src/HOL/Topological_Spaces.thy
 author wenzelm Tue, 06 Oct 2015 17:46:07 +0200 changeset 61342 b98cd131e2b5 parent 61306 9dd394c866fc child 61426 d53db136e8fd permissions -rw-r--r--
isabelle update_cartouches;
```
(*  Title:      HOL/Topological_Spaces.thy
Author:     Brian Huffman
Author:     Johannes Hölzl
*)

section \<open>Topological Spaces\<close>

theory Topological_Spaces
imports Main Conditionally_Complete_Lattices
begin

named_theorems continuous_intros "structural introduction rules for continuity"

subsection \<open>Topological space\<close>

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 .

lemma open_Collect_neg: "closed {x. P x} \<Longrightarrow> open {x. \<not> P x}"
unfolding Collect_neg_eq by (rule open_Compl)

lemma open_Collect_conj: assumes "open {x. P x}" "open {x. Q x}" shows "open {x. P x \<and> Q x}"
using open_Int[OF assms] by (simp add: Int_def)

lemma open_Collect_disj: assumes "open {x. P x}" "open {x. Q x}" shows "open {x. P x \<or> Q x}"
using open_Un[OF assms] by (simp add: Un_def)

lemma open_Collect_ex: "(\<And>i. open {x. P i x}) \<Longrightarrow> open {x. \<exists>i. P i x}"
using open_UN[of UNIV "\<lambda>i. {x. P i x}"] unfolding Collect_ex_eq by simp

lemma open_Collect_imp: "closed {x. P x} \<Longrightarrow> open {x. Q x} \<Longrightarrow> open {x. P x \<longrightarrow> Q x}"
unfolding imp_conv_disj by (intro open_Collect_disj open_Collect_neg)

lemma open_Collect_const: "open {x. P}"
by (cases P) auto

lemma closed_Collect_neg: "open {x. P x} \<Longrightarrow> closed {x. \<not> P x}"
unfolding Collect_neg_eq by (rule closed_Compl)

lemma closed_Collect_conj: assumes "closed {x. P x}" "closed {x. Q x}" shows "closed {x. P x \<and> Q x}"
using closed_Int[OF assms] by (simp add: Int_def)

lemma closed_Collect_disj: assumes "closed {x. P x}" "closed {x. Q x}" shows "closed {x. P x \<or> Q x}"
using closed_Un[OF assms] by (simp add: Un_def)

lemma closed_Collect_all: "(\<And>i. closed {x. P i x}) \<Longrightarrow> closed {x. \<forall>i. P i x}"
using closed_INT[of UNIV "\<lambda>i. {x. P i x}"] unfolding Collect_all_eq by simp

lemma closed_Collect_imp: "open {x. P x} \<Longrightarrow> closed {x. Q x} \<Longrightarrow> closed {x. P x \<longrightarrow> Q x}"
unfolding imp_conv_disj by (intro closed_Collect_disj closed_Collect_neg)

lemma closed_Collect_const: "closed {x. P}"
by (cases P) auto

end

subsection\<open>Hausdorff and other separation properties\<close>

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 \<open>T2 spaces are also known as Hausdorff spaces.\<close>

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 \<open>A perfect space is a topological space with no isolated points.\<close>

class perfect_space = topological_space +
assumes not_open_singleton: "\<not> open {x}"

subsection \<open>Generators for toplogies\<close>

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 standard (auto intro: generate_topology.intros)

subsection \<open>Order topologies\<close>

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}"

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

lemma closed_atLeastAtMost [continuous_intros, simp]: "closed {a::'a::linorder_topology .. b}"
proof -
have "{a .. b} = {a ..} \<inter> {.. b}"
by auto
then show ?thesis
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 \<open>x < y\<close> 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+

subsubsection \<open>Boolean is an order topology\<close>

text \<open>It also is a discrete topology, but don't have a type class for it (yet).\<close>

instantiation bool :: order_topology
begin

definition open_bool :: "bool set \<Rightarrow> bool" where
"open_bool = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"

instance
proof qed (rule open_bool_def)

end

lemma open_bool[simp, intro!]: "open (A::bool set)"
proof -
have *: "{False <..} = {True}" "{..< True} = {False}"
by auto
have "A = UNIV \<or> A = {} \<or> A = {False <..} \<or> A = {..< True}"
using subset_UNIV[of A] unfolding UNIV_bool * by auto
then show "open A"
by auto
qed

subsubsection \<open>Topological filters\<close>

definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
where "nhds a = (INF S:{S. open S \<and> a \<in> S}. principal S)"

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) nhds_generated_topology:
"open = generate_topology T \<Longrightarrow> nhds x = (INF S:{S\<in>T. x \<in> S}. principal S)"
unfolding nhds_def
proof (safe intro!: antisym INF_greatest)
fix S assume "generate_topology T S" "x \<in> S"
then show "(INF S:{S \<in> T. x \<in> S}. principal S) \<le> principal S"
by induction
(auto intro: INF_lower order_trans simp add: inf_principal[symmetric] simp del: inf_principal)
qed (auto intro!: INF_lower intro: generate_topology.intros)

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 by (subst eventually_INF_base) (auto simp: eventually_principal)

lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
unfolding trivial_limit_def eventually_nhds by simp

lemma (in t1_space) t1_space_nhds:
"x \<noteq> y \<Longrightarrow> (\<forall>\<^sub>F x in nhds x. x \<noteq> y)"
by (drule t1_space) (auto simp: eventually_nhds)

lemma at_within_eq: "at x within s = (INF S:{S. open S \<and> x \<in> S}. principal (S \<inter> s - {x}))"
unfolding nhds_def at_within_def by (subst INF_inf_const2[symmetric]) (auto simp add: Diff_Int_distrib)

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_open_NO_MATCH:
"a \<in> s \<Longrightarrow> open s \<Longrightarrow> NO_MATCH UNIV s \<Longrightarrow> at a within s = at a"
by (simp only: at_within_open)

lemma at_within_nhd:
assumes "x \<in> S" "open S" "T \<inter> S - {x} = U \<inter> S - {x}"
shows "at x within T = at x within U"
unfolding filter_eq_iff eventually_at_filter
proof (intro allI eventually_subst)
have "eventually (\<lambda>x. x \<in> S) (nhds x)"
using \<open>x \<in> S\<close> \<open>open S\<close> by (auto simp: eventually_nhds)
then show "\<forall>\<^sub>F n in nhds x. (n \<noteq> x \<longrightarrow> n \<in> T \<longrightarrow> P n) = (n \<noteq> x \<longrightarrow> n \<in> U \<longrightarrow> P n)" for P
by eventually_elim (insert \<open>T \<inter> S - {x} = U \<inter> S - {x}\<close>, blast)
qed

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"

lemma (in order_topology) nhds_order: "nhds x =
inf (INF a:{x <..}. principal {..< a}) (INF a:{..< x}. principal {a <..})"
proof -
have 1: "{S \<in> range lessThan \<union> range greaterThan. x \<in> S} =
(\<lambda>a. {..< a}) ` {x <..} \<union> (\<lambda>a. {a <..}) ` {..< x}"
by auto
show ?thesis
unfolding nhds_generated_topology[OF open_generated_order] INF_union 1 INF_image comp_def ..
qed

lemma (in linorder_topology) at_within_order: "UNIV \<noteq> {x} \<Longrightarrow>
at x within s = inf (INF a:{x <..}. principal ({..< a} \<inter> s - {x}))
(INF a:{..< x}. principal ({a <..} \<inter> s - {x}))"
proof (cases "{x <..} = {}" "{..< x} = {}" rule: case_split[case_product case_split])
assume "UNIV \<noteq> {x}" "{x<..} = {}" "{..< x} = {}"
moreover have "UNIV = {..< x} \<union> {x} \<union> {x <..}"
by auto
ultimately show ?thesis
by auto
qed (auto simp: at_within_def nhds_order Int_Diff inf_principal[symmetric] INF_inf_const2
inf_sup_aci[where 'a="'a filter"]
simp del: inf_principal)

lemma (in linorder_topology) at_left_eq:
"y < x \<Longrightarrow> at_left x = (INF a:{..< x}. principal {a <..< x})"
by (subst at_within_order)
(auto simp: greaterThan_Int_greaterThan greaterThanLessThan_eq[symmetric] min.absorb2 INF_constant
intro!: INF_lower2 inf_absorb2)

lemma (in linorder_topology) eventually_at_left:
"y < x \<Longrightarrow> eventually P (at_left x) \<longleftrightarrow> (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)"
unfolding at_left_eq by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)

lemma (in linorder_topology) at_right_eq:
"x < y \<Longrightarrow> at_right x = (INF a:{x <..}. principal {x <..< a})"
by (subst at_within_order)
(auto simp: lessThan_Int_lessThan greaterThanLessThan_eq[symmetric] max.absorb2 INF_constant Int_commute
intro!: INF_lower2 inf_absorb1)

lemma (in linorder_topology) eventually_at_right:
"x < y \<Longrightarrow> eventually P (at_right x) \<longleftrightarrow> (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)"
unfolding at_right_eq by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)

lemma trivial_limit_at_right_top: "at_right (top::_::{order_top, linorder_topology}) = bot"
unfolding filter_eq_iff eventually_at_topological by auto

lemma trivial_limit_at_left_bot: "at_left (bot::_::{order_bot, linorder_topology}) = bot"
unfolding filter_eq_iff eventually_at_topological by auto

lemma trivial_limit_at_left_real [simp]:
"\<not> trivial_limit (at_left (x::'a::{no_bot, dense_order, linorder_topology}))"
using lt_ex[of x]
by safe (auto simp add: trivial_limit_def eventually_at_left dest: dense)

lemma trivial_limit_at_right_real [simp]:
"\<not> trivial_limit (at_right (x::'a::{no_top, dense_order, linorder_topology}))"
using gt_ex[of x]
by safe (auto simp add: trivial_limit_def eventually_at_right 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)

subsubsection \<open>Tendsto\<close>

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

named_theorems tendsto_intros "introduction rules for tendsto"
setup \<open>
fn context =>
Named_Theorems.get (Context.proof_of context) @{named_theorems tendsto_intros}
|> map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm])))
\<close>

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 nhds_def filterlim_INF filterlim_principal by auto

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 (in order_topology) order_tendsto_iff:
"(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)"
unfolding nhds_order filterlim_inf filterlim_INF filterlim_principal by auto

lemma (in order_topology) order_tendstoI:
"(\<And>a. a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F) \<Longrightarrow> (\<And>a. y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F) \<Longrightarrow>
(f ---> y) F"
unfolding order_tendsto_iff by auto

lemma (in order_topology) order_tendstoD:
assumes "(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 assms unfolding order_tendsto_iff by auto

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

lemma (in linorder_topology) tendsto_max:
assumes X: "(X ---> x) net"
assumes Y: "(Y ---> y) net"
shows "((\<lambda>x. max (X x) (Y x)) ---> max x y) net"
proof (rule order_tendstoI)
fix a assume "a < max x y"
then show "eventually (\<lambda>x. a < max (X x) (Y x)) net"
using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
by (auto simp: less_max_iff_disj elim: eventually_elim1)
next
fix a assume "max x y < a"
then show "eventually (\<lambda>x. max (X x) (Y x) < a) net"
using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
by (auto simp: eventually_conj_iff)
qed

lemma (in linorder_topology) tendsto_min:
assumes X: "(X ---> x) net"
assumes Y: "(Y ---> y) net"
shows "((\<lambda>x. min (X x) (Y x)) ---> min x y) net"
proof (rule order_tendstoI)
fix a assume "a < min x y"
then show "eventually (\<lambda>x. a < min (X x) (Y x)) net"
using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
by (auto simp: eventually_conj_iff)
next
fix a assume "min x y < a"
then show "eventually (\<lambda>x. min (X x) (Y x) < a) net"
using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
by (auto simp: min_less_iff_disj elim: eventually_elim1)
qed

lemma tendsto_ident_at [tendsto_intros, simp, intro]: "((\<lambda>x. x) ---> a) (at a within s)"
unfolding tendsto_def eventually_at_topological by auto

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

lemma (in t2_space) tendsto_unique:
assumes "F \<noteq> bot" 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 \<open>a \<noteq> b\<close>] by fast
have "eventually (\<lambda>x. f x \<in> U) F"
using \<open>(f ---> a) F\<close> \<open>open U\<close> \<open>a \<in> U\<close> by (rule topological_tendstoD)
moreover
have "eventually (\<lambda>x. f x \<in> V) F"
using \<open>(f ---> b) F\<close> \<open>open V\<close> \<open>b \<in> V\<close> 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 \<open>U \<inter> V = {}\<close> show ?case by simp
qed
with \<open>\<not> trivial_limit F\<close> show "False"
qed

lemma (in t2_space) tendsto_const_iff:
assumes "\<not> trivial_limit F" shows "((\<lambda>x. a :: 'a) ---> b) F \<longleftrightarrow> a = b"
by (auto intro!: 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
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])

lemma 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

lemma filterlim_at_bot_at_right:
fixes f :: "'a::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 \<open>P z\<close>]]
unfolding eventually_at_right[OF bound[OF bij(2)[OF \<open>P z\<close>]]] 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 \<open>P z\<close> mono)
qed
qed

lemma filterlim_at_top_at_left:
fixes f :: "'a::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 \<open>P z\<close>]]
unfolding eventually_at_left[OF bound[OF bij(2)[OF \<open>P z\<close>]]] 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 \<open>P z\<close> 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)

lemma eventually_nhds_top:
fixes P :: "'a :: {order_top, linorder_topology} \<Rightarrow> bool"
assumes "(b::'a) < top"
shows "eventually P (nhds top) \<longleftrightarrow> (\<exists>b<top. (\<forall>z. b < z \<longrightarrow> P z))"
unfolding eventually_nhds
proof safe
fix S :: "'a set" assume "open S" "top \<in> S"
note open_left[OF this \<open>b < top\<close>]
moreover assume "\<forall>s\<in>S. P s"
ultimately show "\<exists>b<top. \<forall>z>b. P z"
by (auto simp: subset_eq Ball_def)
next
fix b assume "b < top" "\<forall>z>b. P z"
then show "\<exists>S. open S \<and> top \<in> S \<and> (\<forall>xa\<in>S. P xa)"
by (intro exI[of _ "{b <..}"]) auto
qed

lemma tendsto_at_within_iff_tendsto_nhds:
"(g ---> g l) (at l within S) \<longleftrightarrow> (g ---> g l) (inf (nhds l) (principal S))"
unfolding tendsto_def eventually_at_filter eventually_inf_principal
by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)

subsection \<open>Limits on sequences\<close>

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 \<open>Monotone sequences and subsequences\<close>

definition
monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
--\<open>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.\<close>
"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
--\<open>Definition of subsequence\<close>
"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"

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

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

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

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 \<open>monoseq a\<close>[unfolded monoseq_def] by auto
thus ?thesis by (rule monoI1)
qed

text\<open>Subsequence (alternative definition, (e.g. Hoskins)\<close>

lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
apply (induct_tac k)
apply (auto intro: less_trans)
done

text\<open>for any sequence, there is a monotonic subsequence\<close>
lemma seq_monosub:
fixes s :: "nat => 'a::linorder"
shows "\<exists>f. subseq f \<and> monoseq (\<lambda>n. (s (f n)))"
proof cases
assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. s m \<le> s p"
then have "\<exists>f. \<forall>n. (\<forall>m\<ge>f n. s m \<le> s (f n)) \<and> f n < f (Suc n)"
by (intro dependent_nat_choice) (auto simp: conj_commute)
then obtain f where "subseq f" and mono: "\<And>n m. f n \<le> m \<Longrightarrow> s m \<le> s (f n)"
by (auto simp: subseq_Suc_iff)
moreover
then have "incseq f"
unfolding subseq_Suc_iff incseq_Suc_iff by (auto intro: less_imp_le)
then have "monoseq (\<lambda>n. s (f n))"
by (auto simp add: incseq_def intro!: mono monoI2)
ultimately show ?thesis
by auto
next
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)
have "\<exists>f. \<forall>n. N < f n \<and> f n < f (Suc n) \<and> s (f n) \<le> s (f (Suc n))"
proof (intro dependent_nat_choice)
fix x assume "N < x" with N[of x] show "\<exists>y>N. x < y \<and> s x \<le> s y"
by (auto intro: less_trans)
qed auto
then show ?thesis
by (auto simp: monoseq_iff incseq_Suc_iff subseq_Suc_iff)
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"

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

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

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 \<open>decseq F\<close> 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

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

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

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"
by (rule tendsto_Lim) (rule trivial_limit_sequentially)

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\<open>Increasing and Decreasing Series\<close>

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 \<open>First countable topologies\<close>

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) nhds_countable:
obtains X :: "nat \<Rightarrow> 'a set"
where "decseq X" "\<And>n. open (X n)" "\<And>n. x \<in> X n" "nhds x = (INF n. principal (X n))"
proof -
from first_countable_basis obtain A :: "nat \<Rightarrow> 'a set"
where A: "\<And>n. x \<in> A n" "\<And>n. open (A n)" "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"
by metis
show thesis
proof
show "decseq (\<lambda>n. \<Inter>i\<le>n. A i)"
by (auto simp: decseq_def)
show "\<And>n. x \<in> (\<Inter>i\<le>n. A i)" "\<And>n. open (\<Inter>i\<le>n. A i)"
using A by auto
show "nhds x = (INF n. principal (\<Inter>i\<le>n. A i))"
using A unfolding nhds_def
apply (intro INF_eq)
apply simp_all
apply force
apply (intro exI[of _ "\<Inter>i\<le>n. A i" for n] conjI open_INT)
apply auto
done
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

lemma tendsto_at_iff_sequentially:
fixes f :: "'a :: first_countable_topology \<Rightarrow> _"
shows "(f ---> a) (at x within s) \<longleftrightarrow> (\<forall>X. (\<forall>i. X i \<in> s - {x}) \<longrightarrow> X ----> x \<longrightarrow> ((f \<circ> X) ----> a))"
unfolding filterlim_def[of _ "nhds a"] le_filter_def eventually_filtermap at_within_def eventually_nhds_within_iff_sequentially comp_def
by metis

subsection \<open>Function limit at a point\<close>

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"

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"

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 \<open>Limits are equal for functions equal except at limit point\<close>

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)"

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)

lemma tendsto_compose_filtermap: "((g \<circ> f) ---> T) F \<longleftrightarrow> (g ---> T) (filtermap f F)"
by (simp add: filterlim_def filtermap_filtermap comp_def)

subsubsection \<open>Relation of LIM and LIMSEQ\<close>

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 ..

lemma sequentially_imp_eventually_at_left:
fixes a :: "'a :: {linorder_topology, first_countable_topology}"
assumes b[simp]: "b < a"
assumes *: "\<And>f. (\<And>n. b < f n) \<Longrightarrow> (\<And>n. f n < a) \<Longrightarrow> incseq f \<Longrightarrow> f ----> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
shows "eventually P (at_left a)"
proof (safe intro!: sequentially_imp_eventually_within)
fix X assume X: "\<forall>n. X n \<in> {..< a} \<and> X n \<noteq> a" "X ----> a"
show "eventually (\<lambda>n. P (X n)) sequentially"
proof (rule ccontr)
assume neg: "\<not> eventually (\<lambda>n. P (X n)) sequentially"
have "\<exists>s. \<forall>n. (\<not> P (X (s n)) \<and> b < X (s n)) \<and> (X (s n) \<le> X (s (Suc n)) \<and> Suc (s n) \<le> s (Suc n))"
proof (rule dependent_nat_choice)
have "\<not> eventually (\<lambda>n. b < X n \<longrightarrow> P (X n)) sequentially"
by (intro not_eventually_impI neg order_tendstoD(1) [OF X(2) b])
then show "\<exists>x. \<not> P (X x) \<and> b < X x"
by (auto dest!: not_eventuallyD)
next
fix x n
have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> b < X n \<longrightarrow> X x < X n \<longrightarrow> P (X n)) sequentially"
using X by (intro not_eventually_impI order_tendstoD(1)[OF X(2)] eventually_ge_at_top neg) auto
then show "\<exists>n. (\<not> P (X n) \<and> b < X n) \<and> (X x \<le> X n \<and> Suc x \<le> n)"
by (auto dest!: not_eventuallyD)
qed
then guess s ..
then have "\<And>n. b < X (s n)" "\<And>n. X (s n) < a" "incseq (\<lambda>n. X (s n))" "(\<lambda>n. X (s n)) ----> a" "\<And>n. \<not> P (X (s n))"
using X by (auto simp: subseq_Suc_iff Suc_le_eq incseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X ----> a\<close>, unfolded comp_def])
from *[OF this(1,2,3,4)] this(5) show False by auto
qed
qed

lemma tendsto_at_left_sequentially:
fixes a :: "_ :: {linorder_topology, first_countable_topology}"
assumes "b < a"
assumes *: "\<And>S. (\<And>n. S n < a) \<Longrightarrow> (\<And>n. b < S n) \<Longrightarrow> incseq S \<Longrightarrow> S ----> a \<Longrightarrow> (\<lambda>n. X (S n)) ----> L"
shows "(X ---> L) (at_left a)"
using assms unfolding tendsto_def [where l=L]

lemma sequentially_imp_eventually_at_right:
fixes a :: "'a :: {linorder_topology, first_countable_topology}"
assumes b[simp]: "a < b"
assumes *: "\<And>f. (\<And>n. a < f n) \<Longrightarrow> (\<And>n. f n < b) \<Longrightarrow> decseq f \<Longrightarrow> f ----> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
shows "eventually P (at_right a)"
proof (safe intro!: sequentially_imp_eventually_within)
fix X assume X: "\<forall>n. X n \<in> {a <..} \<and> X n \<noteq> a" "X ----> a"
show "eventually (\<lambda>n. P (X n)) sequentially"
proof (rule ccontr)
assume neg: "\<not> eventually (\<lambda>n. P (X n)) sequentially"
have "\<exists>s. \<forall>n. (\<not> P (X (s n)) \<and> X (s n) < b) \<and> (X (s (Suc n)) \<le> X (s n) \<and> Suc (s n) \<le> s (Suc n))"
proof (rule dependent_nat_choice)
have "\<not> eventually (\<lambda>n. X n < b \<longrightarrow> P (X n)) sequentially"
by (intro not_eventually_impI neg order_tendstoD(2) [OF X(2) b])
then show "\<exists>x. \<not> P (X x) \<and> X x < b"
by (auto dest!: not_eventuallyD)
next
fix x n
have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> X n < b \<longrightarrow> X n < X x \<longrightarrow> P (X n)) sequentially"
using X by (intro not_eventually_impI order_tendstoD(2)[OF X(2)] eventually_ge_at_top neg) auto
then show "\<exists>n. (\<not> P (X n) \<and> X n < b) \<and> (X n \<le> X x \<and> Suc x \<le> n)"
by (auto dest!: not_eventuallyD)
qed
then guess s ..
then have "\<And>n. a < X (s n)" "\<And>n. X (s n) < b" "decseq (\<lambda>n. X (s n))" "(\<lambda>n. X (s n)) ----> a" "\<And>n. \<not> P (X (s n))"
using X by (auto simp: subseq_Suc_iff Suc_le_eq decseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X ----> a\<close>, unfolded comp_def])
from *[OF this(1,2,3,4)] this(5) show False by auto
qed
qed

lemma tendsto_at_right_sequentially:
fixes a :: "_ :: {linorder_topology, first_countable_topology}"
assumes "a < b"
assumes *: "\<And>S. (\<And>n. a < S n) \<Longrightarrow> (\<And>n. S n < b) \<Longrightarrow> decseq S \<Longrightarrow> S ----> a \<Longrightarrow> (\<lambda>n. X (S n)) ----> L"
shows "(X ---> L) (at_right a)"
using assms unfolding tendsto_def [where l=L]

subsection \<open>Continuity\<close>

subsubsection \<open>Continuity on a set\<close>

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 \<open>x \<in> s\<close> \<open>f x \<in> B\<close> 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_open_Un:
"open s \<Longrightarrow> open t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
using continuous_on_open_Union [of "{s,t}"] by 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

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

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)

lemma continuous_on_generate_topology:
assumes *: "open = generate_topology X"
assumes **: "\<And>B. B \<in> X \<Longrightarrow> \<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A"
shows "continuous_on A f"
unfolding continuous_on_open_invariant
proof safe
fix B :: "'a set" assume "open B" then show "\<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A"
unfolding *
proof induction
case (UN K)
then obtain C where "\<And>k. k \<in> K \<Longrightarrow> open (C k)" "\<And>k. k \<in> K \<Longrightarrow> C k \<inter> A = f -` k \<inter> A"
by metis
then show ?case
by (intro exI[of _ "\<Union>k\<in>K. C k"]) blast
qed (auto intro: **)
qed

lemma continuous_onI_mono:
fixes f :: "'a::linorder_topology \<Rightarrow> 'b::{dense_order, linorder_topology}"
assumes "open (f`A)"
assumes mono: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
shows "continuous_on A f"
proof (rule continuous_on_generate_topology[OF open_generated_order], safe)
have monoD: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x < f y \<Longrightarrow> x < y"
by (auto simp: not_le[symmetric] mono)

{ fix a b assume "a \<in> A" "f a < b"
moreover
with open_right[OF \<open>open (f`A)\<close>, of "f a" b] obtain y where "f a < y" "{f a ..< y} \<subseteq> f`A"
by auto
moreover then obtain z where "f a < z" "z < min b y"
using dense[of "f a" "min b y"] \<open>f a < y\<close> \<open>f a < b\<close> by auto
moreover then obtain c where "z = f c" "c \<in> A"
using \<open>{f a ..< y} \<subseteq> f`A\<close>[THEN subsetD, of z] by (auto simp: less_imp_le)
ultimately have "\<exists>x. x \<in> A \<and> f x < b \<and> a < x"
by (auto intro!: exI[of _ c] simp: monoD) }
then show "\<exists>C. open C \<and> C \<inter> A = f -` {..<b} \<inter> A" for b
by (intro exI[of _ "(\<Union>x\<in>{x\<in>A. f x < b}. {..< x})"])
(auto intro: le_less_trans[OF mono] less_imp_le)

{ fix a b assume "a \<in> A" "b < f a"
moreover
with open_left[OF \<open>open (f`A)\<close>, of "f a" b] obtain y where "y < f a" "{y <.. f a} \<subseteq> f`A"
by auto
moreover then obtain z where "max b y < z" "z < f a"
using dense[of "max b y" "f a"] \<open>y < f a\<close> \<open>b < f a\<close> by auto
moreover then obtain c where "z = f c" "c \<in> A"
using \<open>{y <.. f a} \<subseteq> f`A\<close>[THEN subsetD, of z] by (auto simp: less_imp_le)
ultimately have "\<exists>x. x \<in> A \<and> b < f x \<and> x < a"
by (auto intro!: exI[of _ c] simp: monoD) }
then show "\<exists>C. open C \<and> C \<inter> A = f -` {b <..} \<inter> A" for b
by (intro exI[of _ "(\<Union>x\<in>{x\<in>A. b < f x}. {x <..})"])
(auto intro: less_le_trans[OF _ mono] less_imp_le)
qed

subsubsection \<open>Continuity at a point\<close>

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)"

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_imp_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_imp_continuous_at_within simp: continuous_on_eq_continuous_within)

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_imp_continuous_at_within)

lemma filtermap_nhds_open_map:
assumes cont: "isCont f a" and open_map: "\<And>S. open S \<Longrightarrow> open (f`S)"
shows "filtermap f (nhds a) = nhds (f a)"
unfolding filter_eq_iff
proof safe
fix P assume "eventually P (filtermap f (nhds a))"
then guess S unfolding eventually_filtermap eventually_nhds ..
then show "eventually P (nhds (f a))"
unfolding eventually_nhds by (intro exI[of _ "f`S"]) (auto intro!: open_map)
qed (metis filterlim_iff tendsto_at_iff_tendsto_nhds isCont_def eventually_filtermap cont)

lemma continuous_at_split:
"continuous (at (x::'a::linorder_topology)) f = (continuous (at_left x) f \<and> continuous (at_right x) f)"

subsubsection \<open>Open-cover compactness\<close>

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 \<open>closed t\<close> 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 \<open>compact s\<close> 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 \<open>compact U\<close> 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 \<open>?R\<close> 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 \<open>compact s\<close>
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 \<open>compact s\<close>
moreover have "\<forall>u\<in>?C. open u" by simp
moreover have "s \<subseteq> \<Union>?C"
proof
fix x assume "x \<in> s"
with \<open>y \<in> - s\<close> 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 \<open>x \<in> s\<close> 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 \<open>D \<subseteq> ?C\<close> have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto
with \<open>finite D\<close> have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"
with \<open>s \<subseteq> \<Union>D\<close> 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"
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
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 \<open>continuous_on s f\<close> Diff_subset
by (rule continuous_on_subset)
moreover have "compact (s - B)"
using \<open>open B\<close> and \<open>compact s\<close>
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 \<open>x \<in> s\<close> and \<open>x \<in> B\<close> by (simp add: 1)
moreover have "\<forall>y\<in>s. f y \<in> - f ` (s - B) \<longrightarrow> y \<in> B"
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 \<open>compact S\<close> obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {..< t s})"
by (erule compactE_image)
with \<open>S \<noteq> {}\<close> 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 \<open>C \<subseteq> S\<close> 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 \<open>compact S\<close> obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {t s <..})"
by (erule compactE_image)
with \<open>S \<noteq> {}\<close> 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 \<open>C \<subseteq> S\<close> 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 \<open>Connectedness\<close>

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 connected_sing [simp]: "connected {x}"
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_closed:
"connected s \<longleftrightarrow>
~ (\<exists>A B. closed A \<and> closed B \<and> s \<subseteq> A \<union> B \<and> A \<inter> B \<inter> s = {} \<and> A \<inter> s \<noteq> {} \<and> B \<inter> s \<noteq> {})"
apply (simp add: connected_def del: ex_simps, safe)
apply (drule_tac x="-A" in spec)
apply (drule_tac x="-B" in spec)
apply (fastforce simp add: closed_def [symmetric])
apply (drule_tac x="-A" in spec)
apply (drule_tac x="-B" in spec)
apply (fastforce simp add: open_closed [symmetric])
done

lemma connected_Union:
assumes cs: "\<And>s. s \<in> S \<Longrightarrow> connected s" and ne: "\<Inter>S \<noteq> {}"
shows "connected(\<Union>S)"
proof (rule connectedI)
fix A B
assume A: "open A" and B: "open B" and Alap: "A \<inter> \<Union>S \<noteq> {}" and Blap: "B \<inter> \<Union>S \<noteq> {}"
and disj: "A \<inter> B \<inter> \<Union>S = {}" and cover: "\<Union>S \<subseteq> A \<union> B"
have disjs:"\<And>s. s \<in> S \<Longrightarrow> A \<inter> B \<inter> s = {}"
using disj by auto
obtain sa where sa: "sa \<in> S" "A \<inter> sa \<noteq> {}"
using Alap by auto
obtain sb where sb: "sb \<in> S" "B \<inter> sb \<noteq> {}"
using Blap by auto
obtain x where x: "\<And>s. s \<in> S \<Longrightarrow> x \<in> s"
using ne by auto
then have "x \<in> \<Union>S"
using \<open>sa \<in> S\<close> by blast
then have "x \<in> A \<or> x \<in> B"
using cover by auto
then show False
using cs [unfolded connected_def]
by (metis A B IntI Sup_upper sa sb disjs x cover empty_iff subset_trans)
qed

lemma connected_Un: "\<lbrakk>connected s; connected t; s \<inter> t \<noteq> {}\<rbrakk> \<Longrightarrow> connected (s \<union> t)"
using connected_Union [of "{s,t}"] by auto

lemma connected_diff_open_from_closed:
assumes st: "s \<subseteq> t" and tu: "t \<subseteq> u" and s: "open s"
and t: "closed t" and u: "connected u" and ts: "connected (t - s)"
shows "connected(u - s)"
proof (rule connectedI)
fix A B
assume AB: "open A" "open B" "A \<inter> (u - s) \<noteq> {}" "B \<inter> (u - s) \<noteq> {}"
and disj: "A \<inter> B \<inter> (u - s) = {}" and cover: "u - s \<subseteq> A \<union> B"
then consider "A \<inter> (t - s) = {}" | "B \<inter> (t - s) = {}"
using st ts tu connectedD [of "t-s" "A" "B"]
by auto
then show False
proof cases
case 1
then have "(A - t) \<inter> (B \<union> s) \<inter> u = {}"
using disj st by auto
moreover have  "u \<subseteq> (A - t) \<union> (B \<union> s)" using 1 cover by auto
ultimately show False
using connectedD [of u "A - t" "B \<union> s"] AB s t 1 u
by auto
next
case 2
then have "(A \<union> s) \<inter> (B - t) \<inter> u = {}"
using disj st
by auto
moreover have "u \<subseteq> (A \<union> s) \<union> (B - t)" using 2 cover by auto
ultimately show False
using connectedD [of u "A \<union> s" "B - t"] AB s t 2 u
by auto
qed
qed

lemma connected_iff_const:
fixes S :: "'a::topological_space set"
shows "connected S \<longleftrightarrow> (\<forall>P::'a \<Rightarrow> bool. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c))"
proof safe
fix P :: "'a \<Rightarrow> bool" assume "connected S" "continuous_on S P"
then have "\<And>b. \<exists>A. open A \<and> A \<inter> S = P -` {b} \<inter> S"
unfolding continuous_on_open_invariant by simp
from this[of True] this[of False]
obtain t f where "open t" "open f" and *: "f \<inter> S = P -` {False} \<inter> S" "t \<inter> S = P -` {True} \<inter> S"
by auto
then have "t \<inter> S = {} \<or> f \<inter> S = {}"
by (intro connectedD[OF \<open>connected S\<close>])  auto
then show "\<exists>c. \<forall>s\<in>S. P s = c"
proof (rule disjE)
assume "t \<inter> S = {}" then show ?thesis
unfolding * by (intro exI[of _ False]) auto
next
assume "f \<inter> S = {}" then show ?thesis
unfolding * by (intro exI[of _ True]) auto
qed
next
assume P: "\<forall>P::'a \<Rightarrow> bool. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c)"
show "connected S"
proof (rule connectedI)
fix A B assume *: "open A" "open B" "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B"
have "continuous_on S (\<lambda>x. x \<in> A)"
unfolding continuous_on_open_invariant
proof safe
fix C :: "bool set"
have "C = UNIV \<or> C = {True} \<or> C = {False} \<or> C = {}"
using subset_UNIV[of C] unfolding UNIV_bool by auto
with * show "\<exists>T. open T \<and> T \<inter> S = (\<lambda>x. x \<in> A) -` C \<inter> S"
by (intro exI[of _ "(if True \<in> C then A else {}) \<union> (if False \<in> C then B else {})"]) auto
qed
from P[rule_format, OF this] obtain c where "\<And>s. s \<in> S \<Longrightarrow> (s \<in> A) = c" by blast
with * show False
by (cases c) auto
qed
qed

lemma connectedD_const:
fixes P :: "'a::topological_space \<Rightarrow> bool"
shows "connected S \<Longrightarrow> continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c"
unfolding connected_iff_const by auto

lemma connectedI_const:
"(\<And>P::'a::topological_space \<Rightarrow> bool. continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c) \<Longrightarrow> connected S"
unfolding connected_iff_const by auto

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 \<open>connected A\<close> S \<open>a\<in>A\<close>
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 \<open>b\<in>A\<close> show ?thesis by auto
next
assume "?P \<inter> A = {}" then show ?thesis
using \<open>a \<in> A\<close> 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 <..}"])
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_const)
fix P :: "'b \<Rightarrow> bool" assume "continuous_on (f ` s) P"
then have "continuous_on s (P \<circ> f)"
by (rule continuous_on_compose[OF *])
from connectedD_const[OF \<open>connected s\<close> this] show "\<exists>c. \<forall>s\<in>f ` s. P s = c"
by auto
qed

section \<open>Linear Continuum Topologies\<close>

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 \<open>c \<in> A\<close>] 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 \<open>c \<in> A\<close>] 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 \<open>y \<in> B\<close> \<open>x < y\<close> by (auto intro: cInf_lower cInf_greatest)
with \<open>x \<in> U\<close> \<open>y \<in> U\<close> have "?z \<in> U"
by (rule *)
moreover have "?z \<notin> B \<inter> {x <..}"
using \<open>open B\<close> by (intro Inf_notin_open) auto
ultimately have "?z \<in> A"
using \<open>x \<le> ?z\<close> \<open>A \<inter> B \<inter> U = {}\<close> \<open>x \<in> A\<close> \<open>U \<subseteq> A \<union> B\<close> by auto

{ assume "?z < y"
obtain a where "?z < a" "{?z ..< a} \<subseteq> A"
using open_right[OF \<open>open A\<close> \<open>?z \<in> A\<close> \<open>?z < y\<close>] 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"] \<open>?z < a\<close> \<open>?z < y\<close> \<open>x < y\<close> \<open>y \<in> B\<close>
by (auto intro: less_imp_le)
moreover have "?z \<le> b"
using \<open>b \<in> B\<close> \<open>x < b\<close>
by (intro cInf_lower) auto
moreover have "b \<in> U"
using \<open>x \<le> ?z\<close> \<open>?z \<le> b\<close> \<open>b < min a y\<close>
by (intro *[OF \<open>x \<in> U\<close> \<open>y \<in> U\<close>]) (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 \<open>?z \<le> y\<close> \<open>?z \<in> A\<close> \<open>y \<in> B\<close> \<open>y \<in> U\<close> \<open>A \<inter> B \<inter> U = {}\<close> 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 \<open>Intermediate Value Theorem\<close>

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

lemma continuous_at_Sup_mono:
fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder} \<Rightarrow> 'b :: {linorder_topology, conditionally_complete_linorder}"
assumes "mono f"
assumes cont: "continuous (at_left (Sup S)) f"
assumes S: "S \<noteq> {}" "bdd_above S"
shows "f (Sup S) = (SUP s:S. f s)"
proof (rule antisym)
have f: "(f ---> f (Sup S)) (at_left (Sup S))"
using cont unfolding continuous_within .

show "f (Sup S) \<le> (SUP s:S. f s)"
proof cases
assume "Sup S \<in> S" then show ?thesis
by (rule cSUP_upper) (auto intro: bdd_above_image_mono S \<open>mono f\<close>)
next
assume "Sup S \<notin> S"
from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S"
by auto
with \<open>Sup S \<notin> S\<close> S have "s < Sup S"
unfolding less_le by (blast intro: cSup_upper)
show ?thesis
proof (rule ccontr)
assume "\<not> ?thesis"
with order_tendstoD(1)[OF f, of "SUP s:S. f s"] obtain b where "b < Sup S"
and *: "\<And>y. b < y \<Longrightarrow> y < Sup S \<Longrightarrow> (SUP s:S. f s) < f y"
by (auto simp: not_le eventually_at_left[OF \<open>s < Sup S\<close>])
with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "b < c"
using less_cSupD[of S b] by auto
with \<open>Sup S \<notin> S\<close> S have "c < Sup S"
unfolding less_le by (blast intro: cSup_upper)
from *[OF \<open>b < c\<close> \<open>c < Sup S\<close>] cSUP_upper[OF \<open>c \<in> S\<close> bdd_above_image_mono[of f]]
show False
by (auto simp: assms)
qed
qed
qed (intro cSUP_least \<open>mono f\<close>[THEN monoD] cSup_upper S)

lemma continuous_at_Sup_antimono:
fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder} \<Rightarrow> 'b :: {linorder_topology, conditionally_complete_linorder}"
assumes "antimono f"
assumes cont: "continuous (at_left (Sup S)) f"
assumes S: "S \<noteq> {}" "bdd_above S"
shows "f (Sup S) = (INF s:S. f s)"
proof (rule antisym)
have f: "(f ---> f (Sup S)) (at_left (Sup S))"
using cont unfolding continuous_within .

show "(INF s:S. f s) \<le> f (Sup S)"
proof cases
assume "Sup S \<in> S" then show ?thesis
by (intro cINF_lower) (auto intro: bdd_below_image_antimono S \<open>antimono f\<close>)
next
assume "Sup S \<notin> S"
from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S"
by auto
with \<open>Sup S \<notin> S\<close> S have "s < Sup S"
unfolding less_le by (blast intro: cSup_upper)
show ?thesis
proof (rule ccontr)
assume "\<not> ?thesis"
with order_tendstoD(2)[OF f, of "INF s:S. f s"] obtain b where "b < Sup S"
and *: "\<And>y. b < y \<Longrightarrow> y < Sup S \<Longrightarrow> f y < (INF s:S. f s)"
by (auto simp: not_le eventually_at_left[OF \<open>s < Sup S\<close>])
with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "b < c"
using less_cSupD[of S b] by auto
with \<open>Sup S \<notin> S\<close> S have "c < Sup S"
unfolding less_le by (blast intro: cSup_upper)
from *[OF \<open>b < c\<close> \<open>c < Sup S\<close>] cINF_lower[OF bdd_below_image_antimono, of f S c] \<open>c \<in> S\<close>
show False
by (auto simp: assms)
qed
qed
qed (intro cINF_greatest \<open>antimono f\<close>[THEN antimonoD] cSup_upper S)

lemma continuous_at_Inf_mono:
fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder} \<Rightarrow> 'b :: {linorder_topology, conditionally_complete_linorder}"
assumes "mono f"
assumes cont: "continuous (at_right (Inf S)) f"
assumes S: "S \<noteq> {}" "bdd_below S"
shows "f (Inf S) = (INF s:S. f s)"
proof (rule antisym)
have f: "(f ---> f (Inf S)) (at_right (Inf S))"
using cont unfolding continuous_within .

show "(INF s:S. f s) \<le> f (Inf S)"
proof cases
assume "Inf S \<in> S" then show ?thesis
by (rule cINF_lower[rotated]) (auto intro: bdd_below_image_mono S \<open>mono f\<close>)
next
assume "Inf S \<notin> S"
from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S"
by auto
with \<open>Inf S \<notin> S\<close> S have "Inf S < s"
unfolding less_le by (blast intro: cInf_lower)
show ?thesis
proof (rule ccontr)
assume "\<not> ?thesis"
with order_tendstoD(2)[OF f, of "INF s:S. f s"] obtain b where "Inf S < b"
and *: "\<And>y. Inf S < y \<Longrightarrow> y < b \<Longrightarrow> f y < (INF s:S. f s)"
by (auto simp: not_le eventually_at_right[OF \<open>Inf S < s\<close>])
with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "c < b"
using cInf_lessD[of S b] by auto
with \<open>Inf S \<notin> S\<close> S have "Inf S < c"
unfolding less_le by (blast intro: cInf_lower)
from *[OF \<open>Inf S < c\<close> \<open>c < b\<close>] cINF_lower[OF bdd_below_image_mono[of f] \<open>c \<in> S\<close>]
show False
by (auto simp: assms)
qed
qed
qed (intro cINF_greatest \<open>mono f\<close>[THEN monoD] cInf_lower \<open>bdd_below S\<close> \<open>S \<noteq> {}\<close>)

lemma continuous_at_Inf_antimono:
fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder} \<Rightarrow> 'b :: {linorder_topology, conditionally_complete_linorder}"
assumes "antimono f"
assumes cont: "continuous (at_right (Inf S)) f"
assumes S: "S \<noteq> {}" "bdd_below S"
shows "f (Inf S) = (SUP s:S. f s)"
proof (rule antisym)
have f: "(f ---> f (Inf S)) (at_right (Inf S))"
using cont unfolding continuous_within .

show "f (Inf S) \<le> (SUP s:S. f s)"
proof cases
assume "Inf S \<in> S" then show ?thesis
by (rule cSUP_upper) (auto intro: bdd_above_image_antimono S \<open>antimono f\<close>)
next
assume "Inf S \<notin> S"
from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S"
by auto
with \<open>Inf S \<notin> S\<close> S have "Inf S < s"
unfolding less_le by (blast intro: cInf_lower)
show ?thesis
proof (rule ccontr)
assume "\<not> ?thesis"
with order_tendstoD(1)[OF f, of "SUP s:S. f s"] obtain b where "Inf S < b"
and *: "\<And>y. Inf S < y \<Longrightarrow> y < b \<Longrightarrow> (SUP s:S. f s) < f y"
by (auto simp: not_le eventually_at_right[OF \<open>Inf S < s\<close>])
with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "c < b"
using cInf_lessD[of S b] by auto
with \<open>Inf S \<notin> S\<close> S have "Inf S < c"
unfolding less_le by (blast intro: cInf_lower)
from *[OF \<open>Inf S < c\<close> \<open>c < b\<close>] cSUP_upper[OF \<open>c \<in> S\<close> bdd_above_image_antimono[of f]]
show False
by (auto simp: assms)
qed
qed
qed (intro cSUP_least \<open>antimono f\<close>[THEN antimonoD] cInf_lower S)

end
```