(*  Title:      HOL/Topological_Spaces.thy

     Author:     Brian Huffman

     Author:     Johannes Hölzl

 *)

 header {* Topological Spaces *}

 theory Topological_Spaces

 imports Main Conditionally_Complete_Lattices

 begin

 ML {*

 structure Continuous_Intros = Named_Thms

 (

   val name = @{binding continuous_intros}

   val description = "Structural introduction rules for continuity"

 )

 *}

 setup Continuous_Intros.setup

 subsection {* Topological space *}

 class "open" =

   fixes "open" :: "'a set \<Rightarrow> bool"

 class topological_space = "open" +

   assumes open_UNIV [simp, intro]: "open UNIV"

   assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"

   assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union> K)"

 begin

 definition

   closed :: "'a set \<Rightarrow> bool" where

   "closed S \<longleftrightarrow> open (- S)"

 lemma open_empty [continuous_intros, intro, simp]: "open {}"

   using open_Union [of "{}"] by simp

 lemma open_Un [continuous_intros, intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)"

   using open_Union [of "{S, T}"] by simp

 lemma open_UN [continuous_intros, intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)"

   using open_Union [of "B ` A"] by simp

 lemma open_Inter [continuous_intros, intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)"

   by (induct set: finite) auto

 lemma open_INT [continuous_intros, intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)"

   using open_Inter [of "B ` A"] by simp

 lemma openI:

   assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"

   shows "open S"

 proof -

   have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto

   moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)

   ultimately show "open S" by simp

 qed

 lemma closed_empty [continuous_intros, intro, simp]:  "closed {}"

   unfolding closed_def by simp

 lemma closed_Un [continuous_intros, intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)"

   unfolding closed_def by auto

 lemma closed_UNIV [continuous_intros, intro, simp]: "closed UNIV"

   unfolding closed_def by simp

 lemma closed_Int [continuous_intros, intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)"

   unfolding closed_def by auto

 lemma closed_INT [continuous_intros, intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)"

   unfolding closed_def by auto

 lemma closed_Inter [continuous_intros, intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter> K)"

   unfolding closed_def uminus_Inf by auto

 lemma closed_Union [continuous_intros, intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)"

   by (induct set: finite) auto

 lemma closed_UN [continuous_intros, intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"

   using closed_Union [of "B ` A"] by simp

 lemma open_closed: "open S \<longleftrightarrow> closed (- S)"

   unfolding closed_def by simp

 lemma closed_open: "closed S \<longleftrightarrow> open (- S)"

   unfolding closed_def by simp

 lemma open_Diff [continuous_intros, intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)"

   unfolding closed_open Diff_eq by (rule open_Int)

 lemma closed_Diff [continuous_intros, intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)"

   unfolding open_closed Diff_eq by (rule closed_Int)

 lemma open_Compl [continuous_intros, intro]: "closed S \<Longrightarrow> open (- S)"

   unfolding closed_open .

 lemma closed_Compl [continuous_intros, intro]: "open S \<Longrightarrow> closed (- S)"

   unfolding open_closed .

 end

 subsection{* Hausdorff and other separation properties *}

 class t0_space = topological_space +

   assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"

 class t1_space = topological_space +

   assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"

 instance t1_space \<subseteq> t0_space

 proof qed (fast dest: t1_space)

 lemma separation_t1:

   fixes x y :: "'a::t1_space"

   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"

   using t1_space[of x y] by blast

 lemma closed_singleton:

   fixes a :: "'a::t1_space"

   shows "closed {a}"

 proof -

   let ?T = "\<Union>{S. open S \<and> a \<notin> S}"

   have "open ?T" by (simp add: open_Union)

   also have "?T = - {a}"

     by (simp add: set_eq_iff separation_t1, auto)

   finally show "closed {a}" unfolding closed_def .

 qed

 lemma closed_insert [continuous_intros, simp]:

   fixes a :: "'a::t1_space"

   assumes "closed S" shows "closed (insert a S)"

 proof -

   from closed_singleton assms

   have "closed ({a} \<union> S)" by (rule closed_Un)

   thus "closed (insert a S)" by simp

 qed

 lemma finite_imp_closed:

   fixes S :: "'a::t1_space set"

   shows "finite S \<Longrightarrow> closed S"

 by (induct set: finite, simp_all)

 text {* T2 spaces are also known as Hausdorff spaces. *}

 class t2_space = topological_space +

   assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"

 instance t2_space \<subseteq> t1_space

 proof qed (fast dest: hausdorff)

 lemma separation_t2:

   fixes x y :: "'a::t2_space"

   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"

   using hausdorff[of x y] by blast

 lemma separation_t0:

   fixes x y :: "'a::t0_space"

   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"

   using t0_space[of x y] by blast

 text {* A perfect space is a topological space with no isolated points. *}

 class perfect_space = topological_space +

   assumes not_open_singleton: "\<not> open {x}"

 subsection {* Generators for toplogies *}

 inductive generate_topology for S where

   UNIV: "generate_topology S UNIV"

 | Int: "generate_topology S a \<Longrightarrow> generate_topology S b \<Longrightarrow> generate_topology S (a \<inter> b)"

 | UN: "(\<And>k. k \<in> K \<Longrightarrow> generate_topology S k) \<Longrightarrow> generate_topology S (\<Union>K)"

 | Basis: "s \<in> S \<Longrightarrow> generate_topology S s"

 hide_fact (open) UNIV Int UN Basis 

 lemma generate_topology_Union: 

   "(\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)) \<Longrightarrow> generate_topology S (\<Union>k\<in>I. K k)"

   using generate_topology.UN [of "K ` I"] by auto

 lemma topological_space_generate_topology:

   "class.topological_space (generate_topology S)"

   by default (auto intro: generate_topology.intros)

 subsection {* Order topologies *}

 class order_topology = order + "open" +

   assumes open_generated_order: "open = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"

 begin

 subclass topological_space

   unfolding open_generated_order

   by (rule topological_space_generate_topology)

 lemma open_greaterThan [continuous_intros, simp]: "open {a <..}"

   unfolding open_generated_order by (auto intro: generate_topology.Basis)

 lemma open_lessThan [continuous_intros, simp]: "open {..< a}"

   unfolding open_generated_order by (auto intro: generate_topology.Basis)

 lemma open_greaterThanLessThan [continuous_intros, simp]: "open {a <..< b}"

    unfolding greaterThanLessThan_eq by (simp add: open_Int)

 end

 class linorder_topology = linorder + order_topology

 lemma closed_atMost [continuous_intros, simp]: "closed {.. a::'a::linorder_topology}"

   by (simp add: closed_open)

 lemma closed_atLeast [continuous_intros, simp]: "closed {a::'a::linorder_topology ..}"

   by (simp add: closed_open)

 lemma closed_atLeastAtMost [continuous_intros, simp]: "closed {a::'a::linorder_topology .. b}"

 proof -

   have "{a .. b} = {a ..} \<inter> {.. b}"

     by auto

   then show ?thesis

     by (simp add: closed_Int)

 qed

 lemma (in linorder) less_separate:

   assumes "x < y"

   shows "\<exists>a b. x \<in> {..< a} \<and> y \<in> {b <..} \<and> {..< a} \<inter> {b <..} = {}"

 proof (cases "\<exists>z. x < z \<and> z < y")

   case True

   then obtain z where "x < z \<and> z < y" ..

   then have "x \<in> {..< z} \<and> y \<in> {z <..} \<and> {z <..} \<inter> {..< z} = {}"

     by auto

   then show ?thesis by blast

 next

   case False

   with `x < y` have "x \<in> {..< y} \<and> y \<in> {x <..} \<and> {x <..} \<inter> {..< y} = {}"

     by auto

   then show ?thesis by blast

 qed

 instance linorder_topology \<subseteq> t2_space

 proof

   fix x y :: 'a

   from less_separate[of x y] less_separate[of y x]

   show "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"

     by (elim neqE) (metis open_lessThan open_greaterThan Int_commute)+

 qed

 lemma (in linorder_topology) open_right:

   assumes "open S" "x \<in> S" and gt_ex: "x < y" shows "\<exists>b>x. {x ..< b} \<subseteq> S"

   using assms unfolding open_generated_order

 proof induction

   case (Int A B)

   then obtain a b where "a > x" "{x ..< a} \<subseteq> A"  "b > x" "{x ..< b} \<subseteq> B" by auto

   then show ?case by (auto intro!: exI[of _ "min a b"])

 next

   case (Basis S) then show ?case by (fastforce intro: exI[of _ y] gt_ex)

 qed blast+

 lemma (in linorder_topology) open_left:

   assumes "open S" "x \<in> S" and lt_ex: "y < x" shows "\<exists>b<x. {b <.. x} \<subseteq> S"

   using assms unfolding open_generated_order

 proof induction

   case (Int A B)

   then obtain a b where "a < x" "{a <.. x} \<subseteq> A"  "b < x" "{b <.. x} \<subseteq> B" by auto

   then show ?case by (auto intro!: exI[of _ "max a b"])

 next

   case (Basis S) then show ?case by (fastforce intro: exI[of _ y] lt_ex)

 qed blast+

 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 '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 F)"

   using Rep_filter [of F] 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)

 subsubsection {* Eventually *}

 definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"

   where "eventually P F \<longleftrightarrow> 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' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"

   unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..

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

   unfolding eventually_def

   by (rule is_filter.True [OF is_filter_Rep_filter])

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

 proof -

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

   thus "eventually P F" by simp

 qed

 lemma eventually_mono:

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

   unfolding eventually_def

   by (rule is_filter.mono [OF is_filter_Rep_filter])

 lemma eventually_conj:

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

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

   shows "eventually (\<lambda>x. P x \<and> 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 "\<forall>y\<in>A. eventually (\<lambda>x. P x y) net"

   shows "eventually (\<lambda>x. \<forall>y\<in>A. P x y) net"

 using assms by (induct set: finite, simp, simp add: eventually_conj)

 lemma eventually_all_finite:

   fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"

   assumes "\<And>y. eventually (\<lambda>x. P x y) net"

   shows "eventually (\<lambda>x. \<forall>y. P x y) net"

 using eventually_Ball_finite [of UNIV P] assms by simp

 lemma eventually_mp:

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

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

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

 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) F"

     using assms by (rule eventually_conj)

 qed

 lemma eventually_rev_mp:

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

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

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

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

 lemma eventually_conj_iff:

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

   by (auto intro: eventually_conj elim: eventually_rev_mp)

 lemma eventually_elim1:

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

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

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

   using assms by (auto elim!: eventually_rev_mp)

 lemma eventually_elim2:

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

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

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

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

   using assms by (auto elim!: eventually_rev_mp)

 lemma eventually_subst:

   assumes "eventually (\<lambda>n. P n = Q n) F"

   shows "eventually P F = eventually Q F" (is "?L = ?R")

 proof -

   from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"

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

     by (auto elim: eventually_elim1)

   then show ?thesis by (auto elim: eventually_elim2)

 qed

 ML {*

   fun eventually_elim_tac ctxt thms = SUBGOAL_CASES (fn (_, _, st) =>

     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 st)

       val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])])

     in

       CASES cases (rtac raw_elim_thm 1)

     end) 1

 *}

 method_setup eventually_elim = {*

   Scan.succeed (fn ctxt => METHOD_CASES (eventually_elim_tac ctxt))

 *} "elimination of eventually quantifiers"

 subsubsection {* Finer-than relation *}

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

 filter @{term F'}. *}

 instantiation filter :: (type) complete_lattice

 begin

 definition le_filter_def:

   "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"

 definition

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

 definition

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

 definition

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

 definition

   "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"

 definition

   "inf F F' = Abs_filter

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

 definition

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

 definition

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

 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 F F') \<longleftrightarrow> eventually P F \<and> 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') \<longleftrightarrow>

    (\<exists>Q R. eventually Q F \<and> eventually R F' \<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>F\<in>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)

   done

 instance proof

   fix F F' F'' :: "'a filter" and S :: "'a filter set"

   { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"

     by (rule less_filter_def) }

   { show "F \<le> F"

     unfolding le_filter_def by simp }

   { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"

     unfolding le_filter_def by simp }

   { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"

     unfolding le_filter_def filter_eq_iff by fast }

   { show "inf F F' \<le> F" and "inf F F' \<le> F'"

     unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }

   { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"

     unfolding le_filter_def eventually_inf

     by (auto elim!: eventually_mono intro: eventually_conj) }

   { show "F \<le> sup F F'" and "F' \<le> sup F F'"

     unfolding le_filter_def eventually_sup by simp_all }

   { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"

     unfolding le_filter_def eventually_sup by simp }

   { assume "F'' \<in> S" thus "Inf S \<le> F''"

     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }

   { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"

     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }

   { assume "F \<in> S" thus "F \<le> Sup S"

     unfolding le_filter_def eventually_Sup by simp }

   { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"

     unfolding le_filter_def eventually_Sup by simp }

   { show "Inf {} = (top::'a filter)"

     by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)

       (metis (full_types) top_filter_def always_eventually eventually_top) }

   { show "Sup {} = (bot::'a filter)"

     by (auto simp: bot_filter_def Sup_filter_def) }

 qed

 end

 lemma filter_leD:

   "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"

   unfolding le_filter_def by simp

 lemma filter_leI:

   "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"

   unfolding le_filter_def by simp

 lemma eventually_False:

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

   unfolding filter_eq_iff by (auto elim: eventually_rev_mp)

 abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"

   where "trivial_limit F \<equiv> F = bot"

 lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"

   by (rule eventually_False [symmetric])

 lemma eventually_const: "\<not> trivial_limit net \<Longrightarrow> eventually (\<lambda>x. P) net \<longleftrightarrow> P"

   by (cases P) (simp_all add: eventually_False)

 subsubsection {* Map function for filters *}

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

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

 lemma eventually_filtermap:

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

   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) F = F"

   by (simp add: filter_eq_iff eventually_filtermap)

 lemma filtermap_filtermap:

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

   by (simp add: filter_eq_iff eventually_filtermap)

 lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"

   unfolding le_filter_def eventually_filtermap by simp

 lemma 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)

 subsubsection {* Order filters *}

 definition at_top :: "('a::order) filter"

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

 lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"

   unfolding at_top_def

 proof (rule eventually_Abs_filter, rule is_filter.intro)

   fix P Q :: "'a \<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 eventually_ge_at_top:

   "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"

   unfolding eventually_at_top_linorder by auto

 lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::unbounded_dense_linorder. \<forall>n>N. P n)"

   unfolding eventually_at_top_linorder

 proof safe

   fix N assume "\<forall>n\<ge>N. P n"

   then show "\<exists>N. \<forall>n>N. P n" by (auto intro!: exI[of _ N])

 next

   fix N assume "\<forall>n>N. P n"

   moreover obtain y where "N < y" using gt_ex[of N] ..

   ultimately show "\<exists>N. \<forall>n\<ge>N. P n" by (auto intro!: exI[of _ y])

 qed

 lemma eventually_gt_at_top:

   "eventually (\<lambda>x. (c::_::unbounded_dense_linorder) < x) at_top"

   unfolding eventually_at_top_dense by auto

 definition at_bot :: "('a::order) filter"

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

 lemma eventually_at_bot_linorder:

   fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"

   unfolding at_bot_def

 proof (rule eventually_Abs_filter, rule is_filter.intro)

   fix P Q :: "'a \<Rightarrow> bool"

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

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

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

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

 qed auto

 lemma eventually_le_at_bot:

   "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"

   unfolding eventually_at_bot_linorder by auto

 lemma eventually_at_bot_dense:

   fixes P :: "'a::unbounded_dense_linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n<N. P n)"

   unfolding eventually_at_bot_linorder

 proof safe

   fix N assume "\<forall>n\<le>N. P n" then show "\<exists>N. \<forall>n<N. P n" by (auto intro!: exI[of _ N])

 next

   fix N assume "\<forall>n<N. P n" 

   moreover obtain y where "y < N" using lt_ex[of N] ..

   ultimately show "\<exists>N. \<forall>n\<le>N. P n" by (auto intro!: exI[of _ y])

 qed

 lemma eventually_gt_at_bot:

   "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"

   unfolding eventually_at_bot_dense by auto

 lemma trivial_limit_at_bot_linorder: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"

   unfolding trivial_limit_def

   by (metis eventually_at_bot_linorder order_refl)

 lemma trivial_limit_at_top_linorder: "\<not> trivial_limit (at_top ::('a::linorder) filter)"

   unfolding trivial_limit_def

   by (metis eventually_at_top_linorder order_refl)

 subsection {* Sequentially *}

 abbreviation sequentially :: "nat filter"

   where "sequentially == at_top"

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

   unfolding at_top_def by simp

 lemma eventually_sequentially:

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

   by (rule eventually_at_top_linorder)

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

   unfolding filter_eq_iff eventually_sequentially by auto

 lemmas trivial_limit_sequentially = sequentially_bot

 lemma eventually_False_sequentially [simp]:

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

   by (simp add: eventually_False)

 lemma le_sequentially:

   "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> 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 "\<And>x. c \<le> x \<Longrightarrow> P x"

   shows "eventually P sequentially"

 using assms by (auto simp: eventually_sequentially)

 lemma eventually_sequentially_seg:

   "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"

   unfolding eventually_sequentially

   apply safe

    apply (rule_tac x="N + k" in exI)

    apply rule

    apply (erule_tac x="n - k" in allE)

    apply auto []

   apply (rule_tac x=N in exI)

   apply auto []

   done

 subsubsection {* Standard filters *}

 definition principal :: "'a set \<Rightarrow> 'a filter" where

   "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"

 lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"

   unfolding principal_def

   by (rule eventually_Abs_filter, rule is_filter.intro) auto

 lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"

   unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1)

 lemma principal_UNIV[simp]: "principal UNIV = top"

   by (auto simp: filter_eq_iff eventually_principal)

 lemma principal_empty[simp]: "principal {} = bot"

   by (auto simp: filter_eq_iff eventually_principal)

 lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"

   by (auto simp: le_filter_def eventually_principal)

 lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"

   unfolding le_filter_def eventually_principal

   apply safe

   apply (erule_tac x="\<lambda>x. x \<in> A" in allE)

   apply (auto elim: eventually_elim1)

   done

 lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"

   unfolding eq_iff by simp

 lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"

   unfolding filter_eq_iff eventually_sup eventually_principal by auto

 lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"

   unfolding filter_eq_iff eventually_inf eventually_principal

   by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])

 lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"

   unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal)

 lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"

   unfolding filter_eq_iff eventually_filtermap eventually_principal by simp

 subsubsection {* Topological filters *}

 definition (in topological_space) nhds :: "'a \<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 (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter" ("at (_) within (_)" [1000, 60] 60)

   where "at a within s = inf (nhds a) (principal (s - {a}))"

 abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter" ("at") where

   "at x \<equiv> at x within (CONST UNIV)"

 abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter" where

   "at_right x \<equiv> at x within {x <..}"

 abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter" where

   "at_left x \<equiv> at x within {..< x}"

 lemma (in topological_space) 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)" ..

 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)" ..

 qed auto

 lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"

   unfolding trivial_limit_def eventually_nhds by simp

 lemma eventually_at_filter:

   "eventually P (at a within s) \<longleftrightarrow> eventually (\<lambda>x. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x) (nhds a)"

   unfolding at_within_def eventually_inf_principal by (simp add: imp_conjL[symmetric] conj_commute)

 lemma at_le: "s \<subseteq> t \<Longrightarrow> at x within s \<le> at x within t"

   unfolding at_within_def by (intro inf_mono) auto

 lemma eventually_at_topological:

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

   unfolding eventually_nhds eventually_at_filter by simp

 lemma at_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> at a within S = at a"

   unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I)

 lemma at_within_empty [simp]: "at a within {} = bot"

   unfolding at_within_def by simp

 lemma at_within_union: "at x within (S \<union> T) = sup (at x within S) (at x within T)"

   unfolding filter_eq_iff eventually_sup eventually_at_filter

   by (auto elim!: eventually_rev_mp)

 lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> 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) \<noteq> bot"

   by (simp add: at_eq_bot_iff not_open_singleton)

 lemma eventually_at_right:

   fixes x :: "'a :: {no_top, linorder_topology}"

   shows "eventually P (at_right x) \<longleftrightarrow> (\<exists>b. x < b \<and> (\<forall>z. x < z \<and> z < b \<longrightarrow> P z))"

   unfolding eventually_at_topological

 proof safe

   obtain y where "x < y" using gt_ex[of x] ..

   moreover fix S assume "open S" "x \<in> S" note open_right[OF this, of y]

   moreover note gt_ex[of x]

   moreover assume "\<forall>s\<in>S. s \<noteq> x \<longrightarrow> s \<in> {x<..} \<longrightarrow> P s"

   ultimately show "\<exists>b>x. \<forall>z. x < z \<and> z < b \<longrightarrow> P z"

     by (auto simp: subset_eq Ball_def)

 next

   fix b assume "x < b" "\<forall>z. x < z \<and> z < b \<longrightarrow> P z"

   then show "\<exists>S. open S \<and> x \<in> S \<and> (\<forall>xa\<in>S. xa \<noteq> x \<longrightarrow> xa \<in> {x<..} \<longrightarrow> P xa)"

     by (intro exI[of _ "{..< b}"]) auto

 qed

 lemma eventually_at_left:

   fixes x :: "'a :: {no_bot, linorder_topology}"

   shows "eventually P (at_left x) \<longleftrightarrow> (\<exists>b. x > b \<and> (\<forall>z. b < z \<and> z < x \<longrightarrow> P z))"

   unfolding eventually_at_topological

 proof safe

   obtain y where "y < x" using lt_ex[of x] ..

   moreover fix S assume "open S" "x \<in> S" note open_left[OF this, of y]

   moreover assume "\<forall>s\<in>S. s \<noteq> x \<longrightarrow> s \<in> {..<x} \<longrightarrow> P s"

   ultimately show "\<exists>b<x. \<forall>z. b < z \<and> z < x \<longrightarrow> P z"

     by (auto simp: subset_eq Ball_def)

 next

   fix b assume "b < x" "\<forall>z. b < z \<and> z < x \<longrightarrow> P z"

   then show "\<exists>S. open S \<and> x \<in> S \<and> (\<forall>s\<in>S. s \<noteq> x \<longrightarrow> s \<in> {..<x} \<longrightarrow> P s)"

     by (intro exI[of _ "{b <..}"]) auto

 qed

 lemma trivial_limit_at_left_real [simp]:

   "\<not> trivial_limit (at_left (x::'a::{no_bot, unbounded_dense_linorder, linorder_topology}))"

   unfolding trivial_limit_def eventually_at_left by (auto dest: dense)

 lemma trivial_limit_at_right_real [simp]:

   "\<not> trivial_limit (at_right (x::'a::{no_top, unbounded_dense_linorder, linorder_topology}))"

   unfolding trivial_limit_def eventually_at_right by (auto dest: dense)

 lemma at_eq_sup_left_right: "at (x::'a::linorder_topology) = sup (at_left x) (at_right x)"

   by (auto simp: eventually_at_filter filter_eq_iff eventually_sup 

            elim: eventually_elim2 eventually_elim1)

 lemma eventually_at_split:

   "eventually P (at (x::'a::linorder_topology)) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)"

   by (subst at_eq_sup_left_right) (simp add: eventually_sup)

 subsection {* Limits *}

 definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where

   "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"

 syntax

   "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 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) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"

   unfolding filterlim_def le_filter_def eventually_filtermap ..

 lemma filterlim_compose:

   "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"

   unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)

 lemma filterlim_mono:

   "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> 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' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"

   by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)

 lemma filterlim_principal:

   "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"

   unfolding filterlim_def eventually_filtermap le_principal ..

 lemma filterlim_inf:

   "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"

   unfolding filterlim_def by simp

 lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"

   unfolding filterlim_def filtermap_filtermap ..

 lemma filterlim_sup:

   "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"

   unfolding filterlim_def filtermap_sup by auto

 lemma filterlim_Suc: "filterlim Suc sequentially sequentially"

   by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)

 subsubsection {* Tendsto *}

 abbreviation (in topological_space)

   tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where

   "(f ---> l) F \<equiv> filterlim f (nhds l) F"

 definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a" where

   "Lim A f = (THE l. (f ---> l) A)"

 lemma tendsto_eq_rhs: "(f ---> x) F \<Longrightarrow> x = y \<Longrightarrow> (f ---> y) F"

   by simp

 ML {*

 structure Tendsto_Intros = Named_Thms

 (

   val name = @{binding tendsto_intros}

   val description = "introduction rules for tendsto"

 )

 *}

 setup {*

   Tendsto_Intros.setup #>

   Global_Theory.add_thms_dynamic (@{binding tendsto_eq_intros},

     map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm])) o Tendsto_Intros.get o Context.proof_of);

 *}

 lemma (in topological_space) tendsto_def:

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

   unfolding filterlim_def

 proof safe

   fix S assume "open S" "l \<in> S" "filtermap f F \<le> nhds l"

   then show "eventually (\<lambda>x. f x \<in> S) F"

     unfolding eventually_nhds eventually_filtermap le_filter_def

     by (auto elim!: allE[of _ "\<lambda>x. x \<in> S"] eventually_rev_mp)

 qed (auto elim!: eventually_rev_mp simp: eventually_nhds eventually_filtermap le_filter_def)

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

   unfolding tendsto_def le_filter_def by fast

 lemma tendsto_within_subset: "(f ---> l) (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (at x within T)"

   by (blast intro: tendsto_mono at_le)

 lemma filterlim_at:

   "(LIM x F. f x :> at b within s) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> s \<and> f x \<noteq> b) F \<and> (f ---> b) F)"

   by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute)

 lemma (in topological_space) topological_tendstoI:

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

   unfolding tendsto_def by auto

 lemma (in topological_space) topological_tendstoD:

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

   unfolding tendsto_def by auto

 lemma order_tendstoI:

   fixes y :: "_ :: order_topology"

   assumes "\<And>a. a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"

   assumes "\<And>a. y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"

   shows "(f ---> y) F"

 proof (rule topological_tendstoI)

   fix S assume "open S" "y \<in> S"

   then show "eventually (\<lambda>x. f x \<in> S) F"

     unfolding open_generated_order

   proof induct

     case (UN K)

     then obtain k where "y \<in> k" "k \<in> K" by auto

     with UN(2)[of k] show ?case

       by (auto elim: eventually_elim1)

   qed (insert assms, auto elim: eventually_elim2)

 qed

 lemma order_tendstoD:

   fixes y :: "_ :: order_topology"

   assumes y: "(f ---> y) F"

   shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"

     and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"

   using topological_tendstoD[OF y, of "{..< a}"] topological_tendstoD[OF y, of "{a <..}"] by auto

 lemma order_tendsto_iff: 

   fixes f :: "_ \<Rightarrow> 'a :: order_topology"

   shows "(f ---> x) F \<longleftrightarrow>(\<forall>l<x. eventually (\<lambda>x. l < f x) F) \<and> (\<forall>u>x. eventually (\<lambda>x. f x < u) F)"

   by (metis order_tendstoI order_tendstoD)

 lemma tendsto_bot [simp]: "(f ---> a) bot"

   unfolding tendsto_def by simp

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

   unfolding tendsto_def eventually_at_topological by auto

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

   by (simp add: tendsto_def)

 lemma (in t2_space) tendsto_unique:

   assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"

   shows "a = b"

 proof (rule ccontr)

   assume "a \<noteq> b"

   obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"

     using hausdorff [OF `a \<noteq> b`] by fast

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

     using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)

   moreover

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

     using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)

   ultimately

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

   proof eventually_elim

     case (elim x)

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

     with `U \<inter> V = {}` show ?case by simp

   qed

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

     by (simp add: trivial_limit_def)

 qed

 lemma (in t2_space) tendsto_const_iff:

   assumes "\<not> trivial_limit F" shows "((\<lambda>x. a :: 'a) ---> b) F \<longleftrightarrow> a = b"

   by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])

 lemma increasing_tendsto:

   fixes f :: "_ \<Rightarrow> 'a::order_topology"

   assumes bdd: "eventually (\<lambda>n. f n \<le> l) F"

       and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F"

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

   using assms by (intro order_tendstoI) (auto elim!: eventually_elim1)

 lemma decreasing_tendsto:

   fixes f :: "_ \<Rightarrow> 'a::order_topology"

   assumes bdd: "eventually (\<lambda>n. l \<le> f n) F"

       and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F"

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

   using assms by (intro order_tendstoI) (auto elim!: eventually_elim1)

 lemma tendsto_sandwich:

   fixes f g h :: "'a \<Rightarrow> 'b::order_topology"

   assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"

   assumes lim: "(f ---> c) net" "(h ---> c) net"

   shows "(g ---> c) net"

 proof (rule order_tendstoI)

   fix a show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net"

     using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2)

 next

   fix a show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net"

     using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2)

 qed

 lemma tendsto_le:

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

   assumes F: "\<not> trivial_limit F"

   assumes x: "(f ---> x) F" and y: "(g ---> y) F"

   assumes ev: "eventually (\<lambda>x. g x \<le> f x) F"

   shows "y \<le> x"

 proof (rule ccontr)

   assume "\<not> y \<le> x"

   with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a} \<inter> {b<..} = {}"

     by (auto simp: not_le)

   then have "eventually (\<lambda>x. f x < a) F" "eventually (\<lambda>x. b < g x) F"

     using x y by (auto intro: order_tendstoD)

   with ev have "eventually (\<lambda>x. False) F"

     by eventually_elim (insert xy, fastforce)

   with F show False

     by (simp add: eventually_False)

 qed

 lemma tendsto_le_const:

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

   assumes F: "\<not> trivial_limit F"

   assumes x: "(f ---> x) F" and a: "eventually (\<lambda>i. a \<le> f i) F"

   shows "a \<le> x"

   using F x tendsto_const a by (rule tendsto_le)

 lemma tendsto_ge_const:

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

   assumes F: "\<not> trivial_limit F"

   assumes x: "(f ---> x) F" and a: "eventually (\<lambda>i. a \<ge> f i) F"

   shows "a \<ge> x"

   by (rule tendsto_le [OF F tendsto_const x a])

 subsubsection {* Rules about @{const Lim} *}

 lemma (in t2_space) tendsto_Lim:

   "\<not>(trivial_limit net) \<Longrightarrow> (f ---> l) net \<Longrightarrow> Lim net f = l"

   unfolding Lim_def using tendsto_unique[of net f] by auto

 lemma Lim_ident_at: "\<not> trivial_limit (at x within s) \<Longrightarrow> Lim (at x within s) (\<lambda>x. x) = x"

   by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto

 subsection {* Limits to @{const at_top} and @{const at_bot} *}

 lemma filterlim_at_top:

   fixes f :: "'a \<Rightarrow> ('b::linorder)"

   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"

   by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)

 lemma filterlim_at_top_dense:

   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"

   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. 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 \<Rightarrow> ('b::linorder)" and c :: "'b"

   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"

   unfolding filterlim_at_top

 proof safe

   fix Z assume *: "\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F"

   with *[THEN spec, of "max Z c"] show "eventually (\<lambda>x. Z \<le> f x) F"

     by (auto elim!: eventually_elim1)

 qed simp

 lemma filterlim_at_top_at_top:

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

   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"

   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> 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: "\<And>y. x \<le> y \<Longrightarrow> 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 \<le> z"

     with x have "P z" by auto

     have "eventually (\<lambda>x. g z \<le> x) at_top"

       by (rule eventually_ge_at_top)

     with Q show "eventually (\<lambda>x. z \<le> f x) at_top"

       by eventually_elim (metis mono bij `P z`)

   qed

 qed

 lemma filterlim_at_top_gt:

   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"

   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> 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 \<Rightarrow> ('b::linorder)"

   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"

   by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)

 lemma filterlim_at_bot_le:

   fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"

   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"

   unfolding filterlim_at_bot

 proof safe

   fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"

   with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"

     by (auto elim!: eventually_elim1)

 qed simp

 lemma filterlim_at_bot_lt:

   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"

   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> 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 :: "'a::{no_top, linorder_topology} \<Rightarrow> 'b::linorder"

   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"

   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"

   assumes Q: "eventually Q (at_right a)" and bound: "\<And>b. Q b \<Longrightarrow> a < b"

   assumes P: "eventually P at_bot"

   shows "filterlim f at_bot (at_right a)"

 proof -

   from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> 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 \<le> x"

     with x have "P z" by auto

     have "eventually (\<lambda>x. x \<le> g z) (at_right a)"

       using bound[OF bij(2)[OF `P z`]]

       unfolding eventually_at_right by (auto intro!: exI[of _ "g z"])

     with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)"

       by eventually_elim (metis bij `P z` mono)

   qed

 qed

 lemma filterlim_at_top_at_left:

   fixes f :: "'a::{no_bot, linorder_topology} \<Rightarrow> 'b::linorder"

   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"

   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"

   assumes Q: "eventually Q (at_left a)" and bound: "\<And>b. Q b \<Longrightarrow> b < a"

   assumes P: "eventually P at_top"

   shows "filterlim f at_top (at_left a)"

 proof -

   from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> 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 \<le> z"

     with x have "P z" by auto

     have "eventually (\<lambda>x. g z \<le> x) (at_left a)"

       using bound[OF bij(2)[OF `P z`]]

       unfolding eventually_at_left by (auto intro!: exI[of _ "g z"])

     with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)"

       by eventually_elim (metis bij `P z` mono)

   qed

 qed

 lemma filterlim_split_at:

   "filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow> filterlim f F (at (x::'a::linorder_topology))"

   by (subst at_eq_sup_left_right) (rule filterlim_sup)

 lemma filterlim_at_split:

   "filterlim f F (at (x::'a::linorder_topology)) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)"

   by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup)

 subsection {* Limits on sequences *}

 abbreviation (in topological_space)

   LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"

     ("((_)/ ----> (_))" [60, 60] 60) where

   "X ----> L \<equiv> (X ---> L) sequentially"

 abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" where

   "lim X \<equiv> Lim sequentially X"

 definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where

   "convergent X = (\<exists>L. X ----> L)"

 lemma lim_def: "lim X = (THE L. X ----> L)"

   unfolding Lim_def ..

 subsubsection {* Monotone sequences and subsequences *}

 definition

   monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where

     --{*Definition of monotonicity.

         The use of disjunction here complicates proofs considerably.

         One alternative is to add a Boolean argument to indicate the direction.

         Another is to develop the notions of increasing and decreasing first.*}

   "monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) \<or> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"

 abbreviation incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where

   "incseq X \<equiv> mono X"

 lemma incseq_def: "incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<ge> X m)"

   unfolding mono_def ..

 abbreviation decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where

   "decseq X \<equiv> antimono X"

 lemma decseq_def: "decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"

   unfolding antimono_def ..

 definition

   subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" where

     --{*Definition of subsequence*}

   "subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"

 lemma incseq_SucI:

   "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"

   using lift_Suc_mono_le[of X]

   by (auto simp: incseq_def)

 lemma incseqD: "\<And>i j. incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"

   by (auto simp: incseq_def)

 lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)"

   using incseqD[of A i "Suc i"] by auto

 lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"

   by (auto intro: incseq_SucI dest: incseq_SucD)

 lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)"

   unfolding incseq_def by auto

 lemma decseq_SucI:

   "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"

   using order.lift_Suc_mono_le[OF dual_order, of X]

   by (auto simp: decseq_def)

 lemma decseqD: "\<And>i j. decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"

   by (auto simp: decseq_def)

 lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i"

   using decseqD[of A i "Suc i"] by auto

 lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"

   by (auto intro: decseq_SucI dest: decseq_SucD)

 lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)"

   unfolding decseq_def by auto

 lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X"

   unfolding monoseq_def incseq_def decseq_def ..

 lemma monoseq_Suc:

   "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"

   unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..

 lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"

 by (simp add: monoseq_def)

 lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"

 by (simp add: monoseq_def)

 lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"

 by (simp add: monoseq_Suc)

 lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"

 by (simp add: monoseq_Suc)

 lemma monoseq_minus:

   fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"

   assumes "monoseq a"

   shows "monoseq (\<lambda> n. - a n)"

 proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")

   case True

   hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto

   thus ?thesis by (rule monoI2)

 next

   case False

   hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto

   thus ?thesis by (rule monoI1)

 qed

 text{*Subsequence (alternative definition, (e.g. Hoskins)*}

 lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"

 apply (simp add: subseq_def)

 apply (auto dest!: less_imp_Suc_add)

 apply (induct_tac k)

 apply (auto intro: less_trans)

 done

 text{* for any sequence, there is a monotonic subsequence *}

 lemma seq_monosub:

   fixes s :: "nat => 'a::linorder"

   shows "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"

 proof cases

   let "?P p n" = "p > n \<and> (\<forall>m\<ge>p. s m \<le> s p)"

   assume *: "\<forall>n. \<exists>p. ?P p n"

   def f \<equiv> "rec_nat (SOME p. ?P p 0) (\<lambda>_ n. SOME p. ?P p n)"

   have f_0: "f 0 = (SOME p. ?P p 0)" unfolding f_def by simp

   have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat.rec(2) ..

   have P_0: "?P (f 0) 0" unfolding f_0 using *[rule_format] by (rule someI2_ex) auto

   have P_Suc: "\<And>i. ?P (f (Suc i)) (f i)" unfolding f_Suc using *[rule_format] by (rule someI2_ex) auto

   then have "subseq f" unfolding subseq_Suc_iff by auto

   moreover have "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc

   proof (intro disjI2 allI)

     fix n show "s (f (Suc n)) \<le> s (f n)"

     proof (cases n)

       case 0 with P_Suc[of 0] P_0 show ?thesis by auto

     next

       case (Suc m)

       from P_Suc[of n] Suc have "f (Suc m) \<le> f (Suc (Suc m))" by simp

       with P_Suc Suc show ?thesis by simp

     qed

   qed

   ultimately show ?thesis by auto

 next

   let "?P p m" = "m < p \<and> s m < s p"

   assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))"

   then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. s p < s m" by (force simp: not_le le_less)

   def f \<equiv> "rec_nat (SOME p. ?P p (Suc N)) (\<lambda>_ n. SOME p. ?P p n)"

   have f_0: "f 0 = (SOME p. ?P p (Suc N))" unfolding f_def by simp

   have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat.rec(2) ..

   have P_0: "?P (f 0) (Suc N)"

     unfolding f_0 some_eq_ex[of "\<lambda>p. ?P p (Suc N)"] using N[of "Suc N"] by auto

   { fix i have "N < f i \<Longrightarrow> ?P (f (Suc i)) (f i)"

       unfolding f_Suc some_eq_ex[of "\<lambda>p. ?P p (f i)"] using N[of "f i"] . }

   note P' = this

   { fix i have "N < f i \<and> ?P (f (Suc i)) (f i)"

       by (induct i) (insert P_0 P', auto) }

   then have "subseq f" "monoseq (\<lambda>x. s (f x))"

     unfolding subseq_Suc_iff monoseq_Suc by (auto simp: not_le intro: less_imp_le)

   then show ?thesis by auto

 qed

 lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"

 proof(induct n)

   case 0 thus ?case by simp

 next

   case (Suc n)

   from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps

   have "n < f (Suc n)" by arith

   thus ?case by arith

 qed

 lemma eventually_subseq:

   "subseq r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"

   unfolding eventually_sequentially by (metis seq_suble le_trans)

 lemma not_eventually_sequentiallyD:

   assumes P: "\<not> eventually P sequentially"

   shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))"

 proof -

   from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m"

     unfolding eventually_sequentially by (simp add: not_less)

   then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"

     by (auto simp: choice_iff)

   then show ?thesis

     by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"]

              simp: less_eq_Suc_le subseq_Suc_iff)

 qed

 lemma filterlim_subseq: "subseq f \<Longrightarrow> filterlim f sequentially sequentially"

   unfolding filterlim_iff by (metis eventually_subseq)

 lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"

   unfolding subseq_def by simp

 lemma subseq_mono: assumes "subseq r" "m < n" shows "r m < r n"

   using assms by (auto simp: subseq_def)

 lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"

   by (simp add: incseq_def monoseq_def)

 lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"

   by (simp add: decseq_def monoseq_def)

 lemma decseq_eq_incseq:

   fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)" 

   by (simp add: decseq_def incseq_def)

 lemma INT_decseq_offset:

   assumes "decseq F"

   shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"

 proof safe

   fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"

   show "x \<in> F i"

   proof cases

     from x have "x \<in> F n" by auto

     also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i"

       unfolding decseq_def by simp

     finally show ?thesis .

   qed (insert x, simp)

 qed auto

 lemma LIMSEQ_const_iff:

   fixes k l :: "'a::t2_space"

   shows "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"

   using trivial_limit_sequentially by (rule tendsto_const_iff)

 lemma LIMSEQ_SUP:

   "incseq X \<Longrightarrow> X ----> (SUP i. X i :: 'a :: {complete_linorder, linorder_topology})"

   by (intro increasing_tendsto)

      (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans)

 lemma LIMSEQ_INF:

   "decseq X \<Longrightarrow> X ----> (INF i. X i :: 'a :: {complete_linorder, linorder_topology})"

   by (intro decreasing_tendsto)

      (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans)

 lemma LIMSEQ_ignore_initial_segment:

   "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"

   unfolding tendsto_def

   by (subst eventually_sequentially_seg[where k=k])

 lemma LIMSEQ_offset:

   "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"

   unfolding tendsto_def

   by (subst (asm) eventually_sequentially_seg[where k=k])

 lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"

 by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)

 lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"

 by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)

 lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"

 by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)

 lemma LIMSEQ_unique:

   fixes a b :: "'a::t2_space"

   shows "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"

   using trivial_limit_sequentially by (rule tendsto_unique)

 lemma LIMSEQ_le_const:

   "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"

   using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially)

 lemma LIMSEQ_le:

   "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::'a::linorder_topology)"

   using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)

 lemma LIMSEQ_le_const2:

   "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"

   by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) (auto simp: tendsto_const)

 lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"

 by (simp add: convergent_def)

 lemma convergentI: "(X ----> L) ==> convergent X"

 by (auto simp add: convergent_def)

 lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"

 by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)

 lemma convergent_const: "convergent (\<lambda>n. c)"

   by (rule convergentI, rule tendsto_const)

 lemma monoseq_le:

   "monoseq a \<Longrightarrow> a ----> (x::'a::linorder_topology) \<Longrightarrow>

     ((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or> ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"

   by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)

 lemma LIMSEQ_subseq_LIMSEQ:

   "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"

   unfolding comp_def by (rule filterlim_compose[of X, OF _ filterlim_subseq])

 lemma convergent_subseq_convergent:

   "\<lbrakk>convergent X; subseq f\<rbrakk> \<Longrightarrow> convergent (X o f)"

   unfolding convergent_def by (auto intro: LIMSEQ_subseq_LIMSEQ)

 lemma limI: "X ----> L ==> lim X = L"

 apply (simp add: lim_def)

 apply (blast intro: LIMSEQ_unique)

 done

 lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> (x::'a::linorder_topology)) \<Longrightarrow> lim f \<le> x"

   using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff)

 subsubsection{*Increasing and Decreasing Series*}

 lemma incseq_le: "incseq X \<Longrightarrow> X ----> L \<Longrightarrow> X n \<le> (L::'a::linorder_topology)"

   by (metis incseq_def LIMSEQ_le_const)

 lemma decseq_le: "decseq X \<Longrightarrow> X ----> L \<Longrightarrow> (L::'a::linorder_topology) \<le> X n"

   by (metis decseq_def LIMSEQ_le_const2)

 subsection {* First countable topologies *}

 class first_countable_topology = topological_space +

   assumes first_countable_basis:

     "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"

 lemma (in first_countable_topology) countable_basis_at_decseq:

   obtains A :: "nat \<Rightarrow> 'a set" where

     "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"

     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"

 proof atomize_elim

   from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set" where

     nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i"

     and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"  by auto

   def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. A i"

   show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>

       (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"

   proof (safe intro!: exI[of _ F])

     fix i

     show "open (F i)" using nhds(1) by (auto simp: F_def)

     show "x \<in> F i" using nhds(2) by (auto simp: F_def)

   next

     fix S assume "open S" "x \<in> S"

     from incl[OF this] obtain i where "F i \<subseteq> S" unfolding F_def by auto

     moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"

       by (auto simp: F_def)

     ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"

       by (auto simp: eventually_sequentially)

   qed

 qed

 lemma (in first_countable_topology) countable_basis:

   obtains A :: "nat \<Rightarrow> 'a set" where

     "\<And>i. open (A i)" "\<And>i. x \<in> A i"

     "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F ----> x"

 proof atomize_elim

   obtain A :: "nat \<Rightarrow> 'a set" where A:

     "\<And>i. open (A i)"

     "\<And>i. x \<in> A i"

     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"

     by (rule countable_basis_at_decseq) blast

   {

     fix F S assume "\<forall>n. F n \<in> A n" "open S" "x \<in> S"

     with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially"

       by (auto elim: eventually_elim1 simp: subset_eq)

   }

   with A show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> (\<forall>F. (\<forall>n. F n \<in> A n) \<longrightarrow> F ----> x)"

     by (intro exI[of _ A]) (auto simp: tendsto_def)

 qed

 lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within:

   assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"

   shows "eventually P (inf (nhds a) (principal s))"

 proof (rule ccontr)

   obtain A :: "nat \<Rightarrow> 'a set" where A:

     "\<And>i. open (A i)"

     "\<And>i. a \<in> A i"

     "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F ----> a"

     by (rule countable_basis) blast

   assume "\<not> ?thesis"

   with A have P: "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)"

     unfolding eventually_inf_principal eventually_nhds by (intro choice) fastforce

   then obtain F where F0: "\<forall>n. F n \<in> s" and F2: "\<forall>n. F n \<in> A n" and F3: "\<forall>n. \<not> P (F n)"

     by blast

   with A have "F ----> a" by auto

   hence "eventually (\<lambda>n. P (F n)) sequentially"

     using assms F0 by simp

   thus "False" by (simp add: F3)

 qed

 lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially:

   "eventually P (inf (nhds a) (principal s)) \<longleftrightarrow> 

     (\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"

 proof (safe intro!: sequentially_imp_eventually_nhds_within)

   assume "eventually P (inf (nhds a) (principal s))" 

   then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"

     by (auto simp: eventually_inf_principal eventually_nhds)

   moreover fix f assume "\<forall>n. f n \<in> s" "f ----> a"

   ultimately show "eventually (\<lambda>n. P (f n)) sequentially"

     by (auto dest!: topological_tendstoD elim: eventually_elim1)

 qed

 lemma (in first_countable_topology) eventually_nhds_iff_sequentially:

   "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"

   using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp

 subsection {* Function limit at a point *}

 abbreviation

   LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"

         ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where

   "f -- a --> L \<equiv> (f ---> L) (at a)"

 lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f ---> l) (at a within S) \<longleftrightarrow> (f -- a --> l)"

   unfolding tendsto_def by (simp add: at_within_open[where S=S])

 lemma LIM_const_not_eq[tendsto_intros]:

   fixes a :: "'a::perfect_space"

   fixes k L :: "'b::t2_space"

   shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"

   by (simp add: tendsto_const_iff)

 lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]

 lemma LIM_const_eq:

   fixes a :: "'a::perfect_space"

   fixes k L :: "'b::t2_space"

   shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"

   by (simp add: tendsto_const_iff)

 lemma LIM_unique:

   fixes a :: "'a::perfect_space" and L M :: "'b::t2_space"

   shows "f -- a --> L \<Longrightarrow> f -- a --> M \<Longrightarrow> L = M"

   using at_neq_bot by (rule tendsto_unique)

 text {* Limits are equal for functions equal except at limit point *}

 lemma LIM_equal: "\<forall>x. x \<noteq> a --> (f x = g x) \<Longrightarrow> (f -- a --> l) \<longleftrightarrow> (g -- a --> l)"

   unfolding tendsto_def eventually_at_topological by simp

 lemma LIM_cong: "a = b \<Longrightarrow> (\<And>x. x \<noteq> b \<Longrightarrow> f x = g x) \<Longrightarrow> l = m \<Longrightarrow> (f -- a --> l) \<longleftrightarrow> (g -- b --> m)"

   by (simp add: LIM_equal)

 lemma LIM_cong_limit: "f -- x --> L \<Longrightarrow> K = L \<Longrightarrow> f -- x --> K"

   by simp

 lemma tendsto_at_iff_tendsto_nhds:

   "g -- l --> g l \<longleftrightarrow> (g ---> g l) (nhds l)"

   unfolding tendsto_def eventually_at_filter

   by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)

 lemma tendsto_compose:

   "g -- l --> g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"

   unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])

 lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"

   unfolding o_def by (rule tendsto_compose)

 lemma tendsto_compose_eventually:

   "g -- l --> m \<Longrightarrow> (f ---> l) F \<Longrightarrow> eventually (\<lambda>x. f x \<noteq> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> m) F"

   by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)

 lemma LIM_compose_eventually:

   assumes f: "f -- a --> b"

   assumes g: "g -- b --> c"

   assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"

   shows "(\<lambda>x. g (f x)) -- a --> c"

   using g f inj by (rule tendsto_compose_eventually)

 subsubsection {* Relation of LIM and LIMSEQ *}

 lemma (in first_countable_topology) sequentially_imp_eventually_within:

   "(\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow>

     eventually P (at a within s)"

   unfolding at_within_def

   by (intro sequentially_imp_eventually_nhds_within) auto

 lemma (in first_countable_topology) sequentially_imp_eventually_at:

   "(\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> eventually P (at a)"

   using assms sequentially_imp_eventually_within [where s=UNIV] by simp

 lemma LIMSEQ_SEQ_conv1:

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

   assumes f: "f -- a --> l"

   shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"

   using tendsto_compose_eventually [OF f, where F=sequentially] by simp

 lemma LIMSEQ_SEQ_conv2:

   fixes f :: "'a::first_countable_topology \<Rightarrow> 'b::topological_space"

   assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"

   shows "f -- a --> l"

   using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at)

 lemma LIMSEQ_SEQ_conv:

   "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::'a::first_countable_topology) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =

    (X -- a --> (L::'b::topological_space))"

   using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..

 subsection {* Continuity *}

 subsubsection {* Continuity on a set *}

 definition continuous_on :: "'a set \<Rightarrow> ('a :: topological_space \<Rightarrow> 'b :: topological_space) \<Rightarrow> bool" where

   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"

 lemma continuous_on_cong [cong]:

   "s = t \<Longrightarrow> (\<And>x. x \<in> t \<Longrightarrow> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g"

   unfolding continuous_on_def by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter)

 lemma continuous_on_topological:

   "continuous_on s f \<longleftrightarrow>

     (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"

   unfolding continuous_on_def tendsto_def eventually_at_topological by metis

 lemma continuous_on_open_invariant:

   "continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))"

 proof safe

   fix B :: "'b set" assume "continuous_on s f" "open B"

   then have "\<forall>x\<in>f -` B \<inter> s. (\<exists>A. open A \<and> x \<in> A \<and> s \<inter> A \<subseteq> f -` B)"

     by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL)

   then obtain A where "\<forall>x\<in>f -` B \<inter> s. open (A x) \<and> x \<in> A x \<and> s \<inter> A x \<subseteq> f -` B"

     unfolding bchoice_iff ..

   then show "\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s"

     by (intro exI[of _ "\<Union>x\<in>f -` B \<inter> s. A x"]) auto

 next

   assume B: "\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s)"

   show "continuous_on s f"

     unfolding continuous_on_topological

   proof safe

     fix x B assume "x \<in> s" "open B" "f x \<in> B"

     with B obtain A where A: "open A" "A \<inter> s = f -` B \<inter> s" by auto

     with `x \<in> s` `f x \<in> B` show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"

       by (intro exI[of _ A]) auto

   qed

 qed

 lemma continuous_on_open_vimage:

   "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> open (f -` B \<inter> s))"

   unfolding continuous_on_open_invariant

   by (metis open_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])

 corollary continuous_imp_open_vimage:

   assumes "continuous_on s f" "open s" "open B" "f -` B \<subseteq> s"

     shows "open (f -` B)"

 by (metis assms continuous_on_open_vimage le_iff_inf)

 corollary open_vimage[continuous_intros]:

   assumes "open s" and "continuous_on UNIV f"

   shows "open (f -` s)"

   using assms unfolding continuous_on_open_vimage [OF open_UNIV]

   by simp

 lemma continuous_on_closed_invariant:

   "continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))"

 proof -

   have *: "\<And>P Q::'b set\<Rightarrow>bool. (\<And>A. P A \<longleftrightarrow> Q (- A)) \<Longrightarrow> (\<forall>A. P A) \<longleftrightarrow> (\<forall>A. Q A)"

     by (metis double_compl)

   show ?thesis

     unfolding continuous_on_open_invariant by (intro *) (auto simp: open_closed[symmetric])

 qed

 lemma continuous_on_closed_vimage:

   "closed s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> closed (f -` B \<inter> s))"

   unfolding continuous_on_closed_invariant

   by (metis closed_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])

 corollary closed_vimage[continuous_intros]:

   assumes "closed s" and "continuous_on UNIV f"

   shows "closed (f -` s)"

   using assms unfolding continuous_on_closed_vimage [OF closed_UNIV]

   by simp

 lemma continuous_on_open_Union:

   "(\<And>s. s \<in> S \<Longrightarrow> open s) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on s f) \<Longrightarrow> continuous_on (\<Union>S) f"

   unfolding continuous_on_def by safe (metis open_Union at_within_open UnionI)

 lemma continuous_on_open_UN:

   "(\<And>s. s \<in> S \<Longrightarrow> open (A s)) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on (A s) f) \<Longrightarrow> continuous_on (\<Union>s\<in>S. A s) f"

   unfolding Union_image_eq[symmetric] by (rule continuous_on_open_Union) auto

 lemma continuous_on_closed_Un:

   "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"

   by (auto simp add: continuous_on_closed_vimage closed_Un Int_Un_distrib)

 lemma continuous_on_If:

   assumes closed: "closed s" "closed t" and cont: "continuous_on s f" "continuous_on t g"

     and P: "\<And>x. x \<in> s \<Longrightarrow> \<not> P x \<Longrightarrow> f x = g x" "\<And>x. x \<in> t \<Longrightarrow> P x \<Longrightarrow> f x = g x"

   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)" (is "continuous_on _ ?h")

 proof-

   from P have "\<forall>x\<in>s. f x = ?h x" "\<forall>x\<in>t. g x = ?h x"

     by auto

   with cont have "continuous_on s ?h" "continuous_on t ?h"

     by simp_all

   with closed show ?thesis

     by (rule continuous_on_closed_Un)

 qed

 lemma continuous_on_id[continuous_intros]: "continuous_on s (\<lambda>x. x)"

   unfolding continuous_on_def by (fast intro: tendsto_ident_at)

 lemma continuous_on_const[continuous_intros]: "continuous_on s (\<lambda>x. c)"

   unfolding continuous_on_def by (auto intro: tendsto_const)

 lemma continuous_on_compose[continuous_intros]:

   "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"

   unfolding continuous_on_topological by simp metis

 lemma continuous_on_compose2:

   "continuous_on t g \<Longrightarrow> continuous_on s f \<Longrightarrow> t = f ` s \<Longrightarrow> continuous_on s (\<lambda>x. g (f x))"

   using continuous_on_compose[of s f g] by (simp add: comp_def)

 subsubsection {* Continuity at a point *}

 definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where

   "continuous F f \<longleftrightarrow> (f ---> f (Lim F (\<lambda>x. x))) F"

 lemma continuous_bot[continuous_intros, simp]: "continuous bot f"

   unfolding continuous_def by auto

 lemma continuous_trivial_limit: "trivial_limit net \<Longrightarrow> continuous net f"

   by simp

 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f x) (at x within s)"

   by (cases "trivial_limit (at x within s)") (auto simp add: Lim_ident_at continuous_def)

 lemma continuous_within_topological:

   "continuous (at x within s) f \<longleftrightarrow>

     (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"

   unfolding continuous_within tendsto_def eventually_at_topological by metis

 lemma continuous_within_compose[continuous_intros]:

   "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>

   continuous (at x within s) (g o f)"

   by (simp add: continuous_within_topological) metis

 lemma continuous_within_compose2:

   "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>

   continuous (at x within s) (\<lambda>x. g (f x))"

   using continuous_within_compose[of x s f g] by (simp add: comp_def)

 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> f -- x --> f x"

   using continuous_within[of x UNIV f] by simp

 lemma continuous_ident[continuous_intros, simp]: "continuous (at x within S) (\<lambda>x. x)"

   unfolding continuous_within by (rule tendsto_ident_at)

 lemma continuous_const[continuous_intros, simp]: "continuous F (\<lambda>x. c)"

   unfolding continuous_def by (rule tendsto_const)

 lemma continuous_on_eq_continuous_within:

   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. continuous (at x within s) f)"

   unfolding continuous_on_def continuous_within ..

 abbreviation isCont :: "('a::t2_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> bool" where

   "isCont f a \<equiv> continuous (at a) f"

 lemma isCont_def: "isCont f a \<longleftrightarrow> f -- a --> f a"

   by (rule continuous_at)

 lemma continuous_at_within: "isCont f x \<Longrightarrow> continuous (at x within s) f"

   by (auto intro: tendsto_mono at_le simp: continuous_at continuous_within)

 lemma continuous_on_eq_continuous_at: "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. isCont f x)"

   by (simp add: continuous_on_def continuous_at at_within_open[of _ s])

 lemma continuous_on_subset: "continuous_on s f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous_on t f"

   unfolding continuous_on_def by (metis subset_eq tendsto_within_subset)

 lemma continuous_at_imp_continuous_on: "\<forall>x\<in>s. isCont f x \<Longrightarrow> continuous_on s f"

   by (auto intro: continuous_at_within simp: continuous_on_eq_continuous_within)

 lemma isContI_continuous: "continuous (at x within UNIV) f \<Longrightarrow> isCont f x"

   by simp

 lemma isCont_ident[continuous_intros, simp]: "isCont (\<lambda>x. x) a"

   using continuous_ident by (rule isContI_continuous)

 lemmas isCont_const = continuous_const

 lemma isCont_o2: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"

   unfolding isCont_def by (rule tendsto_compose)

 lemma isCont_o[continuous_intros]: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (g \<circ> f) a"

   unfolding o_def by (rule isCont_o2)

 lemma isCont_tendsto_compose: "isCont g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"

   unfolding isCont_def by (rule tendsto_compose)

 lemma continuous_within_compose3:

   "isCont g (f x) \<Longrightarrow> continuous (at x within s) f \<Longrightarrow> continuous (at x within s) (\<lambda>x. g (f x))"

   using continuous_within_compose2[of x s f g] by (simp add: continuous_at_within)

 subsubsection{* Open-cover compactness *}

 context topological_space

 begin

 definition compact :: "'a set \<Rightarrow> bool" where

   compact_eq_heine_borel: -- "This name is used for backwards compatibility"

     "compact S \<longleftrightarrow> (\<forall>C. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union>C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"

 lemma compactI:

   assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union> C \<Longrightarrow> \<exists>C'. C' \<subseteq> C \<and> finite C' \<and> s \<subseteq> \<Union> C'"

   shows "compact s"

   unfolding compact_eq_heine_borel using assms by metis

 lemma compact_empty[simp]: "compact {}"

   by (auto intro!: compactI)

 lemma compactE:

   assumes "compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C"

   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"

   using assms unfolding compact_eq_heine_borel by metis

 lemma compactE_image:

   assumes "compact s" and "\<forall>t\<in>C. open (f t)" and "s \<subseteq> (\<Union>c\<in>C. f c)"

   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> (\<Union>c\<in>C'. f c)"

   using assms unfolding ball_simps[symmetric] SUP_def

   by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s])

 lemma compact_inter_closed [intro]:

   assumes "compact s" and "closed t"

   shows "compact (s \<inter> t)"

 proof (rule compactI)

   fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C"

   from C `closed t` have "\<forall>c\<in>C \<union> {-t}. open c" by auto

   moreover from cover have "s \<subseteq> \<Union>(C \<union> {-t})" by auto

   ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D"

     using `compact s` unfolding compact_eq_heine_borel by auto

   then obtain D where "D \<subseteq> C \<union> {- t} \<and> finite D \<and> s \<subseteq> \<Union>D" ..

   then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D"

     by (intro exI[of _ "D - {-t}"]) auto

 qed

 lemma inj_setminus: "inj_on uminus (A::'a set set)"

   by (auto simp: inj_on_def)

 lemma compact_fip:

   "compact U \<longleftrightarrow>

     (\<forall>A. (\<forall>a\<in>A. closed a) \<longrightarrow> (\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}) \<longrightarrow> U \<inter> \<Inter>A \<noteq> {})"

   (is "_ \<longleftrightarrow> ?R")

 proof (safe intro!: compact_eq_heine_borel[THEN iffD2])

   fix A

   assume "compact U"

     and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"

     and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"

   from A have "(\<forall>a\<in>uminus`A. open a) \<and> U \<subseteq> \<Union>(uminus`A)"

     by auto

   with `compact U` obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<subseteq> \<Union>(uminus`B)"

     unfolding compact_eq_heine_borel by (metis subset_image_iff)

   with fi[THEN spec, of B] show False

     by (auto dest: finite_imageD intro: inj_setminus)

 next

   fix A

   assume ?R

   assume "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

   then have "U \<inter> \<Inter>(uminus`A) = {}" "\<forall>a\<in>uminus`A. closed a"

     by auto

   with `?R` obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<inter> \<Inter>(uminus`B) = {}"

     by (metis subset_image_iff)

   then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

     by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)

 qed

 lemma compact_imp_fip:

   "compact s \<Longrightarrow> \<forall>t \<in> f. closed t \<Longrightarrow> \<forall>f'. finite f' \<and> f' \<subseteq> f \<longrightarrow> (s \<inter> (\<Inter> f') \<noteq> {}) \<Longrightarrow>

     s \<inter> (\<Inter> f) \<noteq> {}"

   unfolding compact_fip by auto

 lemma compact_imp_fip_image:

   assumes "compact s"

     and P: "\<And>i. i \<in> I \<Longrightarrow> closed (f i)"

     and Q: "\<And>I'. finite I' \<Longrightarrow> I' \<subseteq> I \<Longrightarrow> (s \<inter> (\<Inter>i\<in>I'. f i) \<noteq> {})"

   shows "s \<inter> (\<Inter>i\<in>I. f i) \<noteq> {}"

 proof -

   note `compact s`

   moreover from P have "\<forall>i \<in> f ` I. closed i" by blast

   moreover have "\<forall>A. finite A \<and> A \<subseteq> f ` I \<longrightarrow> (s \<inter> (\<Inter>A) \<noteq> {})"

   proof (rule, rule, erule conjE)

     fix A :: "'a set set"

     assume "finite A"

     moreover assume "A \<subseteq> f ` I"

     ultimately obtain B where "B \<subseteq> I" and "finite B" and "A = f ` B"

       using finite_subset_image [of A f I] by blast

     with Q [of B] show "s \<inter> \<Inter>A \<noteq> {}" by simp

   qed

   ultimately have "s \<inter> (\<Inter>(f ` I)) \<noteq> {}" by (rule compact_imp_fip)

   then show ?thesis by simp

 qed

 end

 lemma (in t2_space) compact_imp_closed:

   assumes "compact s" shows "closed s"

 unfolding closed_def

 proof (rule openI)

   fix y assume "y \<in> - s"

   let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}"

   note `compact s`

   moreover have "\<forall>u\<in>?C. open u" by simp

   moreover have "s \<subseteq> \<Union>?C"

   proof

     fix x assume "x \<in> s"

     with `y \<in> - s` have "x \<noteq> y" by clarsimp

     hence "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"

       by (rule hausdorff)

     with `x \<in> s` show "x \<in> \<Union>?C"

       unfolding eventually_nhds by auto

   qed

   ultimately obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D"

     by (rule compactE)

   from `D \<subseteq> ?C` have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto

   with `finite D` have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"

     by (simp add: eventually_Ball_finite)

   with `s \<subseteq> \<Union>D` have "eventually (\<lambda>y. y \<notin> s) (nhds y)"

     by (auto elim!: eventually_mono [rotated])

   thus "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"

     by (simp add: eventually_nhds subset_eq)

 qed

 lemma compact_continuous_image:

   assumes f: "continuous_on s f" and s: "compact s"

   shows "compact (f ` s)"

 proof (rule compactI)

   fix C assume "\<forall>c\<in>C. open c" and cover: "f`s \<subseteq> \<Union>C"

   with f have "\<forall>c\<in>C. \<exists>A. open A \<and> A \<inter> s = f -` c \<inter> s"

     unfolding continuous_on_open_invariant by blast

   then obtain A where A: "\<forall>c\<in>C. open (A c) \<and> A c \<inter> s = f -` c \<inter> s"

     unfolding bchoice_iff ..

   with cover have "\<forall>c\<in>C. open (A c)" "s \<subseteq> (\<Union>c\<in>C. A c)"

     by (fastforce simp add: subset_eq set_eq_iff)+

   from compactE_image[OF s this] obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> (\<Union>c\<in>D. A c)" .

   with A show "\<exists>D \<subseteq> C. finite D \<and> f`s \<subseteq> \<Union>D"

     by (intro exI[of _ D]) (fastforce simp add: subset_eq set_eq_iff)+

 qed

 lemma continuous_on_inv:

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

   assumes "continuous_on s f"  "compact s"  "\<forall>x\<in>s. g (f x) = x"

   shows "continuous_on (f ` s) g"

 unfolding continuous_on_topological

 proof (clarsimp simp add: assms(3))

   fix x :: 'a and B :: "'a set"

   assume "x \<in> s" and "open B" and "x \<in> B"

   have 1: "\<forall>x\<in>s. f x \<in> f ` (s - B) \<longleftrightarrow> x \<in> s - B"

     using assms(3) by (auto, metis)

   have "continuous_on (s - B) f"

     using `continuous_on s f` Diff_subset

     by (rule continuous_on_subset)

   moreover have "compact (s - B)"

     using `open B` and `compact s`

     unfolding Diff_eq by (intro compact_inter_closed closed_Compl)

   ultimately have "compact (f ` (s - B))"

     by (rule compact_continuous_image)

   hence "closed (f ` (s - B))"

     by (rule compact_imp_closed)

   hence "open (- f ` (s - B))"

     by (rule open_Compl)

   moreover have "f x \<in> - f ` (s - B)"

     using `x \<in> s` and `x \<in> B` by (simp add: 1)

   moreover have "\<forall>y\<in>s. f y \<in> - f ` (s - B) \<longrightarrow> y \<in> B"

     by (simp add: 1)

   ultimately show "\<exists>A. open A \<and> f x \<in> A \<and> (\<forall>y\<in>s. f y \<in> A \<longrightarrow> y \<in> B)"

     by fast

 qed

 lemma continuous_on_inv_into:

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

   assumes s: "continuous_on s f" "compact s" and f: "inj_on f s"

   shows "continuous_on (f ` s) (the_inv_into s f)"

   by (rule continuous_on_inv[OF s]) (auto simp: the_inv_into_f_f[OF f])

 lemma (in linorder_topology) compact_attains_sup:

   assumes "compact S" "S \<noteq> {}"

   shows "\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s"

 proof (rule classical)

   assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s)"

   then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. s < t s"

     by (metis not_le)

   then have "\<forall>s\<in>S. open {..< t s}" "S \<subseteq> (\<Union>s\<in>S. {..< t s})"

     by auto

   with `compact S` obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {..< t s})"

     by (erule compactE_image)

   with `S \<noteq> {}` have Max: "Max (t`C) \<in> t`C" and "\<forall>s\<in>t`C. s \<le> Max (t`C)"

     by (auto intro!: Max_in)

   with C have "S \<subseteq> {..< Max (t`C)}"

     by (auto intro: less_le_trans simp: subset_eq)

   with t Max `C \<subseteq> S` show ?thesis

     by fastforce

 qed

 lemma (in linorder_topology) compact_attains_inf:

   assumes "compact S" "S \<noteq> {}"

   shows "\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t"

 proof (rule classical)

   assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t)"

   then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. t s < s"

     by (metis not_le)

   then have "\<forall>s\<in>S. open {t s <..}" "S \<subseteq> (\<Union>s\<in>S. {t s <..})"

     by auto

   with `compact S` obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {t s <..})"

     by (erule compactE_image)

   with `S \<noteq> {}` have Min: "Min (t`C) \<in> t`C" and "\<forall>s\<in>t`C. Min (t`C) \<le> s"

     by (auto intro!: Min_in)

   with C have "S \<subseteq> {Min (t`C) <..}"

     by (auto intro: le_less_trans simp: subset_eq)

   with t Min `C \<subseteq> S` show ?thesis

     by fastforce

 qed

 lemma continuous_attains_sup:

   fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"

   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f \<Longrightarrow> (\<exists>x\<in>s. \<forall>y\<in>s.  f y \<le> f x)"

   using compact_attains_sup[of "f ` s"] compact_continuous_image[of s f] by auto

 lemma continuous_attains_inf:

   fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"

   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f \<Longrightarrow> (\<exists>x\<in>s. \<forall>y\<in>s. f x \<le> f y)"

   using compact_attains_inf[of "f ` s"] compact_continuous_image[of s f] by auto

 subsection {* Connectedness *}

 context topological_space

 begin

 definition "connected S \<longleftrightarrow>

   \<not> (\<exists>A B. open A \<and> open B \<and> S \<subseteq> A \<union> B \<and> A \<inter> B \<inter> S = {} \<and> A \<inter> S \<noteq> {} \<and> B \<inter> S \<noteq> {})"

 lemma connectedI:

   "(\<And>A B. open A \<Longrightarrow> open B \<Longrightarrow> A \<inter> U \<noteq> {} \<Longrightarrow> B \<inter> U \<noteq> {} \<Longrightarrow> A \<inter> B \<inter> U = {} \<Longrightarrow> U \<subseteq> A \<union> B \<Longrightarrow> False)

   \<Longrightarrow> connected U"

   by (auto simp: connected_def)

 lemma connected_empty[simp]: "connected {}"

   by (auto intro!: connectedI)

 lemma connectedD:

   "connected A \<Longrightarrow> open U \<Longrightarrow> open V \<Longrightarrow> U \<inter> V \<inter> A = {} \<Longrightarrow> A \<subseteq> U \<union> V \<Longrightarrow> U \<inter> A = {} \<or> V \<inter> A = {}" 

   by (auto simp: connected_def)

 end

 lemma connected_local_const:

   assumes "connected A" "a \<in> A" "b \<in> A"

   assumes *: "\<forall>a\<in>A. eventually (\<lambda>b. f a = f b) (at a within A)"

   shows "f a = f b"

 proof -

   obtain S where S: "\<And>a. a \<in> A \<Longrightarrow> a \<in> S a" "\<And>a. a \<in> A \<Longrightarrow> open (S a)"

     "\<And>a x. a \<in> A \<Longrightarrow> x \<in> S a \<Longrightarrow> x \<in> A \<Longrightarrow> f a = f x"

     using * unfolding eventually_at_topological by metis

   let ?P = "\<Union>b\<in>{b\<in>A. f a = f b}. S b" and ?N = "\<Union>b\<in>{b\<in>A. f a \<noteq> f b}. S b"

   have "?P \<inter> A = {} \<or> ?N \<inter> A = {}"

     using `connected A` S `a\<in>A`

     by (intro connectedD) (auto, metis)

   then show "f a = f b"

   proof

     assume "?N \<inter> A = {}"

     then have "\<forall>x\<in>A. f a = f x"

       using S(1) by auto

     with `b\<in>A` show ?thesis by auto

   next

     assume "?P \<inter> A = {}" then show ?thesis

       using `a \<in> A` S(1)[of a] by auto

   qed

 qed

 lemma (in linorder_topology) connectedD_interval:

   assumes "connected U" and xy: "x \<in> U" "y \<in> U" and "x \<le> z" "z \<le> y"

   shows "z \<in> U"

 proof -

   have eq: "{..<z} \<union> {z<..} = - {z}"

     by auto

   { assume "z \<notin> U" "x < z" "z < y"

     with xy have "\<not> connected U"

       unfolding connected_def simp_thms

       apply (rule_tac exI[of _ "{..< z}"])

       apply (rule_tac exI[of _ "{z <..}"])

       apply (auto simp add: eq)

       done }

   with assms show "z \<in> U"

     by (metis less_le)

 qed

 lemma connected_continuous_image:

   assumes *: "continuous_on s f"

   assumes "connected s"

   shows "connected (f ` s)"

 proof (rule connectedI)

   fix A B assume A: "open A" "A \<inter> f ` s \<noteq> {}" and B: "open B" "B \<inter> f ` s \<noteq> {}" and

     AB: "A \<inter> B \<inter> f ` s = {}" "f ` s \<subseteq> A \<union> B"

   obtain A' where A': "open A'" "f -` A \<inter> s = A' \<inter> s"

     using * `open A` unfolding continuous_on_open_invariant by metis

   obtain B' where B': "open B'" "f -` B \<inter> s = B' \<inter> s"

     using * `open B` unfolding continuous_on_open_invariant by metis

   have "\<exists>A B. open A \<and> open B \<and> s \<subseteq> A \<union> B \<and> A \<inter> B \<inter> s = {} \<and> A \<inter> s \<noteq> {} \<and> B \<inter> s \<noteq> {}"

   proof (rule exI[of _ A'], rule exI[of _ B'], intro conjI)

     have "s \<subseteq> (f -` A \<inter> s) \<union> (f -` B \<inter> s)" using AB by auto

     then show "s \<subseteq> A' \<union> B'" using A' B' by auto

   next

     have "(f -` A \<inter> s) \<inter> (f -` B \<inter> s) = {}" using AB by auto

     then show "A' \<inter> B' \<inter> s = {}" using A' B' by auto

   qed (insert A' B' A B, auto)

   with `connected s` show False

     unfolding connected_def by blast

 qed

 section {* Connectedness *}

 class linear_continuum_topology = linorder_topology + linear_continuum

 begin

 lemma Inf_notin_open:

   assumes A: "open A" and bnd: "\<forall>a\<in>A. x < a"

   shows "Inf A \<notin> A"

 proof

   assume "Inf A \<in> A"

   then obtain b where "b < Inf A" "{b <.. Inf A} \<subseteq> A"

     using open_left[of A "Inf A" x] assms by auto

   with dense[of b "Inf A"] obtain c where "c < Inf A" "c \<in> A"

     by (auto simp: subset_eq)

   then show False

     using cInf_lower[OF `c \<in> A`] bnd by (metis not_le less_imp_le bdd_belowI)

 qed

 lemma Sup_notin_open:

   assumes A: "open A" and bnd: "\<forall>a\<in>A. a < x"

   shows "Sup A \<notin> A"

 proof

   assume "Sup A \<in> A"

   then obtain b where "Sup A < b" "{Sup A ..< b} \<subseteq> A"

     using open_right[of A "Sup A" x] assms by auto

   with dense[of "Sup A" b] obtain c where "Sup A < c" "c \<in> A"

     by (auto simp: subset_eq)

   then show False

     using cSup_upper[OF `c \<in> A`] bnd by (metis less_imp_le not_le bdd_aboveI)

 qed

 end

 instance linear_continuum_topology \<subseteq> perfect_space

 proof

   fix x :: 'a

   obtain y where "x < y \<or> y < x"

     using ex_gt_or_lt [of x] ..

   with Inf_notin_open[of "{x}" y] Sup_notin_open[of "{x}" y]

   show "\<not> open {x}"

     by auto

 qed

 lemma connectedI_interval:

   fixes U :: "'a :: linear_continuum_topology set"

   assumes *: "\<And>x y z. x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x \<le> z \<Longrightarrow> z \<le> y \<Longrightarrow> z \<in> U"

   shows "connected U"

 proof (rule connectedI)

   { fix A B assume "open A" "open B" "A \<inter> B \<inter> U = {}" "U \<subseteq> A \<union> B"

     fix x y assume "x < y" "x \<in> A" "y \<in> B" "x \<in> U" "y \<in> U"

     let ?z = "Inf (B \<inter> {x <..})"

     have "x \<le> ?z" "?z \<le> y"

       using `y \<in> B` `x < y` by (auto intro: cInf_lower cInf_greatest)

     with `x \<in> U` `y \<in> U` have "?z \<in> U"

       by (rule *)

     moreover have "?z \<notin> B \<inter> {x <..}"

       using `open B` by (intro Inf_notin_open) auto

     ultimately have "?z \<in> A"

       using `x \<le> ?z` `A \<inter> B \<inter> U = {}` `x \<in> A` `U \<subseteq> A \<union> B` by auto

     { assume "?z < y"

       obtain a where "?z < a" "{?z ..< a} \<subseteq> A"

         using open_right[OF `open A` `?z \<in> A` `?z < y`] by auto

       moreover obtain b where "b \<in> B" "x < b" "b < min a y"

         using cInf_less_iff[of "B \<inter> {x <..}" "min a y"] `?z < a` `?z < y` `x < y` `y \<in> B`

         by (auto intro: less_imp_le)

       moreover have "?z \<le> b"

         using `b \<in> B` `x < b`

         by (intro cInf_lower) auto

       moreover have "b \<in> U"

         using `x \<le> ?z` `?z \<le> b` `b < min a y`

         by (intro *[OF `x \<in> U` `y \<in> U`]) (auto simp: less_imp_le)

       ultimately have "\<exists>b\<in>B. b \<in> A \<and> b \<in> U"

         by (intro bexI[of _ b]) auto }

     then have False

       using `?z \<le> y` `?z \<in> A` `y \<in> B` `y \<in> U` `A \<inter> B \<inter> U = {}` unfolding le_less by blast }

   note not_disjoint = this

   fix A B assume AB: "open A" "open B" "U \<subseteq> A \<union> B" "A \<inter> B \<inter> U = {}"

   moreover assume "A \<inter> U \<noteq> {}" then obtain x where x: "x \<in> U" "x \<in> A" by auto

   moreover assume "B \<inter> U \<noteq> {}" then obtain y where y: "y \<in> U" "y \<in> B" by auto

   moreover note not_disjoint[of B A y x] not_disjoint[of A B x y]

   ultimately show False by (cases x y rule: linorder_cases) auto

 qed

 lemma connected_iff_interval:

   fixes U :: "'a :: linear_continuum_topology set"

   shows "connected U \<longleftrightarrow> (\<forall>x\<in>U. \<forall>y\<in>U. \<forall>z. x \<le> z \<longrightarrow> z \<le> y \<longrightarrow> z \<in> U)"

   by (auto intro: connectedI_interval dest: connectedD_interval)

 lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)"

   unfolding connected_iff_interval by auto

 lemma connected_Ioi[simp]: "connected {a::'a::linear_continuum_topology <..}"

   unfolding connected_iff_interval by auto

 lemma connected_Ici[simp]: "connected {a::'a::linear_continuum_topology ..}"

   unfolding connected_iff_interval by auto

 lemma connected_Iio[simp]: "connected {..< a::'a::linear_continuum_topology}"

   unfolding connected_iff_interval by auto

 lemma connected_Iic[simp]: "connected {.. a::'a::linear_continuum_topology}"

   unfolding connected_iff_interval by auto

 lemma connected_Ioo[simp]: "connected {a <..< b::'a::linear_continuum_topology}"

   unfolding connected_iff_interval by auto

 lemma connected_Ioc[simp]: "connected {a <.. b::'a::linear_continuum_topology}"

   unfolding connected_iff_interval by auto

 lemma connected_Ico[simp]: "connected {a ..< b::'a::linear_continuum_topology}"

   unfolding connected_iff_interval by auto

 lemma connected_Icc[simp]: "connected {a .. b::'a::linear_continuum_topology}"

   unfolding connected_iff_interval by auto

 lemma connected_contains_Ioo: 

   fixes A :: "'a :: linorder_topology set"

   assumes A: "connected A" "a \<in> A" "b \<in> A" shows "{a <..< b} \<subseteq> A"

   using connectedD_interval[OF A] by (simp add: subset_eq Ball_def less_imp_le)

 subsection {* Intermediate Value Theorem *}

 lemma IVT':

   fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"

   assumes y: "f a \<le> y" "y \<le> f b" "a \<le> b"

   assumes *: "continuous_on {a .. b} f"

   shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"

 proof -

   have "connected {a..b}"

     unfolding connected_iff_interval by auto

   from connected_continuous_image[OF * this, THEN connectedD_interval, of "f a" "f b" y] y

   show ?thesis

     by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)

 qed

 lemma IVT2':

   fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"

   assumes y: "f b \<le> y" "y \<le> f a" "a \<le> b"

   assumes *: "continuous_on {a .. b} f"

   shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"

 proof -

   have "connected {a..b}"

     unfolding connected_iff_interval by auto

   from connected_continuous_image[OF * this, THEN connectedD_interval, of "f b" "f a" y] y

   show ?thesis

     by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)

 qed

 lemma IVT:

   fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"

   shows "f a \<le> y \<Longrightarrow> y \<le> f b \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow> \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"

   by (rule IVT') (auto intro: continuous_at_imp_continuous_on)

 lemma IVT2:

   fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"

   shows "f b \<le> y \<Longrightarrow> y \<le> f a \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow> \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"

   by (rule IVT2') (auto intro: continuous_at_imp_continuous_on)

 lemma continuous_inj_imp_mono:

   fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"

   assumes x: "a < x" "x < b"

   assumes cont: "continuous_on {a..b} f"

   assumes inj: "inj_on f {a..b}"

   shows "(f a < f x \<and> f x < f b) \<or> (f b < f x \<and> f x < f a)"

 proof -

   note I = inj_on_iff[OF inj]

   { assume "f x < f a" "f x < f b"

     then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f x < f s"

       using IVT'[of f x "min (f a) (f b)" b] IVT2'[of f x "min (f a) (f b)" a] x

       by (auto simp: continuous_on_subset[OF cont] less_imp_le)

     with x I have False by auto }

   moreover

   { assume "f a < f x" "f b < f x"

     then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f s < f x"

       using IVT'[of f a "max (f a) (f b)" x] IVT2'[of f b "max (f a) (f b)" x] x

       by (auto simp: continuous_on_subset[OF cont] less_imp_le)

     with x I have False by auto }

   ultimately show ?thesis

     using I[of a x] I[of x b] x less_trans[OF x] by (auto simp add: le_less less_imp_neq neq_iff)

 qed

 subsection {* Setup @{typ "'a filter"} for lifting and transfer *}

 context begin interpretation lifting_syntax .

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

 where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)"

 lemma rel_filter_eventually:

   "rel_filter R F G \<longleftrightarrow> 

   ((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)"

 by(simp add: rel_filter_def eventually_def)

 lemma filtermap_id [simp, id_simps]: "filtermap id = id"

 by(simp add: fun_eq_iff id_def filtermap_ident)

 lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)"

 using filtermap_id unfolding id_def .

 lemma Quotient_filter [quot_map]:

   assumes Q: "Quotient R Abs Rep T"

   shows "Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)"

 unfolding Quotient_alt_def

 proof(intro conjI strip)

   from Q have *: "\<And>x y. T x y \<Longrightarrow> Abs x = y"

     unfolding Quotient_alt_def by blast

   fix F G

   assume "rel_filter T F G"

   thus "filtermap Abs F = G" unfolding filter_eq_iff

     by(auto simp add: eventually_filtermap rel_filter_eventually * rel_funI del: iffI elim!: rel_funD)

 next

   from Q have *: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by blast

   fix F

   show "rel_filter T (filtermap Rep F) F" 

     by(auto elim: rel_funD intro: * intro!: ext arg_cong[where f="\<lambda>P. eventually P F"] rel_funI

             del: iffI simp add: eventually_filtermap rel_filter_eventually)

 qed(auto simp add: map_fun_def o_def eventually_filtermap filter_eq_iff fun_eq_iff rel_filter_eventually

          fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def])

 lemma eventually_parametric [transfer_rule]:

   "((A ===> op =) ===> rel_filter A ===> op =) eventually eventually"

 by(simp add: rel_fun_def rel_filter_eventually)

 lemma rel_filter_eq [relator_eq]: "rel_filter op = = op ="

 by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff)

 lemma rel_filter_mono [relator_mono]:

   "A \<le> B \<Longrightarrow> rel_filter A \<le> rel_filter B"

 unfolding rel_filter_eventually[abs_def]

 by(rule le_funI)+(intro fun_mono fun_mono[THEN le_funD, THEN le_funD] order.refl)

 lemma rel_filter_conversep [simp]: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>"

 by(auto simp add: rel_filter_eventually fun_eq_iff rel_fun_def)

 lemma is_filter_parametric_aux:

   assumes "is_filter F"

   assumes [transfer_rule]: "bi_total A" "bi_unique A"

   and [transfer_rule]: "((A ===> op =) ===> op =) F G"

   shows "is_filter G"

 proof -

   interpret is_filter F by fact

   show ?thesis

   proof

     have "F (\<lambda>_. True) = G (\<lambda>x. True)" by transfer_prover

     thus "G (\<lambda>x. True)" by(simp add: True)

   next

     fix P' Q'

     assume "G P'" "G Q'"

     moreover

     from bi_total_fun[OF `bi_unique A` bi_total_eq, unfolded bi_total_def]

     obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast

     have "F P = G P'" "F Q = G Q'" by transfer_prover+

     ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj)

     moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover

     ultimately show "G (\<lambda>x. P' x \<and> Q' x)" by simp

   next

     fix P' Q'

     assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'"

     moreover

     from bi_total_fun[OF `bi_unique A` bi_total_eq, unfolded bi_total_def]

     obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast

     have "F P = G P'" by transfer_prover

     moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover

     ultimately have "F Q" by(simp add: mono)

     moreover have "F Q = G Q'" by transfer_prover

     ultimately show "G Q'" by simp

   qed

 qed

 lemma is_filter_parametric [transfer_rule]:

   "\<lbrakk> bi_total A; bi_unique A \<rbrakk>

   \<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter"

 apply(rule rel_funI)

 apply(rule iffI)

  apply(erule (3) is_filter_parametric_aux)

 apply(erule is_filter_parametric_aux[where A="conversep A"])

 apply(auto simp add: rel_fun_def)

 done

 lemma left_total_rel_filter [transfer_rule]:

   assumes [transfer_rule]: "bi_total A" "bi_unique A"

   shows "left_total (rel_filter A)"

 proof(rule left_totalI)

   fix F :: "'a filter"

   from bi_total_fun[OF bi_unique_fun[OF `bi_total A` bi_unique_eq] bi_total_eq]

   obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G" 

     unfolding  bi_total_def by blast

   moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover

   hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter)

   ultimately have "rel_filter A F (Abs_filter G)"

     by(simp add: rel_filter_eventually eventually_Abs_filter)

   thus "\<exists>G. rel_filter A F G" ..

 qed

 lemma right_total_rel_filter [transfer_rule]:

   "\<lbrakk> bi_total A; bi_unique A \<rbrakk> \<Longrightarrow> right_total (rel_filter A)"

 using left_total_rel_filter[of "A\<inverse>\<inverse>"] by simp

 lemma bi_total_rel_filter [transfer_rule]:

   assumes "bi_total A" "bi_unique A"

   shows "bi_total (rel_filter A)"

 unfolding bi_total_alt_def using assms

 by(simp add: left_total_rel_filter right_total_rel_filter)

 lemma left_unique_rel_filter [transfer_rule]:

   assumes "left_unique A"

   shows "left_unique (rel_filter A)"

 proof(rule left_uniqueI)

   fix F F' G

   assume [transfer_rule]: "rel_filter A F G" "rel_filter A F' G"

   show "F = F'"

     unfolding filter_eq_iff

   proof

     fix P :: "'a \<Rightarrow> bool"

     obtain P' where [transfer_rule]: "(A ===> op =) P P'"

       using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast

     have "eventually P F = eventually P' G" 

       and "eventually P F' = eventually P' G" by transfer_prover+

     thus "eventually P F = eventually P F'" by simp

   qed

 qed

 lemma right_unique_rel_filter [transfer_rule]:

   "right_unique A \<Longrightarrow> right_unique (rel_filter A)"

 using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by simp

 lemma bi_unique_rel_filter [transfer_rule]:

   "bi_unique A \<Longrightarrow> bi_unique (rel_filter A)"

 by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter)

 lemma top_filter_parametric [transfer_rule]:

   "bi_total A \<Longrightarrow> (rel_filter A) top top"

 by(simp add: rel_filter_eventually All_transfer)

 lemma bot_filter_parametric [transfer_rule]: "(rel_filter A) bot bot"

 by(simp add: rel_filter_eventually rel_fun_def)

 lemma sup_filter_parametric [transfer_rule]:

   "(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup"

 by(fastforce simp add: rel_filter_eventually[abs_def] eventually_sup dest: rel_funD)

 lemma Sup_filter_parametric [transfer_rule]:

   "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup"

 proof(rule rel_funI)

   fix S T

   assume [transfer_rule]: "rel_set (rel_filter A) S T"

   show "rel_filter A (Sup S) (Sup T)"

     by(simp add: rel_filter_eventually eventually_Sup) transfer_prover

 qed

 lemma principal_parametric [transfer_rule]:

   "(rel_set A ===> rel_filter A) principal principal"

 proof(rule rel_funI)

   fix S S'

   assume [transfer_rule]: "rel_set A S S'"

   show "rel_filter A (principal S) (principal S')"

     by(simp add: rel_filter_eventually eventually_principal) transfer_prover

 qed

 context

   fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"

   assumes [transfer_rule]: "bi_unique A" 

 begin

 lemma le_filter_parametric [transfer_rule]:

   "(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>"

 unfolding le_filter_def[abs_def] by transfer_prover

 lemma less_filter_parametric [transfer_rule]:

   "(rel_filter A ===> rel_filter A ===> op =) op < op <"

 unfolding less_filter_def[abs_def] by transfer_prover

 context

   assumes [transfer_rule]: "bi_total A"

 begin

 lemma Inf_filter_parametric [transfer_rule]:

   "(rel_set (rel_filter A) ===> rel_filter A) Inf Inf"

 unfolding Inf_filter_def[abs_def] by transfer_prover

 lemma inf_filter_parametric [transfer_rule]:

   "(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf"

 proof(intro rel_funI)+

   fix F F' G G'

   assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'"

   have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover

   thus "rel_filter A (inf F G) (inf F' G')" by simp

 qed

 end

 end

 end

 end