlim_iff[of f at_top F])lemma filterlim_at_top_ge:  fixes f :: "'a => ('b::linorder)" and c :: "'b"  shows "(LIM x F. f x :> at_top) <-> (∀Z≥c. eventually (λx. Z ≤ f x) F)"  unfolding filterlim_at_topproof safe  fix Z assume *: "∀Z≥c. eventually (λx. Z ≤ f x) F"  with *[THEN spec, of "max Z c"] show "eventually (λx. Z ≤ f x) F"    by (auto elim!: eventually_elim1)qed simplemma filterlim_at_top_at_top:  fixes f :: "'a::linorder => 'b::linorder"  assumes mono: "!!x y. Q x ==> Q y ==> x ≤ y ==> f x ≤ f y"  assumes bij: "!!x. P x ==> f (g x) = x" "!!x. P x ==> Q (g x)"  assumes Q: "eventually Q at_top"  assumes P: "eventually P at_top"  shows "filterlim f at_top at_top"proof -  from P obtain x where x: "!!y. x ≤ y ==> P y"    unfolding eventually_at_top_linorder by auto  show ?thesis  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)    fix z assume "x ≤ z"    with x have "P z" by auto    have "eventually (λx. g z ≤ x) at_top"      by (rule eventually_ge_at_top)    with Q show "eventually (λx. z ≤ f x) at_top"      by eventually_elim (metis mono bij `P z`)  qedqedlemma filterlim_at_top_gt:  fixes f :: "'a => ('b::dense_linorder)" and c :: "'b"  shows "(LIM x F. f x :> at_top) <-> (∀Z>c. eventually (λx. Z ≤ f x) F)"  by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)lemma filterlim_at_bot:   fixes f :: "'a => ('b::linorder)"  shows "(LIM x F. f x :> at_bot) <-> (∀Z. eventually (λx. f x ≤ Z) F)"  by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)lemma filterlim_at_bot_le:  fixes f :: "'a => ('b::linorder)" and c :: "'b"  shows "(LIM x F. f x :> at_bot) <-> (∀Z≤c. eventually (λx. Z ≥ f x) F)"  unfolding filterlim_at_botproof safe  fix Z assume *: "∀Z≤c. eventually (λx. Z ≥ f x) F"  with *[THEN spec, of "min Z c"] show "eventually (λx. Z ≥ f x) F"    by (auto elim!: eventually_elim1)qed simplemma filterlim_at_bot_lt:  fixes f :: "'a => ('b::dense_linorder)" and c :: "'b"  shows "(LIM x F. f x :> at_bot) <-> (∀Z<c. eventually (λx. Z ≥ f x) F)"  by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)lemma filterlim_at_bot_at_right:  fixes f :: "real => 'b::linorder"  assumes mono: "!!x y. Q x ==> Q y ==> x ≤ y ==> f x ≤ f y"  assumes bij: "!!x. P x ==> f (g x) = x" "!!x. P x ==> Q (g x)"  assumes Q: "eventually Q (at_right a)" and bound: "!!b. Q b ==> a < b"  assumes P: "eventually P at_bot"  shows "filterlim f at_bot (at_right a)"proof -  from P obtain x where x: "!!y. y ≤ x ==> P y"    unfolding eventually_at_bot_linorder by auto  show ?thesis  proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)    fix z assume "z ≤ x"    with x have "P z" by auto    have "eventually (λx. x ≤ g z) (at_right a)"      using bound[OF bij(2)[OF `P z`]]      by (auto simp add: eventually_within_less dist_real_def intro!:  exI[of _ "g z - a"])    with Q show "eventually (λx. f x ≤ z) (at_right a)"      by eventually_elim (metis bij `P z` mono)  qedqedlemma filterlim_at_top_at_left:  fixes f :: "real => 'b::linorder"  assumes mono: "!!x y. Q x ==> Q y ==> x ≤ y ==> f x ≤ f y"  assumes bij: "!!x. P x ==> f (g x) = x" "!!x. P x ==> Q (g x)"  assumes Q: "eventually Q (at_left a)" and bound: "!!b. Q b ==> b < a"  assumes P: "eventually P at_top"  shows "filterlim f at_top (at_left a)"proof -  from P obtain x where x: "!!y. x ≤ y ==> P y"    unfolding eventually_at_top_linorder by auto  show ?thesis  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)    fix z assume "x ≤ z"    with x have "P z" by auto    have "eventually (λx. g z ≤ x) (at_left a)"      using bound[OF bij(2)[OF `P z`]]      by (auto simp add: eventually_within_less dist_real_def intro!:  exI[of _ "a - g z"])    with Q show "eventually (λx. z ≤ f x) (at_left a)"      by eventually_elim (metis bij `P z` mono)  qedqedlemma filterlim_at_infinity:  fixes f :: "_ => 'a::real_normed_vector"  assumes "0 ≤ c"  shows "(LIM x F. f x :> at_infinity) <-> (∀r>c. eventually (λx. r ≤ norm (f x)) F)"  unfolding filterlim_iff eventually_at_infinityproof safe  fix P :: "'a => bool" and b  assume *: "∀r>c. eventually (λx. r ≤ norm (f x)) F"    and P: "∀x. b ≤ norm x --> P x"  have "max b (c + 1) > c" by auto  with * have "eventually (λx. max b (c + 1) ≤ norm (f x)) F"    by auto  then show "eventually (λx. P (f x)) F"  proof eventually_elim    fix x assume "max b (c + 1) ≤ norm (f x)"    with P show "P (f x)" by auto  qedqed forcelemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"  unfolding filterlim_at_top  apply (intro allI)  apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)  apply (auto simp: natceiling_le_eq)  donesubsection {* Relate @{const at}, @{const at_left} and @{const at_right} *}text {*This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and@{term "at_right x"} and also @{term "at_right 0"}.*}lemma at_eq_sup_left_right: "at (x::real) = sup (at_left x) (at_right x)"  by (auto simp: eventually_within at_def filter_eq_iff eventually_sup            elim: eventually_elim2 eventually_elim1)lemma filterlim_split_at_real:  "filterlim f F (at_left x) ==> filterlim f F (at_right x) ==> filterlim f F (at (x::real))"  by (subst at_eq_sup_left_right) (rule filterlim_sup)lemma filtermap_nhds_shift: "filtermap (λx. x - d) (nhds a) = nhds (a - d::real)"  unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric  by (intro allI ex_cong) (auto simp: dist_real_def field_simps)lemma filtermap_nhds_minus: "filtermap (λx. - x) (nhds a) = nhds (- a::real)"  unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric  apply (intro allI ex_cong)  apply (auto simp: dist_real_def field_simps)  apply (erule_tac x="-x" in allE)  apply simp  donelemma filtermap_at_shift: "filtermap (λx. x - d) (at a) = at (a - d::real)"  unfolding at_def filtermap_nhds_shift[symmetric]  by (simp add: filter_eq_iff eventually_filtermap eventually_within)lemma filtermap_at_right_shift: "filtermap (λx. x - d) (at_right a) = at_right (a - d::real)"  unfolding filtermap_at_shift[symmetric]  by (simp add: filter_eq_iff eventually_filtermap eventually_within)lemma at_right_to_0: "at_right (a::real) = filtermap (λx. x + a) (at_right 0)"  using filtermap_at_right_shift[of "-a" 0] by simplemma filterlim_at_right_to_0:  "filterlim f F (at_right (a::real)) <-> filterlim (λx. f (x + a)) F (at_right 0)"  unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..lemma eventually_at_right_to_0:  "eventually P (at_right (a::real)) <-> eventually (λx. P (x + a)) (at_right 0)"  unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)lemma filtermap_at_minus: "filtermap (λx. - x) (at a) = at (- a::real)"  unfolding at_def filtermap_nhds_minus[symmetric]  by (simp add: filter_eq_iff eventually_filtermap eventually_within)lemma at_left_minus: "at_left (a::real) = filtermap (λx. - x) (at_right (- a))"  by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])lemma at_right_minus: "at_right (a::real) = filtermap (λx. - x) (at_left (- a))"  by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])lemma filterlim_at_left_to_right:  "filterlim f F (at_left (a::real)) <-> filterlim (λx. f (- x)) F (at_right (-a))"  unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..lemma eventually_at_left_to_right:  "eventually P (at_left (a::real)) <-> eventually (λx. P (- x)) (at_right (-a))"  unfolding at_left_minus[of a] by (simp add: eventually_filtermap)lemma filterlim_at_split:  "filterlim f F (at (x::real)) <-> filterlim f F (at_left x) ∧ filterlim f F (at_right x)"  by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup)lemma eventually_at_split:  "eventually P (at (x::real)) <-> eventually P (at_left x) ∧ eventually P (at_right x)"  by (subst at_eq_sup_left_right) (simp add: eventually_sup)lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"  unfolding filter_eq_iff eventually_filtermap eventually_at_top_linorder eventually_at_bot_linorder  by (metis le_minus_iff minus_minus)lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"  unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) <-> (LIM x at_bot. f (-x::real) :> F)"  unfolding filterlim_def at_top_mirror filtermap_filtermap ..lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) <-> (LIM x at_top. f (-x::real) :> F)"  unfolding filterlim_def at_bot_mirror filtermap_filtermap ..lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"  unfolding filterlim_at_top eventually_at_bot_dense  by (metis leI minus_less_iff order_less_asym)lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"  unfolding filterlim_at_bot eventually_at_top_dense  by (metis leI less_minus_iff order_less_asym)lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) <-> (LIM x F. - (f x) :: real :> at_bot)"  using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]  using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "λx. - f x" F]  by autolemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) <-> (LIM x F. - (f x) :: real :> at_top)"  unfolding filterlim_uminus_at_top by simplemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"  unfolding filterlim_at_top_gt[where c=0] eventually_within at_defproof safe  fix Z :: real assume [arith]: "0 < Z"  then have "eventually (λx. x < inverse Z) (nhds 0)"    by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "¦inverse Z¦"])  then show "eventually (λx. x ∈ - {0} --> x ∈ {0<..} --> Z ≤ inverse x) (nhds 0)"    by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)qedlemma filterlim_inverse_at_top:  "(f ---> (0 :: real)) F ==> eventually (λx. 0 < f x) F ==> LIM x F. inverse (f x) :> at_top"  by (intro filterlim_compose[OF filterlim_inverse_at_top_right])     (simp add: filterlim_def eventually_filtermap le_within_iff at_def eventually_elim1)lemma filterlim_inverse_at_bot_neg:  "LIM x (at_left (0::real)). inverse x :> at_bot"  by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)lemma filterlim_inverse_at_bot:  "(f ---> (0 :: real)) F ==> eventually (λx. f x < 0) F ==> LIM x F. inverse (f x) :> at_bot"  unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]  by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])lemma tendsto_inverse_0:  fixes x :: "_ => 'a::real_normed_div_algebra"  shows "(inverse ---> (0::'a)) at_infinity"  unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinityproof safe  fix r :: real assume "0 < r"  show "∃b. ∀x. b ≤ norm x --> norm (inverse x :: 'a) < r"  proof (intro exI[of _ "inverse (r / 2)"] allI impI)    fix x :: 'a    from `0 < r` have "0 < inverse (r / 2)" by simp    also assume *: "inverse (r / 2) ≤ norm x"    finally show "norm (inverse x) < r"      using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)  qedqedlemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"proof (rule antisym)  have "(inverse ---> (0::real)) at_top"    by (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)  then show "filtermap inverse at_top ≤ at_right (0::real)"    unfolding at_within_eq    by (intro le_withinI) (simp_all add: eventually_filtermap eventually_gt_at_top filterlim_def)next  have "filtermap inverse (filtermap inverse (at_right (0::real))) ≤ filtermap inverse at_top"    using filterlim_inverse_at_top_right unfolding filterlim_def by (rule filtermap_mono)  then show "at_right (0::real) ≤ filtermap inverse at_top"    by (simp add: filtermap_ident filtermap_filtermap)qedlemma eventually_at_right_to_top:  "eventually P (at_right (0::real)) <-> eventually (λx. P (inverse x)) at_top"  unfolding at_right_to_top eventually_filtermap ..lemma filterlim_at_right_to_top:  "filterlim f F (at_right (0::real)) <-> (LIM x at_top. f (inverse x) :> F)"  unfolding filterlim_def at_right_to_top filtermap_filtermap ..lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"  unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..lemma eventually_at_top_to_right:  "eventually P at_top <-> eventually (λx. P (inverse x)) (at_right (0::real))"  unfolding at_top_to_right eventually_filtermap ..lemma filterlim_at_top_to_right:  "filterlim f F at_top <-> (LIM x (at_right (0::real)). f (inverse x) :> F)"  unfolding filterlim_def at_top_to_right filtermap_filtermap ..lemma filterlim_inverse_at_infinity:  fixes x :: "_ => 'a::{real_normed_div_algebra, division_ring_inverse_zero}"  shows "filterlim inverse at_infinity (at (0::'a))"  unfolding filterlim_at_infinity[OF order_refl]proof safe  fix r :: real assume "0 < r"  then show "eventually (λx::'a. r ≤ norm (inverse x)) (at 0)"    unfolding eventually_at norm_inverse    by (intro exI[of _ "inverse r"])       (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)qedlemma filterlim_inverse_at_iff:  fixes g :: "'a => 'b::{real_normed_div_algebra, division_ring_inverse_zero}"  shows "(LIM x F. inverse (g x) :> at 0) <-> (LIM x F. g x :> at_infinity)"  unfolding filterlim_def filtermap_filtermap[symmetric]proof  assume "filtermap g F ≤ at_infinity"  then have "filtermap inverse (filtermap g F) ≤ filtermap inverse at_infinity"    by (rule filtermap_mono)  also have "… ≤ at 0"    using tendsto_inverse_0    by (auto intro!: le_withinI exI[of _ 1]             simp: eventually_filtermap eventually_at_infinity filterlim_def at_def)  finally show "filtermap inverse (filtermap g F) ≤ at 0" .next  assume "filtermap inverse (filtermap g F) ≤ at 0"  then have "filtermap inverse (filtermap inverse (filtermap g F)) ≤ filtermap inverse (at 0)"    by (rule filtermap_mono)  with filterlim_inverse_at_infinity show "filtermap g F ≤ at_infinity"    by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)qedlemma tendsto_inverse_0_at_top:  "LIM x F. f x :> at_top ==> ((λx. inverse (f x) :: real) ---> 0) F" by (metis at_top_le_at_infinity filterlim_at filterlim_inverse_at_iff filterlim_mono order_refl)text {*We only show rules for multiplication and addition when the functions are either against a realvalue or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.*}lemma filterlim_tendsto_pos_mult_at_top:   assumes f: "(f ---> c) F" and c: "0 < c"  assumes g: "LIM x F. g x :> at_top"  shows "LIM x F. (f x * g x :: real) :> at_top"  unfolding filterlim_at_top_gt[where c=0]proof safe  fix Z :: real assume "0 < Z"  from f `0 < c` have "eventually (λx. c / 2 < f x) F"    by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1             simp: dist_real_def abs_real_def split: split_if_asm)  moreover from g have "eventually (λx. (Z / c * 2) ≤ g x) F"    unfolding filterlim_at_top by auto  ultimately show "eventually (λx. Z ≤ f x * g x) F"  proof eventually_elim    fix x assume "c / 2 < f x" "Z / c * 2 ≤ g x"    with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) ≤ f x * g x"      by (intro mult_mono) (auto simp: zero_le_divide_iff)    with `0 < c` show "Z ≤ f x * g x"       by simp  qedqedlemma filterlim_at_top_mult_at_top:   assumes f: "LIM x F. f x :> at_top"  assumes g: "LIM x F. g x :> at_top"  shows "LIM x F. (f x * g x :: real) :> at_top"  unfolding filterlim_at_top_gt[where c=0]proof safe  fix Z :: real assume "0 < Z"  from f have "eventually (λx. 1 ≤ f x) F"    unfolding filterlim_at_top by auto  moreover from g have "eventually (λx. Z ≤ g x) F"    unfolding filterlim_at_top by auto  ultimately show "eventually (λx. Z ≤ f x * g x) F"  proof eventually_elim    fix x assume "1 ≤ f x" "Z ≤ g x"    with `0 < Z` have "1 * Z ≤ f x * g x"      by (intro mult_mono) (auto simp: zero_le_divide_iff)    then show "Z ≤ f x * g x"       by simp  qedqedlemma filterlim_tendsto_pos_mult_at_bot:  assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"  shows "LIM x F. f x * g x :> at_bot"  using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "λx. - g x"] assms(3)  unfolding filterlim_uminus_at_bot by simplemma filterlim_tendsto_add_at_top:   assumes f: "(f ---> c) F"  assumes g: "LIM x F. g x :> at_top"  shows "LIM x F. (f x + g x :: real) :> at_top"  unfolding filterlim_at_top_gt[where c=0]proof safe  fix Z :: real assume "0 < Z"  from f have "eventually (λx. c - 1 < f x) F"    by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)  moreover from g have "eventually (λx. Z - (c - 1) ≤ g x) F"    unfolding filterlim_at_top by auto  ultimately show "eventually (λx. Z ≤ f x + g x) F"    by eventually_elim simpqedlemma LIM_at_top_divide:  fixes f g :: "'a => real"  assumes f: "(f ---> a) F" "0 < a"  assumes g: "(g ---> 0) F" "eventually (λx. 0 < g x) F"  shows "LIM x F. f x / g x :> at_top"  unfolding divide_inverse  by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])lemma filterlim_at_top_add_at_top:   assumes f: "LIM x F. f x :> at_top"  assumes g: "LIM x F. g x :> at_top"  shows "LIM x F. (f x + g x :: real) :> at_top"  unfolding filterlim_at_top_gt[where c=0]proof safe  fix Z :: real assume "0 < Z"  from f have "eventually (λx. 0 ≤ f x) F"    unfolding filterlim_at_top by auto  moreover from g have "eventually (λx. Z ≤ g x) F"    unfolding filterlim_at_top by auto  ultimately show "eventually (λx. Z ≤ f x + g x) F"    by eventually_elim simpqedlemma tendsto_divide_0:  fixes f :: "_ => 'a::{real_normed_div_algebra, division_ring_inverse_zero}"  assumes f: "(f ---> c) F"  assumes g: "LIM x F. g x :> at_infinity"  shows "((λx. f x / g x) ---> 0) F"  using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)lemma linear_plus_1_le_power:  fixes x :: real  assumes x: "0 ≤ x"  shows "real n * x + 1 ≤ (x + 1) ^ n"proof (induct n)  case (Suc n)  have "real (Suc n) * x + 1 ≤ (x + 1) * (real n * x + 1)"    by (simp add: field_simps real_of_nat_Suc mult_nonneg_nonneg x)  also have "… ≤ (x + 1)^Suc n"    using Suc x by (simp add: mult_left_mono)  finally show ?case .qed simplemma filterlim_realpow_sequentially_gt1:  fixes x :: "'a :: real_normed_div_algebra"  assumes x[arith]: "1 < norm x"  shows "LIM n sequentially. x ^ n :> at_infinity"proof (intro filterlim_at_infinity[THEN iffD2] allI impI)  fix y :: real assume "0 < y"  have "0 < norm x - 1" by simp  then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)  also have "… ≤ real N * (norm x - 1) + 1" by simp  also have "… ≤ (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp  also have "… = norm x ^ N" by simp  finally have "∀n≥N. y ≤ norm x ^ n"    by (metis order_less_le_trans power_increasing order_less_imp_le x)  then show "eventually (λn. y ≤ norm (x ^ n)) sequentially"    unfolding eventually_sequentially    by (auto simp: norm_power)qed simpend`