(* Title: HOL/Topological_Spaces.thy Author: Brian Huffman Author: Johannes Hölzl *) section ‹Topological Spaces› theory Topological_Spaces imports Main begin named_theorems continuous_intros "structural introduction rules for continuity" subsection ‹Topological space› class "open" = fixes "open" :: "'a set ⇒ bool" class topological_space = "open" + assumes open_UNIV [simp, intro]: "open UNIV" assumes open_Int [intro]: "open S ⟹ open T ⟹ open (S ∩ T)" assumes open_Union [intro]: "∀S∈K. open S ⟹ open (⋃K)" begin definition closed :: "'a set ⇒ bool" where "closed S ⟷ open (- S)" lemma open_empty [continuous_intros, intro, simp]: "open {}" using open_Union [of "{}"] by simp lemma open_Un [continuous_intros, intro]: "open S ⟹ open T ⟹ open (S ∪ T)" using open_Union [of "{S, T}"] by simp lemma open_UN [continuous_intros, intro]: "∀x∈A. open (B x) ⟹ open (⋃x∈A. B x)" using open_Union [of "B ` A"] by simp lemma open_Inter [continuous_intros, intro]: "finite S ⟹ ∀T∈S. open T ⟹ open (⋂S)" by (induct set: finite) auto lemma open_INT [continuous_intros, intro]: "finite A ⟹ ∀x∈A. open (B x) ⟹ open (⋂x∈A. B x)" using open_Inter [of "B ` A"] by simp lemma openI: assumes "⋀x. x ∈ S ⟹ ∃T. open T ∧ x ∈ T ∧ T ⊆ S" shows "open S" proof - have "open (⋃{T. open T ∧ T ⊆ S})" by auto moreover have "⋃{T. open T ∧ T ⊆ 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 ⟹ closed T ⟹ closed (S ∪ 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 ⟹ closed T ⟹ closed (S ∩ T)" unfolding closed_def by auto lemma closed_INT [continuous_intros, intro]: "∀x∈A. closed (B x) ⟹ closed (⋂x∈A. B x)" unfolding closed_def by auto lemma closed_Inter [continuous_intros, intro]: "∀S∈K. closed S ⟹ closed (⋂K)" unfolding closed_def uminus_Inf by auto lemma closed_Union [continuous_intros, intro]: "finite S ⟹ ∀T∈S. closed T ⟹ closed (⋃S)" by (induct set: finite) auto lemma closed_UN [continuous_intros, intro]: "finite A ⟹ ∀x∈A. closed (B x) ⟹ closed (⋃x∈A. B x)" using closed_Union [of "B ` A"] by simp lemma open_closed: "open S ⟷ closed (- S)" by (simp add: closed_def) lemma closed_open: "closed S ⟷ open (- S)" by (rule closed_def) lemma open_Diff [continuous_intros, intro]: "open S ⟹ closed T ⟹ open (S - T)" by (simp add: closed_open Diff_eq open_Int) lemma closed_Diff [continuous_intros, intro]: "closed S ⟹ open T ⟹ closed (S - T)" by (simp add: open_closed Diff_eq closed_Int) lemma open_Compl [continuous_intros, intro]: "closed S ⟹ open (- S)" by (simp add: closed_open) lemma closed_Compl [continuous_intros, intro]: "open S ⟹ closed (- S)" by (simp add: open_closed) lemma open_Collect_neg: "closed {x. P x} ⟹ open {x. ¬ 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 ∧ 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 ∨ Q x}" using open_Un[OF assms] by (simp add: Un_def) lemma open_Collect_ex: "(⋀i. open {x. P i x}) ⟹ open {x. ∃i. P i x}" using open_UN[of UNIV "λi. {x. P i x}"] unfolding Collect_ex_eq by simp lemma open_Collect_imp: "closed {x. P x} ⟹ open {x. Q x} ⟹ open {x. P x ⟶ 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} ⟹ closed {x. ¬ 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 ∧ 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 ∨ Q x}" using closed_Un[OF assms] by (simp add: Un_def) lemma closed_Collect_all: "(⋀i. closed {x. P i x}) ⟹ closed {x. ∀i. P i x}" using closed_INT[of UNIV "λi. {x. P i x}"] by (simp add: Collect_all_eq) lemma closed_Collect_imp: "open {x. P x} ⟹ closed {x. Q x} ⟹ closed {x. P x ⟶ 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 ‹Hausdorff and other separation properties› class t0_space = topological_space + assumes t0_space: "x ≠ y ⟹ ∃U. open U ∧ ¬ (x ∈ U ⟷ y ∈ U)" class t1_space = topological_space + assumes t1_space: "x ≠ y ⟹ ∃U. open U ∧ x ∈ U ∧ y ∉ U" instance t1_space ⊆ t0_space by standard (fast dest: t1_space) context t1_space begin lemma separation_t1: "x ≠ y ⟷ (∃U. open U ∧ x ∈ U ∧ y ∉ U)" using t1_space[of x y] by blast lemma closed_singleton [iff]: "closed {a}" proof - let ?T = "⋃{S. open S ∧ a ∉ S}" have "open ?T" by (simp add: open_Union) also have "?T = - {a}" by (auto simp add: set_eq_iff separation_t1) finally show "closed {a}" by (simp only: closed_def) qed lemma closed_insert [continuous_intros, simp]: assumes "closed S" shows "closed (insert a S)" proof - from closed_singleton assms have "closed ({a} ∪ S)" by (rule closed_Un) then show "closed (insert a S)" by simp qed lemma finite_imp_closed: "finite S ⟹ closed S" by (induct pred: finite) simp_all end text ‹T2 spaces are also known as Hausdorff spaces.› class t2_space = topological_space + assumes hausdorff: "x ≠ y ⟹ ∃U V. open U ∧ open V ∧ x ∈ U ∧ y ∈ V ∧ U ∩ V = {}" instance t2_space ⊆ t1_space by standard (fast dest: hausdorff) lemma (in t2_space) separation_t2: "x ≠ y ⟷ (∃U V. open U ∧ open V ∧ x ∈ U ∧ y ∈ V ∧ U ∩ V = {})" using hausdorff [of x y] by blast lemma (in t0_space) separation_t0: "x ≠ y ⟷ (∃U. open U ∧ ¬ (x ∈ U ⟷ y ∈ U))" using t0_space [of x y] by blast text ‹A classical separation axiom for topological space, the T3 axiom -- also called regularity: if a point is not in a closed set, then there are open sets separating them.› class t3_space = t2_space + assumes t3_space: "closed S ⟹ y ∉ S ⟹ ∃U V. open U ∧ open V ∧ y ∈ U ∧ S ⊆ V ∧ U ∩ V = {}" text ‹A classical separation axiom for topological space, the T4 axiom -- also called normality: if two closed sets are disjoint, then there are open sets separating them.› class t4_space = t2_space + assumes t4_space: "closed S ⟹ closed T ⟹ S ∩ T = {} ⟹ ∃U V. open U ∧ open V ∧ S ⊆ U ∧ T ⊆ V ∧ U ∩ V = {}" text ‹T4 is stronger than T3, and weaker than metric.› instance t4_space ⊆ t3_space proof fix S and y::'a assume "closed S" "y ∉ S" then show "∃U V. open U ∧ open V ∧ y ∈ U ∧ S ⊆ V ∧ U ∩ V = {}" using t4_space[of "{y}" S] by auto qed text ‹A perfect space is a topological space with no isolated points.› class perfect_space = topological_space + assumes not_open_singleton: "¬ open {x}" lemma (in perfect_space) UNIV_not_singleton: "UNIV ≠ {x}" for x::'a by (metis (no_types) open_UNIV not_open_singleton) subsection ‹Generators for toplogies› inductive generate_topology :: "'a set set ⇒ 'a set ⇒ bool" for S :: "'a set set" where UNIV: "generate_topology S UNIV" | Int: "generate_topology S (a ∩ b)" if "generate_topology S a" and "generate_topology S b" | UN: "generate_topology S (⋃K)" if "(⋀k. k ∈ K ⟹ generate_topology S k)" | Basis: "generate_topology S s" if "s ∈ S" hide_fact (open) UNIV Int UN Basis lemma generate_topology_Union: "(⋀k. k ∈ I ⟹ generate_topology S (K k)) ⟹ generate_topology S (⋃k∈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 ‹Order topologies› class order_topology = order + "open" + assumes open_generated_order: "open = generate_topology (range (λa. {..< a}) ∪ range (λa. {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}" for a :: "'a::linorder_topology" by (simp add: closed_open) lemma closed_atLeast [continuous_intros, simp]: "closed {a..}" for a :: "'a::linorder_topology" by (simp add: closed_open) lemma closed_atLeastAtMost [continuous_intros, simp]: "closed {a..b}" for a b :: "'a::linorder_topology" proof - have "{a .. b} = {a ..} ∩ {.. b}" by auto then show ?thesis by (simp add: closed_Int) qed lemma (in linorder) less_separate: assumes "x < y" shows "∃a b. x ∈ {..< a} ∧ y ∈ {b <..} ∧ {..< a} ∩ {b <..} = {}" proof (cases "∃z. x < z ∧ z < y") case True then obtain z where "x < z ∧ z < y" .. then have "x ∈ {..< z} ∧ y ∈ {z <..} ∧ {z <..} ∩ {..< z} = {}" by auto then show ?thesis by blast next case False with ‹x < y› have "x ∈ {..< y}" "y ∈ {x <..}" "{x <..} ∩ {..< y} = {}" by auto then show ?thesis by blast qed instance linorder_topology ⊆ t2_space proof fix x y :: 'a show "x ≠ y ⟹ ∃U V. open U ∧ open V ∧ x ∈ U ∧ y ∈ V ∧ U ∩ V = {}" using less_separate [of x y] less_separate [of y x] by (elim neqE; metis open_lessThan open_greaterThan Int_commute) qed lemma (in linorder_topology) open_right: assumes "open S" "x ∈ S" and gt_ex: "x < y" shows "∃b>x. {x ..< b} ⊆ S" using assms unfolding open_generated_order proof induct case UNIV then show ?case by blast next case (Int A B) then obtain a b where "a > x" "{x ..< a} ⊆ A" "b > x" "{x ..< b} ⊆ B" by auto then show ?case by (auto intro!: exI[of _ "min a b"]) next case UN then show ?case by blast next case Basis then show ?case by (fastforce intro: exI[of _ y] gt_ex) qed lemma (in linorder_topology) open_left: assumes "open S" "x ∈ S" and lt_ex: "y < x" shows "∃b<x. {b <.. x} ⊆ S" using assms unfolding open_generated_order proof induction case UNIV then show ?case by blast next case (Int A B) then obtain a b where "a < x" "{a <.. x} ⊆ A" "b < x" "{b <.. x} ⊆ B" by auto then show ?case by (auto intro!: exI[of _ "max a b"]) next case UN then show ?case by blast next case Basis then show ?case by (fastforce intro: exI[of _ y] lt_ex) qed subsection ‹Setup some topologies› subsubsection ‹Boolean is an order topology› class discrete_topology = topological_space + assumes open_discrete: "⋀A. open A" instance discrete_topology < t2_space proof fix x y :: 'a assume "x ≠ y" then show "∃U V. open U ∧ open V ∧ x ∈ U ∧ y ∈ V ∧ U ∩ V = {}" by (intro exI[of _ "{_}"]) (auto intro!: open_discrete) qed instantiation bool :: linorder_topology begin definition open_bool :: "bool set ⇒ bool" where "open_bool = generate_topology (range (λa. {..< a}) ∪ range (λa. {a <..}))" instance by standard (rule open_bool_def) end instance bool :: discrete_topology proof fix A :: "bool set" have *: "{False <..} = {True}" "{..< True} = {False}" by auto have "A = UNIV ∨ A = {} ∨ A = {False <..} ∨ A = {..< True}" using subset_UNIV[of A] unfolding UNIV_bool * by blast then show "open A" by auto qed instantiation nat :: linorder_topology begin definition open_nat :: "nat set ⇒ bool" where "open_nat = generate_topology (range (λa. {..< a}) ∪ range (λa. {a <..}))" instance by standard (rule open_nat_def) end instance nat :: discrete_topology proof fix A :: "nat set" have "open {n}" for n :: nat proof (cases n) case 0 moreover have "{0} = {..<1::nat}" by auto ultimately show ?thesis by auto next case (Suc n') then have "{n} = {..<Suc n} ∩ {n' <..}" by auto with Suc show ?thesis by (auto intro: open_lessThan open_greaterThan) qed then have "open (⋃a∈A. {a})" by (intro open_UN) auto then show "open A" by simp qed instantiation int :: linorder_topology begin definition open_int :: "int set ⇒ bool" where "open_int = generate_topology (range (λa. {..< a}) ∪ range (λa. {a <..}))" instance by standard (rule open_int_def) end instance int :: discrete_topology proof fix A :: "int set" have "{..<i + 1} ∩ {i-1 <..} = {i}" for i :: int by auto then have "open {i}" for i :: int using open_Int[OF open_lessThan[of "i + 1"] open_greaterThan[of "i - 1"]] by auto then have "open (⋃a∈A. {a})" by (intro open_UN) auto then show "open A" by simp qed subsubsection ‹Topological filters› definition (in topological_space) nhds :: "'a ⇒ 'a filter" where "nhds a = (INF S:{S. open S ∧ a ∈ S}. principal S)" definition (in topological_space) at_within :: "'a ⇒ 'a set ⇒ 'a filter" ("at (_)/ within (_)" [1000, 60] 60) where "at a within s = inf (nhds a) (principal (s - {a}))" abbreviation (in topological_space) at :: "'a ⇒ 'a filter" ("at") where "at x ≡ at x within (CONST UNIV)" abbreviation (in order_topology) at_right :: "'a ⇒ 'a filter" where "at_right x ≡ at x within {x <..}" abbreviation (in order_topology) at_left :: "'a ⇒ 'a filter" where "at_left x ≡ at x within {..< x}" lemma (in topological_space) nhds_generated_topology: "open = generate_topology T ⟹ nhds x = (INF S:{S∈T. x ∈ S}. principal S)" unfolding nhds_def proof (safe intro!: antisym INF_greatest) fix S assume "generate_topology T S" "x ∈ S" then show "(INF S:{S ∈ T. x ∈ S}. principal S) ≤ principal S" by induct (auto intro: INF_lower order_trans simp: 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) ⟷ (∃S. open S ∧ a ∈ S ∧ (∀x∈S. P x))" unfolding nhds_def by (subst eventually_INF_base) (auto simp: eventually_principal) lemma eventually_eventually: "eventually (λy. eventually P (nhds y)) (nhds x) = eventually P (nhds x)" by (auto simp: eventually_nhds) lemma (in topological_space) eventually_nhds_in_open: "open s ⟹ x ∈ s ⟹ eventually (λy. y ∈ s) (nhds x)" by (subst eventually_nhds) blast lemma (in topological_space) eventually_nhds_x_imp_x: "eventually P (nhds x) ⟹ P x" by (subst (asm) eventually_nhds) blast lemma (in topological_space) nhds_neq_bot [simp]: "nhds a ≠ bot" by (simp add: trivial_limit_def eventually_nhds) lemma (in t1_space) t1_space_nhds: "x ≠ y ⟹ (∀⇩_{F}x in nhds x. x ≠ y)" by (drule t1_space) (auto simp: eventually_nhds) lemma (in topological_space) nhds_discrete_open: "open {x} ⟹ nhds x = principal {x}" by (auto simp: nhds_def intro!: antisym INF_greatest INF_lower2[of "{x}"]) lemma (in discrete_topology) nhds_discrete: "nhds x = principal {x}" by (simp add: nhds_discrete_open open_discrete) lemma (in discrete_topology) at_discrete: "at x within S = bot" unfolding at_within_def nhds_discrete by simp lemma (in topological_space) at_within_eq: "at x within s = (INF S:{S. open S ∧ x ∈ S}. principal (S ∩ s - {x}))" unfolding nhds_def at_within_def by (subst INF_inf_const2[symmetric]) (auto simp: Diff_Int_distrib) lemma (in topological_space) eventually_at_filter: "eventually P (at a within s) ⟷ eventually (λx. x ≠ a ⟶ x ∈ s ⟶ P x) (nhds a)" by (simp add: at_within_def eventually_inf_principal imp_conjL[symmetric] conj_commute) lemma (in topological_space) at_le: "s ⊆ t ⟹ at x within s ≤ at x within t" unfolding at_within_def by (intro inf_mono) auto lemma (in topological_space) eventually_at_topological: "eventually P (at a within s) ⟷ (∃S. open S ∧ a ∈ S ∧ (∀x∈S. x ≠ a ⟶ x ∈ s ⟶ P x))" by (simp add: eventually_nhds eventually_at_filter) lemma (in topological_space) at_within_open: "a ∈ S ⟹ open S ⟹ at a within S = at a" unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I) lemma (in topological_space) at_within_open_NO_MATCH: "a ∈ s ⟹ open s ⟹ NO_MATCH UNIV s ⟹ at a within s = at a" by (simp only: at_within_open) lemma (in topological_space) at_within_open_subset: "a ∈ S ⟹ open S ⟹ S ⊆ T ⟹ at a within T = at a" by (metis at_le at_within_open dual_order.antisym subset_UNIV) lemma (in topological_space) at_within_nhd: assumes "x ∈ S" "open S" "T ∩ S - {x} = U ∩ 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 (λx. x ∈ S) (nhds x)" using ‹x ∈ S› ‹open S› by (auto simp: eventually_nhds) then show "∀⇩_{F}n in nhds x. (n ≠ x ⟶ n ∈ T ⟶ P n) = (n ≠ x ⟶ n ∈ U ⟶ P n)" for P by eventually_elim (insert ‹T ∩ S - {x} = U ∩ S - {x}›, blast) qed lemma (in topological_space) at_within_empty [simp]: "at a within {} = bot" unfolding at_within_def by simp lemma (in topological_space) at_within_union: "at x within (S ∪ 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 (in topological_space) at_eq_bot_iff: "at a = bot ⟷ open {a}" unfolding trivial_limit_def eventually_at_topological apply safe apply (case_tac "S = {a}") apply simp apply fast apply fast done lemma (in perfect_space) at_neq_bot [simp]: "at a ≠ bot" by (simp add: at_eq_bot_iff not_open_singleton) lemma (in order_topology) nhds_order: "nhds x = inf (INF a:{x <..}. principal {..< a}) (INF a:{..< x}. principal {a <..})" proof - have 1: "{S ∈ range lessThan ∪ range greaterThan. x ∈ S} = (λa. {..< a}) ` {x <..} ∪ (λa. {a <..}) ` {..< x}" by auto show ?thesis by (simp only: nhds_generated_topology[OF open_generated_order] INF_union 1 INF_image comp_def) qed lemma (in topological_space) filterlim_at_within_If: assumes "filterlim f G (at x within (A ∩ {x. P x}))" and "filterlim g G (at x within (A ∩ {x. ¬P x}))" shows "filterlim (λx. if P x then f x else g x) G (at x within A)" proof (rule filterlim_If) note assms(1) also have "at x within (A ∩ {x. P x}) = inf (nhds x) (principal (A ∩ Collect P - {x}))" by (simp add: at_within_def) also have "A ∩ Collect P - {x} = (A - {x}) ∩ Collect P" by blast also have "inf (nhds x) (principal …) = inf (at x within A) (principal (Collect P))" by (simp add: at_within_def inf_assoc) finally show "filterlim f G (inf (at x within A) (principal (Collect P)))" . next note assms(2) also have "at x within (A ∩ {x. ¬ P x}) = inf (nhds x) (principal (A ∩ {x. ¬ P x} - {x}))" by (simp add: at_within_def) also have "A ∩ {x. ¬ P x} - {x} = (A - {x}) ∩ {x. ¬ P x}" by blast also have "inf (nhds x) (principal …) = inf (at x within A) (principal {x. ¬ P x})" by (simp add: at_within_def inf_assoc) finally show "filterlim g G (inf (at x within A) (principal {x. ¬ P x}))" . qed lemma (in topological_space) filterlim_at_If: assumes "filterlim f G (at x within {x. P x})" and "filterlim g G (at x within {x. ¬P x})" shows "filterlim (λx. if P x then f x else g x) G (at x)" using assms by (intro filterlim_at_within_If) simp_all lemma (in linorder_topology) at_within_order: assumes "UNIV ≠ {x}" shows "at x within s = inf (INF a:{x <..}. principal ({..< a} ∩ s - {x})) (INF a:{..< x}. principal ({a <..} ∩ s - {x}))" proof (cases "{x <..} = {}" "{..< x} = {}" rule: case_split [case_product case_split]) case True_True have "UNIV = {..< x} ∪ {x} ∪ {x <..}" by auto with assms True_True show ?thesis by auto qed (auto simp del: inf_principal simp: at_within_def nhds_order Int_Diff inf_principal[symmetric] INF_inf_const2 inf_sup_aci[where 'a="'a filter"]) lemma (in linorder_topology) at_left_eq: "y < x ⟹ 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 ⟹ eventually P (at_left x) ⟷ (∃b<x. ∀y>b. y < x ⟶ 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 ⟹ 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 ⟹ eventually P (at_right x) ⟷ (∃b>x. ∀y>x. y < b ⟶ P y)" unfolding at_right_eq by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def) lemma eventually_at_right_less: "∀⇩_{F}y in at_right (x::'a::{linorder_topology, no_top}). x < y" using gt_ex[of x] eventually_at_right[of x] by auto lemma trivial_limit_at_right_top: "at_right (top::_::{order_top,linorder_topology}) = bot" by (auto simp: filter_eq_iff eventually_at_topological) lemma trivial_limit_at_left_bot: "at_left (bot::_::{order_bot,linorder_topology}) = bot" by (auto simp: filter_eq_iff eventually_at_topological) lemma trivial_limit_at_left_real [simp]: "¬ trivial_limit (at_left x)" for 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]: "¬ trivial_limit (at_right x)" for 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 (in linorder_topology) at_eq_sup_left_right: "at x = sup (at_left x) (at_right x)" by (auto simp: eventually_at_filter filter_eq_iff eventually_sup elim: eventually_elim2 eventually_mono) lemma (in linorder_topology) eventually_at_split: "eventually P (at x) ⟷ eventually P (at_left x) ∧ eventually P (at_right x)" by (subst at_eq_sup_left_right) (simp add: eventually_sup) lemma (in order_topology) eventually_at_leftI: assumes "⋀x. x ∈ {a<..<b} ⟹ P x" "a < b" shows "eventually P (at_left b)" using assms unfolding eventually_at_topological by (intro exI[of _ "{a<..}"]) auto lemma (in order_topology) eventually_at_rightI: assumes "⋀x. x ∈ {a<..<b} ⟹ P x" "a < b" shows "eventually P (at_right a)" using assms unfolding eventually_at_topological by (intro exI[of _ "{..<b}"]) auto lemma eventually_filtercomap_nhds: "eventually P (filtercomap f (nhds x)) ⟷ (∃S. open S ∧ x ∈ S ∧ (∀x. f x ∈ S ⟶ P x))" unfolding eventually_filtercomap eventually_nhds by auto lemma eventually_filtercomap_at_topological: "eventually P (filtercomap f (at A within B)) ⟷ (∃S. open S ∧ A ∈ S ∧ (∀x. f x ∈ S ∩ B - {A} ⟶ P x))" (is "?lhs = ?rhs") unfolding at_within_def filtercomap_inf eventually_inf_principal filtercomap_principal eventually_filtercomap_nhds eventually_principal by blast lemma eventually_at_right_field: "eventually P (at_right x) ⟷ (∃b>x. ∀y>x. y < b ⟶ P y)" for x :: "'a::{linordered_field, linorder_topology}" using linordered_field_no_ub[rule_format, of x] by (auto simp: eventually_at_right) lemma eventually_at_left_field: "eventually P (at_left x) ⟷ (∃b<x. ∀y>b. y < x ⟶ P y)" for x :: "'a::{linordered_field, linorder_topology}" using linordered_field_no_lb[rule_format, of x] by (auto simp: eventually_at_left) subsubsection ‹Tendsto› abbreviation (in topological_space) tendsto :: "('b ⇒ 'a) ⇒ 'a ⇒ 'b filter ⇒ bool" (infixr "⤏" 55) where "(f ⤏ l) F ≡ filterlim f (nhds l) F" definition (in t2_space) Lim :: "'f filter ⇒ ('f ⇒ 'a) ⇒ 'a" where "Lim A f = (THE l. (f ⤏ l) A)" lemma (in topological_space) tendsto_eq_rhs: "(f ⤏ x) F ⟹ x = y ⟹ (f ⤏ y) F" by simp named_theorems tendsto_intros "introduction rules for tendsto" setup ‹ Global_Theory.add_thms_dynamic (\<^binding>‹tendsto_eq_intros›, fn context => Named_Theorems.get (Context.proof_of context) @{named_theorems tendsto_intros} |> map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm]))) › context topological_space begin lemma tendsto_def: "(f ⤏ l) F ⟷ (∀S. open S ⟶ l ∈ S ⟶ eventually (λx. f x ∈ S) F)" unfolding nhds_def filterlim_INF filterlim_principal by auto lemma tendsto_cong: "(f ⤏ c) F ⟷ (g ⤏ c) F" if "eventually (λx. f x = g x) F" by (rule filterlim_cong [OF refl refl that]) lemma tendsto_mono: "F ≤ F' ⟹ (f ⤏ l) F' ⟹ (f ⤏ l) F" unfolding tendsto_def le_filter_def by fast lemma tendsto_ident_at [tendsto_intros, simp, intro]: "((λx. x) ⤏ a) (at a within s)" by (auto simp: tendsto_def eventually_at_topological) lemma tendsto_const [tendsto_intros, simp, intro]: "((λx. k) ⤏ k) F" by (simp add: tendsto_def) lemma filterlim_at: "(LIM x F. f x :> at b within s) ⟷ eventually (λx. f x ∈ s ∧ f x ≠ b) F ∧ (f ⤏ b) F" by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute) lemma (in -) assumes "filterlim f (nhds L) F" shows tendsto_imp_filterlim_at_right: "eventually (λx. f x > L) F ⟹ filterlim f (at_right L) F" and tendsto_imp_filterlim_at_left: "eventually (λx. f x < L) F ⟹ filterlim f (at_left L) F" using assms by (auto simp: filterlim_at elim: eventually_mono) lemma filterlim_at_withinI: assumes "filterlim f (nhds c) F" assumes "eventually (λx. f x ∈ A - {c}) F" shows "filterlim f (at c within A) F" using assms by (simp add: filterlim_at) lemma filterlim_atI: assumes "filterlim f (nhds c) F" assumes "eventually (λx. f x ≠ c) F" shows "filterlim f (at c) F" using assms by (intro filterlim_at_withinI) simp_all lemma topological_tendstoI: "(⋀S. open S ⟹ l ∈ S ⟹ eventually (λx. f x ∈ S) F) ⟹ (f ⤏ l) F" by (auto simp: tendsto_def) lemma topological_tendstoD: "(f ⤏ l) F ⟹ open S ⟹ l ∈ S ⟹ eventually (λx. f x ∈ S) F" by (auto simp: tendsto_def) lemma tendsto_bot [simp]: "(f ⤏ a) bot" by (simp add: tendsto_def) end lemma (in topological_space) filterlim_within_subset: "filterlim f l (at x within S) ⟹ T ⊆ S ⟹ filterlim f l (at x within T)" by (blast intro: filterlim_mono at_le) lemmas tendsto_within_subset = filterlim_within_subset lemma (in order_topology) order_tendsto_iff: "(f ⤏ x) F ⟷ (∀l<x. eventually (λx. l < f x) F) ∧ (∀u>x. eventually (λx. f x < u) F)" by (auto simp: nhds_order filterlim_inf filterlim_INF filterlim_principal) lemma (in order_topology) order_tendstoI: "(⋀a. a < y ⟹ eventually (λx. a < f x) F) ⟹ (⋀a. y < a ⟹ eventually (λx. f x < a) F) ⟹ (f ⤏ y) F" by (auto simp: order_tendsto_iff) lemma (in order_topology) order_tendstoD: assumes "(f ⤏ y) F" shows "a < y ⟹ eventually (λx. a < f x) F" and "y < a ⟹ eventually (λx. f x < a) F" using assms by (auto simp: order_tendsto_iff) lemma (in linorder_topology) tendsto_max[tendsto_intros]: assumes X: "(X ⤏ x) net" and Y: "(Y ⤏ y) net" shows "((λx. max (X x) (Y x)) ⤏ max x y) net" proof (rule order_tendstoI) fix a assume "a < max x y" then show "eventually (λx. 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_mono) next fix a assume "max x y < a" then show "eventually (λx. 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[tendsto_intros]: assumes X: "(X ⤏ x) net" and Y: "(Y ⤏ y) net" shows "((λx. min (X x) (Y x)) ⤏ min x y) net" proof (rule order_tendstoI) fix a assume "a < min x y" then show "eventually (λx. 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 (λx. 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_mono) qed lemma (in order_topology) assumes "a < b" shows at_within_Icc_at_right: "at a within {a..b} = at_right a" and at_within_Icc_at_left: "at b within {a..b} = at_left b" using order_tendstoD(2)[OF tendsto_ident_at assms, of "{a<..}"] using order_tendstoD(1)[OF tendsto_ident_at assms, of "{..<b}"] by (auto intro!: order_class.antisym filter_leI simp: eventually_at_filter less_le elim: eventually_elim2) lemma (in order_topology) at_within_Icc_at: "a < x ⟹ x < b ⟹ at x within {a..b} = at x" by (rule at_within_open_subset[where S="{a<..<b}"]) auto lemma (in t2_space) tendsto_unique: assumes "F ≠ bot" and "(f ⤏ a) F" and "(f ⤏ b) F" shows "a = b" proof (rule ccontr) assume "a ≠ b" obtain U V where "open U" "open V" "a ∈ U" "b ∈ V" "U ∩ V = {}" using hausdorff [OF ‹a ≠ b›] by fast have "eventually (λx. f x ∈ U) F" using ‹(f ⤏ a) F› ‹open U› ‹a ∈ U› by (rule topological_tendstoD) moreover have "eventually (λx. f x ∈ V) F" using ‹(f ⤏ b) F› ‹open V› ‹b ∈ V› by (rule topological_tendstoD) ultimately have "eventually (λx. False) F" proof eventually_elim case (elim x) then have "f x ∈ U ∩ V" by simp with ‹U ∩ V = {}› show ?case by simp qed with ‹¬ trivial_limit F› show "False" by (simp add: trivial_limit_def) qed lemma (in t2_space) tendsto_const_iff: fixes a b :: 'a assumes "¬ trivial_limit F" shows "((λx. a) ⤏ b) F ⟷ a = b" by (auto intro!: tendsto_unique [OF assms tendsto_const]) lemma (in order_topology) increasing_tendsto: assumes bdd: "eventually (λn. f n ≤ l) F" and en: "⋀x. x < l ⟹ eventually (λn. x < f n) F" shows "(f ⤏ l) F" using assms by (intro order_tendstoI) (auto elim!: eventually_mono) lemma (in order_topology) decreasing_tendsto: assumes bdd: "eventually (λn. l ≤ f n) F" and en: "⋀x. l < x ⟹ eventually (λn. f n < x) F" shows "(f ⤏ l) F" using assms by (intro order_tendstoI) (auto elim!: eventually_mono) lemma (in order_topology) tendsto_sandwich: assumes ev: "eventually (λn. f n ≤ g n) net" "eventually (λn. g n ≤ h n) net" assumes lim: "(f ⤏ c) net" "(h ⤏ c) net" shows "(g ⤏ c) net" proof (rule order_tendstoI) fix a show "a < c ⟹ eventually (λx. a < g x) net" using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2) next fix a show "c < a ⟹ eventually (λx. g x < a) net" using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2) qed lemma (in t1_space) limit_frequently_eq: assumes "F ≠ bot" and "frequently (λx. f x = c) F" and "(f ⤏ d) F" shows "d = c" proof (rule ccontr) assume "d ≠ c" from t1_space[OF this] obtain U where "open U" "d ∈ U" "c ∉ U" by blast with assms have "eventually (λx. f x ∈ U) F" unfolding tendsto_def by blast then have "eventually (λx. f x ≠ c) F" by eventually_elim (insert ‹c ∉ U›, blast) with assms(2) show False unfolding frequently_def by contradiction qed lemma (in t1_space) tendsto_imp_eventually_ne: assumes "(f ⤏ c) F" "c ≠ c'" shows "eventually (λz. f z ≠ c') F" proof (cases "F=bot") case True thus ?thesis by auto next case False show ?thesis proof (rule ccontr) assume "¬ eventually (λz. f z ≠ c') F" then have "frequently (λz. f z = c') F" by (simp add: frequently_def) from limit_frequently_eq[OF False this ‹(f ⤏ c) F›] and ‹c ≠ c'› show False by contradiction qed qed lemma (in linorder_topology) tendsto_le: assumes F: "¬ trivial_limit F" and x: "(f ⤏ x) F" and y: "(g ⤏ y) F" and ev: "eventually (λx. g x ≤ f x) F" shows "y ≤ x" proof (rule ccontr) assume "¬ y ≤ x" with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a} ∩ {b<..} = {}" by (auto simp: not_le) then have "eventually (λx. f x < a) F" "eventually (λx. b < g x) F" using x y by (auto intro: order_tendstoD) with ev have "eventually (λx. False) F" by eventually_elim (insert xy, fastforce) with F show False by (simp add: eventually_False) qed lemma (in linorder_topology) tendsto_lowerbound: assumes x: "(f ⤏ x) F" and ev: "eventually (λi. a ≤ f i) F" and F: "¬ trivial_limit F" shows "a ≤ x" using F x tendsto_const ev by (rule tendsto_le) lemma (in linorder_topology) tendsto_upperbound: assumes x: "(f ⤏ x) F" and ev: "eventually (λi. a ≥ f i) F" and F: "¬ trivial_limit F" shows "a ≥ x" by (rule tendsto_le [OF F tendsto_const x ev]) lemma filterlim_at_within_not_equal: fixes f::"'a ⇒ 'b::t2_space" assumes "filterlim f (at a within s) F" shows "eventually (λw. f w∈s ∧ f w ≠b) F" proof (cases "a=b") case True then show ?thesis using assms by (simp add: filterlim_at) next case False from hausdorff[OF this] obtain U V where UV:"open U" "open V" "a ∈ U" "b ∈ V" "U ∩ V = {}" by auto have "(f ⤏ a) F" using assms filterlim_at by auto then have "∀⇩_{F}x in F. f x ∈ U" using UV unfolding tendsto_def by auto moreover have "∀⇩_{F}x in F. f x ∈ s ∧ f x≠a" using assms filterlim_at by auto ultimately show ?thesis apply eventually_elim using UV by auto qed subsubsection ‹Rules about @{const Lim}› lemma tendsto_Lim: "¬ trivial_limit net ⟹ (f ⤏ l) net ⟹ Lim net f = l" unfolding Lim_def using tendsto_unique [of net f] by auto lemma Lim_ident_at: "¬ trivial_limit (at x within s) ⟹ Lim (at x within s) (λx. x) = x" by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto lemma eventually_Lim_ident_at: "(∀⇩_{F}y in at x within X. P (Lim (at x within X) (λx. x)) y) ⟷ (∀⇩_{F}y in at x within X. P x y)" for x::"'a::t2_space" by (cases "at x within X = bot") (auto simp: Lim_ident_at) lemma filterlim_at_bot_at_right: fixes f :: "'a::linorder_topology ⇒ 'b::linorder" assumes mono: "⋀x y. Q x ⟹ Q y ⟹ x ≤ y ⟹ f x ≤ f y" and bij: "⋀x. P x ⟹ f (g x) = x" "⋀x. P x ⟹ Q (g x)" and Q: "eventually Q (at_right a)" and bound: "⋀b. Q b ⟹ a < b" and P: "eventually P at_bot" shows "filterlim f at_bot (at_right a)" proof - from P obtain x where x: "⋀y. y ≤ x ⟹ P y" unfolding eventually_at_bot_linorder by auto show ?thesis proof (intro filterlim_at_bot_le[THEN iffD2] allI impI) fix z assume "z ≤ x" with x have "P z" by auto have "eventually (λx. x ≤ g z) (at_right a)" using bound[OF bij(2)[OF ‹P z›]] unfolding eventually_at_right[OF bound[OF bij(2)[OF ‹P z›]]] by (auto intro!: exI[of _ "g z"]) with Q show "eventually (λx. f x ≤ z) (at_right a)" by eventually_elim (metis bij ‹P z› mono) qed qed lemma filterlim_at_top_at_left: fixes f :: "'a::linorder_topology ⇒ 'b::linorder" assumes mono: "⋀x y. Q x ⟹ Q y ⟹ x ≤ y ⟹ f x ≤ f y" and bij: "⋀x. P x ⟹ f (g x) = x" "⋀x. P x ⟹ Q (g x)" and Q: "eventually Q (at_left a)" and bound: "⋀b. Q b ⟹ b < a" and P: "eventually P at_top" shows "filterlim f at_top (at_left a)" proof - from P obtain x where x: "⋀y. x ≤ y ⟹ P y" unfolding eventually_at_top_linorder by auto show ?thesis proof (intro filterlim_at_top_ge[THEN iffD2] allI impI) fix z assume "x ≤ z" with x have "P z" by auto have "eventually (λx. g z ≤ x) (at_left a)" using bound[OF bij(2)[OF ‹P z›]] unfolding eventually_at_left[OF bound[OF bij(2)[OF ‹P z›]]] by (auto intro!: exI[of _ "g z"]) with Q show "eventually (λx. z ≤ f x) (at_left a)" by eventually_elim (metis bij ‹P z› mono) qed qed lemma filterlim_split_at: "filterlim f F (at_left x) ⟹ filterlim f F (at_right x) ⟹ filterlim f F (at x)" for x :: "'a::linorder_topology" by (subst at_eq_sup_left_right) (rule filterlim_sup) lemma filterlim_at_split: "filterlim f F (at x) ⟷ filterlim f F (at_left x) ∧ filterlim f F (at_right x)" for x :: "'a::linorder_topology" by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup) lemma eventually_nhds_top: fixes P :: "'a :: {order_top,linorder_topology} ⇒ bool" and b :: 'a assumes "b < top" shows "eventually P (nhds top) ⟷ (∃b<top. (∀z. b < z ⟶ P z))" unfolding eventually_nhds proof safe fix S :: "'a set" assume "open S" "top ∈ S" note open_left[OF this ‹b < top›] moreover assume "∀s∈S. P s" ultimately show "∃b<top. ∀z>b. P z" by (auto simp: subset_eq Ball_def) next fix b assume "b < top" "∀z>b. P z" then show "∃S. open S ∧ top ∈ S ∧ (∀xa∈S. P xa)" by (intro exI[of _ "{b <..}"]) auto qed lemma tendsto_at_within_iff_tendsto_nhds: "(g ⤏ g l) (at l within S) ⟷ (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_mono) subsection ‹Limits on sequences› abbreviation (in topological_space) LIMSEQ :: "[nat ⇒ 'a, 'a] ⇒ bool" ("((_)/ ⇢ (_))" [60, 60] 60) where "X ⇢ L ≡ (X ⤏ L) sequentially" abbreviation (in t2_space) lim :: "(nat ⇒ 'a) ⇒ 'a" where "lim X ≡ Lim sequentially X" definition (in topological_space) convergent :: "(nat ⇒ 'a) ⇒ bool" where "convergent X = (∃L. X ⇢ L)" lemma lim_def: "lim X = (THE L. X ⇢ L)" unfolding Lim_def .. subsection ‹Monotone sequences and subsequences› text ‹ 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. › definition monoseq :: "(nat ⇒ 'a::order) ⇒ bool" where "monoseq X ⟷ (∀m. ∀n≥m. X m ≤ X n) ∨ (∀m. ∀n≥m. X n ≤ X m)" abbreviation incseq :: "(nat ⇒ 'a::order) ⇒ bool" where "incseq X ≡ mono X" lemma incseq_def: "incseq X ⟷ (∀m. ∀n≥m. X n ≥ X m)" unfolding mono_def .. abbreviation decseq :: "(nat ⇒ 'a::order) ⇒ bool" where "decseq X ≡ antimono X" lemma decseq_def: "decseq X ⟷ (∀m. ∀n≥m. X n ≤ X m)" unfolding antimono_def .. subsubsection ‹Definition of subsequence.› (* For compatibility with the old "subseq" *) lemma strict_mono_leD: "strict_mono r ⟹ m ≤ n ⟹ r m ≤ r n" by (erule (1) monoD [OF strict_mono_mono]) lemma strict_mono_id: "strict_mono id" by (simp add: strict_mono_def) lemma incseq_SucI: "(⋀n. X n ≤ X (Suc n)) ⟹ incseq X" using lift_Suc_mono_le[of X] by (auto simp: incseq_def) lemma incseqD: "incseq f ⟹ i ≤ j ⟹ f i ≤ f j" by (auto simp: incseq_def) lemma incseq_SucD: "incseq A ⟹ A i ≤ A (Suc i)" using incseqD[of A i "Suc i"] by auto lemma incseq_Suc_iff: "incseq f ⟷ (∀n. f n ≤ f (Suc n))" by (auto intro: incseq_SucI dest: incseq_SucD) lemma incseq_const[simp, intro]: "incseq (λx. k)" unfolding incseq_def by auto lemma decseq_SucI: "(⋀n. X (Suc n) ≤ X n) ⟹ decseq X" using order.lift_Suc_mono_le[OF dual_order, of X] by (auto simp: decseq_def) lemma decseqD: "decseq f ⟹ i ≤ j ⟹ f j ≤ f i" by (auto simp: decseq_def) lemma decseq_SucD: "decseq A ⟹ A (Suc i) ≤ A i" using decseqD[of A i "Suc i"] by auto lemma decseq_Suc_iff: "decseq f ⟷ (∀n. f (Suc n) ≤ f n)" by (auto intro: decseq_SucI dest: decseq_SucD) lemma decseq_const[simp, intro]: "decseq (λx. k)" unfolding decseq_def by auto lemma monoseq_iff: "monoseq X ⟷ incseq X ∨ decseq X" unfolding monoseq_def incseq_def decseq_def .. lemma monoseq_Suc: "monoseq X ⟷ (∀n. X n ≤ X (Suc n)) ∨ (∀n. X (Suc n) ≤ X n)" unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff .. lemma monoI1: "∀m. ∀n ≥ m. X m ≤ X n ⟹ monoseq X" by (simp add: monoseq_def) lemma monoI2: "∀m. ∀n ≥ m. X n ≤ X m ⟹ monoseq X" by (simp add: monoseq_def) lemma mono_SucI1: "∀n. X n ≤ X (Suc n) ⟹ monoseq X" by (simp add: monoseq_Suc) lemma mono_SucI2: "∀n. X (Suc n) ≤ X n ⟹ monoseq X" by (simp add: monoseq_Suc) lemma monoseq_minus: fixes a :: "nat ⇒ 'a::ordered_ab_group_add" assumes "monoseq a" shows "monoseq (λ n. - a n)" proof (cases "∀m. ∀n ≥ m. a m ≤ a n") case True then have "∀m. ∀n ≥ m. - a n ≤ - a m" by auto then show ?thesis by (rule monoI2) next case False then have "∀m. ∀n ≥ m. - a m ≤ - a n" using ‹monoseq a›[unfolded monoseq_def] by auto then show ?thesis by (rule monoI1) qed subsubsection ‹Subsequence (alternative definition, (e.g. Hoskins)› lemma strict_mono_Suc_iff: "strict_mono f ⟷ (∀n. f n < f (Suc n))" proof (intro iffI strict_monoI) assume *: "∀n. f n < f (Suc n)" fix m n :: nat assume "m < n" thus "f m < f n" by (induction rule: less_Suc_induct) (use * in auto) qed (auto simp: strict_mono_def) lemma strict_mono_add: "strict_mono (λn::'a::linordered_semidom. n + k)" by (auto simp: strict_mono_def) text ‹For any sequence, there is a monotonic subsequence.› lemma seq_monosub: fixes s :: "nat ⇒ 'a::linorder" shows "∃f. strict_mono f ∧ monoseq (λn. (s (f n)))" proof (cases "∀n. ∃p>n. ∀m≥p. s m ≤ s p") case True then have "∃f. ∀n. (∀m≥f n. s m ≤ s (f n)) ∧ f n < f (Suc n)" by (intro dependent_nat_choice) (auto simp: conj_commute) then obtain f :: "nat ⇒ nat" where f: "strict_mono f" and mono: "⋀n m. f n ≤ m ⟹ s m ≤ s (f n)" by (auto simp: strict_mono_Suc_iff) then have "incseq f" unfolding strict_mono_Suc_iff incseq_Suc_iff by (auto intro: less_imp_le) then have "monoseq (λn. s (f n))" by (auto simp add: incseq_def intro!: mono monoI2) with f show ?thesis by auto next case False then obtain N where N: "p > N ⟹ ∃m>p. s p < s m" for p by (force simp: not_le le_less) have "∃f. ∀n. N < f n ∧ f n < f (Suc n) ∧ s (f n) ≤ s (f (Suc n))" proof (intro dependent_nat_choice) fix x assume "N < x" with N[of x] show "∃y>N. x < y ∧ s x ≤ s y" by (auto intro: less_trans) qed auto then show ?thesis by (auto simp: monoseq_iff incseq_Suc_iff strict_mono_Suc_iff) qed lemma seq_suble: assumes sf: "strict_mono (f :: nat ⇒ nat)" shows "n ≤ f n" proof (induct n) case 0 show ?case by simp next case (Suc n) with sf [unfolded strict_mono_Suc_iff, rule_format, of n] have "n < f (Suc n)" by arith then show ?case by arith qed lemma eventually_subseq: "strict_mono r ⟹ eventually P sequentially ⟹ eventually (λn. P (r n)) sequentially" unfolding eventually_sequentially by (metis seq_suble le_trans) lemma not_eventually_sequentiallyD: assumes "¬ eventually P sequentially" shows "∃r::nat⇒nat. strict_mono r ∧ (∀n. ¬ P (r n))" proof - from assms have "∀n. ∃m≥n. ¬ P m" unfolding eventually_sequentially by (simp add: not_less) then obtain r where "⋀n. r n ≥ n" "⋀n. ¬ P (r n)" by (auto simp: choice_iff) then show ?thesis by (auto intro!: exI[of _ "λn. r (((Suc ∘ r) ^^ Suc n) 0)"] simp: less_eq_Suc_le strict_mono_Suc_iff) qed lemma sequentially_offset: assumes "eventually (λi. P i) sequentially" shows "eventually (λi. P (i + k)) sequentially" using assms by (rule eventually_sequentially_seg [THEN iffD2]) lemma seq_offset_neg: "(f ⤏ l) sequentially ⟹ ((λi. f(i - k)) ⤏ l) sequentially" apply (erule filterlim_compose) apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially, arith) done lemma filterlim_subseq: "strict_mono f ⟹ filterlim f sequentially sequentially" unfolding filterlim_iff by (metis eventually_subseq) lemma strict_mono_o: "strict_mono r ⟹ strict_mono s ⟹ strict_mono (r ∘ s)" unfolding strict_mono_def by simp lemma strict_mono_compose: "strict_mono r ⟹ strict_mono s ⟹ strict_mono (λx. r (s x))" using strict_mono_o[of r s] by (simp add: o_def) lemma incseq_imp_monoseq: "incseq X ⟹ monoseq X" by (simp add: incseq_def monoseq_def) lemma decseq_imp_monoseq: "decseq X ⟹ monoseq X" by (simp add: decseq_def monoseq_def) lemma decseq_eq_incseq: "decseq X = incseq (λn. - X n)" for X :: "nat ⇒ 'a::ordered_ab_group_add" by (simp add: decseq_def incseq_def) lemma INT_decseq_offset: assumes "decseq F" shows "(⋂i. F i) = (⋂i∈{n..}. F i)" proof safe fix x i assume x: "x ∈ (⋂i∈{n..}. F i)" show "x ∈ F i" proof cases from x have "x ∈ F n" by auto also assume "i ≤ n" with ‹decseq F› have "F n ⊆ F i" unfolding decseq_def by simp finally show ?thesis . qed (insert x, simp) qed auto lemma LIMSEQ_const_iff: "(λn. k) ⇢ l ⟷ k = l" for k l :: "'a::t2_space" using trivial_limit_sequentially by (rule tendsto_const_iff) lemma LIMSEQ_SUP: "incseq X ⟹ 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 ⟹ 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 ⟹ (λn. f (n + k)) ⇢ a" unfolding tendsto_def by (subst eventually_sequentially_seg[where k=k]) lemma LIMSEQ_offset: "(λn. f (n + k)) ⇢ a ⟹ f ⇢ a" unfolding tendsto_def by (subst (asm) eventually_sequentially_seg[where k=k]) lemma LIMSEQ_Suc: "f ⇢ l ⟹ (λn. f (Suc n)) ⇢ l" by (drule LIMSEQ_ignore_initial_segment [where k="Suc 0"]) simp lemma LIMSEQ_imp_Suc: "(λn. f (Suc n)) ⇢ l ⟹ f ⇢ l" by (rule LIMSEQ_offset [where k="Suc 0"]) simp lemma LIMSEQ_Suc_iff: "(λn. f (Suc n)) ⇢ l = f ⇢ l" by (rule filterlim_sequentially_Suc) lemma LIMSEQ_lessThan_iff_atMost: shows "(λn. f {..<n}) ⇢ x ⟷ (λn. f {..n}) ⇢ x" apply (subst LIMSEQ_Suc_iff [symmetric]) apply (simp only: lessThan_Suc_atMost) done lemma LIMSEQ_unique: "X ⇢ a ⟹ X ⇢ b ⟹ a = b" for a b :: "'a::t2_space" using trivial_limit_sequentially by (rule tendsto_unique) lemma LIMSEQ_le_const: "X ⇢ x ⟹ ∃N. ∀n≥N. a ≤ X n ⟹ a ≤ x" for a x :: "'a::linorder_topology" by (simp add: eventually_at_top_linorder tendsto_lowerbound) lemma LIMSEQ_le: "X ⇢ x ⟹ Y ⇢ y ⟹ ∃N. ∀n≥N. X n ≤ Y n ⟹ x ≤ y" for x y :: "'a::linorder_topology" using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially) lemma LIMSEQ_le_const2: "X ⇢ x ⟹ ∃N. ∀n≥N. X n ≤ a ⟹ x ≤ a" for a x :: "'a::linorder_topology" by (rule LIMSEQ_le[of X x "λn. a"]) auto lemma Lim_bounded: "f ⇢ l ⟹ ∀n≥M. f n ≤ C ⟹ l ≤ C" for l :: "'a::linorder_topology" by (intro LIMSEQ_le_const2) auto lemma Lim_bounded2: fixes f :: "nat ⇒ 'a::linorder_topology" assumes lim:"f ⇢ l" and ge: "∀n≥N. f n ≥ C" shows "l ≥ C" using ge by (intro tendsto_le[OF trivial_limit_sequentially lim tendsto_const]) (auto simp: eventually_sequentially) lemma lim_mono: fixes X Y :: "nat ⇒ 'a::linorder_topology" assumes "⋀n. N ≤ n ⟹ X n ≤ Y n" and "X ⇢ x" and "Y ⇢ y" shows "x ≤ y" using assms(1) by (intro LIMSEQ_le[OF assms(2,3)]) auto lemma Sup_lim: fixes a :: "'a::{complete_linorder,linorder_topology}" assumes "⋀n. b n ∈ s" and "b ⇢ a" shows "a ≤ Sup s" by (metis Lim_bounded assms complete_lattice_class.Sup_upper) lemma Inf_lim: fixes a :: "'a::{complete_linorder,linorder_topology}" assumes "⋀n. b n ∈ s" and "b ⇢ a" shows "Inf s ≤ a" by (metis Lim_bounded2 assms complete_lattice_class.Inf_lower) lemma SUP_Lim: fixes X :: "nat ⇒ 'a::{complete_linorder,linorder_topology}" assumes inc: "incseq X" and l: "X ⇢ l" shows "(SUP n. X n) = l" using LIMSEQ_SUP[OF inc] tendsto_unique[OF trivial_limit_sequentially l] by simp lemma INF_Lim: fixes X :: "nat ⇒ 'a::{complete_linorder,linorder_topology}" assumes dec: "decseq X" and l: "X ⇢ l" shows "(INF n. X n) = l" using LIMSEQ_INF[OF dec] tendsto_unique[OF trivial_limit_sequentially l] by simp lemma convergentD: "convergent X ⟹ ∃L. X ⇢ L" by (simp add: convergent_def) lemma convergentI: "X ⇢ L ⟹ convergent X" by (auto simp add: convergent_def) lemma convergent_LIMSEQ_iff: "convergent X ⟷ X ⇢ lim X" by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def) lemma convergent_const: "convergent (λn. c)" by (rule convergentI) (rule tendsto_const) lemma monoseq_le: "monoseq a ⟹ a ⇢ x ⟹ (∀n. a n ≤ x) ∧ (∀m. ∀n≥m. a m ≤ a n) ∨ (∀n. x ≤ a n) ∧ (∀m. ∀n≥m. a n ≤ a m)" for x :: "'a::linorder_topology" by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff) lemma LIMSEQ_subseq_LIMSEQ: "X ⇢ L ⟹ strict_mono f ⟹ (X ∘ f) ⇢ L" unfolding comp_def by (rule filterlim_compose [of X, OF _ filterlim_subseq]) lemma convergent_subseq_convergent: "convergent X ⟹ strict_mono f ⟹ convergent (X ∘ f)" by (auto simp: convergent_def intro: LIMSEQ_subseq_LIMSEQ) lemma limI: "X ⇢ L ⟹ lim X = L" by (rule tendsto_Lim) (rule trivial_limit_sequentially) lemma lim_le: "convergent f ⟹ (⋀n. f n ≤ x) ⟹ lim f ≤ x" for x :: "'a::linorder_topology" using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff) lemma lim_const [simp]: "lim (λm. a) = a" by (simp add: limI) subsubsection ‹Increasing and Decreasing Series› lemma incseq_le: "incseq X ⟹ X ⇢ L ⟹ X n ≤ L" for L :: "'a::linorder_topology" by (metis incseq_def LIMSEQ_le_const) lemma decseq_ge: "decseq X ⟹ X ⇢ L ⟹ L ≤ X n" for L :: "'a::linorder_topology" by (metis decseq_def LIMSEQ_le_const2) subsection ‹First countable topologies› class first_countable_topology = topological_space + assumes first_countable_basis: "∃A::nat ⇒ 'a set. (∀i. x ∈ A i ∧ open (A i)) ∧ (∀S. open S ∧ x ∈ S ⟶ (∃i. A i ⊆ S))" lemma (in first_countable_topology) countable_basis_at_decseq: obtains A :: "nat ⇒ 'a set" where "⋀i. open (A i)" "⋀i. x ∈ (A i)" "⋀S. open S ⟹ x ∈ S ⟹ eventually (λi. A i ⊆ S) sequentially" proof atomize_elim from first_countable_basis[of x] obtain A :: "nat ⇒ 'a set" where nhds: "⋀i. open (A i)" "⋀i. x ∈ A i" and incl: "⋀S. open S ⟹ x ∈ S ⟹ ∃i. A i ⊆ S" by auto define F where "F n = (⋂i≤n. A i)" for n show "∃A. (∀i. open (A i)) ∧ (∀i. x ∈ A i) ∧ (∀S. open S ⟶ x ∈ S ⟶ eventually (λi. A i ⊆ S) sequentially)" proof (safe intro!: exI[of _ F]) fix i show "open (F i)" using nhds(1) by (auto simp: F_def) show "x ∈ F i" using nhds(2) by (auto simp: F_def) next fix S assume "open S" "x ∈ S" from incl[OF this] obtain i where "F i ⊆ S" unfolding F_def by auto moreover have "⋀j. i ≤ j ⟹ F j ⊆ F i" by (simp add: Inf_superset_mono F_def image_mono) ultimately show "eventually (λi. F i ⊆ S) sequentially" by (auto simp: eventually_sequentially) qed qed lemma (in first_countable_topology) nhds_countable: obtains X :: "nat ⇒ 'a set" where "decseq X" "⋀n. open (X n)" "⋀n. x ∈ X n" "nhds x = (INF n. principal (X n))" proof - from first_countable_basis obtain A :: "nat ⇒ 'a set" where *: "⋀n. x ∈ A n" "⋀n. open (A n)" "⋀S. open S ⟹ x ∈ S ⟹ ∃i. A i ⊆ S" by metis show thesis proof show "decseq (λn. ⋂i≤n. A i)" by (simp add: antimono_iff_le_Suc atMost_Suc) show "x ∈ (⋂i≤n. A i)" "⋀n. open (⋂i≤n. A i)" for n using * by auto show "nhds x = (INF n. principal (⋂i≤n. A i))" using * unfolding nhds_def apply - apply (rule INF_eq) apply simp_all apply fastforce apply (intro exI [of _ "⋂i≤n. A i" for n] conjI open_INT) apply auto done qed qed lemma (in first_countable_topology) countable_basis: obtains A :: "nat ⇒ 'a set" where "⋀i. open (A i)" "⋀i. x ∈ A i" "⋀F. (∀n. F n ∈ A n) ⟹ F ⇢ x" proof atomize_elim obtain A :: "nat ⇒ 'a set" where *: "⋀i. open (A i)" "⋀i. x ∈ A i" "⋀S. open S ⟹ x ∈ S ⟹ eventually (λi. A i ⊆ S) sequentially" by (rule countable_basis_at_decseq) blast have "eventually (λn. F n ∈ S) sequentially" if "∀n. F n ∈ A n" "open S" "x ∈ S" for F S using *(3)[of S] that by (auto elim: eventually_mono simp: subset_eq) with * show "∃A. (∀i. open (A i)) ∧ (∀i. x ∈ A i) ∧ (∀F. (∀n. F n ∈ A n) ⟶ F ⇢ x)" by (intro exI[of _ A]) (auto simp: tendsto_def) qed lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within: assumes "∀f. (∀n. f n ∈ s) ∧ f ⇢ a ⟶ eventually (λn. P (f n)) sequentially" shows "eventually P (inf (nhds a) (principal s))" proof (rule ccontr) obtain A :: "nat ⇒ 'a set" where *: "⋀i. open (A i)" "⋀i. a ∈ A i" "⋀F. ∀n. F n ∈ A n ⟹ F ⇢ a" by (rule countable_basis) blast assume "¬ ?thesis" with * have "∃F. ∀n. F n ∈ s ∧ F n ∈ A n ∧ ¬ P (F n)" unfolding eventually_inf_principal eventually_nhds by (intro choice) fastforce then obtain F where F: "∀n. F n ∈ s" and "∀n. F n ∈ A n" and F': "∀n. ¬ P (F n)" by blast with * have "F ⇢ a" by auto then have "eventually (λn. P (F n)) sequentially" using assms F by simp then show False by (simp add: F') qed lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially: "eventually P (inf (nhds a) (principal s)) ⟷ (∀f. (∀n. f n ∈ s) ∧ f ⇢ a ⟶ eventually (λn. 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 ∈ S" "∀x∈S. x ∈ s ⟶ P x" by (auto simp: eventually_inf_principal eventually_nhds) moreover fix f assume "∀n. f n ∈ s" "f ⇢ a" ultimately show "eventually (λn. P (f n)) sequentially" by (auto dest!: topological_tendstoD elim: eventually_mono) qed lemma (in first_countable_topology) eventually_nhds_iff_sequentially: "eventually P (nhds a) ⟷ (∀f. f ⇢ a ⟶ eventually (λn. P (f n)) sequentially)" using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp lemma tendsto_at_iff_sequentially: "(f ⤏ a) (at x within s) ⟷ (∀X. (∀i. X i ∈ s - {x}) ⟶ X ⇢ x ⟶ ((f ∘ X) ⇢ a))" for f :: "'a::first_countable_topology ⇒ _" unfolding filterlim_def[of _ "nhds a"] le_filter_def eventually_filtermap at_within_def eventually_nhds_within_iff_sequentially comp_def by metis lemma approx_from_above_dense_linorder: fixes x::"'a::{dense_linorder, linorder_topology, first_countable_topology}" assumes "x < y" shows "∃u. (∀n. u n > x) ∧ (u ⇢ x)" proof - obtain A :: "nat ⇒ 'a set" where A: "⋀i. open (A i)" "⋀i. x ∈ A i" "⋀F. (∀n. F n ∈ A n) ⟹ F ⇢ x" by (metis first_countable_topology_class.countable_basis) define u where "u = (λn. SOME z. z ∈ A n ∧ z > x)" have "∃z. z ∈ U ∧ x < z" if "x ∈ U" "open U" for U using open_right[OF ‹open U› ‹x ∈ U› ‹x < y›] by (meson atLeastLessThan_iff dense less_imp_le subset_eq) then have *: "u n ∈ A n ∧ x < u n" for n using ‹x ∈ A n› ‹open (A n)› unfolding u_def by (metis (no_types, lifting) someI_ex) then have "u ⇢ x" using A(3) by simp then show ?thesis using * by auto qed lemma approx_from_below_dense_linorder: fixes x::"'a::{dense_linorder, linorder_topology, first_countable_topology}" assumes "x > y" shows "∃u. (∀n. u n < x) ∧ (u ⇢ x)" proof - obtain A :: "nat ⇒ 'a set" where A: "⋀i. open (A i)" "⋀i. x ∈ A i" "⋀F. (∀n. F n ∈ A n) ⟹ F ⇢ x" by (metis first_countable_topology_class.countable_basis) define u where "u = (λn. SOME z. z ∈ A n ∧ z < x)" have "∃z. z ∈ U ∧ z < x" if "x ∈ U" "open U" for U using open_left[OF ‹open U› ‹x ∈ U› ‹x > y›] by (meson dense greaterThanAtMost_iff less_imp_le subset_eq) then have *: "u n ∈ A n ∧ u n < x" for n using ‹x ∈ A n› ‹open (A n)› unfolding u_def by (metis (no_types, lifting) someI_ex) then have "u ⇢ x" using A(3) by simp then show ?thesis using * by auto qed subsection ‹Function limit at a point› abbreviation LIM :: "('a::topological_space ⇒ 'b::topological_space) ⇒ 'a ⇒ 'b ⇒ bool" ("((_)/ ─(_)/→ (_))" [60, 0, 60] 60) where "f ─a→ L ≡ (f ⤏ L) (at a)" lemma tendsto_within_open: "a ∈ S ⟹ open S ⟹ (f ⤏ l) (at a within S) ⟷ (f ─a→ l)" by (simp add: tendsto_def at_within_open[where S = S]) lemma tendsto_within_open_NO_MATCH: "a ∈ S ⟹ NO_MATCH UNIV S ⟹ open S ⟹ (f ⤏ l)(at a within S) ⟷ (f ⤏ l)(at a)" for f :: "'a::topological_space ⇒ 'b::topological_space" using tendsto_within_open by blast lemma LIM_const_not_eq[tendsto_intros]: "k ≠ L ⟹ ¬ (λx. k) ─a→ L" for a :: "'a::perfect_space" and k L :: "'b::t2_space" by (simp add: tendsto_const_iff) lemmas LIM_not_zero = LIM_const_not_eq [where L = 0] lemma LIM_const_eq: "(λx. k) ─a→ L ⟹ k = L" for a :: "'a::perfect_space" and k L :: "'b::t2_space" by (simp add: tendsto_const_iff) lemma LIM_unique: "f ─a→ L ⟹ f ─a→ M ⟹ L = M" for a :: "'a::perfect_space" and L M :: "'b::t2_space" using at_neq_bot by (rule tendsto_unique) text ‹Limits are equal for functions equal except at limit point.› lemma LIM_equal: "∀x. x ≠ a ⟶ f x = g x ⟹ (f ─a→ l) ⟷ (g ─a→ l)" by (simp add: tendsto_def eventually_at_topological) lemma LIM_cong: "a = b ⟹ (⋀x. x ≠ b ⟹ f x = g x) ⟹ l = m ⟹ (f ─a→ l) ⟷ (g ─b→ m)" by (simp add: LIM_equal) lemma LIM_cong_limit: "f ─x→ L ⟹ K = L ⟹ f ─x→ K" by simp lemma tendsto_at_iff_tendsto_nhds: "g ─l→ g l ⟷ (g ⤏ g l) (nhds l)" unfolding tendsto_def eventually_at_filter by (intro ext all_cong imp_cong) (auto elim!: eventually_mono) lemma tendsto_compose: "g ─l→ g l ⟹ (f ⤏ l) F ⟹ ((λx. g (f x)) ⤏ g l) F" unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g]) lemma tendsto_compose_eventually: "g ─l→ m ⟹ (f ⤏ l) F ⟹ eventually (λx. f x ≠ l) F ⟹ ((λx. g (f x)) ⤏ m) F" by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at) lemma LIM_compose_eventually: assumes "f ─a→ b" and "g ─b→ c" and "eventually (λx. f x ≠ b) (at a)" shows "(λx. g (f x)) ─a→ c" using assms(2,1,3) by (rule tendsto_compose_eventually) lemma tendsto_compose_filtermap: "((g ∘ f) ⤏ T) F ⟷ (g ⤏ T) (filtermap f F)" by (simp add: filterlim_def filtermap_filtermap comp_def) lemma tendsto_compose_at: assumes f: "(f ⤏ y) F" and g: "(g ⤏ z) (at y)" and fg: "eventually (λw. f w = y ⟶ g y = z) F" shows "((g ∘ f) ⤏ z) F" proof - have "(∀⇩_{F}a in F. f a ≠ y) ∨ g y = z" using fg by force moreover have "(g ⤏ z) (filtermap f F) ∨ ¬ (∀⇩_{F}a in F. f a ≠ y)" by (metis (no_types) filterlim_atI filterlim_def tendsto_mono f g) ultimately show ?thesis by (metis (no_types) f filterlim_compose filterlim_filtermap g tendsto_at_iff_tendsto_nhds tendsto_compose_filtermap) qed subsubsection ‹Relation of ‹LIM› and ‹LIMSEQ›› lemma (in first_countable_topology) sequentially_imp_eventually_within: "(∀f. (∀n. f n ∈ s ∧ f n ≠ a) ∧ f ⇢ a ⟶ eventually (λn. P (f n)) sequentially) ⟹ 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: "(∀f. (∀n. f n ≠ a) ∧ f ⇢ a ⟶ eventually (λn. P (f n)) sequentially) ⟹ eventually P (at a)" using sequentially_imp_eventually_within [where s=UNIV] by simp lemma LIMSEQ_SEQ_conv1: fixes f :: "'a::topological_space ⇒ 'b::topological_space" assumes f: "f ─a→ l" shows "∀S. (∀n. S n ≠ a) ∧ S ⇢ a ⟶ (λn. 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 ⇒ 'b::topological_space" assumes "∀S. (∀n. S n ≠ a) ∧ S ⇢ a ⟶ (λn. 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: "(∀S. (∀n. S n ≠ a) ∧ S ⇢ a ⟶ (λn. X (S n)) ⇢ L) ⟷ X ─a→ L" for a :: "'a::first_countable_topology" and 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" and *: "⋀f. (⋀n. b < f n) ⟹ (⋀n. f n < a) ⟹ incseq f ⟹ f ⇢ a ⟹ eventually (λn. P (f n)) sequentially" shows "eventually P (at_left a)" proof (safe intro!: sequentially_imp_eventually_within) fix X assume X: "∀n. X n ∈ {..< a} ∧ X n ≠ a" "X ⇢ a" show "eventually (λn. P (X n)) sequentially" proof (rule ccontr) assume neg: "¬ ?thesis" have "∃s. ∀n. (¬ P (X (s n)) ∧ b < X (s n)) ∧ (X (s n) ≤ X (s (Suc n)) ∧ Suc (s n) ≤ s (Suc n))" (is "∃s. ?P s") proof (rule dependent_nat_choice) have "¬ eventually (λn. b < X n ⟶ P (X n)) sequentially" by (intro not_eventually_impI neg order_tendstoD(1) [OF X(2) b]) then show "∃x. ¬ P (X x) ∧ b < X x" by (auto dest!: not_eventuallyD) next fix x n have "¬ eventually (λn. Suc x ≤ n ⟶ b < X n ⟶ X x < X n ⟶ 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 "∃n. (¬ P (X n) ∧ b < X n) ∧ (X x ≤ X n ∧ Suc x ≤ n)" by (auto dest!: not_eventuallyD) qed then obtain s where "?P s" .. with X have "b < X (s n)" and "X (s n) < a" and "incseq (λn. X (s n))" and "(λn. X (s n)) ⇢ a" and "¬ P (X (s n))" for n by (auto simp: strict_mono_Suc_iff Suc_le_eq incseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF ‹X ⇢ a›, 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 b :: "'b::{linorder_topology,first_countable_topology}" assumes "b < a" assumes *: "⋀S. (⋀n. S n < a) ⟹ (⋀n. b < S n) ⟹ incseq S ⟹ S ⇢ a ⟹ (λn. X (S n)) ⇢ L" shows "(X ⤏ L) (at_left a)" using assms by (simp add: tendsto_def [where l=L] sequentially_imp_eventually_at_left) lemma sequentially_imp_eventually_at_right: fixes a b :: "'a::{linorder_topology,first_countable_topology}" assumes b[simp]: "a < b" assumes *: "⋀f. (⋀n. a < f n) ⟹ (⋀n. f n < b) ⟹ decseq f ⟹ f ⇢ a ⟹ eventually (λn. P (f n)) sequentially" shows "eventually P (at_right a)" proof (safe intro!: sequentially_imp_eventually_within) fix X assume X: "∀n. X n ∈ {a <..} ∧ X n ≠ a" "X ⇢ a" show "eventually (λn. P (X n)) sequentially" proof (rule ccontr) assume neg: "¬ ?thesis" have "∃s. ∀n. (¬ P (X (s n)) ∧ X (s n) < b) ∧ (X (s (Suc n)) ≤ X (s n) ∧ Suc (s n) ≤ s (Suc n))" (is "∃s. ?P s") proof (rule dependent_nat_choice) have "¬ eventually (λn. X n < b ⟶ P (X n)) sequentially" by (intro not_eventually_impI neg order_tendstoD(2) [OF X(2) b]) then show "∃x. ¬ P (X x) ∧ X x < b" by (auto dest!: not_eventuallyD) next fix x n have "¬ eventually (λn. Suc x ≤ n ⟶ X n < b ⟶ X n < X x ⟶ 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 "∃n. (¬ P (X n) ∧ X n < b) ∧ (X n ≤ X x ∧ Suc x ≤ n)" by (auto dest!: not_eventuallyD) qed then obtain s where "?P s" .. with X have "a < X (s n)" and "X (s n) < b" and "decseq (λn. X (s n))" and "(λn. X (s n)) ⇢ a" and "¬ P (X (s n))" for n by (auto simp: strict_mono_Suc_iff Suc_le_eq decseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF ‹X ⇢ a›, 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" and *: "⋀S. (⋀n. a < S n) ⟹ (⋀n. S n < b) ⟹ decseq S ⟹ S ⇢ a ⟹ (λn. X (S n)) ⇢ L" shows "(X ⤏ L) (at_right a)" using assms by (simp add: tendsto_def [where l=L] sequentially_imp_eventually_at_right) subsection ‹Continuity› subsubsection ‹Continuity on a set› definition continuous_on :: "'a set ⇒ ('a::topological_space ⇒ 'b::topological_space) ⇒ bool" where "continuous_on s f ⟷ (∀x∈s. (f ⤏ f x) (at x within s))" lemma continuous_on_cong [cong]: "s = t ⟹ (⋀x. x ∈ t ⟹ f x = g x) ⟹ continuous_on s f ⟷ continuous_on t g" unfolding continuous_on_def by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter) lemma continuous_on_strong_cong: "s = t ⟹ (⋀x. x ∈ t =simp=> f x = g x) ⟹ continuous_on s f ⟷ continuous_on t g" unfolding simp_implies_def by (rule continuous_on_cong) lemma continuous_on_topological: "continuous_on s f ⟷ (∀x∈s. ∀B. open B ⟶ f x ∈ B ⟶ (∃A. open A ∧ x ∈ A ∧ (∀y∈s. y ∈ A ⟶ f y ∈ B)))" unfolding continuous_on_def tendsto_def eventually_at_topological by metis lemma continuous_on_open_invariant: "continuous_on s f ⟷ (∀B. open B ⟶ (∃A. open A ∧ A ∩ s = f -` B ∩ s))" proof safe fix B :: "'b set" assume "continuous_on s f" "open B" then have "∀x∈f -` B ∩ s. (∃A. open A ∧ x ∈ A ∧ s ∩ A ⊆ f -` B)" by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL) then obtain A where "∀x∈f -` B ∩ s. open (A x) ∧ x ∈ A x ∧ s ∩ A x ⊆ f -` B" unfolding bchoice_iff .. then show "∃A. open A ∧ A ∩ s = f -` B ∩ s" by (intro exI[of _ "⋃x∈f -` B ∩ s. A x"]) auto next assume B: "∀B. open B ⟶ (∃A. open A ∧ A ∩ s = f -` B ∩ s)" show "continuous_on s f" unfolding continuous_on_topological proof safe fix x B assume "x ∈ s" "open B" "f x ∈ B" with B obtain A where A: "open A" "A ∩ s = f -` B ∩ s" by auto with ‹x ∈ s› ‹f x ∈ B› show "∃A. open A ∧ x ∈ A ∧ (∀y∈s. y ∈ A ⟶ f y ∈ B)" by (intro exI[of _ A]) auto qed qed lemma continuous_on_open_vimage: "open s ⟹ continuous_on s f ⟷ (∀B. open B ⟶ open (f -` B ∩ 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 ⊆ 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 by (simp add: continuous_on_open_vimage [OF open_UNIV]) lemma continuous_on_closed_invariant: "continuous_on s f ⟷ (∀B. closed B ⟶ (∃A. closed A ∧ A ∩ s = f -` B ∩ s))" proof - have *: "(⋀A. P A ⟷ Q (- A)) ⟹ (∀A. P A) ⟷ (∀A. Q A)" for P Q :: "'b set ⇒ bool" 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 ⟹ continuous_on s f ⟷ (∀B. closed B ⟶ closed (f -` B ∩ s))" unfolding continuous_on_closed_invariant by (metis closed_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s]) corollary closed_vimage_Int[continuous_intros]: assumes "closed s" and "continuous_on t f" and t: "closed t" shows "closed (f -` s ∩ t)" using assms by (simp add: continuous_on_closed_vimage [OF t]) corollary closed_vimage[continuous_intros]: assumes "closed s" and "continuous_on UNIV f" shows "closed (f -` s)" using closed_vimage_Int [OF assms] by simp lemma continuous_on_empty [simp]: "continuous_on {} f" by (simp add: continuous_on_def) lemma continuous_on_sing [simp]: "continuous_on {x} f" by (simp add: continuous_on_def at_within_def) lemma continuous_on_open_Union: "(⋀s. s ∈ S ⟹ open s) ⟹ (⋀s. s ∈ S ⟹ continuous_on s f) ⟹ continuous_on (⋃S) f" unfolding continuous_on_def by safe (metis open_Union at_within_open UnionI) lemma continuous_on_open_UN: "(⋀s. s ∈ S ⟹ open (A s)) ⟹ (⋀s. s ∈ S ⟹ continuous_on (A s) f) ⟹ continuous_on (⋃s∈S. A s) f" by (rule continuous_on_open_Union) auto lemma continuous_on_open_Un: "open s ⟹ open t ⟹ continuous_on s f ⟹ continuous_on t f ⟹ continuous_on (s ∪ t) f" using continuous_on_open_Union [of "{s,t}"] by auto lemma continuous_on_closed_Un: "closed s ⟹ closed t ⟹ continuous_on s f ⟹ continuous_on t f ⟹ continuous_on (s ∪ t) f" by (auto simp add: continuous_on_closed_vimage closed_Un Int_Un_distrib) lemma continuous_on_closed_Union: assumes "finite I" "⋀i. i ∈ I ⟹ closed (U i)" "⋀i. i ∈ I ⟹ continuous_on (U i) f" shows "continuous_on (⋃ i ∈ I. U i) f" using assms by (induction I) (auto intro!: continuous_on_closed_Un) lemma continuous_on_If: assumes closed: "closed s" "closed t" and cont: "continuous_on s f" "continuous_on t g" and P: "⋀x. x ∈ s ⟹ ¬ P x ⟹ f x = g x" "⋀x. x ∈ t ⟹ P x ⟹ f x = g x" shows "continuous_on (s ∪ t) (λx. if P x then f x else g x)" (is "continuous_on _ ?h") proof- from P have "∀x∈s. f x = ?h x" "∀x∈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_cases: "closed s ⟹ closed t ⟹ continuous_on s f ⟹ continuous_on t g ⟹ ∀x. (x∈s ∧ ¬ P x) ∨ (x ∈ t ∧ P x) ⟶ f x = g x ⟹ continuous_on (s ∪ t) (λx. if P x then f x else g x)" by (rule continuous_on_If) auto lemma continuous_on_id[continuous_intros,simp]: "continuous_on s (λx. x)" unfolding continuous_on_def by fast lemma continuous_on_id'[continuous_intros,simp]: "continuous_on s id" unfolding continuous_on_def id_def by fast lemma continuous_on_const[continuous_intros,simp]: "continuous_on s (λx. c)" unfolding continuous_on_def by auto lemma continuous_on_subset: "continuous_on s f ⟹ t ⊆ s ⟹ continuous_on t f" unfolding continuous_on_def by (metis subset_eq tendsto_within_subset) lemma continuous_on_compose[continuous_intros]: "continuous_on s f ⟹ continuous_on (f ` s) g ⟹ continuous_on s (g ∘ f)" unfolding continuous_on_topological by simp metis lemma continuous_on_compose2: "continuous_on t g ⟹ continuous_on s f ⟹ f ` s ⊆ t ⟹ continuous_on s (λx. g (f x))" using continuous_on_compose[of s f g] continuous_on_subset by (force simp add: comp_def) lemma continuous_on_generate_topology: assumes *: "open = generate_topology X" and **: "⋀B. B ∈ X ⟹ ∃C. open C ∧ C ∩ A = f -` B ∩ A" shows "continuous_on A f" unfolding continuous_on_open_invariant proof safe fix B :: "'a set" assume "open B" then show "∃C. open C ∧ C ∩ A = f -` B ∩ A" unfolding * proof induct case (UN K) then obtain C where "⋀k. k ∈ K ⟹ open (C k)" "⋀k. k ∈ K ⟹ C k ∩ A = f -` k ∩ A" by metis then show ?case by (intro exI[of _ "⋃k∈K. C k"]) blast qed (auto intro: **) qed lemma continuous_onI_mono: fixes f :: "'a::linorder_topology ⇒ 'b::{dense_order,linorder_topology}" assumes "open (f`A)" and mono: "⋀x y. x ∈ A ⟹ y ∈ A ⟹ x ≤ y ⟹ f x ≤ f y" shows "continuous_on A f" proof (rule continuous_on_generate_topology[OF open_generated_order], safe) have monoD: "⋀x y. x ∈ A ⟹ y ∈ A ⟹ f x < f y ⟹ x < y" by (auto simp: not_le[symmetric] mono) have "∃x. x ∈ A ∧ f x < b ∧ a < x" if a: "a ∈ A" and fa: "f a < b" for a b proof - obtain y where "f a < y" "{f a ..< y} ⊆ f`A" using open_right[OF ‹open (f`A)›, of "f a" b] a fa by auto obtain z where z: "f a < z" "z < min b y" using dense[of "f a" "min b y"] ‹f a < y› ‹f a < b› by auto then obtain c where "z = f c" "c ∈ A" using ‹{f a ..< y} ⊆ f`A›[THEN subsetD, of z] by (auto simp: less_imp_le) with a z show ?thesis by (auto intro!: exI[of _ c] simp: monoD) qed then show "∃C. open C ∧ C ∩ A = f -` {..<b} ∩ A" for b by (intro exI[of _ "(⋃x∈{x∈A. f x < b}. {..< x})"]) (auto intro: le_less_trans[OF mono] less_imp_le) have "∃x. x ∈ A ∧ b < f x ∧ x < a" if a: "a ∈ A" and fa: "b < f a" for a b proof - note a fa moreover obtain y where "y < f a" "{y <.. f a} ⊆ f`A" using open_left[OF ‹open (f`A)›, of "f a" b] a fa by auto then obtain z where z: "max b y < z" "z < f a" using dense[of "max b y" "f a"] ‹y < f a› ‹b < f a› by auto then obtain c where "z = f c" "c ∈ A" using ‹{y <.. f a} ⊆ f`A›[THEN subsetD, of z] by (auto simp: less_imp_le) with a z show ?thesis by (auto intro!: exI[of _ c] simp: monoD) qed then show "∃C. open C ∧ C ∩ A = f -` {b <..} ∩ A" for b by (intro exI[of _ "(⋃x∈{x∈A. b < f x}. {x <..})"]) (auto intro: less_le_trans[OF _ mono] less_imp_le) qed lemma continuous_on_IccI: "⟦(f ⤏ f a) (at_right a); (f ⤏ f b) (at_left b); (⋀x. a < x ⟹ x < b ⟹ f ─x→ f x); a < b⟧ ⟹ continuous_on {a .. b} f" for a::"'a::linorder_topology" using at_within_open[of _ "{a<..<b}"] by (auto simp: continuous_on_def at_within_Icc_at_right at_within_Icc_at_left le_less at_within_Icc_at) lemma fixes a b::"'a::linorder_topology" assumes "continuous_on {a .. b} f" "a < b" shows continuous_on_Icc_at_rightD: "(f ⤏ f a) (at_right a)" and continuous_on_Icc_at_leftD: "(f ⤏ f b) (at_left b)" using assms by (auto simp: at_within_Icc_at_right at_within_Icc_at_left continuous_on_def dest: bspec[where x=a] bspec[where x=b]) subsubsection ‹Continuity at a point› definition continuous :: "'a::t2_space filter ⇒ ('a ⇒ 'b::topological_space) ⇒ bool" where "continuous F f ⟷ (f ⤏ f (Lim F (λx. x))) F" lemma continuous_bot[continuous_intros, simp]: "continuous bot f" unfolding continuous_def by auto lemma continuous_trivial_limit: "trivial_limit net ⟹ continuous net f" by simp lemma continuous_within: "continuous (at x within s) f ⟷ (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 ⟷ (∀B. open B ⟶ f x ∈ B ⟶ (∃A. open A ∧ x ∈ A ∧ (∀y∈s. y ∈ A ⟶ f y ∈ B)))" unfolding continuous_within tendsto_def eventually_at_topological by metis lemma continuous_within_compose[continuous_intros]: "continuous (at x within s) f ⟹ continuous (at (f x) within f ` s) g ⟹ continuous (at x within s) (g ∘ f)" by (simp add: continuous_within_topological) metis lemma continuous_within_compose2: "continuous (at x within s) f ⟹ continuous (at (f x) within f ` s) g ⟹ continuous (at x within s) (λx. g (f x))" using continuous_within_compose[of x s f g] by (simp add: comp_def) lemma continuous_at: "continuous (at x) f ⟷ f ─x→ f x" using continuous_within[of x UNIV f] by simp lemma continuous_ident[continuous_intros, simp]: "continuous (at x within S) (λx. x)" unfolding continuous_within by (rule tendsto_ident_at) lemma continuous_const[continuous_intros, simp]: "continuous F (λx. c)" unfolding continuous_def by (rule tendsto_const) lemma continuous_on_eq_continuous_within: "continuous_on s f ⟷ (∀x∈s. continuous (at x within s) f)" unfolding continuous_on_def continuous_within .. abbreviation isCont :: "('a::t2_space ⇒ 'b::topological_space) ⇒ 'a ⇒ bool" where "isCont f a ≡ continuous (at a) f" lemma isCont_def: "isCont f a ⟷ f ─a→ f a" by (rule continuous_at) lemma isCont_cong: assumes "eventually (λx. f x = g x) (nhds x)" shows "isCont f x ⟷ isCont g x" proof - from assms have [simp]: "f x = g x" by (rule eventually_nhds_x_imp_x) from assms have "eventually (λx. f x = g x) (at x)" by (auto simp: eventually_at_filter elim!: eventually_mono) with assms have "isCont f x ⟷ isCont g x" unfolding isCont_def by (intro filterlim_cong) (auto elim!: eventually_mono) with assms show ?thesis by simp qed lemma continuous_at_imp_continuous_at_within: "isCont f x ⟹ 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 ⟹ continuous_on s f ⟷ (∀x∈s. isCont f x)" by (simp add: continuous_on_def continuous_at at_within_open[of _ s]) lemma continuous_within_open: "a ∈ A ⟹ open A ⟹ continuous (at a within A) f ⟷ isCont f a" by (simp add: at_within_open_NO_MATCH) lemma continuous_at_imp_continuous_on: "∀x∈s. isCont f x ⟹ 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 ⟹ isCont g (f a) ⟹ isCont (λx. g (f x)) a" unfolding isCont_def by (rule tendsto_compose) lemma continuous_at_compose[continuous_intros]: "isCont f a ⟹ isCont g (f a) ⟹ isCont (g ∘ f) a" unfolding o_def by (rule isCont_o2) lemma isCont_tendsto_compose: "isCont g l ⟹ (f ⤏ l) F ⟹ ((λx. g (f x)) ⤏ g l) F" unfolding isCont_def by (rule tendsto_compose) lemma continuous_on_tendsto_compose: assumes f_cont: "continuous_on s f" and g: "(g ⤏ l) F" and l: "l ∈ s" and ev: "∀⇩_{F}x in F. g x ∈ s" shows "((λx. f (g x)) ⤏ f l) F" proof - from f_cont l have f: "(f ⤏ f l) (at l within s)" by (simp add: continuous_on_def) have i: "((λx. if g x = l then f l else f (g x)) ⤏ f l) F" by (rule filterlim_If) (auto intro!: filterlim_compose[OF f] eventually_conj tendsto_mono[OF _ g] simp: filterlim_at eventually_inf_principal eventually_mono[OF ev]) show ?thesis by (rule filterlim_cong[THEN iffD1[OF _ i]]) auto qed lemma continuous_within_compose3: "isCont g (f x) ⟹ continuous (at x within s) f ⟹ continuous (at x within s) (λx. g (f x))" using continuous_at_imp_continuous_at_within continuous_within_compose2 by blast lemma filtermap_nhds_open_map: assumes cont: "isCont f a" and open_map: "⋀S. open S ⟹ 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 obtain S where "open S" "a ∈ S" "∀x∈S. P (f x)" by (auto simp: 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) f ⟷ continuous (at_left x) f ∧ continuous (at_right x) f" for x :: "'a::linorder_topology" by (simp add: continuous_within filterlim_at_split) text ‹ The following open/closed Collect lemmas are ported from Sébastien Gouëzel's ‹Ergodic_Theory›. › lemma open_Collect_neq: fixes f g :: "'a::topological_space ⇒ 'b::t2_space" assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g" shows "open {x. f x ≠ g x}" proof (rule openI) fix t assume "t ∈ {x. f x ≠ g x}" then obtain U V where *: "open U" "open V" "f t ∈ U" "g t ∈ V" "U ∩ V = {}" by (auto simp add: separation_t2) with open_vimage[OF ‹open U› f] open_vimage[OF ‹open V› g] show "∃T. open T ∧ t ∈ T ∧ T ⊆ {x. f x ≠ g x}" by (intro exI[of _ "f -` U ∩ g -` V"]) auto qed lemma closed_Collect_eq: fixes f g :: "'a::topological_space ⇒ 'b::t2_space" assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g" shows "closed {x. f x = g x}" using open_Collect_neq[OF f g] by (simp add: closed_def Collect_neg_eq) lemma open_Collect_less: fixes f g :: "'a::topological_space ⇒ 'b::linorder_topology" assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g" shows "open {x. f x < g x}" proof (rule openI) fix t assume t: "t ∈ {x. f x < g x}" show "∃T. open T ∧ t ∈ T ∧ T ⊆ {x. f x < g x}" proof (cases "∃z. f t < z ∧ z < g t") case True then obtain z where "f t < z ∧ z < g t" by blast then show ?thesis using open_vimage[OF _ f, of "{..< z}"] open_vimage[OF _ g, of "{z <..}"] by (intro exI[of _ "f -` {..<z} ∩ g -` {z<..}"]) auto next case False then have *: "{g t ..} = {f t <..}" "{..< g t} = {.. f t}" using t by (auto intro: leI) show ?thesis using open_vimage[OF _ f, of "{..< g t}"] open_vimage[OF _ g, of "{f t <..}"] t apply (intro exI[of _ "f -` {..< g t} ∩ g -` {f t<..}"]) apply (simp add: open_Int) apply (auto simp add: *) done qed qed lemma closed_Collect_le: fixes f g :: "'a :: topological_space ⇒ 'b::linorder_topology" assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g" shows "closed {x. f x ≤ g x}" using open_Collect_less [OF g f] by (simp add: closed_def Collect_neg_eq[symmetric] not_le) subsubsection ‹Open-cover compactness› context topological_space begin definition compact :: "'a set ⇒ bool" where compact_eq_heine_borel: (* This name is used for backwards compatibility *) "compact S ⟷ (∀C. (∀c∈C. open c) ∧ S ⊆ ⋃C ⟶ (∃D⊆C. finite D ∧ S ⊆ ⋃D))" lemma compactI: assumes "⋀C. ∀t∈C. open t ⟹ s ⊆ ⋃C ⟹ ∃C'. C' ⊆ C ∧ finite C' ∧ s ⊆ ⋃C'" shows "compact s" unfolding compact_eq_heine_borel using assms by metis lemma compact_empty[simp]: "compact {}" by (auto intro!: compactI) lemma compactE: (*related to COMPACT_IMP_HEINE_BOREL in HOL Light*) assumes "compact S" "S ⊆ ⋃𝒯" "⋀B. B ∈ 𝒯 ⟹ open B" obtains 𝒯' where "𝒯' ⊆ 𝒯" "finite 𝒯'" "S ⊆ ⋃𝒯'" by (meson assms compact_eq_heine_borel) lemma compactE_image: assumes "compact S" and opn: "⋀T. T ∈ C ⟹ open (f T)" and S: "S ⊆ (⋃c∈C. f c)" obtains C' where "C' ⊆ C" and "finite C'" and "S ⊆ (⋃c∈C'. f c)" apply (rule compactE[OF ‹compact S› S]) using opn apply force by (metis finite_subset_image) lemma compact_Int_closed [intro]: assumes "compact S" and "closed T" shows "compact (S ∩ T)" proof (rule compactI) fix C assume C: "∀c∈C. open c" assume cover: "S ∩ T ⊆ ⋃C" from C ‹closed T› have "∀c∈C ∪ {- T}. open c" by auto moreover from cover have "S ⊆ ⋃(C ∪ {- T})" by auto ultimately have "∃D⊆C ∪ {- T}. finite D ∧ S ⊆ ⋃D" using ‹compact S› unfolding compact_eq_heine_borel by auto then obtain D where "D ⊆ C ∪ {- T} ∧ finite D ∧ S ⊆ ⋃D" .. then show "∃D⊆C. finite D ∧ S ∩ T ⊆ ⋃D" by (intro exI[of _ "D - {-T}"]) auto qed lemma compact_diff: "⟦compact S; open T⟧ ⟹ compact(S - T)" by (simp add: Diff_eq compact_Int_closed open_closed) lemma inj_setminus: "inj_on uminus (A::'a set set)" by (auto simp: inj_on_def) subsection ‹Finite intersection property› lemma compact_fip: "compact U ⟷ (∀A. (∀a∈A. closed a) ⟶ (∀B ⊆ A. finite B ⟶ U ∩ ⋂B ≠ {}) ⟶ U ∩ ⋂A ≠ {})" (is "_ ⟷ ?R") proof (safe intro!: compact_eq_heine_borel[THEN iffD2]) fix A assume "compact U" assume A: "∀a∈A. closed a" "U ∩ ⋂A = {}" assume fin: "∀B ⊆ A. finite B ⟶ U ∩ ⋂B ≠ {}" from A have "(∀a∈uminus`A. open a) ∧ U ⊆ ⋃(uminus`A)" by auto with ‹compact U› obtain B where "B ⊆ A" "finite (uminus`B)" "U ⊆ ⋃(uminus`B)" unfolding compact_eq_heine_borel by (metis subset_image_iff) with fin[THEN spec, of B] show False by (auto dest: finite_imageD intro: inj_setminus) next fix A assume ?R assume "∀a∈A. open a" "U ⊆ ⋃A" then have "U ∩ ⋂(uminus`A) = {}" "∀a∈uminus`A. closed a" by auto with ‹?R› obtain B where "B ⊆ A" "finite (uminus`B)" "U ∩ ⋂(uminus`B) = {}" by (metis subset_image_iff) then show "∃T⊆A. finite T ∧ U ⊆ ⋃T" by (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD) qed lemma compact_imp_fip: assumes "compact S" and "⋀T. T ∈ F ⟹ closed T" and "⋀F'. finite F' ⟹ F' ⊆ F ⟹ S ∩ (⋂F') ≠ {}" shows "S ∩ (⋂F) ≠ {}" using assms unfolding compact_fip by auto lemma compact_imp_fip_image: assumes "compact s" and P: "⋀i. i ∈ I ⟹ closed (f i)" and Q: "⋀I'. finite I' ⟹ I' ⊆ I ⟹ (s ∩ (⋂i∈I'. f i) ≠ {})" shows "s ∩ (⋂i∈I. f i) ≠ {}" proof - note ‹compact s› moreover from P have "∀i ∈ f ` I. closed i" by blast moreover have "∀A. finite A ∧ A ⊆ f ` I ⟶ (s ∩ (⋂A) ≠ {})" apply rule apply rule apply (erule conjE) proof - fix A :: "'a set set" assume "finite A" and "A ⊆ f ` I" then obtain B where "B ⊆ I" and "finite B" and "A = f ` B" using finite_subset_image [of A f I] by blast with Q [of B] show "s ∩ ⋂A ≠ {}" by simp qed ultimately have "s ∩ (⋂(f ` I)) ≠ {}" by (metis 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 ∈ - s" let ?C = "⋃x∈s. {u. open u ∧ x ∈ u ∧ eventually (λy. y ∉ u) (nhds y)}" have "s ⊆ ⋃?C" proof fix x assume "x ∈ s" with ‹y ∈ - s› have "x ≠ y" by clarsimp then have "∃u v. open u ∧ open v ∧ x ∈ u ∧ y ∈ v ∧ u ∩ v = {}" by (rule hausdorff) with ‹x ∈ s› show "x ∈ ⋃?C" unfolding eventually_nhds by auto qed then obtain D where "D ⊆ ?C" and "finite D" and "s ⊆ ⋃D" by (rule compactE [OF ‹compact s›]) auto from ‹D ⊆ ?C› have "∀x∈D. eventually (λy. y ∉ x) (nhds y)" by auto with ‹finite D› have "eventually (λy. y ∉ ⋃D) (nhds y)" by (simp add: eventually_ball_finite) with ‹s ⊆ ⋃D› have "eventually (λy. y ∉ s) (nhds y)" by (auto elim!: eventually_mono) then show "∃t. open t ∧ y ∈ t ∧ t ⊆ - s" by (simp add: eventually_nhds subset_eq) qed lemma compact_continuous_image: assumes f: "continuous_on s f" and s: "compact s" shows "compact (f ` s)" proof (rule compactI) fix C assume "∀c∈C. open c" and cover: "f`s ⊆ ⋃C" with f have "∀c∈C. ∃A. open A ∧ A ∩ s = f -` c ∩ s" unfolding continuous_on_open_invariant by blast then obtain A where A: "∀c∈C. open (A c) ∧ A c ∩ s = f -` c ∩ s" unfolding bchoice_iff .. with cover have "⋀c. c ∈ C ⟹ open (A c)" "s ⊆ (⋃c∈C. A c)" by (fastforce simp add: subset_eq set_eq_iff)+ from compactE_image[OF s this] obtain D where "D ⊆ C" "finite D" "s ⊆ (⋃c∈D. A c)" . with A show "∃D ⊆ C. finite D ∧ f`s ⊆ ⋃D" by (intro exI[of _ D]) (fastforce simp add: subset_eq set_eq_iff)+ qed lemma continuous_on_inv: fixes f :: "'a::topological_space ⇒ 'b::t2_space" assumes "continuous_on s f" and "compact s" and "∀x∈s. g (f x) = x" shows "continuous_on (f ` s) g" unfolding continuous_on_topological proof (clarsimp simp add: assms(3)) fix x :: 'a and B :: "'a set" assume "x ∈ s" and "open B" and "x ∈ B" have 1: "∀x∈s. f x ∈ f ` (s - B) ⟷ x ∈ s - B" using assms(3) by (auto, metis) have "continuous_on (s - B) f" using ‹continuous_on s f› Diff_subset by (rule continuous_on_subset) moreover have "compact (s - B)" using ‹open B› and ‹compact s› unfolding Diff_eq by (intro compact_Int_closed closed_Compl) ultimately have "compact (f ` (s - B))" by (rule compact_continuous_image) then have "closed (f ` (s - B))" by (rule compact_imp_closed) then have "open (- f ` (s - B))" by (rule open_Compl) moreover have "f x ∈ - f ` (s - B)" using ‹x ∈ s› and ‹x ∈ B› by (simp add: 1) moreover have "∀y∈s. f y ∈ - f ` (s - B) ⟶ y ∈ B" by (simp add: 1) ultimately show "∃A. open A ∧ f x ∈ A ∧ (∀y∈s. f y ∈ A ⟶ y ∈ B)" by fast qed lemma continuous_on_inv_into: fixes f :: "'a::topological_space ⇒ '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 ≠ {}" shows "∃s∈S. ∀t∈S. t ≤ s" proof (rule classical) assume "¬ (∃s∈S. ∀t∈S. t ≤ s)" then obtain t where t: "∀s∈S. t s ∈ S" and "∀s∈S. s < t s" by (metis not_le) then have "⋀s. s∈S ⟹ open {..< t s}" "S ⊆ (⋃s∈S. {..< t s})" by auto with ‹compact S› obtain C where "C ⊆ S" "finite C" and C: "S ⊆ (⋃s∈C. {..< t s})" by (metis compactE_image) with ‹S ≠ {}› have Max: "Max (t`C) ∈ t`C" and "∀s∈t`C. s ≤ Max (t`C)" by (auto intro!: Max_in) with C have "S ⊆ {..< Max (t`C)}" by (auto intro: less_le_trans simp: subset_eq) with t Max ‹C ⊆ S› show ?thesis by fastforce qed lemma (in linorder_topology) compact_attains_inf: assumes "compact S" "S ≠ {}" shows "∃s∈S. ∀t∈S. s ≤ t" proof (rule classical) assume "¬ (∃s∈S. ∀t∈S. s ≤ t)" then obtain t where t: "∀s∈S. t s ∈ S" and "∀s∈S. t s < s" by (metis not_le) then have "⋀s. s∈S ⟹ open {t s <..}" "S ⊆ (⋃s∈S. {t s <..})" by auto with ‹compact S› obtain C where "C ⊆ S" "finite C" and C: "S ⊆ (⋃s∈C. {t s <..})" by (metis compactE_image) with ‹S ≠ {}› have Min: "Min (t`C) ∈ t`C" and "∀s∈t`C. Min (t`C) ≤ s" by (auto intro!: Min_in) with C have "S ⊆ {Min (t`C) <..}" by (auto intro: le_less_trans simp: subset_eq) with t Min ‹C ⊆ S› show ?thesis by fastforce qed lemma continuous_attains_sup: fixes f :: "'a::topological_space ⇒ 'b::linorder_topology" shows "compact s ⟹ s ≠ {} ⟹ continuous_on s f ⟹ (∃x∈s. ∀y∈s. f y ≤ 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 ⇒ 'b::linorder_topology" shows "compact s ⟹ s ≠ {} ⟹ continuous_on s f ⟹ (∃x∈s. ∀y∈s. f x ≤ f y)" using compact_attains_inf[of "f ` s"] compact_continuous_image[of s f] by auto subsection ‹Connectedness› context topological_space begin definition "connected S ⟷ ¬ (∃A B. open A ∧ open B ∧ S ⊆ A ∪ B ∧ A ∩ B ∩ S = {} ∧ A ∩ S ≠ {} ∧ B ∩ S ≠ {})" lemma connectedI: "(⋀A B. open A ⟹ open B ⟹ A ∩ U ≠ {} ⟹ B ∩ U ≠ {} ⟹ A ∩ B ∩ U = {} ⟹ U ⊆ A ∪ B ⟹ False) ⟹ 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 ⟹ open U ⟹ open V ⟹ U ∩ V ∩ A = {} ⟹ A ⊆ U ∪ V ⟹ U ∩ A = {} ∨ V ∩ A = {}" by (auto simp: connected_def) end lemma connected_closed: "connected s ⟷ ¬ (∃A B. closed A ∧ closed B ∧ s ⊆ A ∪ B ∧ A ∩ B ∩ s = {} ∧ A ∩ s ≠ {} ∧ B ∩ s ≠ {})" 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_closedD: "⟦connected s; A ∩ B ∩ s = {}; s ⊆ A ∪ B; closed A; closed B⟧ ⟹ A ∩ s = {} ∨ B ∩ s = {}" by (simp add: connected_closed) lemma connected_Union: assumes cs: "⋀s. s ∈ S ⟹ connected s" and ne: "⋂S ≠ {}" shows "connected(⋃S)" proof (rule connectedI) fix A B assume A: "open A" and B: "open B" and Alap: "A ∩ ⋃S ≠ {}" and Blap: "B ∩ ⋃S ≠ {}" and disj: "A ∩ B ∩ ⋃S = {}" and cover: "⋃S ⊆ A ∪ B" have disjs:"⋀s. s ∈ S ⟹ A ∩ B ∩ s = {}" using disj by auto obtain sa where sa: "sa ∈ S" "A ∩ sa ≠ {}" using Alap by auto obtain sb where sb: "sb ∈ S" "B ∩ sb ≠ {}" using Blap by auto obtain x where x: "⋀s. s ∈ S ⟹ x ∈ s" using ne by auto then have "x ∈ ⋃S" using ‹sa ∈ S› by blast then have "x ∈ A ∨ x ∈ 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: "connected s ⟹ connected t ⟹ s ∩ t ≠ {} ⟹ connected (s ∪ t)" using connected_Union [of "{s,t}"] by auto lemma connected_diff_open_from_closed: assumes st: "s ⊆ t" and tu: "t ⊆ 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 ∩ (u - s) ≠ {}" "B ∩ (u - s) ≠ {}" and disj: "A ∩ B ∩ (u - s) = {}" and cover: "u - s ⊆ A ∪ B" then consider "A ∩ (t - s) = {}" | "B ∩ (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) ∩ (B ∪ s) ∩ u = {}" using disj st by auto moreover have "u ⊆ (A - t) ∪ (B ∪ s)" using 1 cover by auto ultimately show False using connectedD [of u "A - t" "B ∪ s"] AB s t 1 u by auto next case 2 then have "(A ∪ s) ∩ (B - t) ∩ u = {}" using disj st by auto moreover have "u ⊆ (A ∪ s) ∪ (B - t)" using 2 cover by auto ultimately show False using connectedD [of u "A ∪ 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 ⟷ (∀P::'a ⇒ bool. continuous_on S P ⟶ (∃c. ∀s∈S. P s = c))" proof safe fix P :: "'a ⇒ bool" assume "connected S" "continuous_on S P" then have "⋀b. ∃A. open A ∧ A ∩ S = P -` {b} ∩ S" unfolding continuous_on_open_invariant by (simp add: open_discrete) from this[of True] this[of False] obtain t f where "open t" "open f" and *: "f ∩ S = P -` {False} ∩ S" "t ∩ S = P -` {True} ∩ S" by meson then have "t ∩ S = {} ∨ f ∩ S = {}" by (intro connectedD[OF ‹connected S›]) auto then show "∃c. ∀s∈S. P s = c" proof (rule disjE) assume "t ∩ S = {}" then show ?thesis unfolding * by (intro exI[of _ False]) auto next assume "f ∩ S = {}" then show ?thesis unfolding * by (intro exI[of _ True]) auto qed next assume P: "∀P::'a ⇒ bool. continuous_on S P ⟶ (∃c. ∀s∈S. P s = c)" show "connected S" proof (rule connectedI) fix A B assume *: "open A" "open B" "A ∩ S ≠ {}" "B ∩ S ≠ {}" "A ∩ B ∩ S = {}" "S ⊆ A ∪ B" have "continuous_on S (λx. x ∈ A)" unfolding continuous_on_open_invariant proof safe fix C :: "bool set" have "C = UNIV ∨ C = {True} ∨ C = {False} ∨ C = {}" using subset_UNIV[of C] unfolding UNIV_bool by auto with * show "∃T. open T ∧ T ∩ S = (λx. x ∈ A) -` C ∩ S" by (intro exI[of _ "(if True ∈ C then A else {}) ∪ (if False ∈ C then B else {})"]) auto qed from P[rule_format, OF this] obtain c where "⋀s. s ∈ S ⟹ (s ∈ A) = c" by blast with * show False by (cases c) auto qed qed lemma connectedD_const: "connected S ⟹ continuous_on S P ⟹ ∃c. ∀s∈S. P s = c" for P :: "'a::topological_space ⇒ bool" by (auto simp: connected_iff_const) lemma connectedI_const: "(⋀P::'a::topological_space ⇒ bool. continuous_on S P ⟹ ∃c. ∀s∈S. P s = c) ⟹ connected S" by (auto simp: connected_iff_const) lemma connected_local_const: assumes "connected A" "a ∈ A" "b ∈ A" and *: "∀a∈A. eventually (λb. f a = f b) (at a within A)" shows "f a = f b" proof - obtain S where S: "⋀a. a ∈ A ⟹ a ∈ S a" "⋀a. a ∈ A ⟹ open (S a)" "⋀a x. a ∈ A ⟹ x ∈ S a ⟹ x ∈ A ⟹ f a = f x" using * unfolding eventually_at_topological by metis let ?P = "⋃b∈{b∈A. f a = f b}. S b" and ?N = "⋃b∈{b∈A. f a ≠ f b}. S b" have "?P ∩ A = {} ∨ ?N ∩ A = {}" using ‹connected A› S ‹a∈A› by (intro connectedD) (auto, metis) then show "f a = f b" proof assume "?N ∩ A = {}" then have "∀x∈A. f a = f x" using S(1) by auto with ‹b∈A› show ?thesis by auto next assume "?P ∩ A = {}" then show ?thesis using ‹a ∈ A› S(1)[of a] by auto qed qed lemma (in linorder_topology) connectedD_interval: assumes "connected U" and xy: "x ∈ U" "y ∈ U" and "x ≤ z" "z ≤ y" shows "z ∈ U" proof - have eq: "{..<z} ∪ {z<..} = - {z}" by auto have "¬ connected U" if "z ∉ U" "x < z" "z < y" using xy that apply (simp only: connected_def simp_thms) apply (rule_tac exI[of _ "{..< z}"]) apply (rule_tac exI[of _ "{z <..}"]) apply (auto simp add: eq) done with assms show "z ∈ U" by (metis less_le) qed lemma (in linorder_topology) not_in_connected_cases: assumes conn: "connected S" assumes nbdd: "x ∉ S" assumes ne: "S ≠ {}" obtains "bdd_above S" "⋀y. y ∈ S ⟹ x ≥ y" | "bdd_below S" "⋀y. y ∈ S ⟹ x ≤ y" proof - obtain s where "s ∈ S" using ne by blast { assume "s ≤ x" have "False" if "x ≤ y" "y ∈ S" for y using connectedD_interval[OF conn ‹s ∈ S› ‹y ∈ S› ‹s ≤ x› ‹x ≤ y›] ‹x ∉ S› by simp then have wit: "y ∈ S ⟹ x ≥ y" for y using le_cases by blast then have "bdd_above S" by (rule local.bdd_aboveI) note this wit } moreover { assume "x ≤ s" have "False" if "x ≥ y" "y ∈ S" for y using connectedD_interval[OF conn ‹y ∈ S› ‹s ∈ S› ‹x ≥ y› ‹s ≥ x› ] ‹x ∉ S› by simp then have wit: "y ∈ S ⟹ x ≤ y" for y using le_cases by blast then have "bdd_below S" by (rule bdd_belowI) note this wit } ultimately show ?thesis by (meson le_cases that) qed lemma connected_continuous_image: assumes *: "continuous_on s f" and "connected s" shows "connected (f ` s)" proof (rule connectedI_const) fix P :: "'b ⇒ bool" assume "continuous_on (f ` s) P" then have "continuous_on s (P ∘ f)" by (rule continuous_on_compose[OF *]) from connectedD_const[OF ‹connected s› this] show "∃c. ∀s∈f ` s. P s = c" by auto qed section ‹Linear Continuum Topologies› class linear_continuum_topology = linorder_topology + linear_continuum begin lemma Inf_notin_open: assumes A: "open A" and bnd: "∀a∈A. x < a" shows "Inf A ∉ A" proof assume "Inf A ∈ A" then obtain b where "b < Inf A" "{b <.. Inf A} ⊆ 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 ∈ A" by (auto simp: subset_eq) then show False using cInf_lower[OF ‹c ∈ A›] bnd by (metis not_le less_imp_le bdd_belowI) qed lemma Sup_notin_open: assumes A: "open A" and bnd: "∀a∈A. a < x" shows "Sup A ∉ A" proof assume "Sup A ∈ A" with assms obtain b where "Sup A < b" "{Sup A ..< b} ⊆ A" using open_right[of A "Sup A" x] by auto with dense[of "Sup A" b] obtain c where "Sup A < c" "c ∈ A" by (auto simp: subset_eq) then show False using cSup_upper[OF ‹c ∈ A›] bnd by (metis less_imp_le not_le bdd_aboveI) qed end instance linear_continuum_topology ⊆ perfect_space proof fix x :: 'a obtain y where "x < y ∨ y < x" using ex_gt_or_lt [of x] .. with Inf_notin_open[of "{x}" y] Sup_notin_open[of "{x}" y] show "¬ open {x}" by auto qed lemma connectedI_interval: fixes U :: "'a :: linear_continuum_topology set" assumes *: "⋀x y z. x ∈ U ⟹ y ∈ U ⟹ x ≤ z ⟹ z ≤ y ⟹ z ∈ U" shows "connected U" proof (rule connectedI) { fix A B assume "open A" "open B" "A ∩ B ∩ U = {}" "U ⊆ A ∪ B" fix x y assume "x < y" "x ∈ A" "y ∈ B" "x ∈ U" "y ∈ U" let ?z = "Inf (B ∩ {x <..})" have "x ≤ ?z" "?z ≤ y" using ‹y ∈ B› ‹x < y› by (auto intro: cInf_lower cInf_greatest) with ‹x ∈ U› ‹y ∈ U› have "?z ∈ U" by (rule *) moreover have "?z ∉ B ∩ {x <..}" using ‹open B› by (intro Inf_notin_open) auto ultimately have "?z ∈ A" using ‹x ≤ ?z› ‹A ∩ B ∩ U = {}› ‹x ∈ A› ‹U ⊆ A ∪ B› by auto have "∃b∈B. b ∈ A ∧ b ∈ U" if "?z < y" proof - obtain a where "?z < a" "{?z ..< a} ⊆ A" using open_right[OF ‹open A› ‹?z ∈ A› ‹?z < y›] by auto moreover obtain b where "b ∈ B" "x < b" "b < min a y" using cInf_less_iff[of "B ∩ {x <..}" "min a y"] ‹?z < a› ‹?z < y› ‹x < y› ‹y ∈ B› by auto moreover have "?z ≤ b" using ‹b ∈ B› ‹x < b› by (intro cInf_lower) auto moreover have "b ∈ U" using ‹x ≤ ?z› ‹?z ≤ b› ‹b < min a y› by (intro *[OF ‹x ∈ U› ‹y ∈ U›]) (auto simp: less_imp_le) ultimately show ?thesis by (intro bexI[of _ b]) auto qed then have False using ‹?z ≤ y› ‹?z ∈ A› ‹y ∈ B› ‹y ∈ U› ‹A ∩ B ∩ U = {}› unfolding le_less by blast } note not_disjoint = this fix A B assume AB: "open A" "open B" "U ⊆ A ∪ B" "A ∩ B ∩ U = {}" moreover assume "A ∩ U ≠ {}" then obtain x where x: "x ∈ U" "x ∈ A" by auto moreover assume "B ∩ U ≠ {}" then obtain y where y: "y ∈ U" "y ∈ 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: "connected U ⟷ (∀x∈U. ∀y∈U. ∀z. x ≤ z ⟶ z ≤ y ⟶ z ∈ U)" for U :: "'a::linear_continuum_topology set" by (auto intro: connectedI_interval dest: connectedD_interval) lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)" by (simp add: connected_iff_interval) lemma connected_Ioi[simp]: "connected {a<..}" for a :: "'a::linear_continuum_topology" by (auto simp: connected_iff_interval) lemma connected_Ici[simp]: "connected {a..}" for a :: "'a::linear_continuum_topology" by (auto simp: connected_iff_interval) lemma connected_Iio[simp]: "connected {..<a}" for a :: "'a::linear_continuum_topology" by (auto simp: connected_iff_interval) lemma connected_Iic[simp]: "connected {..a}" for a :: "'a::linear_continuum_topology" by (auto simp: connected_iff_interval) lemma connected_Ioo[simp]: "connected {a<..<b}" for a b :: "'a::linear_continuum_topology" unfolding connected_iff_interval by auto lemma connected_Ioc[simp]: "connected {a<..b}" for a b :: "'a::linear_continuum_topology" by (auto simp: connected_iff_interval) lemma connected_Ico[simp]: "connected {a..<b}" for a b :: "'a::linear_continuum_topology" by (auto simp: connected_iff_interval) lemma connected_Icc[simp]: "connected {a..b}" for a b :: "'a::linear_continuum_topology" by (auto simp: connected_iff_interval) lemma connected_contains_Ioo: fixes A :: "'a :: linorder_topology set" assumes "connected A" "a ∈ A" "b ∈ A" shows "{a <..< b} ⊆ A" using connectedD_interval[OF assms] by (simp add: subset_eq Ball_def less_imp_le) lemma connected_contains_Icc: fixes A :: "'a::linorder_topology set" assumes "connected A" "a ∈ A" "b ∈ A" shows "{a..b} ⊆ A" proof fix x assume "x ∈ {a..b}" then have "x = a ∨ x = b ∨ x ∈ {a<..<b}" by auto then show "x ∈ A" using assms connected_contains_Ioo[of A a b] by auto qed subsection ‹Intermediate Value Theorem› lemma IVT': fixes f :: "'a::linear_continuum_topology ⇒ 'b::linorder_topology" assumes y: "f a ≤ y" "y ≤ f b" "a ≤ b" and *: "continuous_on {a .. b} f" shows "∃x. a ≤ x ∧ x ≤ b ∧ 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 ⇒ 'b :: linorder_topology" assumes y: "f b ≤ y" "y ≤ f a" "a ≤ b" and *: "continuous_on {a .. b} f" shows "∃x. a ≤ x ∧ x ≤ b ∧ 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 ⇒ 'b::linorder_topology" shows "f a ≤ y ⟹ y ≤ f b ⟹ a ≤ b ⟹ (∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x) ⟹ ∃x. a ≤ x ∧ x ≤ b ∧ f x = y" by (rule IVT') (auto intro: continuous_at_imp_continuous_on) lemma IVT2: fixes f :: "'a::linear_continuum_topology ⇒ 'b::linorder_topology" shows "f b ≤ y ⟹ y ≤ f a ⟹ a ≤ b ⟹ (∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x) ⟹ ∃x. a ≤ x ∧ x ≤ b ∧ f x = y" by (rule IVT2') (auto intro: continuous_at_imp_continuous_on) lemma continuous_inj_imp_mono: fixes f :: "'a::linear_continuum_topology ⇒ 'b::linorder_topology" assumes x: "a < x" "x < b" and cont: "continuous_on {a..b} f" and inj: "inj_on f {a..b}" shows "(f a < f x ∧ f x < f b) ∨ (f b < f x ∧ f x < f a)" proof - note I = inj_on_eq_iff[OF inj] { assume "f x < f a" "f x < f b" then obtain s t where "x ≤ s" "s ≤ b" "a ≤ t" "t ≤ 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 ≤ s" "s ≤ b" "a ≤ t" "t ≤ 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} ⇒ 'b::{linorder_topology,conditionally_complete_linorder}" assumes "mono f" and cont: "continuous (at_left (Sup S)) f" and S: "S ≠ {}" "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) ≤ (SUP s:S. f s)" proof cases assume "Sup S ∈ S" then show ?thesis by (rule cSUP_upper) (auto intro: bdd_above_image_mono S ‹mono f›) next assume "Sup S ∉ S" from ‹S ≠ {}› obtain s where "s ∈ S" by auto with ‹Sup S ∉ S› S have "s < Sup S" unfolding less_le by (blast intro: cSup_upper) show ?thesis proof (rule ccontr) assume "¬ ?thesis" with order_tendstoD(1)[OF f, of "SUP s:S. f s"] obtain b where "b < Sup S" and *: "⋀y. b < y ⟹ y < Sup S ⟹ (SUP s:S. f s) < f y" by (auto simp: not_le eventually_at_left[OF ‹s < Sup S›]) with ‹S ≠ {}› obtain c where "c ∈ S" "b < c" using less_cSupD[of S b] by auto with ‹Sup S ∉ S› S have "c < Sup S" unfolding less_le by (blast intro: cSup_upper) from *[OF ‹b < c› ‹c < Sup S›] cSUP_upper[OF ‹c ∈ S› bdd_above_image_mono[of f]] show False by (auto simp: assms) qed qed qed (intro cSUP_least ‹mono f›[THEN monoD] cSup_upper S) lemma continuous_at_Sup_antimono: fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} ⇒ 'b::{linorder_topology,conditionally_complete_linorder}" assumes "antimono f" and cont: "continuous (at_left (Sup S)) f" and S: "S ≠ {}" "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) ≤ f (Sup S)" proof cases assume "Sup S ∈ S" then show ?thesis by (intro cINF_lower) (auto intro: bdd_below_image_antimono S ‹antimono f›) next assume "Sup S ∉ S" from ‹S ≠ {}› obtain s where "s ∈ S" by auto with ‹Sup S ∉ S› S have "s < Sup S" unfolding less_le by (blast intro: cSup_upper) show ?thesis proof (rule ccontr) assume "¬ ?thesis" with order_tendstoD(2)[OF f, of "INF s:S. f s"] obtain b where "b < Sup S" and *: "⋀y. b < y ⟹ y < Sup S ⟹ f y < (INF s:S. f s)" by (auto simp: not_le eventually_at_left[OF ‹s < Sup S›]) with ‹S ≠ {}› obtain c where "c ∈ S" "b < c" using less_cSupD[of S b] by auto with ‹Sup S ∉ S› S have "c < Sup S" unfolding less_le by (blast intro: cSup_upper) from *[OF ‹b < c› ‹c < Sup S›] cINF_lower[OF bdd_below_image_antimono, of f S c] ‹c ∈ S› show False by (auto simp: assms) qed qed qed (intro cINF_greatest ‹antimono f›[THEN antimonoD] cSup_upper S) lemma continuous_at_Inf_mono: fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} ⇒ 'b::{linorder_topology,conditionally_complete_linorder}" assumes "mono f" and cont: "continuous (at_right (Inf S)) f" and S: "S ≠ {}" "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) ≤ f (Inf S)" proof cases assume "Inf S ∈ S" then show ?thesis by (rule cINF_lower[rotated]) (auto intro: bdd_below_image_mono S ‹mono f›) next assume "Inf S ∉ S" from ‹S ≠ {}› obtain s where "s ∈ S" by auto with ‹Inf S ∉ S› S have "Inf S < s" unfolding less_le by (blast intro: cInf_lower) show ?thesis proof (rule ccontr) assume "¬ ?thesis" with order_tendstoD(2)[OF f, of "INF s:S. f s"] obtain b where "Inf S < b" and *: "⋀y. Inf S < y ⟹ y < b ⟹ f y < (INF s:S. f s)" by (auto simp: not_le eventually_at_right[OF ‹Inf S < s›]) with ‹S ≠ {}› obtain c where "c ∈ S" "c < b" using cInf_lessD[of S b] by auto with ‹Inf S ∉ S› S have "Inf S < c" unfolding less_le by (blast intro: cInf_lower) from *[OF ‹Inf S < c› ‹c < b›] cINF_lower[OF bdd_below_image_mono[of f] ‹c ∈ S›] show False by (auto simp: assms) qed qed qed (intro cINF_greatest ‹mono f›[THEN monoD] cInf_lower ‹bdd_below S› ‹S ≠ {}›) lemma continuous_at_Inf_antimono: fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} ⇒ 'b::{linorder_topology,conditionally_complete_linorder}" assumes "antimono f" and cont: "continuous (at_right (Inf S)) f" and S: "S ≠ {}" "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) ≤ (SUP s:S. f s)" proof cases assume "Inf S ∈ S" then show ?thesis by (rule cSUP_upper) (auto intro: bdd_above_image_antimono S ‹antimono f›) next assume "Inf S ∉ S" from ‹S ≠ {}› obtain s where "s ∈ S" by auto with ‹Inf S ∉ S› S have "Inf S < s" unfolding less_le by (blast intro: cInf_lower) show ?thesis proof (rule ccontr) assume "¬ ?thesis" with order_tendstoD(1)[OF f, of "SUP s:S. f s"] obtain b where "Inf S < b" and *: "⋀y. Inf S < y ⟹ y < b ⟹ (SUP s:S. f s) < f y" by (auto simp: not_le eventually_at_right[OF ‹Inf S < s›]) with ‹S ≠ {}› obtain c where "c ∈ S" "c < b" using cInf_lessD[of S b] by auto with ‹Inf S ∉ S› S have "Inf S < c" unfolding less_le by (blast intro: cInf_lower) from *[OF ‹Inf S < c› ‹c < b›] cSUP_upper[OF ‹c ∈ S› bdd_above_image_antimono[of f]] show False by (auto simp: assms) qed qed qed (intro cSUP_least ‹antimono f›[THEN antimonoD] cInf_lower S) subsection ‹Uniform spaces› class uniformity = fixes uniformity :: "('a × 'a) filter" begin abbreviation uniformity_on :: "'a set ⇒ ('a × 'a) filter" where "uniformity_on s ≡ inf uniformity (principal (s×s))" end lemma uniformity_Abort: "uniformity = Filter.abstract_filter (λu. Code.abort (STR ''uniformity is not executable'') (λu. uniformity))" by simp class open_uniformity = "open" + uniformity + assumes open_uniformity: "⋀U. open U ⟷ (∀x∈U. eventually (λ(x', y). x' = x ⟶ y ∈ U) uniformity)" begin subclass topological_space by standard (force elim: eventually_mono eventually_elim2 simp: split_beta' open_uniformity)+ end class uniform_space = open_uniformity + assumes uniformity_refl: "eventually E uniformity ⟹ E (x, x)" and uniformity_sym: "eventually E uniformity ⟹ eventually (λ(x, y). E (y, x)) uniformity" and uniformity_trans: "eventually E uniformity ⟹ ∃D. eventually D uniformity ∧ (∀x y z. D (x, y) ⟶ D (y, z) ⟶ E (x, z))" begin lemma uniformity_bot: "uniformity ≠ bot" using uniformity_refl by auto lemma uniformity_trans': "eventually E uniformity ⟹ eventually (λ((x, y), (y', z)). y = y' ⟶ E (x, z)) (uniformity ×⇩_{F}uniformity)" by (drule uniformity_trans) (auto simp add: eventually_prod_same) lemma uniformity_transE: assumes "eventually E uniformity" obtains D where "eventually D uniformity" "⋀x y z. D (x, y) ⟹ D (y, z) ⟹ E (x, z)" using uniformity_trans [OF assms] by auto lemma eventually_nhds_uniformity: "eventually P (nhds x) ⟷ eventually (λ(x', y). x' = x ⟶ P y) uniformity" (is "_ ⟷ ?N P x") unfolding eventually_nhds proof safe assume *: "?N P x" have "?N (?N P) x" if "?N P x" for x proof - from that obtain D where ev: "eventually D uniformity" and D: "D (a, b) ⟹ D (b, c) ⟹ case (a, c) of (x', y) ⇒ x' = x ⟶ P y" for a b c by (rule uniformity_transE) simp from ev show ?thesis by eventually_elim (insert ev D, force elim: eventually_mono split: prod.split) qed then have "open {x. ?N P x}" by (simp add: open_uniformity) then show "∃S. open S ∧ x ∈ S ∧ (∀x∈S. P x)" by (intro exI[of _ "{x. ?N P x}"]) (auto dest: uniformity_refl simp: *) qed (force simp add: open_uniformity elim: eventually_mono) subsubsection ‹Totally bounded sets› definition totally_bounded :: "'a set ⇒ bool" where "totally_bounded S ⟷ (∀E. eventually E uniformity ⟶ (∃X. finite X ∧ (∀s∈S. ∃x∈X. E (x, s))))" lemma totally_bounded_empty[iff]: "totally_bounded {}" by (auto simp add: totally_bounded_def) lemma totally_bounded_subset: "totally_bounded S ⟹ T ⊆ S ⟹ totally_bounded T" by (fastforce simp add: totally_bounded_def) lemma totally_bounded_Union[intro]: assumes M: "finite M" "⋀S. S ∈ M ⟹ totally_bounded S" shows "totally_bounded (⋃M)" unfolding totally_bounded_def proof safe fix E assume "eventually E uniformity" with M obtain X where "∀S∈M. finite (X S) ∧ (∀s∈S. ∃x∈X S. E (x, s))" by (metis totally_bounded_def) with ‹finite M› show "∃X. finite X ∧ (∀s∈⋃M. ∃x∈X. E (x, s))" by (intro exI[of _ "⋃S∈M. X S"]) force qed subsubsection ‹Cauchy filter› definition cauchy_filter :: "'a filter ⇒ bool" where "cauchy_filter F ⟷ F ×⇩_{F}F ≤ uniformity" definition Cauchy :: "(nat ⇒ 'a) ⇒ bool" where Cauchy_uniform: "Cauchy X = cauchy_filter (filtermap X sequentially)" lemma Cauchy_uniform_iff: "Cauchy X ⟷ (∀P. eventually P uniformity ⟶ (∃N. ∀n≥N. ∀m≥N. P (X n, X m)))" unfolding Cauchy_uniform cauchy_filter_def le_filter_def eventually_prod_same eventually_filtermap eventually_sequentially proof safe let ?U = "λP. eventually P uniformity" { fix P assume "?U P" "∀P. ?U P ⟶ (∃Q. (∃N. ∀n≥N. Q (X n)) ∧ (∀x y. Q x ⟶ Q y ⟶ P (x, y)))" then obtain Q N where "⋀n. n ≥ N ⟹ Q (X n)" "⋀x y. Q x ⟹ Q y ⟹ P (x, y)" by metis then show "∃N. ∀n≥N. ∀m≥N. P (X n, X m)" by blast next fix P assume "?U P" and P: "∀P. ?U P ⟶ (∃N. ∀n≥N. ∀m≥N. P (X n, X m))" then obtain Q where "?U Q" and Q: "⋀x y z. Q (x, y) ⟹ Q (y, z) ⟹ P (x, z)" by (auto elim: uniformity_transE) then have "?U (λx. Q x ∧ (λ(x, y). Q (y, x)) x)" unfolding eventually_conj_iff by (simp add: uniformity_sym) from P[rule_format, OF this] obtain N where N: "⋀n m. n ≥ N ⟹ m ≥ N ⟹ Q (X n, X m) ∧ Q (X m, X n)" by auto show "∃Q. (∃N. ∀n≥N. Q (X n)) ∧ (∀x y. Q x ⟶ Q y ⟶ P (x, y))" proof (safe intro!: exI[of _ "λx. ∀n≥N. Q (x, X n) ∧ Q (X n, x)"] exI[of _ N] N) fix x y assume "∀n≥N. Q (x, X n) ∧ Q (X n, x)" "∀n≥N. Q (y, X n) ∧ Q (X n, y)" then have "Q (x, X N)" "Q (X N, y)" by auto then show "P (x, y)" by (rule Q) qed } qed lemma nhds_imp_cauchy_filter: assumes *: "F ≤ nhds x" shows "cauchy_filter F" proof - have "F ×⇩_{F}F ≤ nhds x ×⇩_{F}nhds x" by (intro prod_filter_mono *) also have "… ≤ uniformity" unfolding le_filter_def eventually_nhds_uniformity eventually_prod_same proof safe fix P assume "eventually P uniformity" then obtain Ql where ev: "eventually Ql uniformity" and "Ql (x, y) ⟹ Ql (y, z) ⟹ P (x, z)" for x y z by (rule uniformity_transE) simp with ev[THEN uniformity_sym] show "∃Q. eventually (λ(x', y). x' = x ⟶ Q y) uniformity ∧ (∀x y. Q x ⟶ Q y ⟶ P (x, y))" by (rule_tac exI[of _ "λy. Ql (y, x) ∧ Ql (x, y)"]) (fastforce elim: eventually_elim2) qed finally show ?thesis by (simp add: cauchy_filter_def) qed lemma LIMSEQ_imp_Cauchy: "X ⇢ x ⟹ Cauchy X" unfolding Cauchy_uniform filterlim_def by (intro nhds_imp_cauchy_filter) lemma Cauchy_subseq_Cauchy: assumes "Cauchy X" "strict_mono f" shows "Cauchy (X ∘ f)" unfolding Cauchy_uniform comp_def filtermap_filtermap[symmetric] cauchy_filter_def by (rule order_trans[OF _ ‹Cauchy X›[unfolded Cauchy_uniform cauchy_filter_def]]) (intro prod_filter_mono filtermap_mono filterlim_subseq[OF ‹strict_mono f›, unfolded filterlim_def]) lemma convergent_Cauchy: "convergent X ⟹ Cauchy X" unfolding convergent_def by (erule exE, erule LIMSEQ_imp_Cauchy) definition complete :: "'a set ⇒ bool" where complete_uniform: "complete S ⟷ (∀F ≤ principal S. F ≠ bot ⟶ cauchy_filter F ⟶ (∃x∈S. F ≤ nhds x))" end subsubsection ‹Uniformly continuous functions› definition uniformly_continuous_on :: "'a set ⇒ ('a::uniform_space ⇒ 'b::uniform_space) ⇒ bool" where uniformly_continuous_on_uniformity: "uniformly_continuous_on s f ⟷ (LIM (x, y) (uniformity_on s). (f x, f y) :> uniformity)" lemma uniformly_continuous_onD: "uniformly_continuous_on s f ⟹ eventually E uniformity ⟹ eventually (λ(x, y). x ∈ s ⟶ y ∈ s ⟶ E (f x, f y)) uniformity" by (simp add: uniformly_continuous_on_uniformity filterlim_iff eventually_inf_principal split_beta' mem_Times_iff imp_conjL) lemma uniformly_continuous_on_const[continuous_intros]: "uniformly_continuous_on s (λx. c)" by (auto simp: uniformly_continuous_on_uniformity filterlim_iff uniformity_refl) lemma uniformly_continuous_on_id[continuous_intros]: "uniformly_continuous_on s (λx. x)" by (auto simp: uniformly_continuous_on_uniformity filterlim_def) lemma uniformly_continuous_on_compose[continuous_intros]: "uniformly_continuous_on s g ⟹ uniformly_continuous_on (g`s) f ⟹ uniformly_continuous_on s (λx. f (g x))" using filterlim_compose[of "λ(x, y). (f x, f y)" uniformity "uniformity_on (g`s)" "λ(x, y). (g x, g y)" "uniformity_on s"] by (simp add: split_beta' uniformly_continuous_on_uniformity filterlim_inf filterlim_principal eventually_inf_principal mem_Times_iff) lemma uniformly_continuous_imp_continuous: assumes f: "uniformly_continuous_on s f" shows "continuous_on s f" by (auto simp: filterlim_iff eventually_at_filter eventually_nhds_uniformity continuous_on_def elim: eventually_mono dest!: uniformly_continuous_onD[OF f]) section ‹Product Topology› subsection ‹Product is a topological space› instantiation prod :: (topological_space, topological_space) topological_space begin definition open_prod_def[code del]: "open (S :: ('a × 'b) set) ⟷ (∀x∈S. ∃A B. open A ∧ open B ∧ x ∈ A × B ∧ A × B ⊆ S)" lemma open_prod_elim: assumes "open S" and "x ∈ S" obtains A B where "open A" and "open B" and "x ∈ A × B" and "A × B ⊆ S" using assms unfolding open_prod_def by fast lemma open_prod_intro: assumes "⋀x. x ∈ S ⟹ ∃A B. open A ∧ open B ∧ x ∈ A × B ∧ A × B ⊆ S" shows "open S" using assms unfolding open_prod_def by fast instance proof show "open (UNIV :: ('a × 'b) set)" unfolding open_prod_def by auto next fix S T :: "('a × 'b) set" assume "open S" "open T" show "open (S ∩ T)" proof (rule open_prod_intro) fix x assume x: "x ∈ S ∩ T" from x have "x ∈ S" by simp obtain Sa Sb where A: "open Sa" "open Sb" "x ∈ Sa × Sb" "Sa × Sb ⊆ S" using ‹open S› and ‹x ∈ S› by (rule open_prod_elim) from x have "x ∈ T" by simp obtain Ta Tb where B: "open Ta" "open Tb" "x ∈ Ta × Tb" "Ta × Tb ⊆ T" using ‹open T› and ‹x ∈ T› by (rule open_prod_elim) let ?A = "Sa ∩ Ta" and ?B = "Sb ∩ Tb" have "open ?A ∧ open ?B ∧ x ∈ ?A × ?B ∧ ?A × ?B ⊆ S ∩ T" using A B by (auto simp add: open_Int) then show "∃A B. open A ∧ open B ∧ x ∈ A × B ∧ A × B ⊆ S ∩ T" by fast qed next fix K :: "('a × 'b) set set" assume "∀S∈K. open S" then show "open (⋃K)" unfolding open_prod_def by fast qed end declare [[code abort: "open :: ('a::topological_space × 'b::topological_space) set ⇒ bool"]] lemma open_Times: "open S ⟹ open T ⟹ open (S × T)" unfolding open_prod_def by auto lemma fst_vimage_eq_Times: "fst -` S = S × UNIV" by auto lemma snd_vimage_eq_Times: "snd -` S = UNIV × S" by auto lemma open_vimage_fst: "open S ⟹ open (fst -` S)" by (simp add: fst_vimage_eq_Times open_Times) lemma open_vimage_snd: "open S ⟹ open (snd -` S)" by (simp add: snd_vimage_eq_Times open_Times) lemma closed_vimage_fst: "closed S ⟹ closed (fst -` S)" unfolding closed_open vimage_Compl [symmetric] by (rule open_vimage_fst) lemma closed_vimage_snd: "closed S ⟹ closed (snd -` S)" unfolding closed_open vimage_Compl [symmetric] by (rule open_vimage_snd) lemma closed_Times: "closed S ⟹ closed T ⟹ closed (S × T)" proof - have "S × T = (fst -` S) ∩ (snd -` T)" by auto then show "closed S ⟹ closed T ⟹ closed (S × T)" by (simp add: closed_vimage_fst closed_vimage_snd closed_Int) qed lemma subset_fst_imageI: "A × B ⊆ S ⟹ y ∈ B ⟹ A ⊆ fst ` S" unfolding image_def subset_eq by force lemma subset_snd_imageI: "A × B ⊆ S ⟹ x ∈ A ⟹ B ⊆ snd ` S" unfolding image_def subset_eq by force lemma open_image_fst: assumes "open S" shows "open (fst ` S)" proof (rule openI) fix x assume "x ∈ fst ` S" then obtain y where "(x, y) ∈ S" by auto then obtain A B where "open A" "open B" "x ∈ A" "y ∈ B" "A × B ⊆ S" using ‹open S› unfolding open_prod_def by auto from ‹A × B ⊆ S› ‹y ∈ B› have "A ⊆ fst ` S" by (rule subset_fst_imageI) with ‹open A› ‹x ∈ A› have "open A ∧ x ∈ A ∧ A ⊆ fst ` S" by simp then show "∃T. open T ∧ x ∈ T ∧ T ⊆ fst ` S" .. qed lemma open_image_snd: assumes "open S" shows "open (snd ` S)" proof (rule openI) fix y assume "y ∈ snd ` S" then obtain x where "(x, y) ∈ S" by auto then obtain A B where "open A" "open B" "x ∈ A" "y ∈ B" "A × B ⊆ S" using ‹open S› unfolding open_prod_def by auto from ‹A × B ⊆ S› ‹x ∈ A› have "B ⊆ snd ` S" by (rule subset_snd_imageI) with ‹open B› ‹y ∈ B› have "open B ∧ y ∈ B ∧ B ⊆ snd ` S" by simp then show "∃T. open T ∧ y ∈ T ∧ T ⊆ snd ` S" .. qed lemma nhds_prod: "nhds (a, b) = nhds a ×⇩_{F}nhds b" unfolding nhds_def proof (subst prod_filter_INF, auto intro!: antisym INF_greatest simp: principal_prod_principal) fix S T assume "open S" "a ∈ S" "open T" "b ∈ T" then show "(INF x : {S. open S ∧ (a, b) ∈ S}. principal x) ≤ principal (S × T)" by (intro INF_lower) (auto intro!: open_Times) next fix S' assume "open S'" "(a, b) ∈ S'" then obtain S T where "open S" "a ∈ S" "open T" "b ∈ T" "S × T ⊆ S'" by (auto elim: open_prod_elim) then show "(INF x : {S. open S ∧ a ∈ S}. INF y : {S. open S ∧ b ∈ S}. principal (x × y)) ≤ principal S'" by (auto intro!: INF_lower2) qed subsubsection ‹Continuity of operations› lemma tendsto_fst [tendsto_intros]: assumes "(f ⤏ a) F" shows "((λx. fst (f x)) ⤏ fst a) F" proof (rule topological_tendstoI) fix S assume "open S" and "fst a ∈ S" then have "open (fst -` S)" and "a ∈ fst -` S" by (simp_all add: open_vimage_fst) with assms have "eventually (λx. f x ∈ fst -` S) F" by (rule topological_tendstoD) then show "eventually (λx. fst (f x) ∈ S) F" by simp qed lemma tendsto_snd [tendsto_intros]: assumes "(f ⤏ a) F" shows "((λx. snd (f x)) ⤏ snd a) F" proof (rule topological_tendstoI) fix S assume "open S" and "snd a ∈ S" then have "open (snd -` S)" and "a ∈ snd -` S" by (simp_all add: open_vimage_snd) with assms have "eventually (λx. f x ∈ snd -` S) F" by (rule topological_tendstoD) then show "eventually (λx. snd (f x) ∈ S) F" by simp qed lemma tendsto_Pair [tendsto_intros]: assumes "(f ⤏ a) F" and "(g ⤏ b) F" shows "((λx. (f x, g x)) ⤏ (a, b)) F" unfolding nhds_prod using assms by (rule filterlim_Pair) lemma continuous_fst[continuous_intros]: "continuous F f ⟹ continuous F (λx. fst (f x))" unfolding continuous_def by (rule tendsto_fst) lemma continuous_snd[continuous_intros]: "continuous F f ⟹ continuous F (λx. snd (f x))" unfolding continuous_def by (rule tendsto_snd) lemma continuous_Pair[continuous_intros]: "continuous F f ⟹ continuous F g ⟹ continuous F (λx. (f x, g x))" unfolding continuous_def by (rule tendsto_Pair) lemma continuous_on_fst[continuous_intros]: "continuous_on s f ⟹ continuous_on s (λx. fst (f x))" unfolding continuous_on_def by (auto intro: tendsto_fst) lemma continuous_on_snd[continuous_intros]: "continuous_on s f ⟹ continuous_on s (λx. snd (f x))" unfolding continuous_on_def by (auto intro: tendsto_snd) lemma continuous_on_Pair[continuous_intros]: "continuous_on s f ⟹ continuous_on s g ⟹ continuous_on s (λx. (f x, g x))" unfolding continuous_on_def by (auto intro: tendsto_Pair) lemma continuous_on_swap[continuous_intros]: "continuous_on A prod.swap" by (simp add: prod.swap_def continuous_on_fst continuous_on_snd continuous_on_Pair continuous_on_id) lemma continuous_on_swap_args: assumes "continuous_on (A×B) (λ(x,y). d x y)" shows "continuous_on (B×A) (λ(x,y). d y x)" proof - have "(λ(x,y). d y x) = (λ(x,y). d x y) ∘ prod.swap" by force then show ?thesis apply (rule ssubst) apply (rule continuous_on_compose) apply (force intro: continuous_on_subset [OF continuous_on_swap]) apply (force intro: continuous_on_subset [OF assms]) done qed lemma isCont_fst [simp]: "isCont f a ⟹ isCont (λx. fst (f x)) a" by (fact continuous_fst) lemma isCont_snd [simp]: "isCont f a ⟹ isCont (λx. snd (f x)) a" by (fact continuous_snd) lemma isCont_Pair [simp]: "⟦isCont f a; isCont g a⟧ ⟹ isCont (λx. (f x, g x)) a" by (fact continuous_Pair) lemma continuous_on_compose_Pair: assumes f: "continuous_on (Sigma A B) (λ(a, b). f a b)" assumes g: "continuous_on C g" assumes h: "continuous_on C h" assumes subset: "⋀c. c ∈ C ⟹ g c ∈ A" "⋀c. c ∈ C ⟹ h c ∈ B (g c)" shows "continuous_on C (λc. f (g c) (h c))" using continuous_on_compose2[OF f continuous_on_Pair[OF g h]] subset by auto subsubsection ‹Connectedness of products› proposition connected_Times: assumes S: "connected S" and T: "connected T" shows "connected (S × T)" proof (rule connectedI_const) fix P::"'a × 'b ⇒ bool" assume P[THEN continuous_on_compose2, continuous_intros]: "continuous_on (S × T) P" have "continuous_on S (λs. P (s, t))" if "t ∈ T" for t by (auto intro!: continuous_intros that) from connectedD_const[OF S this] obtain c1 where c1: "⋀s t. t ∈ T ⟹ s ∈ S ⟹ P (s, t) = c1 t" by metis moreover have "continuous_on T (λt. P (s, t))" if "s ∈ S" for s by (auto intro!: continuous_intros that) from connectedD_const[OF T this] obtain c2 where "⋀s t. t ∈ T ⟹ s ∈ S ⟹ P (s, t) = c2 s" by metis ultimately show "∃c. ∀s∈S × T. P s = c" by auto qed corollary connected_Times_eq [simp]: "connected (S × T) ⟷ S = {} ∨ T = {} ∨ connected S ∧ connected T" (is "?lhs = ?rhs") proof assume L: ?lhs show ?rhs proof cases assume "S ≠ {} ∧ T ≠ {}" moreover have "connected (fst ` (S × T))" "connected (snd ` (S × T))" using continuous_on_fst continuous_on_snd continuous_on_id by (blast intro: connected_continuous_image [OF _ L])+ ultimately show ?thesis by auto qed auto qed (auto simp: connected_Times) subsubsection ‹Separation axioms› instance prod :: (t0_space, t0_space) t0_space proof fix x y :: "'a × 'b" assume "x ≠ y" then have "fst x ≠ fst y ∨ snd x ≠ snd y" by (simp add: prod_eq_iff) then show "∃U. open U ∧ (x ∈ U) ≠ (y ∈ U)" by (fast dest: t0_space elim: open_vimage_fst open_vimage_snd) qed instance prod :: (t1_space, t1_space) t1_space proof fix x y :: "'a × 'b" assume "x ≠ y" then have "fst x ≠ fst y ∨ snd x ≠ snd y" by (simp add: prod_eq_iff) then show "∃U. open U ∧ x ∈ U ∧ y ∉ U" by (fast dest: t1_space elim: open_vimage_fst open_vimage_snd) qed instance prod :: (t2_space, t2_space) t2_space proof fix x y :: "'a × 'b" assume "x ≠ y" then have "fst x ≠ fst y ∨ snd x ≠ snd y" by (simp add: prod_eq_iff) then show "∃U V. open U ∧ open V ∧ x ∈ U ∧ y ∈ V ∧ U ∩ V = {}" by (fast dest: hausdorff elim: open_vimage_fst open_vimage_snd) qed lemma isCont_swap[continuous_intros]: "isCont prod.swap a" using continuous_on_eq_continuous_within continuous_on_swap by blast lemma open_diagonal_complement: "open {(x,y) |x y. x ≠ (y::('a::t2_space))}" proof - have "open {(x, y). x ≠ (y::'a)}" unfolding split_def by (intro open_Collect_neq continuous_intros) also have "{(x, y). x ≠ (y::'a)} = {(x, y) |x y. x ≠ (y::'a)}" by auto finally show ?thesis . qed lemma closed_diagonal: "closed {y. ∃ x::('a::t2_space). y = (x,x)}" proof - have "{y. ∃ x::'a. y = (x,x)} = UNIV - {(x,y) | x y. x ≠ y}" by auto then show ?thesis using open_diagonal_complement closed_Diff by auto qed lemma open_superdiagonal: "open {(x,y) | x y. x > (y::'a::{linorder_topology})}" proof - have "open {(x, y). x > (y::'a)}" unfolding split_def by (intro open_Collect_less continuous_intros) also have "{(x, y). x > (y::'a)} = {(x, y) |x y. x > (y::'a)}" by auto finally show ?thesis . qed lemma closed_subdiagonal: "closed {(x,y) | x y. x ≤ (y::'a::{linorder_topology})}" proof - have "{(x,y) | x y. x ≤ (y::'a)} = UNIV - {(x,y) | x y. x > (y::'a)}" by auto then show ?thesis using open_superdiagonal closed_Diff by auto qed lemma open_subdiagonal: "open {(x,y) | x y. x < (y::'a::{linorder_topology})}" proof - have "open {(x, y). x < (y::'a)}" unfolding split_def by (intro open_Collect_less continuous_intros) also have "{(x, y). x < (y::'a)} = {(x, y) |x y. x < (y::'a)}" by auto finally show ?thesis . qed lemma closed_superdiagonal: "closed {(x,y) | x y. x ≥ (y::('a::{linorder_topology}))}" proof - have "{(x,y) | x y. x ≥ (y::'a)} = UNIV - {(x,y) | x y. x < y}" by auto then show ?thesis using open_subdiagonal closed_Diff by auto qed end