(*  Title       : Limits.thy

     Author      : Brian Huffman

 *)

 header {* Filters and Limits *}

 theory Limits

 imports RealVector

 begin

 subsection {* Filters *}

 text {*

   This definition also allows non-proper filters.

 *}

 locale is_filter =

   fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"

   assumes True: "F (\<lambda>x. True)"

   assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"

   assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"

 typedef (open) 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"

 proof

   show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)

 qed

 lemma is_filter_Rep_filter: "is_filter (Rep_filter A)"

   using Rep_filter [of A] by simp

 lemma 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 \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"

   where "eventually P A \<longleftrightarrow> Rep_filter A 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 "A = B \<longleftrightarrow> (\<forall>P. eventually P A = eventually P B)"

   unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..

 lemma eventually_True [simp]: "eventually (\<lambda>x. True) A"

   unfolding eventually_def

   by (rule is_filter.True [OF is_filter_Rep_filter])

 lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P A"

 proof -

   assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)

   thus "eventually P A" by simp

 qed

 lemma eventually_mono:

   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P A \<Longrightarrow> eventually Q A"

   unfolding eventually_def

   by (rule is_filter.mono [OF is_filter_Rep_filter])

 lemma eventually_conj:

   assumes P: "eventually (\<lambda>x. P x) A"

   assumes Q: "eventually (\<lambda>x. Q x) A"

   shows "eventually (\<lambda>x. P x \<and> Q x) A"

   using assms unfolding eventually_def

   by (rule is_filter.conj [OF is_filter_Rep_filter])

 lemma eventually_mp:

   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) A"

   assumes "eventually (\<lambda>x. P x) A"

   shows "eventually (\<lambda>x. Q x) A"

 proof (rule eventually_mono)

   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp

   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) A"

     using assms by (rule eventually_conj)

 qed

 lemma eventually_rev_mp:

   assumes "eventually (\<lambda>x. P x) A"

   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) A"

   shows "eventually (\<lambda>x. Q x) A"

 using assms(2) assms(1) by (rule eventually_mp)

 lemma eventually_conj_iff:

   "eventually (\<lambda>x. P x \<and> Q x) A \<longleftrightarrow> eventually P A \<and> eventually Q A"

   by (auto intro: eventually_conj elim: eventually_rev_mp)

 lemma eventually_elim1:

   assumes "eventually (\<lambda>i. P i) A"

   assumes "\<And>i. P i \<Longrightarrow> Q i"

   shows "eventually (\<lambda>i. Q i) A"

   using assms by (auto elim!: eventually_rev_mp)

 lemma eventually_elim2:

   assumes "eventually (\<lambda>i. P i) A"

   assumes "eventually (\<lambda>i. Q i) A"

   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"

   shows "eventually (\<lambda>i. R i) A"

   using assms by (auto elim!: eventually_rev_mp)

 subsection {* Finer-than relation *}

 text {* @{term "A \<le> B"} means that filter @{term A} is finer than

 filter @{term B}. *}

 instantiation filter :: (type) complete_lattice

 begin

 definition le_filter_def:

   "A \<le> B \<longleftrightarrow> (\<forall>P. eventually P B \<longrightarrow> eventually P A)"

 definition

   "(A :: 'a filter) < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A"

 definition

   "top = Abs_filter (\<lambda>P. \<forall>x. P x)"

 definition

   "bot = Abs_filter (\<lambda>P. True)"

 definition

   "sup A B = Abs_filter (\<lambda>P. eventually P A \<and> eventually P B)"

 definition

   "inf A B = Abs_filter

       (\<lambda>P. \<exists>Q R. eventually Q A \<and> eventually R B \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"

 definition

   "Sup S = Abs_filter (\<lambda>P. \<forall>A\<in>S. eventually P A)"

 definition

   "Inf S = Sup {A::'a filter. \<forall>B\<in>S. A \<le> B}"

 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>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 A B) \<longleftrightarrow> eventually P A \<and> eventually P B"

   unfolding sup_filter_def

   by (rule eventually_Abs_filter, rule is_filter.intro)

      (auto elim!: eventually_rev_mp)

 lemma eventually_inf:

   "eventually P (inf A B) \<longleftrightarrow>

    (\<exists>Q R. eventually Q A \<and> eventually R B \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> 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

   done

 lemma eventually_Sup:

   "eventually P (Sup S) \<longleftrightarrow> (\<forall>A\<in>S. eventually P A)"

   unfolding Sup_filter_def

   apply (rule eventually_Abs_filter, rule is_filter.intro)

   apply (auto intro: eventually_conj elim!: eventually_rev_mp)

   done

 instance proof

   fix A B :: "'a filter" show "A < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A"

     by (rule less_filter_def)

 next

   fix A :: "'a filter" show "A \<le> A"

     unfolding le_filter_def by simp

 next

   fix A B C :: "'a filter" assume "A \<le> B" and "B \<le> C" thus "A \<le> C"

     unfolding le_filter_def by simp

 next

   fix A B :: "'a filter" assume "A \<le> B" and "B \<le> A" thus "A = B"

     unfolding le_filter_def filter_eq_iff by fast

 next

   fix A :: "'a filter" show "A \<le> top"

     unfolding le_filter_def eventually_top by (simp add: always_eventually)

 next

   fix A :: "'a filter" show "bot \<le> A"

     unfolding le_filter_def by simp

 next

   fix A B :: "'a filter" show "A \<le> sup A B" and "B \<le> sup A B"

     unfolding le_filter_def eventually_sup by simp_all

 next

   fix A B C :: "'a filter" assume "A \<le> C" and "B \<le> C" thus "sup A B \<le> C"

     unfolding le_filter_def eventually_sup by simp

 next

   fix A B :: "'a filter" show "inf A B \<le> A" and "inf A B \<le> B"

     unfolding le_filter_def eventually_inf by (auto intro: eventually_True)

 next

   fix A B C :: "'a filter" assume "A \<le> B" and "A \<le> C" thus "A \<le> inf B C"

     unfolding le_filter_def eventually_inf

     by (auto elim!: eventually_mono intro: eventually_conj)

 next

   fix A :: "'a filter" and S assume "A \<in> S" thus "A \<le> Sup S"

     unfolding le_filter_def eventually_Sup by simp

 next

   fix S and B :: "'a filter" assume "\<And>A. A \<in> S \<Longrightarrow> A \<le> B" thus "Sup S \<le> B"

     unfolding le_filter_def eventually_Sup by simp

 next

   fix C :: "'a filter" and S assume "C \<in> S" thus "Inf S \<le> C"

     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp

 next

   fix S and A :: "'a filter" assume "\<And>B. B \<in> S \<Longrightarrow> A \<le> B" thus "A \<le> Inf S"

     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp

 qed

 end

 lemma filter_leD:

   "A \<le> B \<Longrightarrow> eventually P B \<Longrightarrow> eventually P A"

   unfolding le_filter_def by simp

 lemma filter_leI:

   "(\<And>P. eventually P B \<Longrightarrow> eventually P A) \<Longrightarrow> A \<le> B"

   unfolding le_filter_def by simp

 lemma eventually_False:

   "eventually (\<lambda>x. False) A \<longleftrightarrow> A = bot"

   unfolding filter_eq_iff by (auto elim: eventually_rev_mp)

 subsection {* Map function for filters *}

 definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"

   where "filtermap f A = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) A)"

 lemma eventually_filtermap:

   "eventually P (filtermap f A) = eventually (\<lambda>x. P (f x)) A"

   unfolding filtermap_def

   apply (rule eventually_Abs_filter)

   apply (rule is_filter.intro)

   apply (auto elim!: eventually_rev_mp)

   done

 lemma filtermap_ident: "filtermap (\<lambda>x. x) A = A"

   by (simp add: filter_eq_iff eventually_filtermap)

 lemma filtermap_filtermap:

   "filtermap f (filtermap g A) = filtermap (\<lambda>x. f (g x)) A"

   by (simp add: filter_eq_iff eventually_filtermap)

 lemma filtermap_mono: "A \<le> B \<Longrightarrow> filtermap f A \<le> filtermap f B"

   unfolding le_filter_def eventually_filtermap by simp

 lemma filtermap_bot [simp]: "filtermap f bot = bot"

   by (simp add: filter_eq_iff eventually_filtermap)

 subsection {* Sequentially *}

 definition sequentially :: "nat filter"

   where "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"

 lemma eventually_sequentially:

   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"

 unfolding sequentially_def

 proof (rule eventually_Abs_filter, rule is_filter.intro)

   fix P Q :: "nat \<Rightarrow> bool"

   assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"

   then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto

   then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp

   then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..

 qed auto

 lemma sequentially_bot [simp]: "sequentially \<noteq> bot"

   unfolding filter_eq_iff eventually_sequentially by auto

 lemma eventually_False_sequentially [simp]:

   "\<not> eventually (\<lambda>n. False) sequentially"

   by (simp add: eventually_False)

 lemma le_sequentially:

   "A \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) A)"

   unfolding le_filter_def eventually_sequentially

   by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)

 definition trivial_limit :: "'a filter \<Rightarrow> bool"

   where "trivial_limit A \<longleftrightarrow> eventually (\<lambda>x. False) A"

 lemma trivial_limit_sequentially [intro]: "\<not> trivial_limit sequentially"

   by (auto simp add: trivial_limit_def eventually_sequentially)

 subsection {* Standard filters *}

 definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)

   where "A within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) A)"

 definition nhds :: "'a::topological_space \<Rightarrow> 'a filter"

   where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"

 definition at :: "'a::topological_space \<Rightarrow> 'a filter"

   where "at a = nhds a within - {a}"

 lemma eventually_within:

   "eventually P (A within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) A"

   unfolding within_def

   by (rule eventually_Abs_filter, rule is_filter.intro)

      (auto elim!: eventually_rev_mp)

 lemma within_UNIV: "A within UNIV = A"

   unfolding filter_eq_iff eventually_within by simp

 lemma eventually_nhds:

   "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"

 unfolding nhds_def

 proof (rule eventually_Abs_filter, rule is_filter.intro)

   have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp

   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule

 next

   fix P Q

   assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"

      and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"

   then obtain S T where

     "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"

     "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto

   hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"

     by (simp add: open_Int)

   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule

 qed auto

 lemma eventually_nhds_metric:

   "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"

 unfolding eventually_nhds open_dist

 apply safe

 apply fast

 apply (rule_tac x="{x. dist x a < d}" in exI, simp)

 apply clarsimp

 apply (rule_tac x="d - dist x a" in exI, clarsimp)

 apply (simp only: less_diff_eq)

 apply (erule le_less_trans [OF dist_triangle])

 done

 lemma eventually_at_topological:

   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"

 unfolding at_def eventually_within eventually_nhds by simp

 lemma eventually_at:

   fixes a :: "'a::metric_space"

   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"

 unfolding at_def eventually_within eventually_nhds_metric by auto

 subsection {* Boundedness *}

 definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"

   where "Bfun f A = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) A)"

 lemma BfunI:

   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) A" shows "Bfun f A"

 unfolding Bfun_def

 proof (intro exI conjI allI)

   show "0 < max K 1" by simp

 next

   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) A"

     using K by (rule eventually_elim1, simp)

 qed

 lemma BfunE:

   assumes "Bfun f A"

   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) A"

 using assms unfolding Bfun_def by fast

 subsection {* Convergence to Zero *}

 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"

   where "Zfun f A = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) A)"

 lemma ZfunI:

   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) A) \<Longrightarrow> Zfun f A"

   unfolding Zfun_def by simp

 lemma ZfunD:

   "\<lbrakk>Zfun f A; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) A"

   unfolding Zfun_def by simp

 lemma Zfun_ssubst:

   "eventually (\<lambda>x. f x = g x) A \<Longrightarrow> Zfun g A \<Longrightarrow> Zfun f A"

   unfolding Zfun_def by (auto elim!: eventually_rev_mp)

 lemma Zfun_zero: "Zfun (\<lambda>x. 0) A"

   unfolding Zfun_def by simp

 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) A = Zfun (\<lambda>x. f x) A"

   unfolding Zfun_def by simp

 lemma Zfun_imp_Zfun:

   assumes f: "Zfun f A"

   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) A"

   shows "Zfun (\<lambda>x. g x) A"

 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 (\<lambda>x. norm (f x) < r / K) A"

       using ZfunD [OF f] by fast

     with g show "eventually (\<lambda>x. norm (g x) < r) A"

     proof (rule eventually_elim2)

       fix x

       assume *: "norm (g x) \<le> norm (f x) * K"

       assume "norm (f x) < r / K"

       hence "norm (f x) * K < r"

         by (simp add: pos_less_divide_eq K)

       thus "norm (g x) < r"

         by (simp add: order_le_less_trans [OF *])

     qed

   qed

 next

   assume "\<not> 0 < K"

   hence K: "K \<le> 0" by (simp only: not_less)

   show ?thesis

   proof (rule ZfunI)

     fix r :: real

     assume "0 < r"

     from g show "eventually (\<lambda>x. norm (g x) < r) A"

     proof (rule eventually_elim1)

       fix x

       assume "norm (g x) \<le> norm (f x) * K"

       also have "\<dots> \<le> norm (f x) * 0"

         using K norm_ge_zero by (rule mult_left_mono)

       finally show "norm (g x) < r"

         using `0 < r` by simp

     qed

   qed

 qed

 lemma Zfun_le: "\<lbrakk>Zfun g A; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f A"

   by (erule_tac K="1" in Zfun_imp_Zfun, simp)

 lemma Zfun_add:

   assumes f: "Zfun f A" and g: "Zfun g A"

   shows "Zfun (\<lambda>x. f x + g x) A"

 proof (rule ZfunI)

   fix r::real assume "0 < r"

   hence r: "0 < r / 2" by simp

   have "eventually (\<lambda>x. norm (f x) < r/2) A"

     using f r by (rule ZfunD)

   moreover

   have "eventually (\<lambda>x. norm (g x) < r/2) A"

     using g r by (rule ZfunD)

   ultimately

   show "eventually (\<lambda>x. norm (f x + g x) < r) A"

   proof (rule eventually_elim2)

     fix x

     assume *: "norm (f x) < r/2" "norm (g x) < r/2"

     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"

       by (rule norm_triangle_ineq)

     also have "\<dots> < r/2 + r/2"

       using * by (rule add_strict_mono)

     finally show "norm (f x + g x) < r"

       by simp

   qed

 qed

 lemma Zfun_minus: "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. - f x) A"

   unfolding Zfun_def by simp

 lemma Zfun_diff: "\<lbrakk>Zfun f A; Zfun g A\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) A"

   by (simp only: diff_minus Zfun_add Zfun_minus)

 lemma (in bounded_linear) Zfun:

   assumes g: "Zfun g A"

   shows "Zfun (\<lambda>x. f (g x)) A"

 proof -

   obtain K where "\<And>x. norm (f x) \<le> norm x * K"

     using bounded by fast

   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) A"

     by simp

   with g show ?thesis

     by (rule Zfun_imp_Zfun)

 qed

 lemma (in bounded_bilinear) Zfun:

   assumes f: "Zfun f A"

   assumes g: "Zfun g A"

   shows "Zfun (\<lambda>x. f x ** g x) A"

 proof (rule ZfunI)

   fix r::real assume r: "0 < r"

   obtain K where K: "0 < K"

     and norm_le: "\<And>x y. norm (x ** y) \<le> 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 (\<lambda>x. norm (f x) < r) A"

     using f r by (rule ZfunD)

   moreover

   have "eventually (\<lambda>x. norm (g x) < inverse K) A"

     using g K' by (rule ZfunD)

   ultimately

   show "eventually (\<lambda>x. norm (f x ** g x) < r) A"

   proof (rule eventually_elim2)

     fix x

     assume *: "norm (f x) < r" "norm (g x) < inverse K"

     have "norm (f x ** g x) \<le> 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 * K)

     also from K have "r * inverse K * K = r"

       by simp

     finally show "norm (f x ** g x) < r" .

   qed

 qed

 lemma (in bounded_bilinear) Zfun_left:

   "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. f x ** a) A"

   by (rule bounded_linear_left [THEN bounded_linear.Zfun])

 lemma (in bounded_bilinear) Zfun_right:

   "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. a ** f x) A"

   by (rule bounded_linear_right [THEN bounded_linear.Zfun])

 lemmas Zfun_mult = mult.Zfun

 lemmas Zfun_mult_right = mult.Zfun_right

 lemmas Zfun_mult_left = mult.Zfun_left

 subsection {* Limits *}

 definition tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a filter \<Rightarrow> bool"

     (infixr "--->" 55) where

   "(f ---> l) A \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) A)"

 ML {*

 structure Tendsto_Intros = Named_Thms

 (

   val name = "tendsto_intros"

   val description = "introduction rules for tendsto"

 )

 *}

 setup Tendsto_Intros.setup

 lemma tendsto_mono: "A \<le> A' \<Longrightarrow> (f ---> l) A' \<Longrightarrow> (f ---> l) A"

   unfolding tendsto_def le_filter_def by fast

 lemma topological_tendstoI:

   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) A)

     \<Longrightarrow> (f ---> l) A"

   unfolding tendsto_def by auto

 lemma topological_tendstoD:

   "(f ---> l) A \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) A"

   unfolding tendsto_def by auto

 lemma tendstoI:

   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) A"

   shows "(f ---> l) A"

   apply (rule topological_tendstoI)

   apply (simp add: open_dist)

   apply (drule (1) bspec, clarify)

   apply (drule assms)

   apply (erule eventually_elim1, simp)

   done

 lemma tendstoD:

   "(f ---> l) A \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) A"

   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

   done

 lemma tendsto_iff:

   "(f ---> l) A \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) A)"

   using tendstoI tendstoD by fast

 lemma tendsto_Zfun_iff: "(f ---> a) A = Zfun (\<lambda>x. f x - a) A"

   by (simp only: tendsto_iff Zfun_def dist_norm)

 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"

   unfolding tendsto_def eventually_at_topological by auto

 lemma tendsto_ident_at_within [tendsto_intros]:

   "((\<lambda>x. x) ---> a) (at a within S)"

   unfolding tendsto_def eventually_within eventually_at_topological by auto

 lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) A"

   by (simp add: tendsto_def)

 lemma tendsto_const_iff:

   fixes k l :: "'a::metric_space"

   assumes "A \<noteq> bot" shows "((\<lambda>n. k) ---> l) A \<longleftrightarrow> k = l"

   apply (safe intro!: tendsto_const)

   apply (rule ccontr)

   apply (drule_tac e="dist k l" in tendstoD)

   apply (simp add: zero_less_dist_iff)

   apply (simp add: eventually_False assms)

   done

 lemma tendsto_dist [tendsto_intros]:

   assumes f: "(f ---> l) A" and g: "(g ---> m) A"

   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) A"

 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 (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) A"

   proof (rule eventually_elim2)

     fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"

     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

   qed

 qed

 subsubsection {* Norms *}

 lemma norm_conv_dist: "norm x = dist x 0"

   unfolding dist_norm by simp

 lemma tendsto_norm [tendsto_intros]:

   "(f ---> a) A \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) A"

   unfolding norm_conv_dist by (intro tendsto_intros)

 lemma tendsto_norm_zero:

   "(f ---> 0) A \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) A"

   by (drule tendsto_norm, simp)

 lemma tendsto_norm_zero_cancel:

   "((\<lambda>x. norm (f x)) ---> 0) A \<Longrightarrow> (f ---> 0) A"

   unfolding tendsto_iff dist_norm by simp

 lemma tendsto_norm_zero_iff:

   "((\<lambda>x. norm (f x)) ---> 0) A \<longleftrightarrow> (f ---> 0) A"

   unfolding tendsto_iff dist_norm by simp

 lemma tendsto_rabs [tendsto_intros]:

   "(f ---> (l::real)) A \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) A"

   by (fold real_norm_def, rule tendsto_norm)

 lemma tendsto_rabs_zero:

   "(f ---> (0::real)) A \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) A"

   by (fold real_norm_def, rule tendsto_norm_zero)

 lemma tendsto_rabs_zero_cancel:

   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) A \<Longrightarrow> (f ---> 0) A"

   by (fold real_norm_def, rule tendsto_norm_zero_cancel)

 lemma tendsto_rabs_zero_iff:

   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) A \<longleftrightarrow> (f ---> 0) A"

   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 "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) A"

   by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)

 lemma tendsto_add_zero:

   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"

   shows "\<lbrakk>(f ---> 0) A; (g ---> 0) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) A"

   by (drule (1) tendsto_add, simp)

 lemma tendsto_minus [tendsto_intros]:

   fixes a :: "'a::real_normed_vector"

   shows "(f ---> a) A \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) A"

   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)

 lemma tendsto_minus_cancel:

   fixes a :: "'a::real_normed_vector"

   shows "((\<lambda>x. - f x) ---> - a) A \<Longrightarrow> (f ---> a) A"

   by (drule tendsto_minus, simp)

 lemma tendsto_diff [tendsto_intros]:

   fixes a b :: "'a::real_normed_vector"

   shows "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) A"

   by (simp add: diff_minus tendsto_add tendsto_minus)

 lemma tendsto_setsum [tendsto_intros]:

   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"

   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) A"

   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) A"

 proof (cases "finite S")

   assume "finite S" thus ?thesis using assms

     by (induct, simp add: tendsto_const, simp add: tendsto_add)

 next

   assume "\<not> finite S" thus ?thesis

     by (simp add: tendsto_const)

 qed

 subsubsection {* Linear operators and multiplication *}

 lemma (in bounded_linear) tendsto [tendsto_intros]:

   "(g ---> a) A \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) A"

   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)

 lemma (in bounded_linear) tendsto_zero:

   "(g ---> 0) A \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) A"

   by (drule tendsto, simp only: zero)

 lemma (in bounded_bilinear) tendsto [tendsto_intros]:

   "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) A"

   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) A"

   assumes g: "(g ---> 0) A"

   shows "((\<lambda>x. f x ** g x) ---> 0) A"

   using tendsto [OF f g] by (simp add: zero_left)

 lemma (in bounded_bilinear) tendsto_left_zero:

   "(f ---> 0) A \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) A"

   by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])

 lemma (in bounded_bilinear) tendsto_right_zero:

   "(f ---> 0) A \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) A"

   by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])

 lemmas tendsto_mult = mult.tendsto

 lemma tendsto_power [tendsto_intros]:

   fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"

   shows "(f ---> a) A \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) A"

   by (induct n) (simp_all add: tendsto_const tendsto_mult)

 lemma tendsto_setprod [tendsto_intros]:

   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"

   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) A"

   shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) A"

 proof (cases "finite S")

   assume "finite S" thus ?thesis using assms

     by (induct, simp add: tendsto_const, simp add: tendsto_mult)

 next

   assume "\<not> finite S" thus ?thesis

     by (simp add: tendsto_const)

 qed

 subsubsection {* Inverse and division *}

 lemma (in bounded_bilinear) Zfun_prod_Bfun:

   assumes f: "Zfun f A"

   assumes g: "Bfun g A"

   shows "Zfun (\<lambda>x. f x ** g x) A"

 proof -

   obtain K where K: "0 \<le> K"

     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"

     using nonneg_bounded by fast

   obtain B where B: "0 < B"

     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) A"

     using g by (rule BfunE)

   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) A"

   using norm_g proof (rule eventually_elim1)

     fix x

     assume *: "norm (g x) \<le> B"

     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"

       by (rule norm_le)

     also have "\<dots> \<le> norm (f x) * B * K"

       by (intro mult_mono' order_refl norm_g norm_ge_zero

                 mult_nonneg_nonneg K *)

     also have "\<dots> = norm (f x) * (B * K)"

       by (rule mult_assoc)

     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .

   qed

   with f show ?thesis

     by (rule Zfun_imp_Zfun)

 qed

 lemma (in bounded_bilinear) flip:

   "bounded_bilinear (\<lambda>x 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 fast

 lemma (in bounded_bilinear) Bfun_prod_Zfun:

   assumes f: "Bfun f A"

   assumes g: "Zfun g A"

   shows "Zfun (\<lambda>x. f x ** g x) A"

   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)

 lemma Bfun_inverse_lemma:

   fixes x :: "'a::real_normed_div_algebra"

   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"

   apply (subst nonzero_norm_inverse, clarsimp)

   apply (erule (1) le_imp_inverse_le)

   done

 lemma Bfun_inverse:

   fixes a :: "'a::real_normed_div_algebra"

   assumes f: "(f ---> a) A"

   assumes a: "a \<noteq> 0"

   shows "Bfun (\<lambda>x. inverse (f x)) A"

 proof -

   from a have "0 < norm a" by simp

   hence "\<exists>r>0. r < norm a" by (rule dense)

   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast

   have "eventually (\<lambda>x. dist (f x) a < r) A"

     using tendstoD [OF f r1] by fast

   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) A"

   proof (rule eventually_elim1)

     fix x

     assume "dist (f x) a < r"

     hence 1: "norm (f x - a) < r"

       by (simp add: dist_norm)

     hence 2: "f x \<noteq> 0" using r2 by auto

     hence "norm (inverse (f x)) = inverse (norm (f x))"

       by (rule nonzero_norm_inverse)

     also have "\<dots> \<le> 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) \<le> norm (a - f x)"

         by (rule norm_triangle_ineq2)

       also have "\<dots> = norm (f x - a)"

         by (rule norm_minus_commute)

       also have "\<dots> < r" using 1 .

       finally show "norm a - r \<le> norm (f x)" by simp

     qed

     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .

   qed

   thus ?thesis by (rule BfunI)

 qed

 lemma tendsto_inverse_lemma:

   fixes a :: "'a::real_normed_div_algebra"

   shows "\<lbrakk>(f ---> a) A; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) A\<rbrakk>

          \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) A"

   apply (subst tendsto_Zfun_iff)

   apply (rule Zfun_ssubst)

   apply (erule eventually_elim1)

   apply (erule (1) inverse_diff_inverse)

   apply (rule Zfun_minus)

   apply (rule Zfun_mult_left)

   apply (rule mult.Bfun_prod_Zfun)

   apply (erule (1) Bfun_inverse)

   apply (simp add: tendsto_Zfun_iff)

   done

 lemma tendsto_inverse [tendsto_intros]:

   fixes a :: "'a::real_normed_div_algebra"

   assumes f: "(f ---> a) A"

   assumes a: "a \<noteq> 0"

   shows "((\<lambda>x. inverse (f x)) ---> inverse a) A"

 proof -

   from a have "0 < norm a" by simp

   with f have "eventually (\<lambda>x. dist (f x) a < norm a) A"

     by (rule tendstoD)

   then have "eventually (\<lambda>x. f x \<noteq> 0) A"

     unfolding dist_norm by (auto elim!: eventually_elim1)

   with f a show ?thesis

     by (rule tendsto_inverse_lemma)

 qed

 lemma tendsto_divide [tendsto_intros]:

   fixes a b :: "'a::real_normed_field"

   shows "\<lbrakk>(f ---> a) A; (g ---> b) A; b \<noteq> 0\<rbrakk>

     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) A"

   by (simp add: mult.tendsto tendsto_inverse divide_inverse)

 lemma tendsto_sgn [tendsto_intros]:

   fixes l :: "'a::real_normed_vector"

   shows "\<lbrakk>(f ---> l) A; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) A"

   unfolding sgn_div_norm by (simp add: tendsto_intros)

 subsubsection {* Uniqueness *}

 lemma tendsto_unique:

   fixes f :: "'a \<Rightarrow> 'b::t2_space"

   assumes "\<not> trivial_limit A"  "(f ---> l) A"  "(f ---> l') A"

   shows "l = l'"

 proof (rule ccontr)

   assume "l \<noteq> l'"

   obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"

     using hausdorff [OF `l \<noteq> l'`] by fast

   have "eventually (\<lambda>x. f x \<in> U) A"

     using `(f ---> l) A` `open U` `l \<in> U` by (rule topological_tendstoD)

   moreover

   have "eventually (\<lambda>x. f x \<in> V) A"

     using `(f ---> l') A` `open V` `l' \<in> V` by (rule topological_tendstoD)

   ultimately

   have "eventually (\<lambda>x. False) A"

   proof (rule eventually_elim2)

     fix x

     assume "f x \<in> U" "f x \<in> V"

     hence "f x \<in> U \<inter> V" by simp

     with `U \<inter> V = {}` show "False" by simp

   qed

   with `\<not> trivial_limit A` show "False"

     by (simp add: trivial_limit_def)

 qed

 end