imports Indicator_Function Countable_Set FuncSet Linear_Algebra Norm_Arith

(* Author: L C Paulson, University of Cambridge Author: Amine Chaieb, University of Cambridge Author: Robert Himmelmann, TU Muenchen Author: Brian Huffman, Portland State University *) section ‹Elementary topology in Euclidean space› theory Topology_Euclidean_Space imports "HOL-Library.Indicator_Function" "HOL-Library.Countable_Set" "HOL-Library.FuncSet" Linear_Algebra Norm_Arith begin (* FIXME: move elsewhere *) lemma halfspace_Int_eq: "{x. a ∙ x ≤ b} ∩ {x. b ≤ a ∙ x} = {x. a ∙ x = b}" "{x. b ≤ a ∙ x} ∩ {x. a ∙ x ≤ b} = {x. a ∙ x = b}" by auto definition (in monoid_add) support_on :: "'b set ⇒ ('b ⇒ 'a) ⇒ 'b set" where "support_on s f = {x∈s. f x ≠ 0}" lemma in_support_on: "x ∈ support_on s f ⟷ x ∈ s ∧ f x ≠ 0" by (simp add: support_on_def) lemma support_on_simps[simp]: "support_on {} f = {}" "support_on (insert x s) f = (if f x = 0 then support_on s f else insert x (support_on s f))" "support_on (s ∪ t) f = support_on s f ∪ support_on t f" "support_on (s ∩ t) f = support_on s f ∩ support_on t f" "support_on (s - t) f = support_on s f - support_on t f" "support_on (f ` s) g = f ` (support_on s (g ∘ f))" unfolding support_on_def by auto lemma support_on_cong: "(⋀x. x ∈ s ⟹ f x = 0 ⟷ g x = 0) ⟹ support_on s f = support_on s g" by (auto simp: support_on_def) lemma support_on_if: "a ≠ 0 ⟹ support_on A (λx. if P x then a else 0) = {x∈A. P x}" by (auto simp: support_on_def) lemma support_on_if_subset: "support_on A (λx. if P x then a else 0) ⊆ {x ∈ A. P x}" by (auto simp: support_on_def) lemma finite_support[intro]: "finite S ⟹ finite (support_on S f)" unfolding support_on_def by auto (* TODO: is supp_sum really needed? TODO: Generalize to Finite_Set.fold *) definition (in comm_monoid_add) supp_sum :: "('b ⇒ 'a) ⇒ 'b set ⇒ 'a" where "supp_sum f S = (∑x∈support_on S f. f x)" lemma supp_sum_empty[simp]: "supp_sum f {} = 0" unfolding supp_sum_def by auto lemma supp_sum_insert[simp]: "finite (support_on S f) ⟹ supp_sum f (insert x S) = (if x ∈ S then supp_sum f S else f x + supp_sum f S)" by (simp add: supp_sum_def in_support_on insert_absorb) lemma supp_sum_divide_distrib: "supp_sum f A / (r::'a::field) = supp_sum (λn. f n / r) A" by (cases "r = 0") (auto simp: supp_sum_def sum_divide_distrib intro!: sum.cong support_on_cong) (*END OF SUPPORT, ETC.*) lemma image_affinity_interval: fixes c :: "'a::ordered_real_vector" shows "((λx. m *⇩_{R}x + c) ` {a..b}) = (if {a..b}={} then {} else if 0 ≤ m then {m *⇩_{R}a + c .. m *⇩_{R}b + c} else {m *⇩_{R}b + c .. m *⇩_{R}a + c})" (is "?lhs = ?rhs") proof (cases "m=0") case True then show ?thesis by force next case False show ?thesis proof show "?lhs ⊆ ?rhs" by (auto simp: scaleR_left_mono scaleR_left_mono_neg) show "?rhs ⊆ ?lhs" proof (clarsimp, intro conjI impI subsetI) show "⟦0 ≤ m; a ≤ b; x ∈ {m *⇩_{R}a + c..m *⇩_{R}b + c}⟧ ⟹ x ∈ (λx. m *⇩_{R}x + c) ` {a..b}" for x apply (rule_tac x="inverse m *⇩_{R}(x-c)" in rev_image_eqI) using False apply (auto simp: le_diff_eq pos_le_divideRI) using diff_le_eq pos_le_divideR_eq by force show "⟦¬ 0 ≤ m; a ≤ b; x ∈ {m *⇩_{R}b + c..m *⇩_{R}a + c}⟧ ⟹ x ∈ (λx. m *⇩_{R}x + c) ` {a..b}" for x apply (rule_tac x="inverse m *⇩_{R}(x-c)" in rev_image_eqI) apply (auto simp: diff_le_eq neg_le_divideR_eq) using diff_eq_diff_less_eq linordered_field_class.sign_simps(11) minus_diff_eq not_less scaleR_le_cancel_left_neg by fastforce qed qed qed lemma countable_PiE: "finite I ⟹ (⋀i. i ∈ I ⟹ countable (F i)) ⟹ countable (Pi⇩_{E}I F)" by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq) lemma open_sums: fixes T :: "('b::real_normed_vector) set" assumes "open S ∨ open T" shows "open (⋃x∈ S. ⋃y ∈ T. {x + y})" using assms proof assume S: "open S" show ?thesis proof (clarsimp simp: open_dist) fix x y assume "x ∈ S" "y ∈ T" with S obtain e where "e > 0" and e: "⋀x'. dist x' x < e ⟹ x' ∈ S" by (auto simp: open_dist) then have "⋀z. dist z (x + y) < e ⟹ ∃x∈S. ∃y∈T. z = x + y" by (metis ‹y ∈ T› diff_add_cancel dist_add_cancel2) then show "∃e>0. ∀z. dist z (x + y) < e ⟶ (∃x∈S. ∃y∈T. z = x + y)" using ‹0 < e› ‹x ∈ S› by blast qed next assume T: "open T" show ?thesis proof (clarsimp simp: open_dist) fix x y assume "x ∈ S" "y ∈ T" with T obtain e where "e > 0" and e: "⋀x'. dist x' y < e ⟹ x' ∈ T" by (auto simp: open_dist) then have "⋀z. dist z (x + y) < e ⟹ ∃x∈S. ∃y∈T. z = x + y" by (metis ‹x ∈ S› add_diff_cancel_left' add_diff_eq diff_diff_add dist_norm) then show "∃e>0. ∀z. dist z (x + y) < e ⟶ (∃x∈S. ∃y∈T. z = x + y)" using ‹0 < e› ‹y ∈ T› by blast qed qed subsection ‹Topological Basis› context topological_space begin definition%important "topological_basis B ⟷ (∀b∈B. open b) ∧ (∀x. open x ⟶ (∃B'. B' ⊆ B ∧ ⋃B' = x))" lemma topological_basis: "topological_basis B ⟷ (∀x. open x ⟷ (∃B'. B' ⊆ B ∧ ⋃B' = x))" unfolding topological_basis_def apply safe apply fastforce apply fastforce apply (erule_tac x=x in allE, simp) apply (rule_tac x="{x}" in exI, auto) done lemma topological_basis_iff: assumes "⋀B'. B' ∈ B ⟹ open B'" shows "topological_basis B ⟷ (∀O'. open O' ⟶ (∀x∈O'. ∃B'∈B. x ∈ B' ∧ B' ⊆ O'))" (is "_ ⟷ ?rhs") proof safe fix O' and x::'a assume H: "topological_basis B" "open O'" "x ∈ O'" then have "(∃B'⊆B. ⋃B' = O')" by (simp add: topological_basis_def) then obtain B' where "B' ⊆ B" "O' = ⋃B'" by auto then show "∃B'∈B. x ∈ B' ∧ B' ⊆ O'" using H by auto next assume H: ?rhs show "topological_basis B" using assms unfolding topological_basis_def proof safe fix O' :: "'a set" assume "open O'" with H obtain f where "∀x∈O'. f x ∈ B ∧ x ∈ f x ∧ f x ⊆ O'" by (force intro: bchoice simp: Bex_def) then show "∃B'⊆B. ⋃B' = O'" by (auto intro: exI[where x="{f x |x. x ∈ O'}"]) qed qed lemma topological_basisI: assumes "⋀B'. B' ∈ B ⟹ open B'" and "⋀O' x. open O' ⟹ x ∈ O' ⟹ ∃B'∈B. x ∈ B' ∧ B' ⊆ O'" shows "topological_basis B" using assms by (subst topological_basis_iff) auto lemma topological_basisE: fixes O' assumes "topological_basis B" and "open O'" and "x ∈ O'" obtains B' where "B' ∈ B" "x ∈ B'" "B' ⊆ O'" proof atomize_elim from assms have "⋀B'. B'∈B ⟹ open B'" by (simp add: topological_basis_def) with topological_basis_iff assms show "∃B'. B' ∈ B ∧ x ∈ B' ∧ B' ⊆ O'" using assms by (simp add: Bex_def) qed lemma topological_basis_open: assumes "topological_basis B" and "X ∈ B" shows "open X" using assms by (simp add: topological_basis_def) lemma topological_basis_imp_subbasis: assumes B: "topological_basis B" shows "open = generate_topology B" proof (intro ext iffI) fix S :: "'a set" assume "open S" with B obtain B' where "B' ⊆ B" "S = ⋃B'" unfolding topological_basis_def by blast then show "generate_topology B S" by (auto intro: generate_topology.intros dest: topological_basis_open) next fix S :: "'a set" assume "generate_topology B S" then show "open S" by induct (auto dest: topological_basis_open[OF B]) qed lemma basis_dense: fixes B :: "'a set set" and f :: "'a set ⇒ 'a" assumes "topological_basis B" and choosefrom_basis: "⋀B'. B' ≠ {} ⟹ f B' ∈ B'" shows "∀X. open X ⟶ X ≠ {} ⟶ (∃B' ∈ B. f B' ∈ X)" proof (intro allI impI) fix X :: "'a set" assume "open X" and "X ≠ {}" from topological_basisE[OF ‹topological_basis B› ‹open X› choosefrom_basis[OF ‹X ≠ {}›]] obtain B' where "B' ∈ B" "f X ∈ B'" "B' ⊆ X" . then show "∃B'∈B. f B' ∈ X" by (auto intro!: choosefrom_basis) qed end lemma topological_basis_prod: assumes A: "topological_basis A" and B: "topological_basis B" shows "topological_basis ((λ(a, b). a × b) ` (A × B))" unfolding topological_basis_def proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric]) fix S :: "('a × 'b) set" assume "open S" then show "∃X⊆A × B. (⋃(a,b)∈X. a × b) = S" proof (safe intro!: exI[of _ "{x∈A × B. fst x × snd x ⊆ S}"]) fix x y assume "(x, y) ∈ S" from open_prod_elim[OF ‹open S› this] obtain a b where a: "open a""x ∈ a" and b: "open b" "y ∈ b" and "a × b ⊆ S" by (metis mem_Sigma_iff) moreover from A a obtain A0 where "A0 ∈ A" "x ∈ A0" "A0 ⊆ a" by (rule topological_basisE) moreover from B b obtain B0 where "B0 ∈ B" "y ∈ B0" "B0 ⊆ b" by (rule topological_basisE) ultimately show "(x, y) ∈ (⋃(a, b)∈{X ∈ A × B. fst X × snd X ⊆ S}. a × b)" by (intro UN_I[of "(A0, B0)"]) auto qed auto qed (metis A B topological_basis_open open_Times) subsection ‹Countable Basis› locale%important countable_basis = fixes B :: "'a::topological_space set set" assumes is_basis: "topological_basis B" and countable_basis: "countable B" begin lemma open_countable_basis_ex: assumes "open X" shows "∃B' ⊆ B. X = ⋃B'" using assms countable_basis is_basis unfolding topological_basis_def by blast lemma open_countable_basisE: assumes "open X" obtains B' where "B' ⊆ B" "X = ⋃B'" using assms open_countable_basis_ex by atomize_elim simp lemma countable_dense_exists: "∃D::'a set. countable D ∧ (∀X. open X ⟶ X ≠ {} ⟶ (∃d ∈ D. d ∈ X))" proof - let ?f = "(λB'. SOME x. x ∈ B')" have "countable (?f ` B)" using countable_basis by simp with basis_dense[OF is_basis, of ?f] show ?thesis by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI) qed lemma countable_dense_setE: obtains D :: "'a set" where "countable D" "⋀X. open X ⟹ X ≠ {} ⟹ ∃d ∈ D. d ∈ X" using countable_dense_exists by blast end lemma (in first_countable_topology) first_countable_basisE: fixes x :: 'a obtains 𝒜 where "countable 𝒜" "⋀A. A ∈ 𝒜 ⟹ x ∈ A" "⋀A. A ∈ 𝒜 ⟹ open A" "⋀S. open S ⟹ x ∈ S ⟹ (∃A∈𝒜. A ⊆ S)" proof - obtain 𝒜 where 𝒜: "(∀i::nat. x ∈ 𝒜 i ∧ open (𝒜 i))" "(∀S. open S ∧ x ∈ S ⟶ (∃i. 𝒜 i ⊆ S))" using first_countable_basis[of x] by metis show thesis proof show "countable (range 𝒜)" by simp qed (use 𝒜 in auto) qed lemma (in first_countable_topology) first_countable_basis_Int_stableE: obtains 𝒜 where "countable 𝒜" "⋀A. A ∈ 𝒜 ⟹ x ∈ A" "⋀A. A ∈ 𝒜 ⟹ open A" "⋀S. open S ⟹ x ∈ S ⟹ (∃A∈𝒜. A ⊆ S)" "⋀A B. A ∈ 𝒜 ⟹ B ∈ 𝒜 ⟹ A ∩ B ∈ 𝒜" proof atomize_elim obtain ℬ where ℬ: "countable ℬ" "⋀B. B ∈ ℬ ⟹ x ∈ B" "⋀B. B ∈ ℬ ⟹ open B" "⋀S. open S ⟹ x ∈ S ⟹ ∃B∈ℬ. B ⊆ S" by (rule first_countable_basisE) blast define 𝒜 where [abs_def]: "𝒜 = (λN. ⋂((λn. from_nat_into ℬ n) ` N)) ` (Collect finite::nat set set)" then show "∃𝒜. countable 𝒜 ∧ (∀A. A ∈ 𝒜 ⟶ x ∈ A) ∧ (∀A. A ∈ 𝒜 ⟶ open A) ∧ (∀S. open S ⟶ x ∈ S ⟶ (∃A∈𝒜. A ⊆ S)) ∧ (∀A B. A ∈ 𝒜 ⟶ B ∈ 𝒜 ⟶ A ∩ B ∈ 𝒜)" proof (safe intro!: exI[where x=𝒜]) show "countable 𝒜" unfolding 𝒜_def by (intro countable_image countable_Collect_finite) fix A assume "A ∈ 𝒜" then show "x ∈ A" "open A" using ℬ(4)[OF open_UNIV] by (auto simp: 𝒜_def intro: ℬ from_nat_into) next let ?int = "λN. ⋂(from_nat_into ℬ ` N)" fix A B assume "A ∈ 𝒜" "B ∈ 𝒜" then obtain N M where "A = ?int N" "B = ?int M" "finite (N ∪ M)" by (auto simp: 𝒜_def) then show "A ∩ B ∈ 𝒜" by (auto simp: 𝒜_def intro!: image_eqI[where x="N ∪ M"]) next fix S assume "open S" "x ∈ S" then obtain a where a: "a∈ℬ" "a ⊆ S" using ℬ by blast then show "∃a∈𝒜. a ⊆ S" using a ℬ by (intro bexI[where x=a]) (auto simp: 𝒜_def intro: image_eqI[where x="{to_nat_on ℬ a}"]) qed qed lemma (in topological_space) first_countableI: assumes "countable 𝒜" and 1: "⋀A. A ∈ 𝒜 ⟹ x ∈ A" "⋀A. A ∈ 𝒜 ⟹ open A" and 2: "⋀S. open S ⟹ x ∈ S ⟹ ∃A∈𝒜. A ⊆ S" shows "∃𝒜::nat ⇒ 'a set. (∀i. x ∈ 𝒜 i ∧ open (𝒜 i)) ∧ (∀S. open S ∧ x ∈ S ⟶ (∃i. 𝒜 i ⊆ S))" proof (safe intro!: exI[of _ "from_nat_into 𝒜"]) fix i have "𝒜 ≠ {}" using 2[of UNIV] by auto show "x ∈ from_nat_into 𝒜 i" "open (from_nat_into 𝒜 i)" using range_from_nat_into_subset[OF ‹𝒜 ≠ {}›] 1 by auto next fix S assume "open S" "x∈S" from 2[OF this] show "∃i. from_nat_into 𝒜 i ⊆ S" using subset_range_from_nat_into[OF ‹countable 𝒜›] by auto qed instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology proof fix x :: "'a × 'b" obtain 𝒜 where 𝒜: "countable 𝒜" "⋀a. a ∈ 𝒜 ⟹ fst x ∈ a" "⋀a. a ∈ 𝒜 ⟹ open a" "⋀S. open S ⟹ fst x ∈ S ⟹ ∃a∈𝒜. a ⊆ S" by (rule first_countable_basisE[of "fst x"]) blast obtain B where B: "countable B" "⋀a. a ∈ B ⟹ snd x ∈ a" "⋀a. a ∈ B ⟹ open a" "⋀S. open S ⟹ snd x ∈ S ⟹ ∃a∈B. a ⊆ S" by (rule first_countable_basisE[of "snd x"]) blast show "∃𝒜::nat ⇒ ('a × 'b) set. (∀i. x ∈ 𝒜 i ∧ open (𝒜 i)) ∧ (∀S. open S ∧ x ∈ S ⟶ (∃i. 𝒜 i ⊆ S))" proof (rule first_countableI[of "(λ(a, b). a × b) ` (𝒜 × B)"], safe) fix a b assume x: "a ∈ 𝒜" "b ∈ B" show "x ∈ a × b" by (simp add: 𝒜(2) B(2) mem_Times_iff x) show "open (a × b)" by (simp add: 𝒜(3) B(3) open_Times x) next fix S assume "open S" "x ∈ S" then obtain a' b' where a'b': "open a'" "open b'" "x ∈ a' × b'" "a' × b' ⊆ S" by (rule open_prod_elim) moreover from a'b' 𝒜(4)[of a'] B(4)[of b'] obtain a b where "a ∈ 𝒜" "a ⊆ a'" "b ∈ B" "b ⊆ b'" by auto ultimately show "∃a∈(λ(a, b). a × b) ` (𝒜 × B). a ⊆ S" by (auto intro!: bexI[of _ "a × b"] bexI[of _ a] bexI[of _ b]) qed (simp add: 𝒜 B) qed class second_countable_topology = topological_space + assumes ex_countable_subbasis: "∃B::'a::topological_space set set. countable B ∧ open = generate_topology B" begin lemma ex_countable_basis: "∃B::'a set set. countable B ∧ topological_basis B" proof - from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B" by blast let ?B = "Inter ` {b. finite b ∧ b ⊆ B }" show ?thesis proof (intro exI conjI) show "countable ?B" by (intro countable_image countable_Collect_finite_subset B) { fix S assume "open S" then have "∃B'⊆{b. finite b ∧ b ⊆ B}. (⋃b∈B'. ⋂b) = S" unfolding B proof induct case UNIV show ?case by (intro exI[of _ "{{}}"]) simp next case (Int a b) then obtain x y where x: "a = UNION x Inter" "⋀i. i ∈ x ⟹ finite i ∧ i ⊆ B" and y: "b = UNION y Inter" "⋀i. i ∈ y ⟹ finite i ∧ i ⊆ B" by blast show ?case unfolding x y Int_UN_distrib2 by (intro exI[of _ "{i ∪ j| i j. i ∈ x ∧ j ∈ y}"]) (auto dest: x(2) y(2)) next case (UN K) then have "∀k∈K. ∃B'⊆{b. finite b ∧ b ⊆ B}. UNION B' Inter = k" by auto then obtain k where "∀ka∈K. k ka ⊆ {b. finite b ∧ b ⊆ B} ∧ UNION (k ka) Inter = ka" unfolding bchoice_iff .. then show "∃B'⊆{b. finite b ∧ b ⊆ B}. UNION B' Inter = ⋃K" by (intro exI[of _ "UNION K k"]) auto next case (Basis S) then show ?case by (intro exI[of _ "{{S}}"]) auto qed then have "(∃B'⊆Inter ` {b. finite b ∧ b ⊆ B}. ⋃B' = S)" unfolding subset_image_iff by blast } then show "topological_basis ?B" unfolding topological_space_class.topological_basis_def by (safe intro!: topological_space_class.open_Inter) (simp_all add: B generate_topology.Basis subset_eq) qed qed end sublocale second_countable_topology < countable_basis "SOME B. countable B ∧ topological_basis B" using someI_ex[OF ex_countable_basis] by unfold_locales safe instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology proof obtain A :: "'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto moreover obtain B :: "'b set set" where "countable B" "topological_basis B" using ex_countable_basis by auto ultimately show "∃B::('a × 'b) set set. countable B ∧ open = generate_topology B" by (auto intro!: exI[of _ "(λ(a, b). a × b) ` (A × B)"] topological_basis_prod topological_basis_imp_subbasis) qed instance second_countable_topology ⊆ first_countable_topology proof fix x :: 'a define B :: "'a set set" where "B = (SOME B. countable B ∧ topological_basis B)" then have B: "countable B" "topological_basis B" using countable_basis is_basis by (auto simp: countable_basis is_basis) then show "∃A::nat ⇒ 'a set. (∀i. x ∈ A i ∧ open (A i)) ∧ (∀S. open S ∧ x ∈ S ⟶ (∃i. A i ⊆ S))" by (intro first_countableI[of "{b∈B. x ∈ b}"]) (fastforce simp: topological_space_class.topological_basis_def)+ qed instance nat :: second_countable_topology proof show "∃B::nat set set. countable B ∧ open = generate_topology B" by (intro exI[of _ "range lessThan ∪ range greaterThan"]) (auto simp: open_nat_def) qed lemma countable_separating_set_linorder1: shows "∃B::('a::{linorder_topology, second_countable_topology} set). countable B ∧ (∀x y. x < y ⟶ (∃b ∈ B. x < b ∧ b ≤ y))" proof - obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto define B1 where "B1 = {(LEAST x. x ∈ U)| U. U ∈ A}" then have "countable B1" using ‹countable A› by (simp add: Setcompr_eq_image) define B2 where "B2 = {(SOME x. x ∈ U)| U. U ∈ A}" then have "countable B2" using ‹countable A› by (simp add: Setcompr_eq_image) have "∃b ∈ B1 ∪ B2. x < b ∧ b ≤ y" if "x < y" for x y proof (cases) assume "∃z. x < z ∧ z < y" then obtain z where z: "x < z ∧ z < y" by auto define U where "U = {x<..<y}" then have "open U" by simp moreover have "z ∈ U" using z U_def by simp ultimately obtain V where "V ∈ A" "z ∈ V" "V ⊆ U" using topological_basisE[OF ‹topological_basis A›] by auto define w where "w = (SOME x. x ∈ V)" then have "w ∈ V" using ‹z ∈ V› by (metis someI2) then have "x < w ∧ w ≤ y" using ‹w ∈ V› ‹V ⊆ U› U_def by fastforce moreover have "w ∈ B1 ∪ B2" using w_def B2_def ‹V ∈ A› by auto ultimately show ?thesis by auto next assume "¬(∃z. x < z ∧ z < y)" then have *: "⋀z. z > x ⟹ z ≥ y" by auto define U where "U = {x<..}" then have "open U" by simp moreover have "y ∈ U" using ‹x < y› U_def by simp ultimately obtain "V" where "V ∈ A" "y ∈ V" "V ⊆ U" using topological_basisE[OF ‹topological_basis A›] by auto have "U = {y..}" unfolding U_def using * ‹x < y› by auto then have "V ⊆ {y..}" using ‹V ⊆ U› by simp then have "(LEAST w. w ∈ V) = y" using ‹y ∈ V› by (meson Least_equality atLeast_iff subsetCE) then have "y ∈ B1 ∪ B2" using ‹V ∈ A› B1_def by auto moreover have "x < y ∧ y ≤ y" using ‹x < y› by simp ultimately show ?thesis by auto qed moreover have "countable (B1 ∪ B2)" using ‹countable B1› ‹countable B2› by simp ultimately show ?thesis by auto qed lemma countable_separating_set_linorder2: shows "∃B::('a::{linorder_topology, second_countable_topology} set). countable B ∧ (∀x y. x < y ⟶ (∃b ∈ B. x ≤ b ∧ b < y))" proof - obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto define B1 where "B1 = {(GREATEST x. x ∈ U) | U. U ∈ A}" then have "countable B1" using ‹countable A› by (simp add: Setcompr_eq_image) define B2 where "B2 = {(SOME x. x ∈ U)| U. U ∈ A}" then have "countable B2" using ‹countable A› by (simp add: Setcompr_eq_image) have "∃b ∈ B1 ∪ B2. x ≤ b ∧ b < y" if "x < y" for x y proof (cases) assume "∃z. x < z ∧ z < y" then obtain z where z: "x < z ∧ z < y" by auto define U where "U = {x<..<y}" then have "open U" by simp moreover have "z ∈ U" using z U_def by simp ultimately obtain "V" where "V ∈ A" "z ∈ V" "V ⊆ U" using topological_basisE[OF ‹topological_basis A›] by auto define w where "w = (SOME x. x ∈ V)" then have "w ∈ V" using ‹z ∈ V› by (metis someI2) then have "x ≤ w ∧ w < y" using ‹w ∈ V› ‹V ⊆ U› U_def by fastforce moreover have "w ∈ B1 ∪ B2" using w_def B2_def ‹V ∈ A› by auto ultimately show ?thesis by auto next assume "¬(∃z. x < z ∧ z < y)" then have *: "⋀z. z < y ⟹ z ≤ x" using leI by blast define U where "U = {..<y}" then have "open U" by simp moreover have "x ∈ U" using ‹x < y› U_def by simp ultimately obtain "V" where "V ∈ A" "x ∈ V" "V ⊆ U" using topological_basisE[OF ‹topological_basis A›] by auto have "U = {..x}" unfolding U_def using * ‹x < y› by auto then have "V ⊆ {..x}" using ‹V ⊆ U› by simp then have "(GREATEST x. x ∈ V) = x" using ‹x ∈ V› by (meson Greatest_equality atMost_iff subsetCE) then have "x ∈ B1 ∪ B2" using ‹V ∈ A› B1_def by auto moreover have "x ≤ x ∧ x < y" using ‹x < y› by simp ultimately show ?thesis by auto qed moreover have "countable (B1 ∪ B2)" using ‹countable B1› ‹countable B2› by simp ultimately show ?thesis by auto qed lemma countable_separating_set_dense_linorder: shows "∃B::('a::{linorder_topology, dense_linorder, second_countable_topology} set). countable B ∧ (∀x y. x < y ⟶ (∃b ∈ B. x < b ∧ b < y))" proof - obtain B::"'a set" where B: "countable B" "⋀x y. x < y ⟹ (∃b ∈ B. x < b ∧ b ≤ y)" using countable_separating_set_linorder1 by auto have "∃b ∈ B. x < b ∧ b < y" if "x < y" for x y proof - obtain z where "x < z" "z < y" using ‹x < y› dense by blast then obtain b where "b ∈ B" "x < b ∧ b ≤ z" using B(2) by auto then have "x < b ∧ b < y" using ‹z < y› by auto then show ?thesis using ‹b ∈ B› by auto qed then show ?thesis using B(1) by auto qed subsection%important ‹Polish spaces› text ‹Textbooks define Polish spaces as completely metrizable. We assume the topology to be complete for a given metric.› class polish_space = complete_space + second_countable_topology subsection ‹General notion of a topology as a value› definition%important "istopology L ⟷ L {} ∧ (∀S T. L S ⟶ L T ⟶ L (S ∩ T)) ∧ (∀K. Ball K L ⟶ L (⋃K))" typedef%important 'a topology = "{L::('a set) ⇒ bool. istopology L}" morphisms "openin" "topology" unfolding istopology_def by blast lemma istopology_openin[intro]: "istopology(openin U)" using openin[of U] by blast lemma topology_inverse': "istopology U ⟹ openin (topology U) = U" using topology_inverse[unfolded mem_Collect_eq] . lemma topology_inverse_iff: "istopology U ⟷ openin (topology U) = U" using topology_inverse[of U] istopology_openin[of "topology U"] by auto lemma topology_eq: "T1 = T2 ⟷ (∀S. openin T1 S ⟷ openin T2 S)" proof assume "T1 = T2" then show "∀S. openin T1 S ⟷ openin T2 S" by simp next assume H: "∀S. openin T1 S ⟷ openin T2 S" then have "openin T1 = openin T2" by (simp add: fun_eq_iff) then have "topology (openin T1) = topology (openin T2)" by simp then show "T1 = T2" unfolding openin_inverse . qed text‹Infer the "universe" from union of all sets in the topology.› definition "topspace T = ⋃{S. openin T S}" subsubsection ‹Main properties of open sets› proposition openin_clauses: fixes U :: "'a topology" shows "openin U {}" "⋀S T. openin U S ⟹ openin U T ⟹ openin U (S∩T)" "⋀K. (∀S ∈ K. openin U S) ⟹ openin U (⋃K)" using openin[of U] unfolding istopology_def mem_Collect_eq by fast+ lemma openin_subset[intro]: "openin U S ⟹ S ⊆ topspace U" unfolding topspace_def by blast lemma openin_empty[simp]: "openin U {}" by (rule openin_clauses) lemma openin_Int[intro]: "openin U S ⟹ openin U T ⟹ openin U (S ∩ T)" by (rule openin_clauses) lemma openin_Union[intro]: "(⋀S. S ∈ K ⟹ openin U S) ⟹ openin U (⋃K)" using openin_clauses by blast lemma openin_Un[intro]: "openin U S ⟹ openin U T ⟹ openin U (S ∪ T)" using openin_Union[of "{S,T}" U] by auto lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (force simp: openin_Union topspace_def) lemma openin_subopen: "openin U S ⟷ (∀x ∈ S. ∃T. openin U T ∧ x ∈ T ∧ T ⊆ S)" (is "?lhs ⟷ ?rhs") proof assume ?lhs then show ?rhs by auto next assume H: ?rhs let ?t = "⋃{T. openin U T ∧ T ⊆ S}" have "openin U ?t" by (force simp: openin_Union) also have "?t = S" using H by auto finally show "openin U S" . qed lemma openin_INT [intro]: assumes "finite I" "⋀i. i ∈ I ⟹ openin T (U i)" shows "openin T ((⋂i ∈ I. U i) ∩ topspace T)" using assms by (induct, auto simp: inf_sup_aci(2) openin_Int) lemma openin_INT2 [intro]: assumes "finite I" "I ≠ {}" "⋀i. i ∈ I ⟹ openin T (U i)" shows "openin T (⋂i ∈ I. U i)" proof - have "(⋂i ∈ I. U i) ⊆ topspace T" using ‹I ≠ {}› openin_subset[OF assms(3)] by auto then show ?thesis using openin_INT[of _ _ U, OF assms(1) assms(3)] by (simp add: inf.absorb2 inf_commute) qed lemma openin_Inter [intro]: assumes "finite ℱ" "ℱ ≠ {}" "⋀X. X ∈ ℱ ⟹ openin T X" shows "openin T (⋂ℱ)" by (metis (full_types) assms openin_INT2 image_ident) subsubsection ‹Closed sets› definition%important "closedin U S ⟷ S ⊆ topspace U ∧ openin U (topspace U - S)" lemma closedin_subset: "closedin U S ⟹ S ⊆ topspace U" by (metis closedin_def) lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def) lemma closedin_topspace[intro, simp]: "closedin U (topspace U)" by (simp add: closedin_def) lemma closedin_Un[intro]: "closedin U S ⟹ closedin U T ⟹ closedin U (S ∪ T)" by (auto simp: Diff_Un closedin_def) lemma Diff_Inter[intro]: "A - ⋂S = ⋃{A - s|s. s∈S}" by auto lemma closedin_Union: assumes "finite S" "⋀T. T ∈ S ⟹ closedin U T" shows "closedin U (⋃S)" using assms by induction auto lemma closedin_Inter[intro]: assumes Ke: "K ≠ {}" and Kc: "⋀S. S ∈K ⟹ closedin U S" shows "closedin U (⋂K)" using Ke Kc unfolding closedin_def Diff_Inter by auto lemma closedin_INT[intro]: assumes "A ≠ {}" "⋀x. x ∈ A ⟹ closedin U (B x)" shows "closedin U (⋂x∈A. B x)" apply (rule closedin_Inter) using assms apply auto done lemma closedin_Int[intro]: "closedin U S ⟹ closedin U T ⟹ closedin U (S ∩ T)" using closedin_Inter[of "{S,T}" U] by auto lemma openin_closedin_eq: "openin U S ⟷ S ⊆ topspace U ∧ closedin U (topspace U - S)" apply (auto simp: closedin_def Diff_Diff_Int inf_absorb2) apply (metis openin_subset subset_eq) done lemma openin_closedin: "S ⊆ topspace U ⟹ (openin U S ⟷ closedin U (topspace U - S))" by (simp add: openin_closedin_eq) lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)" proof - have "S - T = S ∩ (topspace U - T)" using openin_subset[of U S] oS cT by (auto simp: topspace_def openin_subset) then show ?thesis using oS cT by (auto simp: closedin_def) qed lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)" proof - have "S - T = S ∩ (topspace U - T)" using closedin_subset[of U S] oS cT by (auto simp: topspace_def) then show ?thesis using oS cT by (auto simp: openin_closedin_eq) qed subsubsection ‹Subspace topology› definition%important "subtopology U V = topology (λT. ∃S. T = S ∩ V ∧ openin U S)" lemma istopology_subtopology: "istopology (λT. ∃S. T = S ∩ V ∧ openin U S)" (is "istopology ?L") proof - have "?L {}" by blast { fix A B assume A: "?L A" and B: "?L B" from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa ∩ V" and Sb: "openin U Sb" "B = Sb ∩ V" by blast have "A ∩ B = (Sa ∩ Sb) ∩ V" "openin U (Sa ∩ Sb)" using Sa Sb by blast+ then have "?L (A ∩ B)" by blast } moreover { fix K assume K: "K ⊆ Collect ?L" have th0: "Collect ?L = (λS. S ∩ V) ` Collect (openin U)" by blast from K[unfolded th0 subset_image_iff] obtain Sk where Sk: "Sk ⊆ Collect (openin U)" "K = (λS. S ∩ V) ` Sk" by blast have "⋃K = (⋃Sk) ∩ V" using Sk by auto moreover have "openin U (⋃Sk)" using Sk by (auto simp: subset_eq) ultimately have "?L (⋃K)" by blast } ultimately show ?thesis unfolding subset_eq mem_Collect_eq istopology_def by auto qed lemma openin_subtopology: "openin (subtopology U V) S ⟷ (∃T. openin U T ∧ S = T ∩ V)" unfolding subtopology_def topology_inverse'[OF istopology_subtopology] by auto lemma topspace_subtopology: "topspace (subtopology U V) = topspace U ∩ V" by (auto simp: topspace_def openin_subtopology) lemma closedin_subtopology: "closedin (subtopology U V) S ⟷ (∃T. closedin U T ∧ S = T ∩ V)" unfolding closedin_def topspace_subtopology by (auto simp: openin_subtopology) lemma openin_subtopology_refl: "openin (subtopology U V) V ⟷ V ⊆ topspace U" unfolding openin_subtopology by auto (metis IntD1 in_mono openin_subset) lemma subtopology_superset: assumes UV: "topspace U ⊆ V" shows "subtopology U V = U" proof - { fix S { fix T assume T: "openin U T" "S = T ∩ V" from T openin_subset[OF T(1)] UV have eq: "S = T" by blast have "openin U S" unfolding eq using T by blast } moreover { assume S: "openin U S" then have "∃T. openin U T ∧ S = T ∩ V" using openin_subset[OF S] UV by auto } ultimately have "(∃T. openin U T ∧ S = T ∩ V) ⟷ openin U S" by blast } then show ?thesis unfolding topology_eq openin_subtopology by blast qed lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U" by (simp add: subtopology_superset) lemma subtopology_UNIV[simp]: "subtopology U UNIV = U" by (simp add: subtopology_superset) lemma openin_subtopology_empty: "openin (subtopology U {}) S ⟷ S = {}" by (metis Int_empty_right openin_empty openin_subtopology) lemma closedin_subtopology_empty: "closedin (subtopology U {}) S ⟷ S = {}" by (metis Int_empty_right closedin_empty closedin_subtopology) lemma closedin_subtopology_refl [simp]: "closedin (subtopology U X) X ⟷ X ⊆ topspace U" by (metis closedin_def closedin_topspace inf.absorb_iff2 le_inf_iff topspace_subtopology) lemma openin_imp_subset: "openin (subtopology U S) T ⟹ T ⊆ S" by (metis Int_iff openin_subtopology subsetI) lemma closedin_imp_subset: "closedin (subtopology U S) T ⟹ T ⊆ S" by (simp add: closedin_def topspace_subtopology) lemma openin_subtopology_Un: "⟦openin (subtopology X T) S; openin (subtopology X U) S⟧ ⟹ openin (subtopology X (T ∪ U)) S" by (simp add: openin_subtopology) blast lemma closedin_subtopology_Un: "⟦closedin (subtopology X T) S; closedin (subtopology X U) S⟧ ⟹ closedin (subtopology X (T ∪ U)) S" by (simp add: closedin_subtopology) blast subsubsection ‹The standard Euclidean topology› definition%important euclidean :: "'a::topological_space topology" where "euclidean = topology open" lemma open_openin: "open S ⟷ openin euclidean S" unfolding euclidean_def apply (rule cong[where x=S and y=S]) apply (rule topology_inverse[symmetric]) apply (auto simp: istopology_def) done declare open_openin [symmetric, simp] lemma topspace_euclidean [simp]: "topspace euclidean = UNIV" by (force simp: topspace_def) lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S" by (simp add: topspace_subtopology) lemma closed_closedin: "closed S ⟷ closedin euclidean S" by (simp add: closed_def closedin_def Compl_eq_Diff_UNIV) declare closed_closedin [symmetric, simp] lemma open_subopen: "open S ⟷ (∀x∈S. ∃T. open T ∧ x ∈ T ∧ T ⊆ S)" using openI by auto lemma openin_subtopology_self [simp]: "openin (subtopology euclidean S) S" by (metis openin_topspace topspace_euclidean_subtopology) text ‹Basic "localization" results are handy for connectedness.› lemma openin_open: "openin (subtopology euclidean U) S ⟷ (∃T. open T ∧ (S = U ∩ T))" by (auto simp: openin_subtopology) lemma openin_Int_open: "⟦openin (subtopology euclidean U) S; open T⟧ ⟹ openin (subtopology euclidean U) (S ∩ T)" by (metis open_Int Int_assoc openin_open) lemma openin_open_Int[intro]: "open S ⟹ openin (subtopology euclidean U) (U ∩ S)" by (auto simp: openin_open) lemma open_openin_trans[trans]: "open S ⟹ open T ⟹ T ⊆ S ⟹ openin (subtopology euclidean S) T" by (metis Int_absorb1 openin_open_Int) lemma open_subset: "S ⊆ T ⟹ open S ⟹ openin (subtopology euclidean T) S" by (auto simp: openin_open) lemma closedin_closed: "closedin (subtopology euclidean U) S ⟷ (∃T. closed T ∧ S = U ∩ T)" by (simp add: closedin_subtopology Int_ac) lemma closedin_closed_Int: "closed S ⟹ closedin (subtopology euclidean U) (U ∩ S)" by (metis closedin_closed) lemma closed_subset: "S ⊆ T ⟹ closed S ⟹ closedin (subtopology euclidean T) S" by (auto simp: closedin_closed) lemma closedin_closed_subset: "⟦closedin (subtopology euclidean U) V; T ⊆ U; S = V ∩ T⟧ ⟹ closedin (subtopology euclidean T) S" by (metis (no_types, lifting) Int_assoc Int_commute closedin_closed inf.orderE) lemma finite_imp_closedin: fixes S :: "'a::t1_space set" shows "⟦finite S; S ⊆ T⟧ ⟹ closedin (subtopology euclidean T) S" by (simp add: finite_imp_closed closed_subset) lemma closedin_singleton [simp]: fixes a :: "'a::t1_space" shows "closedin (subtopology euclidean U) {a} ⟷ a ∈ U" using closedin_subset by (force intro: closed_subset) lemma openin_euclidean_subtopology_iff: fixes S U :: "'a::metric_space set" shows "openin (subtopology euclidean U) S ⟷ S ⊆ U ∧ (∀x∈S. ∃e>0. ∀x'∈U. dist x' x < e ⟶ x'∈ S)" (is "?lhs ⟷ ?rhs") proof assume ?lhs then show ?rhs unfolding openin_open open_dist by blast next define T where "T = {x. ∃a∈S. ∃d>0. (∀y∈U. dist y a < d ⟶ y ∈ S) ∧ dist x a < d}" have 1: "∀x∈T. ∃e>0. ∀y. dist y x < e ⟶ y ∈ T" unfolding T_def apply clarsimp apply (rule_tac x="d - dist x a" in exI) apply (clarsimp simp add: less_diff_eq) by (metis dist_commute dist_triangle_lt) assume ?rhs then have 2: "S = U ∩ T" unfolding T_def by auto (metis dist_self) from 1 2 show ?lhs unfolding openin_open open_dist by fast qed lemma connected_openin: "connected S ⟷ ~(∃E1 E2. openin (subtopology euclidean S) E1 ∧ openin (subtopology euclidean S) E2 ∧ S ⊆ E1 ∪ E2 ∧ E1 ∩ E2 = {} ∧ E1 ≠ {} ∧ E2 ≠ {})" apply (simp add: connected_def openin_open disjoint_iff_not_equal, safe) apply (simp_all, blast+) (* SLOW *) done lemma connected_openin_eq: "connected S ⟷ ~(∃E1 E2. openin (subtopology euclidean S) E1 ∧ openin (subtopology euclidean S) E2 ∧ E1 ∪ E2 = S ∧ E1 ∩ E2 = {} ∧ E1 ≠ {} ∧ E2 ≠ {})" apply (simp add: connected_openin, safe, blast) by (metis Int_lower1 Un_subset_iff openin_open subset_antisym) lemma connected_closedin: "connected S ⟷ (∄E1 E2. closedin (subtopology euclidean S) E1 ∧ closedin (subtopology euclidean S) E2 ∧ S ⊆ E1 ∪ E2 ∧ E1 ∩ E2 = {} ∧ E1 ≠ {} ∧ E2 ≠ {})" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs by (auto simp add: connected_closed closedin_closed) next assume R: ?rhs then show ?lhs proof (clarsimp simp add: connected_closed closedin_closed) fix A B assume s_sub: "S ⊆ A ∪ B" "B ∩ S ≠ {}" and disj: "A ∩ B ∩ S = {}" and cl: "closed A" "closed B" have "S ∩ (A ∪ B) = S" using s_sub(1) by auto have "S - A = B ∩ S" using Diff_subset_conv Un_Diff_Int disj s_sub(1) by auto then have "S ∩ A = {}" by (metis Diff_Diff_Int Diff_disjoint Un_Diff_Int R cl closedin_closed_Int inf_commute order_refl s_sub(2)) then show "A ∩ S = {}" by blast qed qed lemma connected_closedin_eq: "connected S ⟷ ~(∃E1 E2. closedin (subtopology euclidean S) E1 ∧ closedin (subtopology euclidean S) E2 ∧ E1 ∪ E2 = S ∧ E1 ∩ E2 = {} ∧ E1 ≠ {} ∧ E2 ≠ {})" apply (simp add: connected_closedin, safe, blast) by (metis Int_lower1 Un_subset_iff closedin_closed subset_antisym) text ‹These "transitivity" results are handy too› lemma openin_trans[trans]: "openin (subtopology euclidean T) S ⟹ openin (subtopology euclidean U) T ⟹ openin (subtopology euclidean U) S" unfolding open_openin openin_open by blast lemma openin_open_trans: "openin (subtopology euclidean T) S ⟹ open T ⟹ open S" by (auto simp: openin_open intro: openin_trans) lemma closedin_trans[trans]: "closedin (subtopology euclidean T) S ⟹ closedin (subtopology euclidean U) T ⟹ closedin (subtopology euclidean U) S" by (auto simp: closedin_closed closed_Inter Int_assoc) lemma closedin_closed_trans: "closedin (subtopology euclidean T) S ⟹ closed T ⟹ closed S" by (auto simp: closedin_closed intro: closedin_trans) lemma openin_subtopology_Int_subset: "⟦openin (subtopology euclidean u) (u ∩ S); v ⊆ u⟧ ⟹ openin (subtopology euclidean v) (v ∩ S)" by (auto simp: openin_subtopology) lemma openin_open_eq: "open s ⟹ (openin (subtopology euclidean s) t ⟷ open t ∧ t ⊆ s)" using open_subset openin_open_trans openin_subset by fastforce subsection ‹Open and closed balls› definition%important ball :: "'a::metric_space ⇒ real ⇒ 'a set" where "ball x e = {y. dist x y < e}" definition%important cball :: "'a::metric_space ⇒ real ⇒ 'a set" where "cball x e = {y. dist x y ≤ e}" definition%important sphere :: "'a::metric_space ⇒ real ⇒ 'a set" where "sphere x e = {y. dist x y = e}" lemma mem_ball [simp]: "y ∈ ball x e ⟷ dist x y < e" by (simp add: ball_def) lemma mem_cball [simp]: "y ∈ cball x e ⟷ dist x y ≤ e" by (simp add: cball_def) lemma mem_sphere [simp]: "y ∈ sphere x e ⟷ dist x y = e" by (simp add: sphere_def) lemma ball_trivial [simp]: "ball x 0 = {}" by (simp add: ball_def) lemma cball_trivial [simp]: "cball x 0 = {x}" by (simp add: cball_def) lemma sphere_trivial [simp]: "sphere x 0 = {x}" by (simp add: sphere_def) lemma mem_ball_0 [simp]: "x ∈ ball 0 e ⟷ norm x < e" for x :: "'a::real_normed_vector" by (simp add: dist_norm) lemma mem_cball_0 [simp]: "x ∈ cball 0 e ⟷ norm x ≤ e" for x :: "'a::real_normed_vector" by (simp add: dist_norm) lemma disjoint_ballI: "dist x y ≥ r+s ⟹ ball x r ∩ ball y s = {}" using dist_triangle_less_add not_le by fastforce lemma disjoint_cballI: "dist x y > r + s ⟹ cball x r ∩ cball y s = {}" by (metis add_mono disjoint_iff_not_equal dist_triangle2 dual_order.trans leD mem_cball) lemma mem_sphere_0 [simp]: "x ∈ sphere 0 e ⟷ norm x = e" for x :: "'a::real_normed_vector" by (simp add: dist_norm) lemma sphere_empty [simp]: "r < 0 ⟹ sphere a r = {}" for a :: "'a::metric_space" by auto lemma centre_in_ball [simp]: "x ∈ ball x e ⟷ 0 < e" by simp lemma centre_in_cball [simp]: "x ∈ cball x e ⟷ 0 ≤ e" by simp lemma ball_subset_cball [simp, intro]: "ball x e ⊆ cball x e" by (simp add: subset_eq) lemma mem_ball_imp_mem_cball: "x ∈ ball y e ⟹ x ∈ cball y e" by (auto simp: mem_ball mem_cball) lemma sphere_cball [simp,intro]: "sphere z r ⊆ cball z r" by force lemma cball_diff_sphere: "cball a r - sphere a r = ball a r" by auto lemma subset_ball[intro]: "d ≤ e ⟹ ball x d ⊆ ball x e" by (simp add: subset_eq) lemma subset_cball[intro]: "d ≤ e ⟹ cball x d ⊆ cball x e" by (simp add: subset_eq) lemma mem_ball_leI: "x ∈ ball y e ⟹ e ≤ f ⟹ x ∈ ball y f" by (auto simp: mem_ball mem_cball) lemma mem_cball_leI: "x ∈ cball y e ⟹ e ≤ f ⟹ x ∈ cball y f" by (auto simp: mem_ball mem_cball) lemma cball_trans: "y ∈ cball z b ⟹ x ∈ cball y a ⟹ x ∈ cball z (b + a)" unfolding mem_cball proof - have "dist z x ≤ dist z y + dist y x" by (rule dist_triangle) also assume "dist z y ≤ b" also assume "dist y x ≤ a" finally show "dist z x ≤ b + a" by arith qed lemma ball_max_Un: "ball a (max r s) = ball a r ∪ ball a s" by (simp add: set_eq_iff) arith lemma ball_min_Int: "ball a (min r s) = ball a r ∩ ball a s" by (simp add: set_eq_iff) lemma cball_max_Un: "cball a (max r s) = cball a r ∪ cball a s" by (simp add: set_eq_iff) arith lemma cball_min_Int: "cball a (min r s) = cball a r ∩ cball a s" by (simp add: set_eq_iff) lemma cball_diff_eq_sphere: "cball a r - ball a r = sphere a r" by (auto simp: cball_def ball_def dist_commute) lemma image_add_ball [simp]: fixes a :: "'a::real_normed_vector" shows "(+) b ` ball a r = ball (a+b) r" apply (intro equalityI subsetI) apply (force simp: dist_norm) apply (rule_tac x="x-b" in image_eqI) apply (auto simp: dist_norm algebra_simps) done lemma image_add_cball [simp]: fixes a :: "'a::real_normed_vector" shows "(+) b ` cball a r = cball (a+b) r" apply (intro equalityI subsetI) apply (force simp: dist_norm) apply (rule_tac x="x-b" in image_eqI) apply (auto simp: dist_norm algebra_simps) done lemma open_ball [intro, simp]: "open (ball x e)" proof - have "open (dist x -` {..<e})" by (intro open_vimage open_lessThan continuous_intros) also have "dist x -` {..<e} = ball x e" by auto finally show ?thesis . qed lemma open_contains_ball: "open S ⟷ (∀x∈S. ∃e>0. ball x e ⊆ S)" by (simp add: open_dist subset_eq mem_ball Ball_def dist_commute) lemma openI [intro?]: "(⋀x. x∈S ⟹ ∃e>0. ball x e ⊆ S) ⟹ open S" by (auto simp: open_contains_ball) lemma openE[elim?]: assumes "open S" "x∈S" obtains e where "e>0" "ball x e ⊆ S" using assms unfolding open_contains_ball by auto lemma open_contains_ball_eq: "open S ⟹ x∈S ⟷ (∃e>0. ball x e ⊆ S)" by (metis open_contains_ball subset_eq centre_in_ball) lemma openin_contains_ball: "openin (subtopology euclidean t) s ⟷ s ⊆ t ∧ (∀x ∈ s. ∃e. 0 < e ∧ ball x e ∩ t ⊆ s)" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs apply (simp add: openin_open) apply (metis Int_commute Int_mono inf.cobounded2 open_contains_ball order_refl subsetCE) done next assume ?rhs then show ?lhs apply (simp add: openin_euclidean_subtopology_iff) by (metis (no_types) Int_iff dist_commute inf.absorb_iff2 mem_ball) qed lemma openin_contains_cball: "openin (subtopology euclidean t) s ⟷ s ⊆ t ∧ (∀x ∈ s. ∃e. 0 < e ∧ cball x e ∩ t ⊆ s)" apply (simp add: openin_contains_ball) apply (rule iffI) apply (auto dest!: bspec) apply (rule_tac x="e/2" in exI, force+) done lemma ball_eq_empty[simp]: "ball x e = {} ⟷ e ≤ 0" unfolding mem_ball set_eq_iff apply (simp add: not_less) apply (metis zero_le_dist order_trans dist_self) done lemma ball_empty: "e ≤ 0 ⟹ ball x e = {}" by simp lemma closed_cball [iff]: "closed (cball x e)" proof - have "closed (dist x -` {..e})" by (intro closed_vimage closed_atMost continuous_intros) also have "dist x -` {..e} = cball x e" by auto finally show ?thesis . qed lemma open_contains_cball: "open S ⟷ (∀x∈S. ∃e>0. cball x e ⊆ S)" proof - { fix x and e::real assume "x∈S" "e>0" "ball x e ⊆ S" then have "∃d>0. cball x d ⊆ S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto) } moreover { fix x and e::real assume "x∈S" "e>0" "cball x e ⊆ S" then have "∃d>0. ball x d ⊆ S" unfolding subset_eq apply (rule_tac x="e/2" in exI, auto) done } ultimately show ?thesis unfolding open_contains_ball by auto qed lemma open_contains_cball_eq: "open S ⟹ (∀x. x ∈ S ⟷ (∃e>0. cball x e ⊆ S))" by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball) lemma euclidean_dist_l2: fixes x y :: "'a :: euclidean_space" shows "dist x y = L2_set (λi. dist (x ∙ i) (y ∙ i)) Basis" unfolding dist_norm norm_eq_sqrt_inner L2_set_def by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left) lemma norm_nth_le: "norm (x ∙ i) ≤ norm x" if "i ∈ Basis" proof - have "(x ∙ i)⇧^{2}= (∑i∈{i}. (x ∙ i)⇧^{2})" by simp also have "… ≤ (∑i∈Basis. (x ∙ i)⇧^{2})" by (intro sum_mono2) (auto simp: that) finally show ?thesis unfolding norm_conv_dist euclidean_dist_l2[of x] L2_set_def by (auto intro!: real_le_rsqrt) qed lemma eventually_nhds_ball: "d > 0 ⟹ eventually (λx. x ∈ ball z d) (nhds z)" by (rule eventually_nhds_in_open) simp_all lemma eventually_at_ball: "d > 0 ⟹ eventually (λt. t ∈ ball z d ∧ t ∈ A) (at z within A)" unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute) lemma eventually_at_ball': "d > 0 ⟹ eventually (λt. t ∈ ball z d ∧ t ≠ z ∧ t ∈ A) (at z within A)" unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute) lemma at_within_ball: "e > 0 ⟹ dist x y < e ⟹ at y within ball x e = at y" by (subst at_within_open) auto lemma atLeastAtMost_eq_cball: fixes a b::real shows "{a .. b} = cball ((a + b)/2) ((b - a)/2)" by (auto simp: dist_real_def field_simps mem_cball) lemma greaterThanLessThan_eq_ball: fixes a b::real shows "{a <..< b} = ball ((a + b)/2) ((b - a)/2)" by (auto simp: dist_real_def field_simps mem_ball) subsection ‹Boxes› abbreviation One :: "'a::euclidean_space" where "One ≡ ∑Basis" lemma One_non_0: assumes "One = (0::'a::euclidean_space)" shows False proof - have "dependent (Basis :: 'a set)" apply (simp add: dependent_finite) apply (rule_tac x="λi. 1" in exI) using SOME_Basis apply (auto simp: assms) done with independent_Basis show False by force qed corollary One_neq_0[iff]: "One ≠ 0" by (metis One_non_0) corollary Zero_neq_One[iff]: "0 ≠ One" by (metis One_non_0) definition%important (in euclidean_space) eucl_less (infix "<e" 50) where "eucl_less a b ⟷ (∀i∈Basis. a ∙ i < b ∙ i)" definition%important box_eucl_less: "box a b = {x. a <e x ∧ x <e b}" definition%important "cbox a b = {x. ∀i∈Basis. a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i}" lemma box_def: "box a b = {x. ∀i∈Basis. a ∙ i < x ∙ i ∧ x ∙ i < b ∙ i}" and in_box_eucl_less: "x ∈ box a b ⟷ a <e x ∧ x <e b" and mem_box: "x ∈ box a b ⟷ (∀i∈Basis. a ∙ i < x ∙ i ∧ x ∙ i < b ∙ i)" "x ∈ cbox a b ⟷ (∀i∈Basis. a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i)" by (auto simp: box_eucl_less eucl_less_def cbox_def) lemma cbox_Pair_eq: "cbox (a, c) (b, d) = cbox a b × cbox c d" by (force simp: cbox_def Basis_prod_def) lemma cbox_Pair_iff [iff]: "(x, y) ∈ cbox (a, c) (b, d) ⟷ x ∈ cbox a b ∧ y ∈ cbox c d" by (force simp: cbox_Pair_eq) lemma cbox_Complex_eq: "cbox (Complex a c) (Complex b d) = (λ(x,y). Complex x y) ` (cbox a b × cbox c d)" apply (auto simp: cbox_def Basis_complex_def) apply (rule_tac x = "(Re x, Im x)" in image_eqI) using complex_eq by auto lemma cbox_Pair_eq_0: "cbox (a, c) (b, d) = {} ⟷ cbox a b = {} ∨ cbox c d = {}" by (force simp: cbox_Pair_eq) lemma swap_cbox_Pair [simp]: "prod.swap ` cbox (c, a) (d, b) = cbox (a,c) (b,d)" by auto lemma mem_box_real[simp]: "(x::real) ∈ box a b ⟷ a < x ∧ x < b" "(x::real) ∈ cbox a b ⟷ a ≤ x ∧ x ≤ b" by (auto simp: mem_box) lemma box_real[simp]: fixes a b:: real shows "box a b = {a <..< b}" "cbox a b = {a .. b}" by auto lemma box_Int_box: fixes a :: "'a::euclidean_space" shows "box a b ∩ box c d = box (∑i∈Basis. max (a∙i) (c∙i) *⇩_{R}i) (∑i∈Basis. min (b∙i) (d∙i) *⇩_{R}i)" unfolding set_eq_iff and Int_iff and mem_box by auto lemma rational_boxes: fixes x :: "'a::euclidean_space" assumes "e > 0" shows "∃a b. (∀i∈Basis. a ∙ i ∈ ℚ ∧ b ∙ i ∈ ℚ) ∧ x ∈ box a b ∧ box a b ⊆ ball x e" proof - define e' where "e' = e / (2 * sqrt (real (DIM ('a))))" then have e: "e' > 0" using assms by (auto simp: DIM_positive) have "∀i. ∃y. y ∈ ℚ ∧ y < x ∙ i ∧ x ∙ i - y < e'" (is "∀i. ?th i") proof fix i from Rats_dense_in_real[of "x ∙ i - e'" "x ∙ i"] e show "?th i" by auto qed from choice[OF this] obtain a where a: "∀xa. a xa ∈ ℚ ∧ a xa < x ∙ xa ∧ x ∙ xa - a xa < e'" .. have "∀i. ∃y. y ∈ ℚ ∧ x ∙ i < y ∧ y - x ∙ i < e'" (is "∀i. ?th i") proof fix i from Rats_dense_in_real[of "x ∙ i" "x ∙ i + e'"] e show "?th i" by auto qed from choice[OF this] obtain b where b: "∀xa. b xa ∈ ℚ ∧ x ∙ xa < b xa ∧ b xa - x ∙ xa < e'" .. let ?a = "∑i∈Basis. a i *⇩_{R}i" and ?b = "∑i∈Basis. b i *⇩_{R}i" show ?thesis proof (rule exI[of _ ?a], rule exI[of _ ?b], safe) fix y :: 'a assume *: "y ∈ box ?a ?b" have "dist x y = sqrt (∑i∈Basis. (dist (x ∙ i) (y ∙ i))⇧^{2})" unfolding L2_set_def[symmetric] by (rule euclidean_dist_l2) also have "… < sqrt (∑(i::'a)∈Basis. e^2 / real (DIM('a)))" proof (rule real_sqrt_less_mono, rule sum_strict_mono) fix i :: "'a" assume i: "i ∈ Basis" have "a i < y∙i ∧ y∙i < b i" using * i by (auto simp: box_def) moreover have "a i < x∙i" "x∙i - a i < e'" using a by auto moreover have "x∙i < b i" "b i - x∙i < e'" using b by auto ultimately have "¦x∙i - y∙i¦ < 2 * e'" by auto then have "dist (x ∙ i) (y ∙ i) < e/sqrt (real (DIM('a)))" unfolding e'_def by (auto simp: dist_real_def) then have "(dist (x ∙ i) (y ∙ i))⇧^{2}< (e/sqrt (real (DIM('a))))⇧^{2}" by (rule power_strict_mono) auto then show "(dist (x ∙ i) (y ∙ i))⇧^{2}< e⇧^{2}/ real DIM('a)" by (simp add: power_divide) qed auto also have "… = e" using ‹0 < e› by simp finally show "y ∈ ball x e" by (auto simp: ball_def) qed (insert a b, auto simp: box_def) qed lemma open_UNION_box: fixes M :: "'a::euclidean_space set" assumes "open M" defines "a' ≡ λf :: 'a ⇒ real × real. (∑(i::'a)∈Basis. fst (f i) *⇩_{R}i)" defines "b' ≡ λf :: 'a ⇒ real × real. (∑(i::'a)∈Basis. snd (f i) *⇩_{R}i)" defines "I ≡ {f∈Basis →⇩_{E}ℚ × ℚ. box (a' f) (b' f) ⊆ M}" shows "M = (⋃f∈I. box (a' f) (b' f))" proof - have "x ∈ (⋃f∈I. box (a' f) (b' f))" if "x ∈ M" for x proof - obtain e where e: "e > 0" "ball x e ⊆ M" using openE[OF ‹open M› ‹x ∈ M›] by auto moreover obtain a b where ab: "x ∈ box a b" "∀i ∈ Basis. a ∙ i ∈ ℚ" "∀i∈Basis. b ∙ i ∈ ℚ" "box a b ⊆ ball x e" using rational_boxes[OF e(1)] by metis ultimately show ?thesis by (intro UN_I[of "λi∈Basis. (a ∙ i, b ∙ i)"]) (auto simp: euclidean_representation I_def a'_def b'_def) qed then show ?thesis by (auto simp: I_def) qed corollary open_countable_Union_open_box: fixes S :: "'a :: euclidean_space set" assumes "open S" obtains 𝒟 where "countable 𝒟" "𝒟 ⊆ Pow S" "⋀X. X ∈ 𝒟 ⟹ ∃a b. X = box a b" "⋃𝒟 = S" proof - let ?a = "λf. (∑(i::'a)∈Basis. fst (f i) *⇩_{R}i)" let ?b = "λf. (∑(i::'a)∈Basis. snd (f i) *⇩_{R}i)" let ?I = "{f∈Basis →⇩_{E}ℚ × ℚ. box (?a f) (?b f) ⊆ S}" let ?𝒟 = "(λf. box (?a f) (?b f)) ` ?I" show ?thesis proof have "countable ?I" by (simp add: countable_PiE countable_rat) then show "countable ?𝒟" by blast show "⋃?𝒟 = S" using open_UNION_box [OF assms] by metis qed auto qed lemma rational_cboxes: fixes x :: "'a::euclidean_space" assumes "e > 0" shows "∃a b. (∀i∈Basis. a ∙ i ∈ ℚ ∧ b ∙ i ∈ ℚ) ∧ x ∈ cbox a b ∧ cbox a b ⊆ ball x e" proof - define e' where "e' = e / (2 * sqrt (real (DIM ('a))))" then have e: "e' > 0" using assms by auto have "∀i. ∃y. y ∈ ℚ ∧ y < x ∙ i ∧ x ∙ i - y < e'" (is "∀i. ?th i") proof fix i from Rats_dense_in_real[of "x ∙ i - e'" "x ∙ i"] e show "?th i" by auto qed from choice[OF this] obtain a where a: "∀u. a u ∈ ℚ ∧ a u < x ∙ u ∧ x ∙ u - a u < e'" .. have "∀i. ∃y. y ∈ ℚ ∧ x ∙ i < y ∧ y - x ∙ i < e'" (is "∀i. ?th i") proof fix i from Rats_dense_in_real[of "x ∙ i" "x ∙ i + e'"] e show "?th i" by auto qed from choice[OF this] obtain b where b: "∀u. b u ∈ ℚ ∧ x ∙ u < b u ∧ b u - x ∙ u < e'" .. let ?a = "∑i∈Basis. a i *⇩_{R}i" and ?b = "∑i∈Basis. b i *⇩_{R}i" show ?thesis proof (rule exI[of _ ?a], rule exI[of _ ?b], safe) fix y :: 'a assume *: "y ∈ cbox ?a ?b" have "dist x y = sqrt (∑i∈Basis. (dist (x ∙ i) (y ∙ i))⇧^{2})" unfolding L2_set_def[symmetric] by (rule euclidean_dist_l2) also have "… < sqrt (∑(i::'a)∈Basis. e^2 / real (DIM('a)))" proof (rule real_sqrt_less_mono, rule sum_strict_mono) fix i :: "'a" assume i: "i ∈ Basis" have "a i ≤ y∙i ∧ y∙i ≤ b i" using * i by (auto simp: cbox_def) moreover have "a i < x∙i" "x∙i - a i < e'" using a by auto moreover have "x∙i < b i" "b i - x∙i < e'" using b by auto ultimately have "¦x∙i - y∙i¦ < 2 * e'" by auto then have "dist (x ∙ i) (y ∙ i) < e/sqrt (real (DIM('a)))" unfolding e'_def by (auto simp: dist_real_def) then have "(dist (x ∙ i) (y ∙ i))⇧^{2}< (e/sqrt (real (DIM('a))))⇧^{2}" by (rule power_strict_mono) auto then show "(dist (x ∙ i) (y ∙ i))⇧^{2}< e⇧^{2}/ real DIM('a)" by (simp add: power_divide) qed auto also have "… = e" using ‹0 < e› by simp finally show "y ∈ ball x e" by (auto simp: ball_def) next show "x ∈ cbox (∑i∈Basis. a i *⇩_{R}i) (∑i∈Basis. b i *⇩_{R}i)" using a b less_imp_le by (auto simp: cbox_def) qed (use a b cbox_def in auto) qed lemma open_UNION_cbox: fixes M :: "'a::euclidean_space set" assumes "open M" defines "a' ≡ λf. (∑(i::'a)∈Basis. fst (f i) *⇩_{R}i)" defines "b' ≡ λf. (∑(i::'a)∈Basis. snd (f i) *⇩_{R}i)" defines "I ≡ {f∈Basis →⇩_{E}ℚ × ℚ. cbox (a' f) (b' f) ⊆ M}" shows "M = (⋃f∈I. cbox (a' f) (b' f))" proof - have "x ∈ (⋃f∈I. cbox (a' f) (b' f))" if "x ∈ M" for x proof - obtain e where e: "e > 0" "ball x e ⊆ M" using openE[OF ‹open M› ‹x ∈ M›] by auto moreover obtain a b where ab: "x ∈ cbox a b" "∀i ∈ Basis. a ∙ i ∈ ℚ" "∀i ∈ Basis. b ∙ i ∈ ℚ" "cbox a b ⊆ ball x e" using rational_cboxes[OF e(1)] by metis ultimately show ?thesis by (intro UN_I[of "λi∈Basis. (a ∙ i, b ∙ i)"]) (auto simp: euclidean_representation I_def a'_def b'_def) qed then show ?thesis by (auto simp: I_def) qed corollary open_countable_Union_open_cbox: fixes S :: "'a :: euclidean_space set" assumes "open S" obtains 𝒟 where "countable 𝒟" "𝒟 ⊆ Pow S" "⋀X. X ∈ 𝒟 ⟹ ∃a b. X = cbox a b" "⋃𝒟 = S" proof - let ?a = "λf. (∑(i::'a)∈Basis. fst (f i) *⇩_{R}i)" let ?b = "λf. (∑(i::'a)∈Basis. snd (f i) *⇩_{R}i)" let ?I = "{f∈Basis →⇩_{E}ℚ × ℚ. cbox (?a f) (?b f) ⊆ S}" let ?𝒟 = "(λf. cbox (?a f) (?b f)) ` ?I" show ?thesis proof have "countable ?I" by (simp add: countable_PiE countable_rat) then show "countable ?𝒟" by blast show "⋃?𝒟 = S" using open_UNION_cbox [OF assms] by metis qed auto qed lemma box_eq_empty: fixes a :: "'a::euclidean_space" shows "(box a b = {} ⟷ (∃i∈Basis. b∙i ≤ a∙i))" (is ?th1) and "(cbox a b = {} ⟷ (∃i∈Basis. b∙i < a∙i))" (is ?th2) proof - { fix i x assume i: "i∈Basis" and as:"b∙i ≤ a∙i" and x:"x∈box a b" then have "a ∙ i < x ∙ i ∧ x ∙ i < b ∙ i" unfolding mem_box by (auto simp: box_def) then have "a∙i < b∙i" by auto then have False using as by auto } moreover { assume as: "∀i∈Basis. ¬ (b∙i ≤ a∙i)" let ?x = "(1/2) *⇩_{R}(a + b)" { fix i :: 'a assume i: "i ∈ Basis" have "a∙i < b∙i" using as[THEN bspec[where x=i]] i by auto then have "a∙i < ((1/2) *⇩_{R}(a+b)) ∙ i" "((1/2) *⇩_{R}(a+b)) ∙ i < b∙i" by (auto simp: inner_add_left) } then have "box a b ≠ {}" using mem_box(1)[of "?x" a b] by auto } ultimately show ?th1 by blast { fix i x assume i: "i ∈ Basis" and as:"b∙i < a∙i" and x:"x∈cbox a b" then have "a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i" unfolding mem_box by auto then have "a∙i ≤ b∙i" by auto then have False using as by auto } moreover { assume as:"∀i∈Basis. ¬ (b∙i < a∙i)" let ?x = "(1/2) *⇩_{R}(a + b)" { fix i :: 'a assume i:"i ∈ Basis" have "a∙i ≤ b∙i" using as[THEN bspec[where x=i]] i by auto then have "a∙i ≤ ((1/2) *⇩_{R}(a+b)) ∙ i" "((1/2) *⇩_{R}(a+b)) ∙ i ≤ b∙i" by (auto simp: inner_add_left) } then have "cbox a b ≠ {}" using mem_box(2)[of "?x" a b] by auto } ultimately show ?th2 by blast qed lemma box_ne_empty: fixes a :: "'a::euclidean_space" shows "cbox a b ≠ {} ⟷ (∀i∈Basis. a∙i ≤ b∙i)" and "box a b ≠ {} ⟷ (∀i∈Basis. a∙i < b∙i)" unfolding box_eq_empty[of a b] by fastforce+ lemma fixes a :: "'a::euclidean_space" shows cbox_sing [simp]: "cbox a a = {a}" and box_sing [simp]: "box a a = {}" unfolding set_eq_iff mem_box eq_iff [symmetric] by (auto intro!: euclidean_eqI[where 'a='a]) (metis all_not_in_conv nonempty_Basis) lemma subset_box_imp: fixes a :: "'a::euclidean_space" shows "(∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i) ⟹ cbox c d ⊆ cbox a b" and "(∀i∈Basis. a∙i < c∙i ∧ d∙i < b∙i) ⟹ cbox c d ⊆ box a b" and "(∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i) ⟹ box c d ⊆ cbox a b" and "(∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i) ⟹ box c d ⊆ box a b" unfolding subset_eq[unfolded Ball_def] unfolding mem_box by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+ lemma box_subset_cbox: fixes a :: "'a::euclidean_space" shows "box a b ⊆ cbox a b" unfolding subset_eq [unfolded Ball_def] mem_box by (fast intro: less_imp_le) lemma subset_box: fixes a :: "'a::euclidean_space" shows "cbox c d ⊆ cbox a b ⟷ (∀i∈Basis. c∙i ≤ d∙i) ⟶ (∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i)" (is ?th1) and "cbox c d ⊆ box a b ⟷ (∀i∈Basis. c∙i ≤ d∙i) ⟶ (∀i∈Basis. a∙i < c∙i ∧ d∙i < b∙i)" (is ?th2) and "box c d ⊆ cbox a b ⟷ (∀i∈Basis. c∙i < d∙i) ⟶ (∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i)" (is ?th3) and "box c d ⊆ box a b ⟷ (∀i∈Basis. c∙i < d∙i) ⟶ (∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i)" (is ?th4) proof - let ?lesscd = "∀i∈Basis. c∙i < d∙i" let ?lerhs = "∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i" show ?th1 ?th2 by (fastforce simp: mem_box)+ have acdb: "a∙i ≤ c∙i ∧ d∙i ≤ b∙i" if i: "i ∈ Basis" and box: "box c d ⊆ cbox a b" and cd: "⋀i. i ∈ Basis ⟹ c∙i < d∙i" for i proof - have "box c d ≠ {}" using that unfolding box_eq_empty by force { let ?x = "(∑j∈Basis. (if j=i then ((min (a∙j) (d∙j))+c∙j)/2 else (c∙j+d∙j)/2) *⇩_{R}j)::'a" assume *: "a∙i > c∙i" then have "c ∙ j < ?x ∙ j ∧ ?x ∙ j < d ∙ j" if "j ∈ Basis" for j using cd that by (fastforce simp add: i *) then have "?x ∈ box c d" unfolding mem_box by auto moreover have "?x ∉ cbox a b" using i cd * by (force simp: mem_box) ultimately have False using box by auto } then have "a∙i ≤ c∙i" by force moreover { let ?x = "(∑j∈Basis. (if j=i then ((max (b∙j) (c∙j))+d∙j)/2 else (c∙j+d∙j)/2) *⇩_{R}j)::'a" assume *: "b∙i < d∙i" then have "d ∙ j > ?x ∙ j ∧ ?x ∙ j > c ∙ j" if "j ∈ Basis" for j using cd that by (fastforce simp add: i *) then have "?x ∈ box c d" unfolding mem_box by auto moreover have "?x ∉ cbox a b" using i cd * by (force simp: mem_box) ultimately have False using box by auto } then have "b∙i ≥ d∙i" by (rule ccontr) auto ultimately show ?thesis by auto qed show ?th3 using acdb by (fastforce simp add: mem_box) have acdb': "a∙i ≤ c∙i ∧ d∙i ≤ b∙i" if "i ∈ Basis" "box c d ⊆ box a b" "⋀i. i ∈ Basis ⟹ c∙i < d∙i" for i using box_subset_cbox[of a b] that acdb by auto show ?th4 using acdb' by (fastforce simp add: mem_box) qed lemma eq_cbox: "cbox a b = cbox c d ⟷ cbox a b = {} ∧ cbox c d = {} ∨ a = c ∧ b = d" (is "?lhs = ?rhs") proof assume ?lhs then have "cbox a b ⊆ cbox c d" "cbox c d ⊆ cbox a b" by auto then show ?rhs by (force simp: subset_box box_eq_empty intro: antisym euclidean_eqI) next assume ?rhs then show ?lhs by force qed lemma eq_cbox_box [simp]: "cbox a b = box c d ⟷ cbox a b = {} ∧ box c d = {}" (is "?lhs ⟷ ?rhs") proof assume L: ?lhs then have "cbox a b ⊆ box c d" "box c d ⊆ cbox a b" by auto then show ?rhs apply (simp add: subset_box) using L box_ne_empty box_sing apply (fastforce simp add:) done qed force lemma eq_box_cbox [simp]: "box a b = cbox c d ⟷ box a b = {} ∧ cbox c d = {}" by (metis eq_cbox_box) lemma eq_box: "box a b = box c d ⟷ box a b = {} ∧ box c d = {} ∨ a = c ∧ b = d" (is "?lhs ⟷ ?rhs") proof assume L: ?lhs then have "box a b ⊆ box c d" "box c d ⊆ box a b" by auto then show ?rhs apply (simp add: subset_box) using box_ne_empty(2) L apply auto apply (meson euclidean_eqI less_eq_real_def not_less)+ done qed force lemma subset_box_complex: "cbox a b ⊆ cbox c d ⟷ (Re a ≤ Re b ∧ Im a ≤ Im b) ⟶ Re a ≥ Re c ∧ Im a ≥ Im c ∧ Re b ≤ Re d ∧ Im b ≤ Im d" "cbox a b ⊆ box c d ⟷ (Re a ≤ Re b ∧ Im a ≤ Im b) ⟶ Re a > Re c ∧ Im a > Im c ∧ Re b < Re d ∧ Im b < Im d" "box a b ⊆ cbox c d ⟷ (Re a < Re b ∧ Im a < Im b) ⟶ Re a ≥ Re c ∧ Im a ≥ Im c ∧ Re b ≤ Re d ∧ Im b ≤ Im d" "box a b ⊆ box c d ⟷ (Re a < Re b ∧ Im a < Im b) ⟶ Re a ≥ Re c ∧ Im a ≥ Im c ∧ Re b ≤ Re d ∧ Im b ≤ Im d" by (subst subset_box; force simp: Basis_complex_def)+ lemma Int_interval: fixes a :: "'a::euclidean_space" shows "cbox a b ∩ cbox c d = cbox (∑i∈Basis. max (a∙i) (c∙i) *⇩_{R}i) (∑i∈Basis. min (b∙i) (d∙i) *⇩_{R}i)" unfolding set_eq_iff and Int_iff and mem_box by auto lemma disjoint_interval: fixes a::"'a::euclidean_space" shows "cbox a b ∩ cbox c d = {} ⟷ (∃i∈Basis. (b∙i < a∙i ∨ d∙i < c∙i ∨ b∙i < c∙i ∨ d∙i < a∙i))" (is ?th1) and "cbox a b ∩ box c d = {} ⟷ (∃i∈Basis. (b∙i < a∙i ∨ d∙i ≤ c∙i ∨ b∙i ≤ c∙i ∨ d∙i ≤ a∙i))" (is ?th2) and "box a b ∩ cbox c d = {} ⟷ (∃i∈Basis. (b∙i ≤ a∙i ∨ d∙i < c∙i ∨ b∙i ≤ c∙i ∨ d∙i ≤ a∙i))" (is ?th3) and "box a b ∩ box c d = {} ⟷ (∃i∈Basis. (b∙i ≤ a∙i ∨ d∙i ≤ c∙i ∨ b∙i ≤ c∙i ∨ d∙i ≤ a∙i))" (is ?th4) proof - let ?z = "(∑i∈Basis. (((max (a∙i) (c∙i)) + (min (b∙i) (d∙i))) / 2) *⇩_{R}i)::'a" have **: "⋀P Q. (⋀i :: 'a. i ∈ Basis ⟹ Q ?z i ⟹ P i) ⟹ (⋀i x :: 'a. i ∈ Basis ⟹ P i ⟹ Q x i) ⟹ (∀x. ∃i∈Basis. Q x i) ⟷ (∃i∈Basis. P i)" by blast note * = set_eq_iff Int_iff empty_iff mem_box ball_conj_distrib[symmetric] eq_False ball_simps(10) show ?th1 unfolding * by (intro **) auto show ?th2 unfolding * by (intro **) auto show ?th3 unfolding * by (intro **) auto show ?th4 unfolding * by (intro **) auto qed lemma UN_box_eq_UNIV: "(⋃i::nat. box (- (real i *⇩_{R}One)) (real i *⇩_{R}One)) = UNIV" proof - have "¦x ∙ b¦ < real_of_int (⌈Max ((λb. ¦x ∙ b¦)`Basis)⌉ + 1)" if [simp]: "b ∈ Basis" for x b :: 'a proof - have "¦x ∙ b¦ ≤ real_of_int ⌈¦x ∙ b¦⌉" by (rule le_of_int_ceiling) also have "… ≤ real_of_int ⌈Max ((λb. ¦x ∙ b¦)`Basis)⌉" by (auto intro!: ceiling_mono) also have "… < real_of_int (⌈Max ((λb. ¦x ∙ b¦)`Basis)⌉ + 1)" by simp finally show ?thesis . qed then have "∃n::nat. ∀b∈Basis. ¦x ∙ b¦ < real n" for x :: 'a by (metis order.strict_trans reals_Archimedean2) moreover have "⋀x b::'a. ⋀n::nat. ¦x ∙ b¦ < real n ⟷ - real n < x ∙ b ∧ x ∙ b < real n" by auto ultimately show ?thesis by (auto simp: box_def inner_sum_left inner_Basis sum.If_cases) qed subsection ‹Intervals in general, including infinite and mixtures of open and closed› definition%important "is_interval (s::('a::euclidean_space) set) ⟷ (∀a∈s. ∀b∈s. ∀x. (∀i∈Basis. ((a∙i ≤ x∙i ∧ x∙i ≤ b∙i) ∨ (b∙i ≤ x∙i ∧ x∙i ≤ a∙i))) ⟶ x ∈ s)" lemma is_interval_1: "is_interval (s::real set) ⟷ (∀a∈s. ∀b∈s. ∀ x. a ≤ x ∧ x ≤ b ⟶ x ∈ s)" unfolding is_interval_def by auto lemma is_interval_inter: "is_interval X ⟹ is_interval Y ⟹ is_interval (X ∩ Y)" unfolding is_interval_def by blast lemma is_interval_cbox [simp]: "is_interval (cbox a (b::'a::euclidean_space))" (is ?th1) and is_interval_box [simp]: "is_interval (box a b)" (is ?th2) unfolding is_interval_def mem_box Ball_def atLeastAtMost_iff by (meson order_trans le_less_trans less_le_trans less_trans)+ lemma is_interval_empty [iff]: "is_interval {}" unfolding is_interval_def by simp lemma is_interval_univ [iff]: "is_interval UNIV" unfolding is_interval_def by simp lemma mem_is_intervalI: assumes "is_interval s" and "a ∈ s" "b ∈ s" and "⋀i. i ∈ Basis ⟹ a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i ∨ b ∙ i ≤ x ∙ i ∧ x ∙ i ≤ a ∙ i" shows "x ∈ s" by (rule assms(1)[simplified is_interval_def, rule_format, OF assms(2,3,4)]) lemma interval_subst: fixes S::"'a::euclidean_space set" assumes "is_interval S" and "x ∈ S" "y j ∈ S" and "j ∈ Basis" shows "(∑i∈Basis. (if i = j then y i ∙ i else x ∙ i) *⇩_{R}i) ∈ S" by (rule mem_is_intervalI[OF assms(1,2)]) (auto simp: assms) lemma mem_box_componentwiseI: fixes S::"'a::euclidean_space set" assumes "is_interval S" assumes "⋀i. i ∈ Basis ⟹ x ∙ i ∈ ((λx. x ∙ i) ` S)" shows "x ∈ S" proof - from assms have "∀i ∈ Basis. ∃s ∈ S. x ∙ i = s ∙ i" by auto with finite_Basis obtain s and bs::"'a list" where s: "⋀i. i ∈ Basis ⟹ x ∙ i = s i ∙ i" "⋀i. i ∈ Basis ⟹ s i ∈ S" and bs: "set bs = Basis" "distinct bs" by (metis finite_distinct_list) from nonempty_Basis s obtain j where j: "j ∈ Basis" "s j ∈ S" by blast define y where "y = rec_list (s j) (λj _ Y. (∑i∈Basis. (if i = j then s i ∙ i else Y ∙ i) *⇩_{R}i))" have "x = (∑i∈Basis. (if i ∈ set bs then s i ∙ i else s j ∙ i) *⇩_{R}i)" using bs by (auto simp: s(1)[symmetric] euclidean_representation) also have [symmetric]: "y bs = …" using bs(2) bs(1)[THEN equalityD1] by (induct bs) (auto simp: y_def euclidean_representation intro!: euclidean_eqI[where 'a='a]) also have "y bs ∈ S" using bs(1)[THEN equalityD1] apply (induct bs) apply (auto simp: y_def j) apply (rule interval_subst[OF assms(1)]) apply (auto simp: s) done finally show ?thesis . qed lemma cbox01_nonempty [simp]: "cbox 0 One ≠ {}" by (simp add: box_ne_empty inner_Basis inner_sum_left sum_nonneg) lemma box01_nonempty [simp]: "box 0 One ≠ {}" by (simp add: box_ne_empty inner_Basis inner_sum_left) lemma empty_as_interval: "{} = cbox One (0::'a::euclidean_space)" using nonempty_Basis box01_nonempty box_eq_empty(1) box_ne_empty(1) by blast lemma interval_subset_is_interval: assumes "is_interval S" shows "cbox a b ⊆ S ⟷ cbox a b = {} ∨ a ∈ S ∧ b ∈ S" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs using box_ne_empty(1) mem_box(2) by fastforce next assume ?rhs have "cbox a b ⊆ S" if "a ∈ S" "b ∈ S" using assms unfolding is_interval_def apply (clarsimp simp add: mem_box) using that by blast with ‹?rhs› show ?lhs by blast qed lemma is_real_interval_union: "is_interval (X ∪ Y)" if X: "is_interval X" and Y: "is_interval Y" and I: "(X ≠ {} ⟹ Y ≠ {} ⟹ X ∩ Y ≠ {})" for X Y::"real set" proof - consider "X ≠ {}" "Y ≠ {}" | "X = {}" | "Y = {}" by blast then show ?thesis proof cases case 1 then obtain r where "r ∈ X ∨ X ∩ Y = {}" "r ∈ Y ∨ X ∩ Y = {}" by blast then show ?thesis using I 1 X Y unfolding is_interval_1 by (metis (full_types) Un_iff le_cases) qed (use that in auto) qed lemma is_interval_translationI: assumes "is_interval X" shows "is_interval ((+) x ` X)" unfolding is_interval_def proof safe fix b d e assume "b ∈ X" "d ∈ X" "∀i∈Basis. (x + b) ∙ i ≤ e ∙ i ∧ e ∙ i ≤ (x + d) ∙ i ∨ (x + d) ∙ i ≤ e ∙ i ∧ e ∙ i ≤ (x + b) ∙ i" hence "e - x ∈ X" by (intro mem_is_intervalI[OF assms ‹b ∈ X› ‹d ∈ X›, of "e - x"]) (auto simp: algebra_simps) thus "e ∈ (+) x ` X" by force qed lemma is_interval_uminusI: assumes "is_interval X" shows "is_interval (uminus ` X)" unfolding is_interval_def proof safe fix b d e assume "b ∈ X" "d ∈ X" "∀i∈Basis. (- b) ∙ i ≤ e ∙ i ∧ e ∙ i ≤ (- d) ∙ i ∨ (- d) ∙ i ≤ e ∙ i ∧ e ∙ i ≤ (- b) ∙ i" hence "- e ∈ X" by (intro mem_is_intervalI[OF assms ‹b ∈ X› ‹d ∈ X›, of "- e"]) (auto simp: algebra_simps) thus "e ∈ uminus ` X" by force qed lemma is_interval_uminus[simp]: "is_interval (uminus ` x) = is_interval x" using is_interval_uminusI[of x] is_interval_uminusI[of "uminus ` x"] by (auto simp: image_image) lemma is_interval_neg_translationI: assumes "is_interval X" shows "is_interval ((-) x ` X)" proof - have "(-) x ` X = (+) x ` uminus ` X" by (force simp: algebra_simps) also have "is_interval …" by (metis is_interval_uminusI is_interval_translationI assms) finally show ?thesis . qed lemma is_interval_translation[simp]: "is_interval ((+) x ` X) = is_interval X" using is_interval_neg_translationI[of "(+) x ` X" x] by (auto intro!: is_interval_translationI simp: image_image) lemma is_interval_minus_translation[simp]: shows "is_interval ((-) x ` X) = is_interval X" proof - have "(-) x ` X = (+) x ` uminus ` X" by (force simp: algebra_simps) also have "is_interval … = is_interval X" by simp finally show ?thesis . qed lemma is_interval_minus_translation'[simp]: shows "is_interval ((λx. x - c) ` X) = is_interval X" using is_interval_translation[of "-c" X] by (metis image_cong uminus_add_conv_diff) subsection ‹Limit points› definition%important (in topological_space) islimpt:: "'a ⇒ 'a set ⇒ bool" (infixr "islimpt" 60) where "x islimpt S ⟷ (∀T. x∈T ⟶ open T ⟶ (∃y∈S. y∈T ∧ y≠x))" lemma islimptI: assumes "⋀T. x ∈ T ⟹ open T ⟹ ∃y∈S. y ∈ T ∧ y ≠ x" shows "x islimpt S" using assms unfolding islimpt_def by auto lemma islimptE: assumes "x islimpt S" and "x ∈ T" and "open T" obtains y where "y ∈ S" and "y ∈ T" and "y ≠ x" using assms unfolding islimpt_def by auto lemma islimpt_iff_eventually: "x islimpt S ⟷ ¬ eventually (λy. y ∉ S) (at x)" unfolding islimpt_def eventually_at_topological by auto lemma islimpt_subset: "x islimpt S ⟹ S ⊆ T ⟹ x islimpt T" unfolding islimpt_def by fast lemma islimpt_approachable: fixes x :: "'a::metric_space" shows "x islimpt S ⟷ (∀e>0. ∃x'∈S. x' ≠ x ∧ dist x' x < e)" unfolding islimpt_iff_eventually eventually_at by fast lemma islimpt_approachable_le: "x islimpt S ⟷ (∀e>0. ∃x'∈ S. x' ≠ x ∧ dist x' x ≤ e)" for x :: "'a::metric_space" unfolding islimpt_approachable using approachable_lt_le [where f="λy. dist y x" and P="λy. y ∉ S ∨ y = x", THEN arg_cong [where f=Not]] by (simp add: Bex_def conj_commute conj_left_commute) lemma islimpt_UNIV_iff: "x islimpt UNIV ⟷ ¬ open {x}" unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast) lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})" unfolding islimpt_def by blast text ‹A perfect space has no isolated points.› lemma islimpt_UNIV [simp, intro]: "x islimpt UNIV" for x :: "'a::perfect_space" unfolding islimpt_UNIV_iff by (rule not_open_singleton) lemma perfect_choose_dist: "0 < r ⟹ ∃a. a ≠ x ∧ dist a x < r" for x :: "'a::{perfect_space,metric_space}" using islimpt_UNIV [of x] by (simp add: islimpt_approachable) lemma closed_limpt: "closed S ⟷ (∀x. x islimpt S ⟶ x ∈ S)" unfolding closed_def apply (subst open_subopen) apply (simp add: islimpt_def subset_eq) apply (metis ComplE ComplI) done lemma islimpt_EMPTY[simp]: "¬ x islimpt {}" by (auto simp: islimpt_def) lemma finite_ball_include: fixes a :: "'a::metric_space" assumes "finite S" shows "∃e>0. S ⊆ ball a e" using assms proof induction case (insert x S) then obtain e0 where "e0>0" and e0:"S ⊆ ball a e0" by auto define e where "e = max e0 (2 * dist a x)" have "e>0" unfolding e_def using ‹e0>0› by auto moreover have "insert x S ⊆ ball a e" using e0 ‹e>0› unfolding e_def by auto ultimately show ?case by auto qed (auto intro: zero_less_one) lemma finite_set_avoid: fixes a :: "'a::metric_space" assumes "finite S" shows "∃d>0. ∀x∈S. x ≠ a ⟶ d ≤ dist a x" using assms proof induction case (insert x S) then obtain d where "d > 0" and d: "∀x∈S. x ≠ a ⟶ d ≤ dist a x" by blast show ?case proof (cases "x = a") case True with ‹d > 0 ›d show ?thesis by auto next case False let ?d = "min d (dist a x)" from False ‹d > 0› have dp: "?d > 0" by auto from d have d': "∀x∈S. x ≠ a ⟶ ?d ≤ dist a x" by auto with dp False show ?thesis by (metis insert_iff le_less min_less_iff_conj not_less) qed qed (auto intro: zero_less_one) lemma islimpt_Un: "x islimpt (S ∪ T) ⟷ x islimpt S ∨ x islimpt T" by (simp add: islimpt_iff_eventually eventually_conj_iff) lemma discrete_imp_closed: fixes S :: "'a::metric_space set" assumes e: "0 < e" and d: "∀x ∈ S. ∀y ∈ S. dist y x < e ⟶ y = x" shows "closed S" proof - have False if C: "⋀e. e>0 ⟹ ∃x'∈S. x' ≠ x ∧ dist x' x < e" for x proof - from e have e2: "e/2 > 0" by arith from C[rule_format, OF e2] obtain y where y: "y ∈ S" "y ≠ x" "dist y x < e/2" by blast let ?m = "min (e/2) (dist x y) " from e2 y(2) have mp: "?m > 0" by simp from C[OF mp] obtain z where z: "z ∈ S" "z ≠ x" "dist z x < ?m" by blast from z y have "dist z y < e" by (intro dist_triangle_lt [where z=x]) simp from d[rule_format, OF y(1) z(1) this] y z show ?thesis by (auto simp: dist_commute) qed then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a]) qed lemma closed_of_nat_image: "closed (of_nat ` A :: 'a::real_normed_algebra_1 set)" by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat) lemma closed_of_int_image: "closed (of_int ` A :: 'a::real_normed_algebra_1 set)" by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_int) lemma closed_Nats [simp]: "closed (ℕ :: 'a :: real_normed_algebra_1 set)" unfolding Nats_def by (rule closed_of_nat_image) lemma closed_Ints [simp]: "closed (ℤ :: 'a :: real_normed_algebra_1 set)" unfolding Ints_def by (rule closed_of_int_image) lemma closed_subset_Ints: fixes A :: "'a :: real_normed_algebra_1 set" assumes "A ⊆ ℤ" shows "closed A" proof (intro discrete_imp_closed[OF zero_less_one] ballI impI, goal_cases) case (1 x y) with assms have "x ∈ ℤ" and "y ∈ ℤ" by auto with ‹dist y x < 1› show "y = x" by (auto elim!: Ints_cases simp: dist_of_int) qed subsection ‹Interior of a Set› definition%important "interior S = ⋃{T. open T ∧ T ⊆ S}" lemma interiorI [intro?]: assumes "open T" and "x ∈ T" and "T ⊆ S" shows "x ∈ interior S" using assms unfolding interior_def by fast lemma interiorE [elim?]: assumes "x ∈ interior S" obtains T where "open T" and "x ∈ T" and "T ⊆ S" using assms unfolding interior_def by fast lemma open_interior [simp, intro]: "open (interior S)" by (simp add: interior_def open_Union) lemma interior_subset: "interior S ⊆ S" by (auto simp: interior_def) lemma interior_maximal: "T ⊆ S ⟹ open T ⟹ T ⊆ interior S" by (auto simp: interior_def) lemma interior_open: "open S ⟹ interior S = S" by (intro equalityI interior_subset interior_maximal subset_refl) lemma interior_eq: "interior S = S ⟷ open S" by (metis open_interior interior_open) lemma open_subset_interior: "open S ⟹ S ⊆ interior T ⟷ S ⊆ T" by (metis interior_maximal interior_subset subset_trans) lemma interior_empty [simp]: "interior {} = {}" using open_empty by (rule interior_open) lemma interior_UNIV [simp]: "interior UNIV = UNIV" using open_UNIV by (rule interior_open) lemma interior_interior [simp]: "interior (interior S) = interior S" using open_interior by (rule interior_open) lemma interior_mono: "S ⊆ T ⟹ interior S ⊆ interior T" by (auto simp: interior_def) lemma interior_unique: assumes "T ⊆ S" and "open T" assumes "⋀T'. T' ⊆ S ⟹ open T' ⟹ T' ⊆ T" shows "interior S = T" by (intro equalityI assms interior_subset open_interior interior_maximal) lemma interior_singleton [simp]: "interior {a} = {}" for a :: "'a::perfect_space" apply (rule interior_unique, simp_all) using not_open_singleton subset_singletonD apply fastforce done lemma interior_Int [simp]: "interior (S ∩ T) = interior S ∩ interior T" by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1 Int_lower2 interior_maximal interior_subset open_Int open_interior) lemma mem_interior: "x ∈ interior S ⟷ (∃e>0. ball x e ⊆ S)" using open_contains_ball_eq [where S="interior S"] by (simp add: open_subset_interior) lemma eventually_nhds_in_nhd: "x ∈ interior s ⟹ eventually (λy. y ∈ s) (nhds x)" using interior_subset[of s] by (subst eventually_nhds) blast lemma interior_limit_point [intro]: fixes x :: "'a::perfect_space" assumes x: "x ∈ interior S" shows "x islimpt S" using x islimpt_UNIV [of x] unfolding interior_def islimpt_def apply (clarsimp, rename_tac T T') apply (drule_tac x="T ∩ T'" in spec) apply (auto simp: open_Int) done lemma interior_closed_Un_empty_interior: assumes cS: "closed S" and iT: "interior T = {}" shows "interior (S ∪ T) = interior S" proof show "interior S ⊆ interior (S ∪ T)" by (rule interior_mono) (rule Un_upper1) show "interior (S ∪ T) ⊆ interior S" proof fix x assume "x ∈ interior (S ∪ T)" then obtain R where "open R" "x ∈ R" "R ⊆ S ∪ T" .. show "x ∈ interior S" proof (rule ccontr) assume "x ∉ interior S" with ‹x ∈ R› ‹open R› obtain y where "y ∈ R - S" unfolding interior_def by fast from ‹open R› ‹closed S› have "open (R - S)" by (rule open_Diff) from ‹R ⊆ S ∪ T› have "R - S ⊆ T" by fast from ‹y ∈ R - S› ‹open (R - S)› ‹R - S ⊆ T› ‹interior T = {}› show False unfolding interior_def by fast qed qed qed lemma interior_Times: "interior (A × B) = interior A × interior B" proof (rule interior_unique) show "interior A × interior B ⊆ A × B" by (intro Sigma_mono interior_subset) show "open (interior A × interior B)" by (intro open_Times open_interior) fix T assume "T ⊆ A × B" and "open T" then show "T ⊆ interior A × interior B" proof safe fix x y assume "(x, y) ∈ T" then obtain C D where "open C" "open D" "C × D ⊆ T" "x ∈ C" "y ∈ D" using ‹open T› unfolding open_prod_def by fast then have "open C" "open D" "C ⊆ A" "D ⊆ B" "x ∈ C" "y ∈ D" using ‹T ⊆ A × B› by auto then show "x ∈ interior A" and "y ∈ interior B" by (auto intro: interiorI) qed qed lemma interior_Ici: fixes x :: "'a :: {dense_linorder,linorder_topology}" assumes "b < x" shows "interior {x ..} = {x <..}" proof (rule interior_unique) fix T assume "T ⊆ {x ..}" "open T" moreover have "x ∉ T" proof assume "x ∈ T" obtain y where "y < x" "{y <.. x} ⊆ T" using open_left[OF ‹open T› ‹x ∈ T› ‹b < x›] by auto with dense[OF ‹y < x›] obtain z where "z ∈ T" "z < x" by (auto simp: subset_eq Ball_def) with ‹T ⊆ {x ..}› show False by auto qed ultimately show "T ⊆ {x <..}" by (auto simp: subset_eq less_le) qed auto lemma interior_Iic: fixes x :: "'a ::{dense_linorder,linorder_topology}" assumes "x < b" shows "interior {.. x} = {..< x}" proof (rule interior_unique) fix T assume "T ⊆ {.. x}" "open T" moreover have "x ∉ T" proof assume "x ∈ T" obtain y where "x < y" "{x ..< y} ⊆ T" using open_right[OF ‹open T› ‹x ∈ T› ‹x < b›] by auto with dense[OF ‹x < y›] obtain z where "z ∈ T" "x < z" by (auto simp: subset_eq Ball_def less_le) with ‹T ⊆ {.. x}› show False by auto qed ultimately show "T ⊆ {..< x}" by (auto simp: subset_eq less_le) qed auto subsection ‹Closure of a Set› definition%important "closure S = S ∪ {x | x. x islimpt S}" lemma interior_closure: "interior S = - (closure (- S))" by (auto simp: interior_def closure_def islimpt_def) lemma closure_interior: "closure S = - interior (- S)" by (simp add: interior_closure) lemma closed_closure[simp, intro]: "closed (closure S)" by (simp add: closure_interior closed_Compl) lemma closure_subset: "S ⊆ closure S" by (simp add: closure_def) lemma closure_hull: "closure S = closed hull S" by (auto simp: hull_def closure_interior interior_def) lemma closure_eq: "closure S = S ⟷ closed S" unfolding closure_hull using closed_Inter by (rule hull_eq) lemma closure_closed [simp]: "closed S ⟹ closure S = S" by (simp only: closure_eq) lemma closure_closure [simp]: "closure (closure S) = closure S" unfolding closure_hull by (rule hull_hull) lemma closure_mono: "S ⊆ T ⟹ closure S ⊆ closure T" unfolding closure_hull by (rule hull_mono) lemma closure_minimal: "S ⊆ T ⟹ closed T ⟹ closure S ⊆ T" unfolding closure_hull by (rule hull_minimal) lemma closure_unique: assumes "S ⊆ T" and "closed T" and "⋀T'. S ⊆ T' ⟹ closed T' ⟹ T ⊆ T'" shows "closure S = T" using assms unfolding closure_hull by (rule hull_unique) lemma closure_empty [simp]: "closure {} = {}" using closed_empty by (rule closure_closed) lemma closure_UNIV [simp]: "closure UNIV = UNIV" using closed_UNIV by (rule closure_closed) lemma closure_Un [simp]: "closure (S ∪ T) = closure S ∪ closure T" by (simp add: closure_interior) lemma closure_eq_empty [iff]: "closure S = {} ⟷ S = {}" using closure_empty closure_subset[of S] by blast lemma closure_subset_eq: "closure S ⊆ S ⟷ closed S" using closure_eq[of S] closure_subset[of S] by simp lemma open_Int_closure_eq_empty: "open S ⟹ (S ∩ closure T) = {} ⟷ S ∩ T = {}" using open_subset_interior[of S "- T"] using interior_subset[of "- T"] by (auto simp: closure_interior) lemma open_Int_closure_subset: "open S ⟹ S ∩ closure T ⊆ closure (S ∩ T)" proof fix x assume *: "open S" "x ∈ S ∩ closure T" have "x islimpt (S ∩ T)" if **: "x islimpt T" proof (rule islimptI) fix A assume "x ∈ A" "open A" with * have "x ∈ A ∩ S" "open (A ∩ S)" by (simp_all add: open_Int) with ** obtain y where "y ∈ T" "y ∈ A ∩ S" "y ≠ x" by (rule islimptE) then have "y ∈ S ∩ T" "y ∈ A ∧ y ≠ x" by simp_all then show "∃y∈(S ∩ T). y ∈ A ∧ y ≠ x" .. qed with * show "x ∈ closure (S ∩ T)" unfolding closure_def by blast qed lemma closure_complement: "closure (- S) = - interior S" by (simp add: closure_interior) lemma interior_complement: "interior (- S) = - closure S" by (simp add: closure_interior) lemma interior_diff: "interior(S - T) = interior S - closure T" by (simp add: Diff_eq interior_complement) lemma closure_Times: "closure (A × B) = closure A × closure B" proof (rule closure_unique) show "A × B ⊆ closure A × closure B" by (intro Sigma_mono closure_subset) show "closed (closure A × closure B)" by (intro closed_Times closed_closure) fix T assume "A × B ⊆ T" and "closed T" then show "closure A × closure B ⊆ T" apply (simp add: closed_def open_prod_def, clarify) apply (rule ccontr) apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D) apply (simp add: closure_interior interior_def) apply (drule_tac x=C in spec) apply (drule_tac x=D in spec, auto) done qed lemma closure_openin_Int_closure: assumes ope: "openin (subtopology euclidean U) S" and "T ⊆ U" shows "closure(S ∩ closure T) = closure(S ∩ T)" proof obtain V where "open V" and S: "S = U ∩ V" using ope using openin_open by metis show "closure (S ∩ closure T) ⊆ closure (S ∩ T)" proof (clarsimp simp: S) fix x assume "x ∈ closure (U ∩ V ∩ closure T)" then have "V ∩ closure T ⊆ A ⟹ x ∈ closure A" for A by (metis closure_mono subsetD inf.coboundedI2 inf_assoc) then have "x ∈ closure (T ∩ V)" by (metis ‹open V› closure_closure inf_commute open_Int_closure_subset) then show "x ∈ closure (U ∩ V ∩ T)" by (metis ‹T ⊆ U› inf.absorb_iff2 inf_assoc inf_commute) qed next show "closure (S ∩ T) ⊆ closure (S ∩ closure T)" by (meson Int_mono closure_mono closure_subset order_refl) qed lemma islimpt_in_closure: "(x islimpt S) = (x∈closure(S-{x}))" unfolding closure_def using islimpt_punctured by blast lemma connected_imp_connected_closure: "connected S ⟹ connected (closure S)" by (rule connectedI) (meson closure_subset open_Int open_Int_closure_eq_empty subset_trans connectedD) lemma limpt_of_limpts: "x islimpt {y. y islimpt S} ⟹ x islimpt S" for x :: "'a::metric_space" apply (clarsimp simp add: islimpt_approachable) apply (drule_tac x="e/2" in spec) apply (auto simp: simp del: less_divide_eq_numeral1) apply (drule_tac x="dist x' x" in spec) apply (auto simp: zero_less_dist_iff simp del: less_divide_eq_numeral1) apply (erule rev_bexI) apply (metis dist_commute dist_triangle_half_r less_trans less_irrefl) done lemma closed_limpts: "closed {x::'a::metric_space. x islimpt S}" using closed_limpt limpt_of_limpts by blast lemma limpt_of_closure: "x islimpt closure S ⟷ x islimpt S" for x :: "'a::metric_space" by (auto simp: closure_def islimpt_Un dest: limpt_of_limpts) lemma closedin_limpt: "closedin (subtopology euclidean T) S ⟷ S ⊆ T ∧ (∀x. x islimpt S ∧ x ∈ T ⟶ x ∈ S)" apply (simp add: closedin_closed, safe) apply (simp add: closed_limpt islimpt_subset) apply (rule_tac x="closure S" in exI, simp) apply (force simp: closure_def) done lemma closedin_closed_eq: "closed S ⟹ closedin (subtopology euclidean S) T ⟷ closed T ∧ T ⊆ S" by (meson closedin_limpt closed_subset closedin_closed_trans) lemma connected_closed_set: "closed S ⟹ connected S ⟷ (∄A B. closed A ∧ closed B ∧ A ≠ {} ∧ B ≠ {} ∧ A ∪ B = S ∧ A ∩ B = {})" unfolding connected_closedin_eq closedin_closed_eq connected_closedin_eq by blast text ‹If a connnected set is written as the union of two nonempty closed sets, then these sets have to intersect.› lemma connected_as_closed_union: assumes "connected C" "C = A ∪ B" "closed A" "closed B" "A ≠ {}" "B ≠ {}" shows "A ∩ B ≠ {}" by (metis assms closed_Un connected_closed_set) lemma closedin_subset_trans: "closedin (subtopology euclidean U) S ⟹ S ⊆ T ⟹ T ⊆ U ⟹ closedin (subtopology euclidean T) S" by (meson closedin_limpt subset_iff) lemma openin_subset_trans: "openin (subtopology euclidean U) S ⟹ S ⊆ T ⟹ T ⊆ U ⟹ openin (subtopology euclidean T) S" by (auto simp: openin_open) lemma openin_Times: "openin (subtopology euclidean S) S' ⟹ openin (subtopology euclidean T) T' ⟹ openin (subtopology euclidean (S × T)) (S' × T')" unfolding openin_open using open_Times by blast lemma Times_in_interior_subtopology: fixes U :: "('a::metric_space × 'b::metric_space) set" assumes "(x, y) ∈ U" "openin (subtopology euclidean (S × T)) U" obtains V W where "openin (subtopology euclidean S) V" "x ∈ V" "openin (subtopology euclidean T) W" "y ∈ W" "(V × W) ⊆ U" proof - from assms obtain e where "e > 0" and "U ⊆ S × T" and e: "⋀x' y'. ⟦x'∈S; y'∈T; dist (x', y') (x, y) < e⟧ ⟹ (x', y') ∈ U" by (force simp: openin_euclidean_subtopology_iff) with assms have "x ∈ S" "y ∈ T" by auto show ?thesis proof show "openin (subtopology euclidean S) (ball x (e/2) ∩ S)" by (simp add: Int_commute openin_open_Int) show "x ∈ ball x (e / 2) ∩ S" by (simp add: ‹0 < e› ‹x ∈ S›) show "openin (subtopology euclidean T) (ball y (e/2) ∩ T)" by (simp add: Int_commute openin_open_Int) show "y ∈ ball y (e / 2) ∩ T" by (simp add: ‹0 < e› ‹y ∈ T›) show "(ball x (e / 2) ∩ S) × (ball y (e / 2) ∩ T) ⊆ U" by clarify (simp add: e dist_Pair_Pair ‹0 < e› dist_commute sqrt_sum_squares_half_less) qed qed lemma openin_Times_eq: fixes S :: "'a::metric_space set" and T :: "'b::metric_space set" shows "openin (subtopology euclidean (S × T)) (S' × T') ⟷ S' = {} ∨ T' = {} ∨ openin (subtopology euclidean S) S' ∧ openin (subtopology euclidean T) T'" (is "?lhs = ?rhs") proof (cases "S' = {} ∨ T' = {}") case True then show ?thesis by auto next case False then obtain x y where "x ∈ S'" "y ∈ T'" by blast show ?thesis proof assume ?lhs have "openin (subtopology euclidean S) S'" apply (subst openin_subopen, clarify) apply (rule Times_in_interior_subtopology [OF _ ‹?lhs›]) using ‹y ∈ T'› apply auto done moreover have "openin (subtopology euclidean T) T'" apply (subst openin_subopen, clarify) apply (rule Times_in_interior_subtopology [OF _ ‹?lhs›]) using ‹x ∈ S'› apply auto done ultimately show ?rhs by simp next assume ?rhs with False show ?lhs by (simp add: openin_Times) qed qed lemma closedin_Times: "closedin (subtopology euclidean S) S' ⟹ closedin (subtopology euclidean T) T' ⟹ closedin (subtopology euclidean (S × T)) (S' × T')" unfolding closedin_closed using closed_Times by blast lemma bdd_below_closure: fixes A :: "real set" assumes "bdd_below A" shows "bdd_below (closure A)" proof - from assms obtain m where "⋀x. x ∈ A ⟹ m ≤ x" by (auto simp: bdd_below_def) then have "A ⊆ {m..}" by auto then have "closure A ⊆ {m..}" using closed_real_atLeast by (rule closure_minimal) then show ?thesis by (auto simp: bdd_below_def) qed subsection ‹Frontier (also known as boundary)› definition%important "frontier S = closure S - interior S" lemma frontier_closed [iff]: "closed (frontier S)" by (simp add: frontier_def closed_Diff) lemma frontier_closures: "frontier S = closure S ∩ closure (- S)" by (auto simp: frontier_def interior_closure) lemma frontier_Int: "frontier(S ∩ T) = closure(S ∩ T) ∩ (frontier S ∪ frontier T)" proof - have "closure (S ∩ T) ⊆ closure S" "closure (S ∩ T) ⊆ closure T" by (simp_all add: closure_mono) then show ?thesis by (auto simp: frontier_closures) qed lemma frontier_Int_subset: "frontier(S ∩ T) ⊆ frontier S ∪ frontier T" by (auto simp: frontier_Int) lemma frontier_Int_closed: assumes "closed S" "closed T" shows "frontier(S ∩ T) = (frontier S ∩ T) ∪ (S ∩ frontier T)" proof - have "closure (S ∩ T) = T ∩ S" using assms by (simp add: Int_commute closed_Int) moreover have "T ∩ (closure S ∩ closure (- S)) = frontier S ∩ T" by (simp add: Int_commute frontier_closures) ultimately show ?thesis by (simp add: Int_Un_distrib Int_assoc Int_left_commute assms frontier_closures) qed lemma frontier_straddle: fixes a :: "'a::metric_space" shows "a ∈ frontier S ⟷ (∀e>0. (∃x∈S. dist a x < e) ∧ (∃x. x ∉ S ∧ dist a x < e))" unfolding frontier_def closure_interior by (auto simp: mem_interior subset_eq ball_def) lemma frontier_subset_closed: "closed S ⟹ frontier S ⊆ S" by (metis frontier_def closure_closed Diff_subset) lemma frontier_empty [simp]: "frontier {} = {}" by (simp add: frontier_def) lemma frontier_subset_eq: "frontier S ⊆ S ⟷ closed S" proof - { assume "frontier S ⊆ S" then have "closure S ⊆ S" using interior_subset unfolding frontier_def by auto then have "closed S" using closure_subset_eq by auto } then show ?thesis using frontier_subset_closed[of S] .. qed lemma frontier_complement [simp]: "frontier (- S) = frontier S" by (auto simp: frontier_def closure_complement interior_complement) lemma frontier_Un_subset: "frontier(S ∪ T) ⊆ frontier S ∪ frontier T" by (metis compl_sup frontier_Int_subset frontier_complement) lemma frontier_disjoint_eq: "frontier S ∩ S = {} ⟷ open S" using frontier_complement frontier_subset_eq[of "- S"] unfolding open_closed by auto lemma frontier_UNIV [simp]: "frontier UNIV = {}" using frontier_complement frontier_empty by fastforce lemma frontier_interiors: "frontier s = - interior(s) - interior(-s)" by (simp add: Int_commute frontier_def interior_closure) lemma frontier_interior_subset: "frontier(interior S) ⊆ frontier S" by (simp add: Diff_mono frontier_interiors interior_mono interior_subset) lemma connected_Int_frontier: "⟦connected s; s ∩ t ≠ {}; s - t ≠ {}⟧ ⟹ (s ∩ frontier t ≠ {})" apply (simp add: frontier_interiors connected_openin, safe) apply (drule_tac x="s ∩ interior t" in spec, safe) apply (drule_tac [2] x="s ∩ interior (-t)" in spec) apply (auto simp: disjoint_eq_subset_Compl dest: interior_subset [THEN subsetD]) done lemma closure_Un_frontier: "closure S = S ∪ frontier S" proof - have "S ∪ interior S = S" using interior_subset by auto then show ?thesis using closure_subset by (auto simp: frontier_def) qed subsection%unimportant ‹Filters and the ``eventually true'' quantifier› definition indirection :: "'a::real_normed_vector ⇒ 'a ⇒ 'a filter" (infixr "indirection" 70) where "a indirection v = at a within {b. ∃c≥0. b - a = scaleR c v}" text ‹Identify Trivial limits, where we can't approach arbitrarily closely.› lemma trivial_limit_within: "trivial_limit (at a within S) ⟷ ¬ a islimpt S" proof assume "trivial_limit (at a within S)" then show "¬ a islimpt S" unfolding trivial_limit_def unfolding eventually_at_topological unfolding islimpt_def apply (clarsimp simp add: set_eq_iff) apply (rename_tac T, rule_tac x=T in exI) apply (clarsimp, drule_tac x=y in bspec, simp_all) done next assume "¬ a islimpt S" then show "trivial_limit (at a within S)" unfolding trivial_limit_def eventually_at_topological islimpt_def by metis qed lemma trivial_limit_at_iff: "trivial_limit (at a) ⟷ ¬ a islimpt UNIV" using trivial_limit_within [of a UNIV] by simp lemma trivial_limit_at: "¬ trivial_limit (at a)" for a :: "'a::perfect_space" by (rule at_neq_bot) lemma trivial_limit_at_infinity: "¬ trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)" unfolding trivial_limit_def eventually_at_infinity apply clarsimp apply (subgoal_tac "∃x::'a. x ≠ 0", clarify) apply (rule_tac x="scaleR (b / norm x) x" in exI, simp) apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def]) apply (drule_tac x=UNIV in spec, simp) done lemma not_trivial_limit_within: "¬ trivial_limit (at x within S) = (x ∈ closure (S - {x}))" using islimpt_in_closure by (metis trivial_limit_within) lemma not_in_closure_trivial_limitI: "x ∉ closure s ⟹ trivial_limit (at x within s)" using not_trivial_limit_within[of x s] by safe (metis Diff_empty Diff_insert0 closure_subset contra_subsetD) lemma filterlim_at_within_closure_implies_filterlim: "filterlim f l (at x within s)" if "x ∈ closure s ⟹ filterlim f l (at x within s)" by (metis bot.extremum filterlim_filtercomap filterlim_mono not_in_closure_trivial_limitI that) lemma at_within_eq_bot_iff: "at c within A = bot ⟷ c ∉ closure (A - {c})" using not_trivial_limit_within[of c A] by blast text ‹Some property holds "sufficiently close" to the limit point.› lemma trivial_limit_eventually: "trivial_limit net ⟹ eventually P net" by simp lemma trivial_limit_eq: "trivial_limit net ⟷ (∀P. eventually P net)" by (simp add: filter_eq_iff) subsection ‹Limits› proposition Lim: "(f ⤏ l) net ⟷ trivial_limit net ∨ (∀e>0. eventually (λx. dist (f x) l < e) net)" by (auto simp: tendsto_iff trivial_limit_eq) text ‹Show that they yield usual definitions in the various cases.› proposition Lim_within_le: "(f ⤏ l)(at a within S) ⟷ (∀e>0. ∃d>0. ∀x∈S. 0 < dist x a ∧ dist x a ≤ d ⟶ dist (f x) l < e)" by (auto simp: tendsto_iff eventually_at_le) proposition Lim_within: "(f ⤏ l) (at a within S) ⟷ (∀e >0. ∃d>0. ∀x ∈ S. 0 < dist x a ∧ dist x a < d ⟶ dist (f x) l < e)" by (auto simp: tendsto_iff eventually_at) corollary Lim_withinI [intro?]: assumes "⋀e. e > 0 ⟹ ∃d>0. ∀x ∈ S. 0 < dist x a ∧ dist x a < d ⟶ dist (f x) l ≤ e" shows "(f ⤏ l) (at a within S)" apply (simp add: Lim_within, clarify) apply (rule ex_forward [OF assms [OF half_gt_zero]], auto) done proposition Lim_at: "(f ⤏ l) (at a) ⟷ (∀e >0. ∃d>0. ∀x. 0 < dist x a ∧ dist x a < d ⟶ dist (f x) l < e)" by (auto simp: tendsto_iff eventually_at) proposition Lim_at_infinity: "(f ⤏ l) at_infinity ⟷ (∀e>0. ∃b. ∀x. norm x ≥ b ⟶ dist (f x) l < e)" by (auto simp: tendsto_iff eventually_at_infinity) corollary Lim_at_infinityI [intro?]: assumes "⋀e. e > 0 ⟹ ∃B. ∀x. norm x ≥ B ⟶ dist (f x) l ≤ e" shows "(f ⤏ l) at_infinity" apply (simp add: Lim_at_infinity, clarify) apply (rule ex_forward [OF assms [OF half_gt_zero]], auto) done lemma Lim_eventually: "eventually (λx. f x = l) net ⟹ (f ⤏ l) net" by (rule topological_tendstoI) (auto elim: eventually_mono) lemma Lim_transform_within_set: fixes a :: "'a::metric_space" and l :: "'b::metric_space" shows "⟦(f ⤏ l) (at a within S); eventually (λx. x ∈ S ⟷ x ∈ T) (at a)⟧ ⟹ (f ⤏ l) (at a within T)" apply (clarsimp simp: eventually_at Lim_within) apply (drule_tac x=e in spec, clarify) apply (rename_tac k) apply (rule_tac x="min d k" in exI, simp) done lemma Lim_transform_within_set_eq: fixes a l :: "'a::real_normed_vector" shows "eventually (λx. x ∈ s ⟷ x ∈ t) (at a) ⟹ ((f ⤏ l) (at a within s) ⟷ (f ⤏ l) (at a within t))" by (force intro: Lim_transform_within_set elim: eventually_mono) lemma Lim_transform_within_openin: fixes a :: "'a::metric_space" assumes f: "(f ⤏ l) (at a within T)" and "openin (subtopology euclidean T) S" "a ∈ S" and eq: "⋀x. ⟦x ∈ S; x ≠ a⟧ ⟹ f x = g x" shows "(g ⤏ l) (at a within T)" proof - obtain ε where "0 < ε" and ε: "ball a ε ∩ T ⊆ S" using assms by (force simp: openin_contains_ball) then have "a ∈ ball a ε" by simp show ?thesis by (rule Lim_transform_within [OF f ‹0 < ε› eq]) (use ε in ‹auto simp: dist_commute subset_iff›) qed lemma continuous_transform_within_openin: fixes a :: "'a::metric_space" assumes "continuous (at a within T) f" and "openin (subtopology euclidean T) S" "a ∈ S" and eq: "⋀x. x ∈ S ⟹ f x = g x" shows "continuous (at a within T) g" using assms by (simp add: Lim_transform_within_openin continuous_within) text ‹The expected monotonicity property.› lemma Lim_Un: assumes "(f ⤏ l) (at x within S)" "(f ⤏ l) (at x within T)" shows "(f ⤏ l) (at x within (S ∪ T))" using assms unfolding at_within_union by (rule filterlim_sup) lemma Lim_Un_univ: "(f ⤏ l) (at x within S) ⟹ (f ⤏ l) (at x within T) ⟹ S ∪ T = UNIV ⟹ (f ⤏ l) (at x)" by (metis Lim_Un) text ‹Interrelations between restricted and unrestricted limits.› lemma Lim_at_imp_Lim_at_within: "(f ⤏ l) (at x) ⟹ (f ⤏ l) (at x within S)" by (metis order_refl filterlim_mono subset_UNIV at_le) lemma eventually_within_interior: assumes "x ∈ interior S" shows "eventually P (at x within S) ⟷ eventually P (at x)" (is "?lhs = ?rhs") proof from assms obtain T where T: "open T" "x ∈ T" "T ⊆ S" .. { assume ?lhs then obtain A where "open A" and "x ∈ A" and "∀y∈A. y ≠ x ⟶ y ∈ S ⟶ P y" by (auto simp: eventually_at_topological) with T have "open (A ∩ T)" and "x ∈ A ∩ T" and "∀y ∈ A ∩ T. y ≠ x ⟶ P y" by auto then show ?rhs by (auto simp: eventually_at_topological) next assume ?rhs then show ?lhs by (auto elim: eventually_mono simp: eventually_at_filter) } qed lemma at_within_interior: "x ∈ interior S ⟹ at x within S = at x" unfolding filter_eq_iff by (intro allI eventually_within_interior) lemma Lim_within_LIMSEQ: fixes a :: "'a::first_countable_topology" assumes "∀S. (∀n. S n ≠ a ∧ S n ∈ T) ∧ S ⇢ a ⟶ (λn. X (S n)) ⇢ L" shows "(X ⤏ L) (at a within T)" using assms unfolding tendsto_def [where l=L] by (simp add: sequentially_imp_eventually_within) lemma Lim_right_bound: fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} ⇒ 'b::{linorder_topology, conditionally_complete_linorder}" assumes mono: "⋀a b. a ∈ I ⟹ b ∈ I ⟹ x < a ⟹ a ≤ b ⟹ f a ≤ f b" and bnd: "⋀a. a ∈ I ⟹ x < a ⟹ K ≤ f a" shows "(f ⤏ Inf (f ` ({x<..} ∩ I))) (at x within ({x<..} ∩ I))" proof (cases "{x<..} ∩ I = {}") case True then show ?thesis by simp next case False show ?thesis proof (rule order_tendstoI) fix a assume a: "a < Inf (f ` ({x<..} ∩ I))" { fix y assume "y ∈ {x<..} ∩ I" with False bnd have "Inf (f ` ({x<..} ∩ I)) ≤ f y" by (auto intro!: cInf_lower bdd_belowI2) with a have "a < f y" by (blast intro: less_le_trans) } then show "eventually (λx. a < f x) (at x within ({x<..} ∩ I))" by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one) next fix a assume "Inf (f ` ({x<..} ∩ I)) < a" from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y ∈ I" "f y < a" by auto then have "eventually (λx. x ∈ I ⟶ f x < a) (at_right x)" unfolding eventually_at_right[OF ‹x < y›] by (metis less_imp_le le_less_trans mono) then show "eventually (λx. f x < a) (at x within ({x<..} ∩ I))" unfolding eventually_at_filter by eventually_elim simp qed qed text ‹Another limit point characterization.› lemma limpt_sequential_inj: fixes x :: "'a::metric_space" shows "x islimpt S ⟷ (∃f. (∀n::nat. f n ∈ S - {x}) ∧ inj f ∧ (f ⤏ x) sequentially)" (is "?lhs = ?rhs") proof assume ?lhs then have "∀e>0. ∃x'∈S. x' ≠ x ∧ dist x' x < e" by (force simp: islimpt_approachable) then obtain y where y: "⋀e. e>0 ⟹ y e ∈ S ∧ y e ≠ x ∧ dist (y e) x < e" by metis define f where "f ≡ rec_nat (y 1) (λn fn. y (min (inverse(2 ^ (Suc n))) (dist fn x)))" have [simp]: "f 0 = y 1" "f(Suc n) = y (min (inverse(2 ^ (Suc n))) (dist (f n) x))" for n by (simp_all add: f_def) have f: "f n ∈ S ∧ (f n ≠ x) ∧ dist (f n) x < inverse(2 ^ n)" for n proof (induction n) case 0 show ?case by (simp add: y) next case (Suc n) then show ?case apply (auto simp: y) by (metis half_gt_zero_iff inverse_positive_iff_positive less_divide_eq_numeral1(1) min_less_iff_conj y zero_less_dist_iff zero_less_numeral zero_less_power) qed show ?rhs proof (rule_tac x=f in exI, intro conjI allI) show "⋀n. f n ∈ S - {x}" using f by blast have "dist (f n) x < dist (f m) x" if "m < n" for m n using that proof (induction n) case 0 then show ?case by simp next case (Suc n) then consider "m < n" | "m = n" using less_Suc_eq by blast then show ?case proof cases assume "m < n" have "dist (f(Suc n)) x = dist (y (min (inverse(2 ^ (Suc n))) (dist (f n) x))) x" by simp also have "… < dist (f n) x" by (metis dist_pos_lt f min.strict_order_iff min_less_iff_conj y) also have "… < dist (f m) x" using Suc.IH ‹m < n› by blast finally show ?thesis . next assume "m = n" then show ?case by simp (metis dist_pos_lt f half_gt_zero_iff inverse_positive_iff_positive min_less_iff_conj y zero_less_numeral zero_less_power) qed qed then show "inj f" by (metis less_irrefl linorder_injI) show "f ⇢ x" apply (rule tendstoI) apply (rule_tac c="nat (ceiling(1/e))" in eventually_sequentiallyI) apply (rule less_trans [OF f [THEN conjunct2, THEN conjunct2]]) apply (simp add: field_simps) by (meson le_less_trans mult_less_cancel_left not_le of_nat_less_two_power) qed next assume ?rhs then show ?lhs by (fastforce simp add: islimpt_approachable lim_sequentially) qed (*could prove directly from islimpt_sequential_inj, but only for metric spaces*) lemma islimpt_sequential: fixes x :: "'a::first_countable_topology" shows "x islimpt S ⟷ (∃f. (∀n::nat. f n ∈ S - {x}) ∧ (f ⤏ x) sequentially)" (is "?lhs = ?rhs") proof assume ?lhs from countable_basis_at_decseq[of x] obtain A where A: "⋀i. open (A i)" "⋀i. x ∈ A i" "⋀S. open S ⟹ x ∈ S ⟹ eventually (λi. A i ⊆ S) sequentially" by blast define f where "f n = (SOME y. y ∈ S ∧ y ∈ A n ∧ x ≠ y)" for n { fix n from ‹?lhs› have "∃y. y ∈ S ∧ y ∈ A n ∧ x ≠ y" unfolding islimpt_def using A(1,2)[of n] by auto then have "f n ∈ S ∧ f n ∈ A n ∧ x ≠ f n" unfolding f_def by (rule someI_ex) then have "f n ∈ S" "f n ∈ A n" "x ≠ f n" by auto } then have "∀n. f n ∈ S - {x}" by auto moreover have "(λn. f n) ⇢ x" proof (rule topological_tendstoI) fix S assume "open S" "x ∈ S" from A(3)[OF this] ‹⋀n. f n ∈ A n› show "eventually (λx. f x ∈ S) sequentially" by (auto elim!: eventually_mono) qed ultimately show ?rhs by fast next assume ?rhs then obtain f :: "nat ⇒ 'a" where f: "⋀n. f n ∈ S - {x}" and lim: "f ⇢ x" by auto show ?lhs unfolding islimpt_def proof safe fix T assume "open T" "x ∈ T" from lim[THEN topological_tendstoD, OF this] f show "∃y∈S. y ∈ T ∧ y ≠ x" unfolding eventually_sequentially by auto qed qed lemma Lim_null: fixes f :: "'a ⇒ 'b::real_normed_vector" shows "(f ⤏ l) net ⟷ ((λx. f(x) - l) ⤏ 0) net" by (simp add: Lim dist_norm) lemma Lim_null_comparison: fixes f :: "'a ⇒ 'b::real_normed_vector" assumes "eventually (λx. norm (f x) ≤ g x) net" "(g ⤏ 0) net" shows "(f ⤏ 0) net" using assms(2) proof (rule metric_tendsto_imp_tendsto) show "eventually (λx. dist (f x) 0 ≤ dist (g x) 0) net" using assms(1) by (rule eventually_mono) (simp add: dist_norm) qed lemma Lim_transform_bound: fixes f :: "'a ⇒ 'b::real_normed_vector" and g :: "'a ⇒ 'c::real_normed_vector" assumes "eventually (λn. norm (f n) ≤ norm (g n)) net" and "(g ⤏ 0) net" shows "(f ⤏ 0) net" using assms(1) tendsto_norm_zero [OF assms(2)] by (rule Lim_null_comparison) lemma lim_null_mult_right_bounded: fixes f :: "'a ⇒ 'b::real_normed_div_algebra" assumes f: "(f ⤏ 0) F" and g: "eventually (λx. norm(g x) ≤ B) F" shows "((λz. f z * g z) ⤏ 0) F" proof - have *: "((λx. norm (f x) * B) ⤏ 0) F" by (simp add: f tendsto_mult_left_zero tendsto_norm_zero) have "((λx. norm (f x) * norm (g x)) ⤏ 0) F" apply (rule Lim_null_comparison [OF _ *]) apply (simp add: eventually_mono [OF g] mult_left_mono) done then show ?thesis by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult) qed lemma lim_null_mult_left_bounded: fixes f :: "'a ⇒ 'b::real_normed_div_algebra" assumes g: "eventually (λx. norm(g x) ≤ B) F" and f: "(f ⤏ 0) F" shows "((λz. g z * f z) ⤏ 0) F" proof - have *: "((λx. B * norm (f x)) ⤏ 0) F" by (simp add: f tendsto_mult_right_zero tendsto_norm_zero) have "((λx. norm (g x) * norm (f x)) ⤏ 0) F" apply (rule Lim_null_comparison [OF _ *]) apply (simp add: eventually_mono [OF g] mult_right_mono) done then show ?thesis by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult) qed lemma lim_null_scaleR_bounded: assumes f: "(f ⤏ 0) net" and gB: "eventually (λa. f a = 0 ∨ norm(g a) ≤ B) net" shows "((λn. f n *⇩_{R}g n) ⤏ 0) net" proof fix ε::real assume "0 < ε" then have B: "0 < ε / (abs B + 1)" by simp have *: "¦f x¦ * norm (g x) < ε" if f: "¦f x¦ * (¦B¦ + 1) < ε" and g: "norm (g x) ≤ B" for x proof - have "¦f x¦ * norm (g x) ≤ ¦f x¦ * B" by (simp add: mult_left_mono g) also have "… ≤ ¦f x¦ * (¦B¦ + 1)" by (simp add: mult_left_mono) also have "… < ε" by (rule f) finally show ?thesis . qed show "∀⇩_{F}x in net. dist (f x *⇩_{R}g x) 0 < ε" apply (rule eventually_mono [OF eventually_conj [OF tendstoD [OF f B] gB] ]) apply (auto simp: ‹0 < ε› divide_simps * split: if_split_asm) done qed text‹Deducing things about the limit from the elements.› lemma Lim_in_closed_set: assumes "closed S" and "eventually (λx. f(x) ∈ S) net" and "¬ trivial_limit net" "(f ⤏ l) net" shows "l ∈ S" proof (rule ccontr) assume "l ∉ S" with ‹closed S› have "open (- S)" "l ∈ - S" by (simp_all add: open_Compl) with assms(4) have "eventually (λx. f x ∈ - S) net" by (rule topological_tendstoD) with assms(2) have "eventually (λx. False) net" by (rule eventually_elim2) simp with assms(3) show "False" by (simp add: eventually_False) qed text‹Need to prove closed(cball(x,e)) before deducing this as a corollary.› lemma Lim_dist_ubound: assumes "¬(trivial_limit net)" and "(f ⤏ l) net" and "eventually (λx. dist a (f x) ≤ e) net" shows "dist a l ≤ e" using assms by (fast intro: tendsto_le tendsto_intros) lemma Lim_norm_ubound: fixes f :: "'a ⇒ 'b::real_normed_vector" assumes "¬(trivial_limit net)" "(f ⤏ l) net" "eventually (λx. norm(f x) ≤ e) net" shows "norm(l) ≤ e" using assms by (fast intro: tendsto_le tendsto_intros) lemma Lim_norm_lbound: fixes f :: "'a ⇒ 'b::real_normed_vector" assumes "¬ trivial_limit net" and "(f ⤏ l) net" and "eventually (λx. e ≤ norm (f x)) net" shows "e ≤ norm l" using assms by (fast intro: tendsto_le tendsto_intros) text‹Limit under bilinear function› lemma Lim_bilinear: assumes "(f ⤏ l) net" and "(g ⤏ m) net" and "bounded_bilinear h" shows "((λx. h (f x) (g x)) ⤏ (h l m)) net" using ‹bounded_bilinear h› ‹(f ⤏ l) net› ‹(g ⤏ m) net› by (rule bounded_bilinear.tendsto) text‹These are special for limits out of the same vector space.› lemma Lim_within_id: "(id ⤏ a) (at a within s)" unfolding id_def by (rule tendsto_ident_at) lemma Lim_at_id: "(id ⤏ a) (at a)" unfolding id_def by (rule tendsto_ident_at) lemma Lim_at_zero: fixes a :: "'a::real_normed_vector" and l :: "'b::topological_space" shows "(f ⤏ l) (at a) ⟷ ((λx. f(a + x)) ⤏ l) (at 0)" using LIM_offset_zero LIM_offset_zero_cancel .. text‹It's also sometimes useful to extract the limit point from the filter.› abbreviation netlimit :: "'a::t2_space filter ⇒ 'a" where "netlimit F ≡ Lim F (λx. x)" lemma netlimit_within: "¬ trivial_limit (at a within S) ⟹ netlimit (at a within S) = a" by (rule tendsto_Lim) (auto intro: tendsto_intros) lemma netlimit_at [simp]: fixes a :: "'a::{perfect_space,t2_space}" shows "netlimit (at a) = a" using netlimit_within [of a UNIV] by simp lemma lim_within_interior: "x ∈ interior S ⟹ (f ⤏ l) (at x within S) ⟷ (f ⤏ l) (at x)" by (metis at_within_interior) lemma netlimit_within_interior: fixes x :: "'a::{t2_space,perfect_space}" assumes "x ∈ interior S" shows "netlimit (at x within S) = x" using assms by (metis at_within_interior netlimit_at) lemma netlimit_at_vector: fixes a :: "'a::real_normed_vector" shows "netlimit (at a) = a" proof (cases "∃x. x ≠ a") case True then obtain x where x: "x ≠ a" .. have "¬ trivial_limit (at a)" unfolding trivial_limit_def eventually_at dist_norm apply clarsimp apply (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI) apply (simp add: norm_sgn sgn_zero_iff x) done then show ?thesis by (rule netlimit_within [of a UNIV]) qed simp text‹Useful lemmas on closure and set of possible sequential limits.› lemma closure_sequential: fixes l :: "'a::first_countable_topology" shows "l ∈ closure S ⟷ (∃x. (∀n. x n ∈ S) ∧ (x ⤏ l) sequentially)" (is "?lhs = ?rhs") proof assume "?lhs" moreover { assume "l ∈ S" then have "?rhs" using tendsto_const[of l sequentially] by auto } moreover { assume "l islimpt S" then have "?rhs" unfolding islimpt_sequential by auto } ultimately show "?rhs" unfolding closure_def by auto next assume "?rhs" then show "?lhs" unfolding closure_def islimpt_sequential by auto qed lemma closed_sequential_limits: fixes S :: "'a::first_countable_topology set" shows "closed S ⟷ (∀x l. (∀n. x n ∈ S) ∧ (x ⤏ l) sequentially ⟶ l ∈ S)" by (metis closure_sequential closure_subset_eq subset_iff) lemma closure_approachable: fixes S :: "'a::metric_space set" shows "x ∈ closure S ⟷ (∀e>0. ∃y∈S. dist y x < e)" apply (auto simp: closure_def islimpt_approachable) apply (metis dist_self) done lemma closure_approachable_le: fixes S :: "'a::metric_space set" shows "x ∈ closure S ⟷ (∀e>0. ∃y∈S. dist y x ≤ e)" unfolding closure_approachable using dense by force lemma closure_approachableD: assumes "x ∈ closure S" "e>0" shows "∃y∈S. dist x y < e" using assms unfolding closure_approachable by (auto simp: dist_commute) lemma closed_approachable: fixes S :: "'a::metric_space set" shows "closed S ⟹ (∀e>0. ∃y∈S. dist y x < e) ⟷ x ∈ S" by (metis closure_closed closure_approachable) lemma closure_contains_Inf: fixes S :: "real set" assumes "S ≠ {}" "bdd_below S" shows "Inf S ∈ closure S" proof - have *: "∀x∈S. Inf S ≤ x" using cInf_lower[of _ S] assms by metis { fix e :: real assume "e > 0" then have "Inf S < Inf S + e" by simp with assms obtain x where "x ∈ S" "x < Inf S + e" by (subst (asm) cInf_less_iff) auto with * have "∃x∈S. dist x (Inf S) < e" by (intro bexI[of _ x]) (auto simp: dist_real_def) } then show ?thesis unfolding closure_approachable by auto qed lemma closure_Int_ballI: fixes S :: "'a :: metric_space set" assumes "⋀U. ⟦openin (subtopology euclidean S) U; U ≠ {}⟧ ⟹ T ∩ U ≠ {}" shows "S ⊆ closure T" proof (clarsimp simp: closure_approachable dist_commute) fix x and e::real assume "x ∈ S" "0 < e" with assms [of "S ∩ ball x e"] show "∃y∈T. dist x y < e" by force qed lemma closed_contains_Inf: fixes S :: "real set" shows "S ≠ {} ⟹ bdd_below S ⟹ closed S ⟹ Inf S ∈ S" by (metis closure_contains_Inf closure_closed) lemma closed_subset_contains_Inf: fixes A C :: "real set" shows "closed C ⟹ A ⊆ C ⟹ A ≠ {} ⟹ bdd_below A ⟹ Inf A ∈ C" by (metis closure_contains_Inf closure_minimal subset_eq) lemma atLeastAtMost_subset_contains_Inf: fixes A :: "real set" and a b :: real shows "A ≠ {} ⟹ a ≤ b ⟹ A ⊆ {a..b} ⟹ Inf A ∈ {a..b}" by (rule closed_subset_contains_Inf) (auto intro: closed_real_atLeastAtMost intro!: bdd_belowI[of A a]) lemma not_trivial_limit_within_ball: "¬ trivial_limit (at x within S) ⟷ (∀e>0. S ∩ ball x e - {x} ≠ {})" (is "?lhs ⟷ ?rhs") proof show ?rhs if ?lhs proof - { fix e :: real assume "e > 0" then obtain y where "y ∈ S - {x}" and "dist y x < e" using ‹?lhs› not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto then have "y ∈ S ∩ ball x e - {x}" unfolding ball_def by (simp add: dist_commute) then have "S ∩ ball x e - {x} ≠ {}" by blast } then show ?thesis by auto qed show ?lhs if ?rhs proof - { fix e :: real assume "e > 0" then obtain y where "y ∈ S ∩ ball x e - {x}" using ‹?rhs› by blast then have "y ∈ S - {x}" and "dist y x < e" unfolding ball_def by (simp_all add: dist_commute) then have "∃y ∈ S - {x}. dist y x < e" by auto } then show ?thesis using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto qed qed lemma tendsto_If_within_closures: assumes f: "x ∈ s ∪ (closure s ∩ closure t) ⟹ (f ⤏ l x) (at x within s ∪ (closure s ∩ closure t))" assumes g: "x ∈ t ∪ (closure s ∩ closure t) ⟹ (g ⤏ l x) (at x within t ∪ (closure s ∩ closure t))" assumes "x ∈ s ∪ t" shows "((λx. if x ∈ s then f x else g x) ⤏ l x) (at x within s ∪ t)" proof - have *: "(s ∪ t) ∩ {x. x ∈ s} = s" "(s ∪ t) ∩ {x. x ∉ s} = t - s" by auto have "(f ⤏ l x) (at x within s)" by (rule filterlim_at_within_closure_implies_filterlim) (use ‹x ∈ _› in ‹auto simp: inf_commute closure_def intro: tendsto_within_subset[OF f]›) moreover have "(g ⤏ l x) (at x within t - s)" by (rule filterlim_at_within_closure_implies_filterlim) (use ‹x ∈ _› in ‹auto intro!: tendsto_within_subset[OF g] simp: closure_def intro: islimpt_subset›) ultimately show ?thesis by (intro filterlim_at_within_If) (simp_all only: *) qed subsection ‹Boundedness› (* FIXME: This has to be unified with BSEQ!! *) definition%important (in metric_space) bounded :: "'a set ⇒ bool" where "bounded S ⟷ (∃x e. ∀y∈S. dist x y ≤ e)" lemma bounded_subset_cball: "bounded S ⟷ (∃e x. S ⊆ cball x e ∧ 0 ≤ e)" unfolding bounded_def subset_eq by auto (meson order_trans zero_le_dist) lemma bounded_any_center: "bounded S ⟷ (∃e. ∀y∈S. dist a y ≤ e)" unfolding bounded_def by auto (metis add.commute add_le_cancel_right dist_commute dist_triangle_le) lemma bounded_iff: "bounded S ⟷ (∃a. ∀x∈S. norm x ≤ a)" unfolding bounded_any_center [where a=0] by (simp add: dist_norm) lemma bdd_above_norm: "bdd_above (norm ` X) ⟷ bounded X" by (simp add: bounded_iff bdd_above_def) lemma bounded_norm_comp: "bounded ((λx. norm (f x)) ` S) = bounded (f ` S)" by (simp add: bounded_iff) lemma boundedI: assumes "⋀x. x ∈ S ⟹ norm x ≤ B" shows "bounded S" using assms bounded_iff by blast lemma bounded_empty [simp]: "bounded {}" by (simp add: bounded_def) lemma bounded_subset: "bounded T ⟹ S ⊆ T ⟹ bounded S" by (metis bounded_def subset_eq) lemma bounded_interior[intro]: "bounded S ⟹ bounded(interior S)" by (metis bounded_subset interior_subset) lemma bounded_closure[intro]: assumes "bounded S" shows "bounded (closure S)" proof - from assms obtain x and a where a: "∀y∈S. dist x y ≤ a" unfolding bounded_def by auto { fix y assume "y ∈ closure S" then obtain f where f: "∀n. f n ∈ S" "(f ⤏ y) sequentially" unfolding closure_sequential by auto have "∀n. f n ∈ S ⟶ dist x (f n) ≤ a" using a by simp then have "eventually (λn. dist x (f n) ≤ a) sequentially" by (simp add: f(1)) have "dist x y ≤ a" apply (rule Lim_dist_ubound [of sequentially f]) apply (rule trivial_limit_sequentially) apply (rule f(2)) apply fact done } then show ?thesis unfolding bounded_def by auto qed lemma bounded_closure_image: "bounded (f ` closure S) ⟹ bounded (f ` S)" by (simp add: bounded_subset closure_subset image_mono) lemma bounded_cball[simp,intro]: "bounded (cball x e)" apply (simp add: bounded_def) apply (rule_tac x=x in exI) apply (rule_tac x=e in exI, auto) done lemma bounded_ball[simp,intro]: "bounded (ball x e)" by (metis ball_subset_cball bounded_cball bounded_subset) lemma bounded_Un[simp]: "bounded (S ∪ T) ⟷ bounded S ∧ bounded T" by (auto simp: bounded_def) (metis Un_iff bounded_any_center le_max_iff_disj) lemma bounded_Union[intro]: "finite F ⟹ ∀S∈F. bounded S ⟹ bounded (⋃F)" by (induct rule: finite_induct[of F]) auto lemma bounded_UN [intro]: "finite A ⟹ ∀x∈A. bounded (B x) ⟹ bounded (⋃x∈A. B x)" by (induct set: finite) auto lemma bounded_insert [simp]: "bounded (insert x S) ⟷ bounded S" proof - have "∀y∈{x}. dist x y ≤ 0" by simp then have "bounded {x}" unfolding bounded_def by fast then show ?thesis by (metis insert_is_Un bounded_Un) qed lemma bounded_subset_ballI: "S ⊆ ball x r ⟹ bounded S" by (meson bounded_ball bounded_subset) lemma bounded_subset_ballD: assumes "bounded S" shows "∃r. 0 < r ∧ S ⊆ ball x r" proof - obtain e::real and y where "S ⊆ cball y e" "0 ≤ e" using assms by (auto simp: bounded_subset_cball) then show ?thesis apply (rule_tac x="dist x y + e + 1" in exI) apply (simp add: add.commute add_pos_nonneg) apply (erule subset_trans) apply (clarsimp simp add: cball_def) by (metis add_le_cancel_right add_strict_increasing dist_commute dist_triangle_le zero_less_one) qed lemma finite_imp_bounded [intro]: "finite S ⟹ bounded S" by (induct set: finite) simp_all lemma bounded_pos: "bounded S ⟷ (∃b>0. ∀x∈ S. norm x ≤ b)" apply (simp add: bounded_iff) apply (subgoal_tac "⋀x (y::real). 0 < 1 + ¦y¦ ∧ (x ≤ y ⟶ x ≤ 1 + ¦y¦)") apply metis apply arith done lemma bounded_pos_less: "bounded S ⟷ (∃b>0. ∀x∈ S. norm x < b)" apply (simp add: bounded_pos) apply (safe; rule_tac x="b+1" in exI; force) done lemma Bseq_eq_bounded: fixes f :: "nat ⇒ 'a::real_normed_vector" shows "Bseq f ⟷ bounded (range f)" unfolding Bseq_def bounded_pos by auto lemma bounded_Int[intro]: "bounded S ∨ bounded T ⟹ bounded (S ∩ T)" by (metis Int_lower1 Int_lower2 bounded_subset) lemma bounded_diff[intro]: "bounded S ⟹ bounded (S - T)" by (metis Diff_subset bounded_subset) lemma not_bounded_UNIV[simp]: "¬ bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)" proof (auto simp: bounded_pos not_le) obtain x :: 'a where "x ≠ 0" using perfect_choose_dist [OF zero_less_one] by fast fix b :: real assume b: "b >0" have b1: "b +1 ≥ 0" using b by simp with ‹x ≠ 0› have "b < norm (scaleR (b + 1) (sgn x))" by (simp add: norm_sgn) then show "∃x::'a. b < norm x" .. qed corollary cobounded_imp_unbounded: fixes S :: "'a::{real_normed_vector, perfect_space} set" shows "bounded (- S) ⟹ ~ (bounded S)" using bounded_Un [of S "-S"] by (simp add: sup_compl_top) lemma bounded_dist_comp: assumes "bounded (f ` S)" "bounded (g ` S)" shows "bounded ((λx. dist (f x) (g x)) ` S)" proof - from assms obtain M1 M2 where *: "dist (f x) undefined ≤ M1" "dist undefined (g x) ≤ M2" if "x ∈ S" for x by (auto simp: bounded_any_center[of _ undefined] dist_commute) have "dist (f x) (g x) ≤ M1 + M2" if "x ∈ S" for x using *[OF that] by (rule order_trans[OF dist_triangle add_mono]) then show ?thesis by (auto intro!: boundedI) qed lemma bounded_linear_image: assumes "bounded S" and "bounded_linear f" shows "bounded (f ` S)" proof - from assms(1) obtain b where "b > 0" and b: "∀x∈S. norm x ≤ b" unfolding bounded_pos by auto from assms(2) obtain B where B: "B > 0" "∀x. norm (f x) ≤ B * norm x" using bounded_linear.pos_bounded by (auto simp: ac_simps) show ?thesis unfolding bounded_pos proof (intro exI, safe) show "norm (f x) ≤ B * b" if "x ∈ S" for x by (meson B b less_imp_le mult_left_mono order_trans that) qed (use ‹b > 0› ‹B > 0› in auto) qed lemma bounded_scaling: fixes S :: "'a::real_normed_vector set" shows "bounded S ⟹ bounded ((λx. c *⇩_{R}x) ` S)" apply (rule bounded_linear_image, assumption) apply (rule bounded_linear_scaleR_right) done lemma bounded_scaleR_comp: fixes f :: "'a ⇒ 'b::real_normed_vector" assumes "bounded (f ` S)" shows "bounded ((λx. r *⇩_{R}f x) ` S)" using bounded_scaling[of "f ` S" r] assms by (auto simp: image_image) lemma bounded_translation: fixes S :: "'a::real_normed_vector set" assumes "bounded S" shows "bounded ((λx. a + x) ` S)" proof - from assms obtain b where b: "b > 0" "∀x∈S. norm x ≤ b" unfolding bounded_pos by auto { fix x assume "x ∈ S" then have "norm (a + x) ≤ b + norm a" using norm_triangle_ineq[of a x] b by auto } then show ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"] by (auto intro!: exI[of _ "b + norm a"]) qed lemma bounded_translation_minus: fixes S :: "'a::real_normed_vector set" shows "bounded S ⟹ bounded ((λx. x - a) ` S)" using bounded_translation [of S "-a"] by simp lemma bounded_uminus [simp]: fixes X :: "'a::real_normed_vector set" shows "bounded (uminus ` X) ⟷ bounded X" by (auto simp: bounded_def dist_norm; rule_tac x="-x" in exI; force simp: add.commute norm_minus_commute) lemma uminus_bounded_comp [simp]: fixes f :: "'a ⇒ 'b::real_normed_vector" shows "bounded ((λx. - f x) ` S) ⟷ bounded (f ` S)" using bounded_uminus[of "f ` S"] by (auto simp: image_image) lemma bounded_plus_comp: fixes f g::"'a ⇒ 'b::real_normed_vector" assumes "bounded (f ` S)" assumes "bounded (g ` S)" shows "bounded ((λx. f x + g x) ` S)" proof - { fix B C assume "⋀x. x∈S ⟹ norm (f x) ≤ B" "⋀x. x∈S ⟹ norm (g x) ≤ C" then have "⋀x. x ∈ S ⟹ norm (f x + g x) ≤ B + C" by (auto intro!: norm_triangle_le add_mono) } then show ?thesis using assms by (fastforce simp: bounded_iff) qed lemma bounded_plus: fixes S ::"'a::real_normed_vector set" assumes "bounded S" "bounded T" shows "bounded ((λ(x,y). x + y) ` (S × T))" using bounded_plus_comp [of fst "S × T" snd] assms by (auto simp: split_def split: if_split_asm) lemma bounded_minus_comp: "bounded (f ` S) ⟹ bounded (g ` S) ⟹ bounded ((λx. f x - g x) ` S)" for f g::"'a ⇒ 'b::real_normed_vector" using bounded_plus_comp[of "f" S "λx. - g x"] by auto lemma bounded_minus: fixes S ::"'a::real_normed_vector set" assumes "bounded S" "bounded T" shows "bounded ((λ(x,y). x - y) ` (S × T))" using bounded_minus_comp [of fst "S × T" snd] assms by (auto simp: split_def split: if_split_asm) subsection ‹Compactness› subsubsection ‹Bolzano-Weierstrass property› proposition heine_borel_imp_bolzano_weierstrass: assumes "compact s" and "infinite t" and "t ⊆ s" shows "∃x ∈ s. x islimpt t" proof (rule ccontr) assume "¬ (∃x ∈ s. x islimpt t)" then obtain f where f: "∀x∈s. x ∈ f x ∧ open (f x) ∧ (∀y∈t. y ∈ f x ⟶ y = x)" unfolding islimpt_def using bchoice[of s "λ x T. x ∈ T ∧ open T ∧ (∀y∈t. y ∈ T ⟶ y = x)"] by auto obtain g where g: "g ⊆ {t. ∃x. x ∈ s ∧ t = f x}" "finite g" "s ⊆ ⋃g" using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. ∃x. x∈s ∧ t = f x}"]] using f by auto from g(1,3) have g':"∀x∈g. ∃xa ∈ s. x = f xa" by auto { fix x y assume "x ∈ t" "y ∈ t" "f x = f y" then have "x ∈ f x" "y ∈ f x ⟶ y = x" using f[THEN bspec[where x=x]] and ‹t ⊆ s› by auto then have "x = y" using ‹f x = f y› and f[THEN bspec[where x=y]] and ‹y ∈ t› and ‹t ⊆ s› by auto } then have "inj_on f t" unfolding inj_on_def by simp then have "infinite (f ` t)" using assms(2) using finite_imageD by auto moreover { fix x assume "x ∈ t" "f x ∉ g" from g(3) assms(3) ‹x ∈ t› obtain h where "h ∈ g" and "x ∈ h" by auto then obtain y where "y ∈ s" "h = f y" using g'[THEN bspec[where x=h]] by auto then have "y = x" using f[THEN bspec[where x=y]] and ‹x∈t› and ‹x∈h›[unfolded ‹h = f y›] by auto then have False using ‹f x ∉ g› ‹h ∈ g› unfolding ‹h = f y› by auto } then have "f ` t ⊆ g" by auto ultimately show False using g(2) using finite_subset by auto qed lemma acc_point_range_imp_convergent_subsequence: fixes l :: "'a :: first_countable_topology" assumes l: "∀U. l∈U ⟶ open U ⟶ infinite (U ∩ range f)" shows "∃r::nat⇒nat. strict_mono r ∧ (f ∘ r) ⇢ l" proof - from countable_basis_at_decseq[of l] obtain A where A: "⋀i. open (A i)" "⋀i. l ∈ A i" "⋀S. open S ⟹ l ∈ S ⟹ eventually (λi. A i ⊆ S) sequentially" by blast define s where "s n i = (SOME j. i < j ∧ f j ∈ A (Suc n))" for n i { fix n i have "infinite (A (Suc n) ∩ range f - f`{.. i})" using l A by auto then have "∃x. x ∈ A (Suc n) ∩ range f - f`{.. i}" unfolding ex_in_conv by (intro notI) simp then have "∃j. f j ∈ A (Suc n) ∧ j ∉ {.. i}" by auto then have "∃a. i < a ∧ f a ∈ A (Suc n)" by (auto simp: not_le) then have "i < s n i" "f (s n i) ∈ A (Suc n)" unfolding s_def by (auto intro: someI2_ex) } note s = this define r where "r = rec_nat (s 0 0) s" have "strict_mono r" by (auto simp: r_def s strict_mono_Suc_iff) moreover have "(λn. f (r n)) ⇢ l" proof (rule topological_tendstoI) fix S assume "open S" "l ∈ S" with A(3) have "eventually (λi. A i ⊆ S) sequentially" by auto moreover { fix i assume "Suc 0 ≤ i" then have "f (r i) ∈ A i" by (cases i) (simp_all add: r_def s) } then have "eventually (λi. f (r i) ∈ A i) sequentially" by (auto simp: eventually_sequentially) ultimately show "eventually (λi. f (r i) ∈ S) sequentially" by eventually_elim auto qed ultimately show "∃r::nat⇒nat. strict_mono r ∧ (f ∘ r) ⇢ l" by (auto simp: convergent_def comp_def) qed lemma sequence_infinite_lemma: fixes f :: "nat ⇒ 'a::t1_space" assumes "∀n. f n ≠ l" and "(f ⤏ l) sequentially" shows "infinite (range f)" proof assume "finite (range f)" then have "closed (range f)" by (rule finite_imp_closed) then have "open (- range f)" by (rule open_Compl) from assms(1) have "l ∈ - range f" by auto from assms(2) have "eventually (λn. f n ∈ - range f) sequentially" using ‹open (- range f)› ‹l ∈ - range f› by (rule topological_tendstoD) then show False unfolding eventually_sequentially by auto qed lemma closure_insert: fixes x :: "'a::t1_space" shows "closure (insert x s) = insert x (closure s)" apply (rule closure_unique) apply (rule insert_mono [OF closure_subset]) apply (rule closed_insert [OF closed_closure]) apply (simp add: closure_minimal) done lemma islimpt_insert: fixes x :: "'a::t1_space" shows "x islimpt (insert a s) ⟷ x islimpt s" proof assume *: "x islimpt (insert a s)" show "x islimpt s" proof (rule islimptI) fix t assume t: "x ∈ t" "open t" show "∃y∈s. y ∈ t ∧ y ≠ x" proof (cases "x = a") case True obtain y where "y ∈ insert a s" "y ∈ t" "y ≠ x" using * t by (rule islimptE) with ‹x = a› show ?thesis by auto next case False with t have t': "x ∈ t - {a}" "open (t - {a})" by (simp_all add: open_Diff) obtain y where "y ∈ insert a s" "y ∈ t - {a}" "y ≠ x" using * t' by (rule islimptE) then show ?thesis by auto qed qed next assume "x islimpt s" then show "x islimpt (insert a s)" by (rule islimpt_subset) auto qed lemma islimpt_finite: fixes x :: "'a::t1_space" shows "finite s ⟹ ¬ x islimpt s" by (induct set: finite) (simp_all add: islimpt_insert) lemma islimpt_Un_finite: fixes x :: "'a::t1_space" shows "finite s ⟹ x islimpt (s ∪ t) ⟷ x islimpt t" by (simp add: islimpt_Un islimpt_finite) lemma islimpt_eq_acc_point: fixes l :: "'a :: t1_space" shows "l islimpt S ⟷ (∀U. l∈U ⟶ open U ⟶ infinite (U ∩ S))" proof (safe intro!: islimptI) fix U assume "l islimpt S" "l ∈ U" "open U" "finite (U ∩ S)" then have "l islimpt S" "l ∈ (U - (U ∩ S - {l}))" "open (U - (U ∩ S - {l}))" by (auto intro: finite_imp_closed) then show False by (rule islimptE) auto next fix T assume *: "∀U. l∈U ⟶ open U ⟶ infinite (U ∩ S)" "l ∈ T" "open T" then have "infinite (T ∩ S - {l})" by auto then have "∃x. x ∈ (T ∩ S - {l})" unfolding ex_in_conv by (intro notI) simp then show "∃y∈S. y ∈ T ∧ y ≠ l" by auto qed corollary infinite_openin: fixes S :: "'a :: t1_space set" shows "⟦openin (subtopology euclidean U) S; x ∈ S; x islimpt U⟧ ⟹ infinite S" by (clarsimp simp add: openin_open islimpt_eq_acc_point inf_commute) lemma islimpt_range_imp_convergent_subsequence: fixes l :: "'a :: {t1_space, first_countable_topology}" assumes l: "l islimpt (range f)" shows "∃r::nat⇒nat. strict_mono r ∧ (f ∘ r) ⇢ l" using l unfolding islimpt_eq_acc_point by (rule acc_point_range_imp_convergent_subsequence) lemma islimpt_eq_infinite_ball: "x islimpt S ⟷ (∀e>0. infinite(S ∩ ball x e))" apply (simp add: islimpt_eq_acc_point, safe) apply (metis Int_commute open_ball centre_in_ball) by (metis open_contains_ball Int_mono finite_subset inf_commute subset_refl) lemma islimpt_eq_infinite_cball: "x islimpt S ⟷ (∀e>0. infinite(S ∩ cball x e))" apply (simp add: islimpt_eq_infinite_ball, safe) apply (meson Int_mono ball_subset_cball finite_subset order_refl) by (metis open_ball centre_in_ball finite_Int inf.absorb_iff2 inf_assoc open_contains_cball_eq) lemma sequence_unique_limpt: fixes f :: "nat ⇒ 'a::t2_space" assumes "(f ⤏ l) sequentially" and "l' islimpt (range f)" shows "l' = l" proof (rule ccontr) assume "l' ≠ l" obtain s t where "open s" "open t" "l' ∈ s" "l ∈ t" "s ∩ t = {}" using hausdorff [OF ‹l' ≠ l›] by auto have "eventually (λn. f n ∈ t) sequentially" using assms(1) ‹open t› ‹l ∈ t› by (rule topological_tendstoD) then obtain N where "∀n≥N. f n ∈ t" unfolding eventually_sequentially by auto have "UNIV = {..<N} ∪ {N..}" by auto then have "l' islimpt (f ` ({..<N} ∪ {N..}))" using assms(2) by simp then have "l' islimpt (f ` {..<N} ∪ f ` {N..})" by (simp add: image_Un) then have "l' islimpt (f ` {N..})" by (simp add: islimpt_Un_finite) then obtain y where "y ∈ f ` {N..}" "y ∈ s" "y ≠ l'" using ‹l' ∈ s› ‹open s› by (rule islimptE) then obtain n where "N ≤ n" "f n ∈ s" "f n ≠ l'" by auto with ‹∀n≥N. f n ∈ t› have "f n ∈ s ∩ t" by simp with ‹s ∩ t = {}› show False by simp qed lemma bolzano_weierstrass_imp_closed: fixes s :: "'a::{first_countable_topology,t2_space} set" assumes "∀t. infinite t ∧ t ⊆ s --> (∃x ∈ s. x islimpt t)" shows "closed s" proof - { fix x l assume as: "∀n::nat. x n ∈ s" "(x ⤏ l) sequentially" then have "l ∈ s" proof (cases "∀n. x n ≠ l") case False then show "l∈s" using as(1) by auto next case True note cas = this with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto then obtain l' where "l'∈s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto then show "l∈s" using sequence_unique_limpt[of x l l'] using as cas by auto qed } then show ?thesis unfolding closed_sequential_limits by fast qed lemma compact_imp_bounded: assumes "compact U" shows "bounded U" proof - have "compact U" "∀x∈U. open (ball x 1)" "U ⊆ (⋃x∈U. ball x 1)" using assms by auto then obtain D where D: "D ⊆ U" "finite D" "U ⊆ (⋃x∈D. ball x 1)" by (metis compactE_image) from ‹finite D› have "bounded (⋃x∈D. ball x 1)" by (simp add: bounded_UN) then show "bounded U" using ‹U ⊆ (⋃x∈D. ball x 1)› by (rule bounded_subset) qed text‹In particular, some common special cases.› lemma compact_Un [intro]: assumes "compact s" and "compact t" shows " compact (s ∪ t)" proof (rule compactI) fix f assume *: "Ball f open" "s ∪ t ⊆ ⋃f" from * ‹compact s› obtain s' where "s' ⊆ f ∧ finite s' ∧ s ⊆ ⋃s'" unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) moreover from * ‹compact t› obtain t' where "t' ⊆ f ∧ finite t' ∧ t ⊆ ⋃t'" unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) ultimately show "∃f'⊆f. finite f' ∧ s ∪ t ⊆ ⋃f'" by (auto intro!: exI[of _ "s' ∪ t'"]) qed lemma compact_Union [intro]: "finite S ⟹ (⋀T. T ∈ S ⟹ compact T) ⟹ compact (⋃S)" by (induct set: finite) auto lemma compact_UN [intro]: "finite A ⟹ (⋀x. x ∈ A ⟹ compact (B x)) ⟹ compact (⋃x∈A. B x)" by (rule compact_Union) auto lemma closed_Int_compact [intro]: assumes "closed s" and "compact t" shows "compact (s ∩ t)" using compact_Int_closed [of t s] assms by (simp add: Int_commute) lemma compact_Int [intro]: fixes s t :: "'a :: t2_space set" assumes "compact s" and "compact t" shows "compact (s ∩ t)" using assms by (intro compact_Int_closed compact_imp_closed) lemma compact_sing [simp]: "compact {a}" unfolding compact_eq_heine_borel by auto lemma compact_insert [simp]: assumes "compact s" shows "compact (insert x s)" proof - have "compact ({x} ∪ s)" using compact_sing assms by (rule compact_Un) then show ?thesis by simp qed lemma finite_imp_compact: "finite s ⟹ compact s" by (induct set: finite) simp_all lemma open_delete: fixes s :: "'a::t1_space set" shows "open s ⟹ open (s - {x})" by (simp add: open_Diff) lemma openin_delete: fixes a :: "'a :: t1_space" shows "openin (subtopology euclidean u) s ⟹ openin (subtopology euclidean u) (s - {a})" by (metis Int_Diff open_delete openin_open) text‹Compactness expressed with filters› lemma closure_iff_nhds_not_empty: "x ∈ closure X ⟷ (∀A. ∀S⊆A. open S ⟶ x ∈ S ⟶ X ∩ A ≠ {})" proof safe assume x: "x ∈ closure X" fix S A assume "open S" "x ∈ S" "X ∩ A = {}" "S ⊆ A" then have "x ∉ closure (-S)" by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI) with x have "x ∈ closure X - closure (-S)" by auto also have "… ⊆ closure (X ∩ S)" using ‹open S› open_Int_closure_subset[of S X] by (simp add: closed_Compl ac_simps) finally have "X ∩ S ≠ {}" by auto then show False using ‹X ∩ A = {}› ‹S ⊆ A› by auto next assume "∀A S. S ⊆ A ⟶ open S ⟶ x ∈ S ⟶ X ∩ A ≠ {}" from this[THEN spec, of "- X", THEN spec, of "- closure X"] show "x ∈ closure X" by (simp add: closure_subset open_Compl) qed corollary closure_Int_ball_not_empty: assumes "S ⊆ closure T" "x ∈ S" "r > 0" shows "T ∩ ball x r ≠ {}" using assms centre_in_ball closure_iff_nhds_not_empty by blast lemma compact_filter: "compact U ⟷ (∀F. F ≠ bot ⟶ eventually (λx. x ∈ U) F ⟶ (∃x∈U. inf (nhds x) F ≠ bot))" proof (intro allI iffI impI compact_fip[THEN iffD2] notI) fix F assume "compact U" assume F: "F ≠ bot" "eventually (λx. x ∈ U) F" then have "U ≠ {}" by (auto simp: eventually_False) define Z where "Z = closure ` {A. eventually (λx. x ∈ A) F}" then have "∀z∈Z. closed z" by auto moreover have ev_Z: "⋀z. z ∈ Z ⟹ eventually (λx. x ∈ z) F" unfolding Z_def by (auto elim: eventually_mono intro: set_mp[OF closure_subset]) have "(∀B ⊆ Z. finite B ⟶ U ∩ ⋂B ≠ {})" proof (intro allI impI) fix B assume "finite B" "B ⊆ Z" with ‹finite B› ev_Z F(2) have "eventually (λx. x ∈ U ∩ (⋂B)) F" by (auto simp: eventually_ball_finite_distrib eventually_conj_iff) with F show "U ∩ ⋂B ≠ {}" by (intro notI) (simp add: eventually_False) qed ultimately have "U ∩ ⋂Z ≠ {}" using ‹compact U› unfolding compact_fip by blast then obtain x where "x ∈ U" and x: "⋀z. z ∈ Z ⟹ x ∈ z" by auto have "⋀P. eventually P (inf (nhds x) F) ⟹ P ≠ bot" unfolding eventually_inf eventually_nhds proof safe fix P Q R S assume "eventually R F" "open S" "x ∈ S" with open_Int_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"] have "S ∩ {x. R x} ≠ {}" by (auto simp: Z_def) moreover assume "Ball S Q" "∀x. Q x ∧ R x ⟶ bot x" ultimately show False by (auto simp: set_eq_iff) qed with ‹x ∈ U› show "∃x∈U. inf (nhds x) F ≠ bot" by (metis eventually_bot) next fix A assume A: "∀a∈A. closed a" "∀B⊆A. finite B ⟶ U ∩ ⋂B ≠ {}" "U ∩ ⋂A = {}" define F where "F = (INF a:insert U A. principal a)" have "F ≠ bot" unfolding F_def proof (rule INF_filter_not_bot) fix X assume X: "X ⊆ insert U A" "finite X" with A(2)[THEN spec, of "X - {U}"] have "U ∩ ⋂(X - {U}) ≠ {}" by auto with X show "(INF a:X. principal a) ≠ bot" by (auto simp: INF_principal_finite principal_eq_bot_iff) qed moreover have "F ≤ principal U" unfolding F_def by auto then have "eventually (λx. x ∈ U) F" by (auto simp: le_filter_def eventually_principal) moreover assume "∀F. F ≠ bot ⟶ eventually (λx. x ∈ U) F ⟶ (∃x∈U. inf (nhds x) F ≠ bot)" ultimately obtain x where "x ∈ U" and x: "inf (nhds x) F ≠ bot" by auto { fix V assume "V ∈ A" then have "F ≤ principal V" unfolding F_def by (intro INF_lower2[of V]) auto then have V: "eventually (λx. x ∈ V) F" by (auto simp: le_filter_def eventually_principal) have "x ∈ closure V" unfolding closure_iff_nhds_not_empty proof (intro impI allI) fix S A assume "open S"