imports Countable_Set FuncSet Linear_Algebra Norm_Arith

(* title: HOL/Library/Topology_Euclidian_Space.thy 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 Complex_Main "~~/src/HOL/Library/Countable_Set" "~~/src/HOL/Library/FuncSet" Linear_Algebra Norm_Arith begin lemma dist_0_norm: fixes x :: "'a::real_normed_vector" shows "dist 0 x = norm x" unfolding dist_norm by simp lemma dist_double: "dist x y < d / 2 ==> dist x z < d / 2 ==> dist y z < d" using dist_triangle[of y z x] by (simp add: dist_commute) (* LEGACY *) lemma lim_subseq: "subseq r ==> s ----> l ==> (s o r) ----> l" by (rule LIMSEQ_subseq_LIMSEQ) lemma countable_PiE: "finite I ==> (!!i. i ∈ I ==> countable (F i)) ==> countable (PiE I F)" by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq) lemma Lim_within_open: fixes f :: "'a::topological_space => 'b::topological_space" shows "a ∈ S ==> open S ==> (f ---> l)(at a within S) <-> (f ---> l)(at a)" by (fact tendsto_within_open) lemma continuous_on_union: "closed s ==> closed t ==> continuous_on s f ==> continuous_on t f ==> continuous_on (s ∪ t) f" by (fact continuous_on_closed_Un) lemma continuous_on_cases: "closed s ==> closed t ==> continuous_on s f ==> continuous_on t g ==> ∀x. (x∈s ∧ ¬ P x) ∨ (x ∈ t ∧ P x) --> f x = g x ==> continuous_on (s ∪ t) (λx. if P x then f x else g x)" by (rule continuous_on_If) auto subsection {* Topological Basis *} context topological_space begin definition "topological_basis B <-> (∀b∈B. open b) ∧ (∀x. open x --> (∃B'. B' ⊆ B ∧ \<Union>B' = x))" lemma topological_basis: "topological_basis B <-> (∀x. open x <-> (∃B'. B' ⊆ B ∧ \<Union>B' = x))" unfolding topological_basis_def apply safe apply fastforce apply fastforce apply (erule_tac x="x" in allE) apply simp apply (rule_tac x="{x}" in exI) apply 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. \<Union>B' = O')" by (simp add: topological_basis_def) then obtain B' where "B' ⊆ B" "O' = \<Union>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. \<Union>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 = \<Union>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. (\<Union>(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) ∈ (\<Union>(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 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 = Union 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 = Union 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: obtains A where "countable A" "!!a. a ∈ A ==> x ∈ a" "!!a. a ∈ A ==> open a" "!!S. open S ==> x ∈ S ==> (∃a∈A. a ⊆ S)" using first_countable_basis[of x] apply atomize_elim apply (elim exE) apply (rule_tac x="range A" in exI) apply auto done 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 A' where A': "countable A'" "!!a. a ∈ A' ==> x ∈ a" "!!a. a ∈ A' ==> open a" "!!S. open S ==> x ∈ S ==> ∃a∈A'. a ⊆ S" by (rule first_countable_basisE) blast def A ≡ "(λN. \<Inter>((λn. from_nat_into A' 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 A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into) next let ?int = "λN. \<Inter>(from_nat_into A' ` 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∈A'" "a ⊆ S" using A' by blast then show "∃a∈A. a ⊆ S" using a A' by (intro bexI[where x=a]) (auto simp: A_def intro: image_eqI[where x="{to_nat_on A' 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" with A(2, 3)[of a] B(2, 3)[of b] show "x ∈ a × b" and "open (a × b)" unfolding mem_Times_iff by (auto intro: open_Times) 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}. (\<Union>b∈B'. \<Inter>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 = \<Union>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}. \<Union>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 def B ≡ "SOME B::'a set set. 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 subsection {* 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 "istopology L <-> L {} ∧ (∀S T. L S --> L T --> L (S ∩ T)) ∧ (∀K. Ball K L --> L (\<Union> K))" typedef 'a topology = "{L::('a set) => bool. istopology L}" morphisms "openin" "topology" unfolding istopology_def by blast lemma istopology_open_in[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_open_in[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 = \<Union>{S. openin T S}" subsubsection {* Main properties of open sets *} lemma 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 (\<Union>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 (simp add: openin_clauses) lemma openin_Int[intro]: "openin U S ==> openin U T ==> openin U (S ∩ T)" using openin_clauses by simp lemma openin_Union[intro]: "(∀S ∈K. openin U S) ==> openin U (\<Union> K)" using openin_clauses by simp 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 (simp add: 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 = "\<Union>{T. openin U T ∧ T ⊆ S}" have "openin U ?t" by (simp add: openin_Union) also have "?t = S" using H by auto finally show "openin U S" . qed subsubsection {* Closed sets *} definition "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 add: Diff_Un closedin_def) lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s∈S}" by auto lemma closedin_Inter[intro]: assumes Ke: "K ≠ {}" and Kc: "∀S ∈K. closedin U S" shows "closedin U (\<Inter> K)" using Ke Kc unfolding closedin_def Diff_Inter by auto lemma closedin_Int[intro]: "closedin U S ==> closedin U T ==> closedin U (S ∩ T)" using closedin_Inter[of "{S,T}" U] by auto lemma Diff_Diff_Int: "A - (A - B) = A ∩ B" by blast lemma openin_closedin_eq: "openin U S <-> S ⊆ topspace U ∧ closedin U (topspace U - S)" apply (auto simp add: 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 add: topspace_def openin_subset) then show ?thesis using oS cT by (auto simp add: 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 add: topspace_def) then show ?thesis using oS cT by (auto simp add: openin_closedin_eq) qed subsubsection {* Subspace topology *} definition "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 "\<Union>K = (\<Union>Sk) ∩ V" using Sk by auto moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq) ultimately have "?L (\<Union>K)" by blast } ultimately show ?thesis unfolding subset_eq mem_Collect_eq istopology_def by blast 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 add: 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 add: 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) subsubsection {* The standard Euclidean topology *} definition 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 add: istopology_def) done lemma topspace_euclidean: "topspace euclidean = UNIV" apply (simp add: topspace_def) apply (rule set_eqI) apply (auto simp add: open_openin[symmetric]) done lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S" by (simp add: topspace_euclidean topspace_subtopology) lemma closed_closedin: "closed S <-> closedin euclidean S" by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV) lemma open_subopen: "open S <-> (∀x∈S. ∃T. open T ∧ x ∈ T ∧ T ⊆ S)" by (simp add: open_openin openin_subopen[symmetric]) 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 add: openin_subtopology open_openin[symmetric]) lemma openin_open_Int[intro]: "open S ==> openin (subtopology euclidean U) (U ∩ S)" by (auto simp add: 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 add: openin_open) lemma closedin_closed: "closedin (subtopology euclidean U) S <-> (∃T. closed T ∧ S = U ∩ T)" by (simp add: closedin_subtopology closed_closedin Int_ac) lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U ∩ S)" by (metis closedin_closed) lemma closed_closedin_trans: "closed S ==> closed T ==> T ⊆ S ==> closedin (subtopology euclidean S) T" by (metis closedin_closed inf.absorb2) lemma closed_subset: "S ⊆ T ==> closed S ==> closedin (subtopology euclidean T) S" by (auto simp add: closedin_closed) 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 def 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 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 add: 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 add: closedin_closed closed_closedin closed_Inter Int_assoc) lemma closedin_closed_trans: "closedin (subtopology euclidean T) S ==> closed T ==> closed S" by (auto simp add: closedin_closed intro: closedin_trans) subsection {* Open and closed balls *} definition ball :: "'a::metric_space => real => 'a set" where "ball x e = {y. dist x y < e}" definition cball :: "'a::metric_space => real => 'a set" where "cball 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_ball_0: fixes x :: "'a::real_normed_vector" shows "x ∈ ball 0 e <-> norm x < e" by (simp add: dist_norm) lemma mem_cball_0: fixes x :: "'a::real_normed_vector" shows "x ∈ cball 0 e <-> norm x ≤ e" by (simp add: dist_norm) lemma centre_in_ball: "x ∈ ball x e <-> 0 < e" by simp lemma centre_in_cball: "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 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 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 diff_less_iff: "(a::real) - b > 0 <-> a > b" "(a::real) - b < 0 <-> a < b" "a - b < c <-> a < c + b" "a - b > c <-> a > c + b" by arith+ lemma diff_le_iff: "(a::real) - b ≥ 0 <-> a ≥ b" "(a::real) - b ≤ 0 <-> a ≤ b" "a - b ≤ c <-> a ≤ c + b" "a - b ≥ c <-> a ≥ c + b" by arith+ 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)" unfolding open_dist subset_eq mem_ball Ball_def dist_commute .. 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. x∈S <-> (∃e>0. ball x e ⊆ S)" by (metis open_contains_ball subset_eq centre_in_ball) 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[intro]: "e ≤ 0 ==> ball x e = {}" by simp lemma euclidean_dist_l2: fixes x y :: "'a :: euclidean_space" shows "dist x y = setL2 (λi. dist (x • i) (y • i)) Basis" unfolding dist_norm norm_eq_sqrt_inner setL2_def by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left) subsection {* Boxes *} abbreviation One :: "'a::euclidean_space" where "One ≡ ∑Basis" definition (in euclidean_space) eucl_less (infix "<e" 50) where "eucl_less a b <-> (∀i∈Basis. a • i < b • i)" definition box_eucl_less: "box a b = {x. a <e x ∧ x <e b}" definition "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 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 ∈ \<rat> ∧ b • i ∈ \<rat> ) ∧ x ∈ box a b ∧ box a b ⊆ ball x e" proof - def e' ≡ "e / (2 * sqrt (real (DIM ('a))))" then have e: "e' > 0" using assms by (auto simp: DIM_positive) have "∀i. ∃y. y ∈ \<rat> ∧ 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 ∈ \<rat> ∧ a xa < x • xa ∧ x • xa - a xa < e'" .. have "∀i. ∃y. y ∈ \<rat> ∧ 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 ∈ \<rat> ∧ 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 setL2_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 setsum_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 add: real_eq_of_nat) 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}\<rat> × \<rat>. box (a' f) (b' f) ⊆ M}" shows "M = (\<Union>f∈I. box (a' f) (b' f))" proof - { fix x assume "x ∈ M" 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 ∈ \<rat>" "∀i∈Basis. b • i ∈ \<rat>" "box a b ⊆ ball x e" using rational_boxes[OF e(1)] by metis ultimately have "x ∈ (\<Union>f∈I. box (a' f) (b' f))" by (intro UN_I[of "λi∈Basis. (a • i, b • i)"]) (auto simp: euclidean_representation I_def a'_def b'_def) } then show ?thesis by (auto simp: I_def) 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: "cbox a a = {a}" and box_sing: "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 - show ?th1 unfolding subset_eq and Ball_def and mem_box by (auto intro: order_trans) show ?th2 unfolding subset_eq and Ball_def and mem_box by (auto intro: le_less_trans less_le_trans order_trans less_imp_le) { assume as: "box c d ⊆ cbox a b" "∀i∈Basis. c•i < d•i" then have "box c d ≠ {}" unfolding box_eq_empty by auto fix i :: 'a assume i: "i ∈ Basis" (** TODO combine the following two parts as done in the HOL_light version. **) { 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 as2: "a•i > c•i" { fix j :: 'a assume j: "j ∈ Basis" then have "c • j < ?x • j ∧ ?x • j < d • j" apply (cases "j = i") using as(2)[THEN bspec[where x=j]] i apply (auto simp add: as2) done } then have "?x∈box c d" using i unfolding mem_box by auto moreover have "?x ∉ cbox a b" unfolding mem_box apply auto apply (rule_tac x=i in bexI) using as(2)[THEN bspec[where x=i]] and as2 i apply auto done ultimately have False using as by auto } then have "a•i ≤ c•i" by (rule ccontr) auto 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 as2: "b•i < d•i" { fix j :: 'a assume "j∈Basis" then have "d • j > ?x • j ∧ ?x • j > c • j" apply (cases "j = i") using as(2)[THEN bspec[where x=j]] apply (auto simp add: as2) done } then have "?x∈box c d" unfolding mem_box by auto moreover have "?x∉cbox a b" unfolding mem_box apply auto apply (rule_tac x=i in bexI) using as(2)[THEN bspec[where x=i]] and as2 using i apply auto done ultimately have False using as by auto } then have "b•i ≥ d•i" by (rule ccontr) auto ultimately have "a•i ≤ c•i ∧ d•i ≤ b•i" by auto } note part1 = this show ?th3 unfolding subset_eq and Ball_def and mem_box apply (rule, rule, rule, rule) apply (rule part1) unfolding subset_eq and Ball_def and mem_box prefer 4 apply auto apply (erule_tac x=xa in allE, erule_tac x=xa in allE, fastforce)+ done { assume as: "box c d ⊆ box a b" "∀i∈Basis. c•i < d•i" fix i :: 'a assume i:"i∈Basis" from as(1) have "box c d ⊆ cbox a b" using box_subset_cbox[of a b] by auto then have "a•i ≤ c•i ∧ d•i ≤ b•i" using part1 and as(2) using i by auto } note * = this show ?th4 unfolding subset_eq and Ball_def and mem_box apply (rule, rule, rule, rule) apply (rule *) unfolding subset_eq and Ball_def and mem_box prefer 4 apply auto apply (erule_tac x=xa in allE, simp)+ done qed lemma inter_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: "(\<Union>i::nat. box (- (real i *⇩_{R}One)) (real i *⇩_{R}One)) = UNIV" proof - { fix x b :: 'a assume [simp]: "b ∈ Basis" have "¦x • b¦ ≤ real (ceiling ¦x • b¦)" by (rule real_of_int_ceiling_ge) also have "… ≤ real (ceiling (Max ((λb. ¦x • b¦)`Basis)))" by (auto intro!: ceiling_mono) also have "… < real (ceiling (Max ((λb. ¦x • b¦)`Basis)) + 1)" by simp finally have "¦x • b¦ < real (ceiling (Max ((λb. ¦x • b¦)`Basis)) + 1)" . } then have "!!x::'a. ∃n::nat. ∀b∈Basis. ¦x • b¦ < real n" 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_setsum_left inner_Basis setsum.If_cases) qed text {* Intervals in general, including infinite and mixtures of open and closed. *} definition "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_cbox: "is_interval (cbox a (b::'a::euclidean_space))" (is ?th1) and is_interval_box: "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: "is_interval {}" unfolding is_interval_def by simp lemma is_interval_univ: "is_interval UNIV" unfolding is_interval_def by simp lemma mem_is_intervalI: assumes "is_interval s" assumes "a ∈ s" "b ∈ s" assumes "!!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" assumes "x ∈ S" "y j ∈ S" assumes "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 def 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 add: 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 subsection{* Connectedness *} lemma connected_local: "connected S <-> ¬ (∃e1 e2. openin (subtopology euclidean S) e1 ∧ openin (subtopology euclidean S) e2 ∧ S ⊆ e1 ∪ e2 ∧ e1 ∩ e2 = {} ∧ e1 ≠ {} ∧ e2 ≠ {})" unfolding connected_def openin_open by safe blast+ lemma exists_diff: fixes P :: "'a set => bool" shows "(∃S. P(- S)) <-> (∃S. P S)" (is "?lhs <-> ?rhs") proof - { assume "?lhs" then have ?rhs by blast } moreover { fix S assume H: "P S" have "S = - (- S)" by auto with H have "P (- (- S))" by metis } ultimately show ?thesis by metis qed lemma connected_clopen: "connected S <-> (∀T. openin (subtopology euclidean S) T ∧ closedin (subtopology euclidean S) T --> T = {} ∨ T = S)" (is "?lhs <-> ?rhs") proof - have "¬ connected S <-> (∃e1 e2. open e1 ∧ open (- e2) ∧ S ⊆ e1 ∪ (- e2) ∧ e1 ∩ (- e2) ∩ S = {} ∧ e1 ∩ S ≠ {} ∧ (- e2) ∩ S ≠ {})" unfolding connected_def openin_open closedin_closed by (metis double_complement) then have th0: "connected S <-> ¬ (∃e2 e1. closed e2 ∧ open e1 ∧ S ⊆ e1 ∪ (- e2) ∧ e1 ∩ (- e2) ∩ S = {} ∧ e1 ∩ S ≠ {} ∧ (- e2) ∩ S ≠ {})" (is " _ <-> ¬ (∃e2 e1. ?P e2 e1)") apply (simp add: closed_def) apply metis done have th1: "?rhs <-> ¬ (∃t' t. closed t'∧t = S∩t' ∧ t≠{} ∧ t≠S ∧ (∃t'. open t' ∧ t = S ∩ t'))" (is "_ <-> ¬ (∃t' t. ?Q t' t)") unfolding connected_def openin_open closedin_closed by auto { fix e2 { fix e1 have "?P e2 e1 <-> (∃t. closed e2 ∧ t = S∩e2 ∧ open e1 ∧ t = S∩e1 ∧ t≠{} ∧ t ≠ S)" by auto } then have "(∃e1. ?P e2 e1) <-> (∃t. ?Q e2 t)" by metis } then have "∀e2. (∃e1. ?P e2 e1) <-> (∃t. ?Q e2 t)" by blast then show ?thesis unfolding th0 th1 by simp qed subsection{* Limit points *} definition (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: fixes x :: "'a::metric_space" shows "x islimpt S <-> (∀e>0. ∃x'∈ S. x' ≠ x ∧ dist x' x ≤ e)" 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::'a::perfect_space) islimpt UNIV" unfolding islimpt_UNIV_iff by (rule not_open_singleton) lemma perfect_choose_dist: fixes x :: "'a::{perfect_space, metric_space}" shows "0 < r ==> ∃a. a ≠ x ∧ dist a x < r" 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 {}" unfolding islimpt_def by auto lemma finite_set_avoid: fixes a :: "'a::metric_space" assumes fS: "finite S" shows "∃d>0. ∀x∈S. x ≠ a --> d ≤ dist a x" proof (induct rule: finite_induct[OF fS]) case 1 then show ?case by (auto intro: zero_less_one) next case (2 x F) from 2 obtain d where d: "d >0" "∀x∈F. x≠a --> d ≤ dist a x" by blast show ?case proof (cases "x = a") case True then show ?thesis using d by auto next case False let ?d = "min d (dist a x)" have dp: "?d > 0" using False d(1) using dist_nz by auto from d have d': "∀x∈F. x≠a --> ?d ≤ dist a x" by auto with dp False show ?thesis by (auto intro!: exI[where x="?d"]) qed qed 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 - { fix x assume C: "∀e>0. ∃x'∈S. x' ≠ x ∧ dist x' x < e" 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 add: dist_nz[symmetric]) from C[rule_format, OF mp] obtain z where z: "z ∈ S" "z ≠ x" "dist z x < ?m" by blast have th: "dist z y < e" using z y by (intro dist_triangle_lt [where z=x], simp) from d[rule_format, OF y(1) z(1) th] y z have False by (auto simp add: dist_commute)} then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a]) qed subsection {* Interior of a Set *} definition "interior S = \<Union>{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 add: interior_def) lemma interior_maximal: "T ⊆ S ==> open T ==> T ⊆ interior S" by (auto simp add: 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 add: 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_inter [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 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 add: 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 subsection {* Closure of a Set *} definition "closure S = S ∪ {x | x. x islimpt S}" lemma interior_closure: "interior S = - (closure (- S))" unfolding interior_def closure_def islimpt_def by auto lemma closure_interior: "closure S = - interior (- S)" unfolding interior_closure by simp lemma closed_closure[simp, intro]: "closed (closure S)" unfolding closure_interior by (simp add: closed_Compl) lemma closure_subset: "S ⊆ closure S" unfolding closure_def by simp lemma closure_hull: "closure S = closed hull S" unfolding hull_def closure_interior interior_def by auto 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" unfolding 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_union [simp]: "closure (S ∪ T) = closure S ∪ closure T" unfolding closure_interior by simp lemma closure_eq_empty: "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_inter_closure_eq_empty: "open S ==> (S ∩ closure T) = {} <-> S ∩ T = {}" using open_subset_interior[of S "- T"] using interior_subset[of "- T"] unfolding closure_interior by auto lemma open_inter_closure_subset: "open S ==> (S ∩ (closure T)) ⊆ closure(S ∩ T)" proof fix x assume as: "open S" "x ∈ S ∩ closure T" { assume *: "x islimpt T" have "x islimpt (S ∩ T)" proof (rule islimptI) fix A assume "x ∈ A" "open A" with as 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 } then show "x ∈ closure (S ∩ T)" using as unfolding closure_def by blast qed lemma closure_complement: "closure (- S) = - interior S" unfolding closure_interior by simp lemma interior_complement: "interior (- S) = - closure S" unfolding closure_interior by simp 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) apply auto done qed lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))" unfolding closure_def using islimpt_punctured by blast subsection {* Frontier (aka boundary) *} definition "frontier S = closure S - interior S" lemma frontier_closed: "closed (frontier S)" by (simp add: frontier_def closed_Diff) lemma frontier_closures: "frontier S = (closure S) ∩ (closure(- S))" by (auto simp add: frontier_def interior_closure) 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 add: 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: "frontier (- S) = frontier S" by (auto simp add: frontier_def closure_complement interior_complement) lemma frontier_disjoint_eq: "frontier S ∩ S = {} <-> open S" using frontier_complement frontier_subset_eq[of "- S"] unfolding open_closed by auto subsection {* 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: fixes a :: "'a::perfect_space" shows "¬ trivial_limit (at a)" 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) text {* Some property holds "sufficiently close" to the limit point. *} lemma eventually_at2: "eventually P (at a) <-> (∃d>0. ∀x. 0 < dist x a ∧ dist x a < d --> P x)" unfolding eventually_at dist_nz by auto lemma eventually_happens: "eventually P net ==> trivial_limit net ∨ (∃x. P x)" unfolding trivial_limit_def by (auto elim: eventually_rev_mp) 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) text{* Combining theorems for "eventually" *} lemma eventually_rev_mono: "eventually P net ==> (∀x. P x --> Q x) ==> eventually Q net" using eventually_mono [of P Q] by fast lemma not_eventually: "(∀x. ¬ P x ) ==> ¬ trivial_limit net ==> ¬ eventually (λx. P x) net" by (simp add: eventually_False) subsection {* Limits *} lemma Lim: "(f ---> l) net <-> trivial_limit net ∨ (∀e>0. eventually (λx. dist (f x) l < e) net)" unfolding tendsto_iff trivial_limit_eq by auto text{* Show that they yield usual definitions in the various cases. *} lemma 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 add: tendsto_iff eventually_at_le dist_nz) lemma 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 add: tendsto_iff eventually_at dist_nz) lemma 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 add: tendsto_iff eventually_at2) lemma Lim_at_infinity: "(f ---> l) at_infinity <-> (∀e>0. ∃b. ∀x. norm x ≥ b --> dist (f x) l < e)" by (auto simp add: tendsto_iff eventually_at_infinity) lemma Lim_eventually: "eventually (λx. f x = l) net ==> (f ---> l) net" by (rule topological_tendstoI, auto elim: eventually_rev_mono) 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_within: (* FIXME: rename *) "(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" unfolding eventually_at_topological by auto with T have "open (A ∩ T)" and "x ∈ A ∩ T" and "∀y ∈ A ∩ T. y ≠ x --> P y" by auto then show "?rhs" unfolding eventually_at_topological by auto next assume "?rhs" then show "?lhs" by (auto elim: eventually_elim1 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 simp del: Inf_image_eq) 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 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 def f ≡ "λn. SOME y. y ∈ S ∧ y ∈ A n ∧ x ≠ y" { 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_elim1) 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_elim1) (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) 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: 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) text{* Transformation of limit. *} lemma Lim_transform: fixes f g :: "'a::type => 'b::real_normed_vector" assumes "((λx. f x - g x) ---> 0) net" "(f ---> l) net" shows "(g ---> l) net" using tendsto_diff [OF assms(2) assms(1)] by simp lemma Lim_transform_eventually: "eventually (λx. f x = g x) net ==> (f ---> l) net ==> (g ---> l) net" apply (rule topological_tendstoI) apply (drule (2) topological_tendstoD) apply (erule (1) eventually_elim2, simp) done lemma Lim_transform_within: assumes "0 < d" and "∀x'∈S. 0 < dist x' x ∧ dist x' x < d --> f x' = g x'" and "(f ---> l) (at x within S)" shows "(g ---> l) (at x within S)" proof (rule Lim_transform_eventually) show "eventually (λx. f x = g x) (at x within S)" using assms(1,2) by (auto simp: dist_nz eventually_at) show "(f ---> l) (at x within S)" by fact qed lemma Lim_transform_at: assumes "0 < d" and "∀x'. 0 < dist x' x ∧ dist x' x < d --> f x' = g x'" and "(f ---> l) (at x)" shows "(g ---> l) (at x)" using _ assms(3) proof (rule Lim_transform_eventually) show "eventually (λx. f x = g x) (at x)" unfolding eventually_at2 using assms(1,2) by auto qed text{* Common case assuming being away from some crucial point like 0. *} lemma Lim_transform_away_within: fixes a b :: "'a::t1_space" assumes "a ≠ b" and "∀x∈S. x ≠ a ∧ x ≠ b --> f x = g x" and "(f ---> l) (at a within S)" shows "(g ---> l) (at a within S)" proof (rule Lim_transform_eventually) show "(f ---> l) (at a within S)" by fact show "eventually (λx. f x = g x) (at a within S)" unfolding eventually_at_topological by (rule exI [where x="- {b}"], simp add: open_Compl assms) qed lemma Lim_transform_away_at: fixes a b :: "'a::t1_space" assumes ab: "a≠b" and fg: "∀x. x ≠ a ∧ x ≠ b --> f x = g x" and fl: "(f ---> l) (at a)" shows "(g ---> l) (at a)" using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp text{* Alternatively, within an open set. *} lemma Lim_transform_within_open: assumes "open S" and "a ∈ S" and "∀x∈S. x ≠ a --> f x = g x" and "(f ---> l) (at a)" shows "(g ---> l) (at a)" proof (rule Lim_transform_eventually) show "eventually (λx. f x = g x) (at a)" unfolding eventually_at_topological using assms(1,2,3) by auto show "(f ---> l) (at a)" by fact qed text{* A congruence rule allowing us to transform limits assuming not at point. *} (* FIXME: Only one congruence rule for tendsto can be used at a time! *) lemma Lim_cong_within(*[cong add]*): assumes "a = b" and "x = y" and "S = T" and "!!x. x ≠ b ==> x ∈ T ==> f x = g x" shows "(f ---> x) (at a within S) <-> (g ---> y) (at b within T)" unfolding tendsto_def eventually_at_topological using assms by simp lemma Lim_cong_at(*[cong add]*): assumes "a = b" "x = y" and "!!x. x ≠ a ==> f x = g x" shows "((λx. f x) ---> x) (at a) <-> ((g ---> y) (at a))" unfolding tendsto_def eventually_at_topological using assms by 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 add: closure_def islimpt_approachable) apply (metis dist_self) done 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 add: dist_real_def) } then show ?thesis unfolding closure_approachable by auto 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 assms) lemma not_trivial_limit_within_ball: "¬ trivial_limit (at x within S) <-> (∀e>0. S ∩ ball x e - {x} ≠ {})" (is "?lhs = ?rhs") proof - { assume "?lhs" { 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 have "?rhs" by auto } moreover { assume "?rhs" { 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 have "?lhs" using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto } ultimately show ?thesis by auto qed subsection {* Infimum Distance *} definition "infdist x A = (if A = {} then 0 else INF a:A. dist x a)" lemma bdd_below_infdist[intro, simp]: "bdd_below (dist x`A)" by (auto intro!: zero_le_dist) lemma infdist_notempty: "A ≠ {} ==> infdist x A = (INF a:A. dist x a)" by (simp add: infdist_def) lemma infdist_nonneg: "0 ≤ infdist x A" by (auto simp add: infdist_def intro: cINF_greatest) lemma infdist_le: "a ∈ A ==> infdist x A ≤ dist x a" by (auto intro: cINF_lower simp add: infdist_def) lemma infdist_le2: "a ∈ A ==> dist x a ≤ d ==> infdist x A ≤ d" by (auto intro!: cINF_lower2 simp add: infdist_def) lemma infdist_zero[simp]: "a ∈ A ==> infdist a A = 0" by (auto intro!: antisym infdist_nonneg infdist_le2) lemma infdist_triangle: "infdist x A ≤ infdist y A + dist x y" proof (cases "A = {}") case True then show ?thesis by (simp add: infdist_def) next case False then obtain a where "a ∈ A" by auto have "infdist x A ≤ Inf {dist x y + dist y a |a. a ∈ A}" proof (rule cInf_greatest) from `A ≠ {}` show "{dist x y + dist y a |a. a ∈ A} ≠ {}" by simp fix d assume "d ∈ {dist x y + dist y a |a. a ∈ A}" then obtain a where d: "d = dist x y + dist y a" "a ∈ A" by auto show "infdist x A ≤ d" unfolding infdist_notempty[OF `A ≠ {}`] proof (rule cINF_lower2) show "a ∈ A" by fact show "dist x a ≤ d" unfolding d by (rule dist_triangle) qed simp qed also have "… = dist x y + infdist y A" proof (rule cInf_eq, safe) fix a assume "a ∈ A" then show "dist x y + infdist y A ≤ dist x y + dist y a" by (auto intro: infdist_le) next fix i assume inf: "!!d. d ∈ {dist x y + dist y a |a. a ∈ A} ==> i ≤ d" then have "i - dist x y ≤ infdist y A" unfolding infdist_notempty[OF `A ≠ {}`] using `a ∈ A` by (intro cINF_greatest) (auto simp: field_simps) then show "i ≤ dist x y + infdist y A" by simp qed finally show ?thesis by simp qed lemma in_closure_iff_infdist_zero: assumes "A ≠ {}" shows "x ∈ closure A <-> infdist x A = 0" proof assume "x ∈ closure A" show "infdist x A = 0" proof (rule ccontr) assume "infdist x A ≠ 0" with infdist_nonneg[of x A] have "infdist x A > 0" by auto then have "ball x (infdist x A) ∩ closure A = {}" apply auto apply (metis `x ∈ closure A` closure_approachable dist_commute infdist_le not_less) done then have "x ∉ closure A" by (metis `0 < infdist x A` centre_in_ball disjoint_iff_not_equal) then show False using `x ∈ closure A` by simp qed next assume x: "infdist x A = 0" then obtain a where "a ∈ A" by atomize_elim (metis all_not_in_conv assms) show "x ∈ closure A" unfolding closure_approachable apply safe proof (rule ccontr) fix e :: real assume "e > 0" assume "¬ (∃y∈A. dist y x < e)" then have "infdist x A ≥ e" using `a ∈ A` unfolding infdist_def by (force simp: dist_commute intro: cINF_greatest) with x `e > 0` show False by auto qed qed lemma in_closed_iff_infdist_zero: assumes "closed A" "A ≠ {}" shows "x ∈ A <-> infdist x A = 0" proof - have "x ∈ closure A <-> infdist x A = 0" by (rule in_closure_iff_infdist_zero) fact with assms show ?thesis by simp qed lemma tendsto_infdist [tendsto_intros]: assumes f: "(f ---> l) F" shows "((λx. infdist (f x) A) ---> infdist l A) F" proof (rule tendstoI) fix e ::real assume "e > 0" from tendstoD[OF f this] show "eventually (λx. dist (infdist (f x) A) (infdist l A) < e) F" proof (eventually_elim) fix x from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l] have "dist (infdist (f x) A) (infdist l A) ≤ dist (f x) l" by (simp add: dist_commute dist_real_def) also assume "dist (f x) l < e" finally show "dist (infdist (f x) A) (infdist l A) < e" . qed qed text{* Some other lemmas about sequences. *} lemma sequentially_offset: (* TODO: move to Topological_Spaces.thy *) assumes "eventually (λi. P i) sequentially" shows "eventually (λi. P (i + k)) sequentially" using assms by (rule eventually_sequentially_seg [THEN iffD2]) lemma seq_offset_neg: (* TODO: move to Topological_Spaces.thy *) "(f ---> l) sequentially ==> ((λi. f(i - k)) ---> l) sequentially" apply (erule filterlim_compose) apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially) apply arith done lemma seq_harmonic: "((λn. inverse (real n)) ---> 0) sequentially" using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc) (* TODO: move to Limits.thy *) subsection {* More properties of closed balls *} lemma closed_vimage: (* TODO: move to Topological_Spaces.thy *) assumes "closed s" and "continuous_on UNIV f" shows "closed (vimage f s)" using assms unfolding continuous_on_closed_vimage [OF closed_UNIV] by simp lemma closed_cball: "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) apply 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 mem_interior_cball: "x ∈ interior S <-> (∃e>0. cball x e ⊆ S)" apply (simp add: interior_def, safe) apply (force simp add: open_contains_cball) apply (rule_tac x="ball x e" in exI) apply (simp add: subset_trans [OF ball_subset_cball]) done lemma islimpt_ball: fixes x y :: "'a::{real_normed_vector,perfect_space}" shows "y islimpt ball x e <-> 0 < e ∧ y ∈ cball x e" (is "?lhs = ?rhs") proof assume "?lhs" { assume "e ≤ 0" then have *:"ball x e = {}" using ball_eq_empty[of x e] by auto have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto } then have "e > 0" by (metis not_less) moreover have "y ∈ cball x e" using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"] ball_subset_cball[of x e] `?lhs` unfolding closed_limpt by auto ultimately show "?rhs" by auto next assume "?rhs" then have "e > 0" by auto { fix d :: real assume "d > 0" have "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d" proof (cases "d ≤ dist x y") case True then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d" proof (cases "x = y") case True then have False using `d ≤ dist x y` `d>0` by auto then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d" by auto next case False have "dist x (y - (d / (2 * dist y x)) *⇩_{R}(y - x)) = norm (x - y + (d / (2 * norm (y - x))) *⇩_{R}(y - x))" unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric] by auto also have "… = ¦- 1 + d / (2 * norm (x - y))¦ * norm (x - y)" using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"] unfolding scaleR_minus_left scaleR_one by (auto simp add: norm_minus_commute) also have "… = ¦- norm (x - y) + d / 2¦" unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]] unfolding distrib_right using `x≠y`[unfolded dist_nz, unfolded dist_norm] by auto also have "… ≤ e - d/2" using `d ≤ dist x y` and `d>0` and `?rhs` by (auto simp add: dist_norm) finally have "y - (d / (2 * dist y x)) *⇩_{R}(y - x) ∈ ball x e" using `d>0` by auto moreover have "(d / (2*dist y x)) *⇩_{R}(y - x) ≠ 0" using `x≠y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute) moreover have "dist (y - (d / (2 * dist y x)) *⇩_{R}(y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel using `d > 0` `x≠y`[unfolded dist_nz] dist_commute[of x y] unfolding dist_norm apply auto done ultimately show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d" apply (rule_tac x = "y - (d / (2*dist y x)) *⇩_{R}(y - x)" in bexI) apply auto done qed next case False then have "d > dist x y" by auto show "∃x' ∈ ball x e. x' ≠ y ∧ dist x' y < d" proof (cases "x = y") case True obtain z where **: "z ≠ y" "dist z y < min e d" using perfect_choose_dist[of "min e d" y] using `d > 0` `e>0` by auto show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d" unfolding `x = y` using `z ≠ y` ** apply (rule_tac x=z in bexI) apply (auto simp add: dist_commute) done next case False then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d" using `d>0` `d > dist x y` `?rhs` apply (rule_tac x=x in bexI) apply auto done qed qed } then show "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto qed lemma closure_ball_lemma: fixes x y :: "'a::real_normed_vector" assumes "x ≠ y" shows "y islimpt ball x (dist x y)" proof (rule islimptI) fix T assume "y ∈ T" "open T" then obtain r where "0 < r" "∀z. dist z y < r --> z ∈ T" unfolding open_dist by fast (* choose point between x and y, within distance r of y. *) def k ≡ "min 1 (r / (2 * dist x y))" def z ≡ "y + scaleR k (x - y)" have z_def2: "z = x + scaleR (1 - k) (y - x)" unfolding z_def by (simp add: algebra_simps) have "dist z y < r" unfolding z_def k_def using `0 < r` by (simp add: dist_norm min_def) then have "z ∈ T" using `∀z. dist z y < r --> z ∈ T` by simp have "dist x z < dist x y" unfolding z_def2 dist_norm apply (simp add: norm_minus_commute) apply (simp only: dist_norm [symmetric]) apply (subgoal_tac "¦1 - k¦ * dist x y < 1 * dist x y", simp) apply (rule mult_strict_right_mono) apply (simp add: k_def zero_less_dist_iff `0 < r` `x ≠ y`) apply (simp add: zero_less_dist_iff `x ≠ y`) done then have "z ∈ ball x (dist x y)" by simp have "z ≠ y" unfolding z_def k_def using `x ≠ y` `0 < r` by (simp add: min_def) show "∃z∈ball x (dist x y). z ∈ T ∧ z ≠ y" using `z ∈ ball x (dist x y)` `z ∈ T` `z ≠ y` by fast qed lemma closure_ball: fixes x :: "'a::real_normed_vector" shows "0 < e ==> closure (ball x e) = cball x e" apply (rule equalityI) apply (rule closure_minimal) apply (rule ball_subset_cball) apply (rule closed_cball) apply (rule subsetI, rename_tac y) apply (simp add: le_less [where 'a=real]) apply (erule disjE) apply (rule subsetD [OF closure_subset], simp) apply (simp add: closure_def) apply clarify apply (rule closure_ball_lemma) apply (simp add: zero_less_dist_iff) done (* In a trivial vector space, this fails for e = 0. *) lemma interior_cball: fixes x :: "'a::{real_normed_vector, perfect_space}" shows "interior (cball x e) = ball x e" proof (cases "e ≥ 0") case False note cs = this from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover { fix y assume "y ∈ cball x e" then have False unfolding mem_cball using dist_nz[of x y] cs by auto } then have "cball x e = {}" by auto then have "interior (cball x e) = {}" using interior_empty by auto ultimately show ?thesis by blast next case True note cs = this have "ball x e ⊆ cball x e" using ball_subset_cball by auto moreover { fix S y assume as: "S ⊆ cball x e" "open S" "y∈S" then obtain d where "d>0" and d: "∀x'. dist x' y < d --> x' ∈ S" unfolding open_dist by blast then obtain xa where xa_y: "xa ≠ y" and xa: "dist xa y < d" using perfect_choose_dist [of d] by auto have "xa ∈ S" using d[THEN spec[where x = xa]] using xa by (auto simp add: dist_commute) then have xa_cball: "xa ∈ cball x e" using as(1) by auto then have "y ∈ ball x e" proof (cases "x = y") case True then have "e > 0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute) then show "y ∈ ball x e" using `x = y ` by simp next case False have "dist (y + (d / 2 / dist y x) *⇩_{R}(y - x)) y < d" unfolding dist_norm using `d>0` norm_ge_zero[of "y - x"] `x ≠ y` by auto then have *: "y + (d / 2 / dist y x) *⇩_{R}(y - x) ∈ cball x e" using d as(1)[unfolded subset_eq] by blast have "y - x ≠ 0" using `x ≠ y` by auto hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[symmetric] using `d>0` by auto have "dist (y + (d / 2 / dist y x) *⇩_{R}(y - x)) x = norm (y + (d / (2 * norm (y - x))) *⇩_{R}y - (d / (2 * norm (y - x))) *⇩_{R}x - x)" by (auto simp add: dist_norm algebra_simps) also have "… = norm ((1 + d / (2 * norm (y - x))) *⇩_{R}(y - x))" by (auto simp add: algebra_simps) also have "… = ¦1 + d / (2 * norm (y - x))¦ * norm (y - x)" using ** by auto also have "… = (dist y x) + d/2" using ** by (auto simp add: distrib_right dist_norm) finally have "e ≥ dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute) then show "y ∈ ball x e" unfolding mem_ball using `d>0` by auto qed } then have "∀S ⊆ cball x e. open S --> S ⊆ ball x e" by auto ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto qed lemma frontier_ball: fixes a :: "'a::real_normed_vector" shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}" apply (simp add: frontier_def closure_ball interior_open order_less_imp_le) apply (simp add: set_eq_iff) apply arith done lemma frontier_cball: fixes a :: "'a::{real_normed_vector, perfect_space}" shows "frontier (cball a e) = {x. dist a x = e}" apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le) apply (simp add: set_eq_iff) apply arith done lemma cball_eq_empty: "cball x e = {} <-> e < 0" apply (simp add: set_eq_iff not_le) apply (metis zero_le_dist dist_self order_less_le_trans) done lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty) lemma cball_eq_sing: fixes x :: "'a::{metric_space,perfect_space}" shows "cball x e = {x} <-> e = 0" proof (rule linorder_cases) assume e: "0 < e" obtain a where "a ≠ x" "dist a x < e" using perfect_choose_dist [OF e] by auto then have "a ≠ x" "dist x a ≤ e" by (auto simp add: dist_commute) with e show ?thesis by (auto simp add: set_eq_iff) qed auto lemma cball_sing: fixes x :: "'a::metric_space" shows "e = 0 ==> cball x e = {x}" by (auto simp add: set_eq_iff) subsection {* Boundedness *} (* FIXME: This has to be unified with BSEQ!! *) definition (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)" unfolding bounded_def subset_eq by auto 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 bounded_realI: assumes "∀x∈s. abs (x::real) ≤ B" shows "bounded s" unfolding bounded_def dist_real_def by (metis abs_minus_commute assms diff_0_right) 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 (rule eventually_mono, 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_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) apply 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" apply (auto simp add: bounded_def) by (metis Un_iff add_le_cancel_left dist_triangle le_max_iff_disj max.order_iff) lemma bounded_Union[intro]: "finite F ==> ∀S∈F. bounded S ==> bounded (\<Union>F)" by (induct rule: finite_induct[of F]) auto lemma bounded_UN [intro]: "finite A ==> ∀x∈A. bounded (B x) ==> bounded (\<Union>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 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 + abs y ∧ (x ≤ y --> x ≤ 1 + abs y)") apply metis apply arith 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, intro]: "¬ bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)" proof (auto simp add: 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 lemma bounded_linear_image: assumes "bounded S" and "bounded_linear f" shows "bounded (f ` S)" proof - from assms(1) obtain b where b: "b > 0" "∀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 add: ac_simps) { fix x assume "x ∈ S" then have "norm x ≤ b" using b by auto then have "norm (f x) ≤ B * b" using B(2) apply (erule_tac x=x in allE) apply (metis B(1) B(2) order_trans mult_le_cancel_left_pos) done } then show ?thesis unfolding bounded_pos apply (rule_tac x="b*B" in exI) using b B by (auto simp add: mult.commute) 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) apply assumption apply (rule bounded_linear_scaleR_right) done 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 text{* Some theorems on sups and infs using the notion "bounded". *} lemma bounded_real: "bounded (S::real set) <-> (∃a. ∀x∈S. ¦x¦ ≤ a)" by (simp add: bounded_iff) lemma bounded_imp_bdd_above: "bounded S ==> bdd_above (S :: real set)" by (auto simp: bounded_def bdd_above_def dist_real_def) (metis abs_le_D1 abs_minus_commute diff_le_eq) lemma bounded_imp_bdd_below: "bounded S ==> bdd_below (S :: real set)" by (auto simp: bounded_def bdd_below_def dist_real_def) (metis abs_le_D1 add.commute diff_le_eq) (* TODO: remove the following lemmas about Inf and Sup, is now in conditionally complete lattice *) lemma bounded_has_Sup: fixes S :: "real set" assumes "bounded S" and "S ≠ {}" shows "∀x∈S. x ≤ Sup S" and "∀b. (∀x∈S. x ≤ b) --> Sup S ≤ b" proof show "∀b. (∀x∈S. x ≤ b) --> Sup S ≤ b" using assms by (metis cSup_least) qed (metis cSup_upper assms(1) bounded_imp_bdd_above) lemma Sup_insert: fixes S :: "real set" shows "bounded S ==> Sup (insert x S) = (if S = {} then x else max x (Sup S))" by (auto simp: bounded_imp_bdd_above sup_max cSup_insert_If) lemma Sup_insert_finite: fixes S :: "real set" shows "finite S ==> Sup (insert x S) = (if S = {} then x else max x (Sup S))" apply (rule Sup_insert) apply (rule finite_imp_bounded) apply simp done lemma bounded_has_Inf: fixes S :: "real set" assumes "bounded S" and "S ≠ {}" shows "∀x∈S. x ≥ Inf S" and "∀b. (∀x∈S. x ≥ b) --> Inf S ≥ b" proof show "∀b. (∀x∈S. x ≥ b) --> Inf S ≥ b" using assms by (metis cInf_greatest) qed (metis cInf_lower assms(1) bounded_imp_bdd_below) lemma Inf_insert: fixes S :: "real set" shows "bounded S ==> Inf (insert x S) = (if S = {} then x else min x (Inf S))" by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_If) lemma Inf_insert_finite: fixes S :: "real set" shows "finite S ==> Inf (insert x S) = (if S = {} then x else min x (Inf S))" apply (rule Inf_insert) apply (rule finite_imp_bounded) apply simp done subsection {* Compactness *} subsubsection {* Bolzano-Weierstrass property *} lemma 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 ⊆ \<Union>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. subseq r ∧ (f o 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 def s ≡ "λn i. SOME j. i < j ∧ f j ∈ A (Suc n)" { 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 def r ≡ "rec_nat (s 0 0) s" have "subseq r" by (auto simp: r_def s subseq_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. subseq r ∧ (f o 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_union_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 lemma islimpt_range_imp_convergent_subsequence: fixes l :: "'a :: {t1_space, first_countable_topology}" assumes l: "l islimpt (range f)" shows "∃r. subseq r ∧ (f o r) ----> l" using l unfolding islimpt_eq_acc_point by (rule acc_point_range_imp_convergent_subsequence) 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_union_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 ⊆ (\<Union>x∈U. ball x 1)" using assms by auto then obtain D where D: "D ⊆ U" "finite D" "U ⊆ (\<Union>x∈D. ball x 1)" by (rule compactE_image) from `finite D` have "bounded (\<Union>x∈D. ball x 1)" by (simp add: bounded_UN) then show "bounded U" using `U ⊆ (\<Union>x∈D. ball x 1)` by (rule bounded_subset) qed text{* In particular, some common special cases. *} lemma compact_union [intro]: assumes "compact s" and "compact t" shows " compact (s ∪ t)" proof (rule compactI) fix f assume *: "Ball f open" "s ∪ t ⊆ \<Union>f" from * `compact s` obtain s' where "s' ⊆ f ∧ finite s' ∧ s ⊆ \<Union>s'" unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) moreover from * `compact t` obtain t' where "t' ⊆ f ∧ finite t' ∧ t ⊆ \<Union>t'" unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) ultimately show "∃f'⊆f. finite f' ∧ s ∪ t ⊆ \<Union>f'" by (auto intro!: exI[of _ "s' ∪ t'"]) qed lemma compact_Union [intro]: "finite S ==> (!!T. T ∈ S ==> compact T) ==> compact (\<Union>S)" by (induct set: finite) auto lemma compact_UN [intro]: "finite A ==> (!!x. x ∈ A ==> compact (B x)) ==> compact (\<Union>x∈A. B x)" unfolding SUP_def by (rule compact_Union) auto lemma closed_inter_compact [intro]: assumes "closed s" and "compact t" shows "compact (s ∩ t)" using compact_inter_closed [of t s] assms by (simp add: Int_commute) lemma compact_inter [intro]: fixes s t :: "'a :: t2_space set" assumes "compact s" and "compact t" shows "compact (s ∩ t)" using assms by (intro compact_inter_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_union) 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) 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_inter_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 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) def 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_elim1 intro: set_mp[OF closure_subset]) have "(∀B ⊆ Z. finite B --> U ∩ \<Inter>B ≠ {})" proof (intro allI impI) fix B assume "finite B" "B ⊆ Z" with `finite B` ev_Z F(2) have "eventually (λx. x ∈ U ∩ (\<Inter>B)) F" by (auto simp: eventually_ball_finite_distrib eventually_conj_iff) with F show "U ∩ \<Inter>B ≠ {}" by (intro notI) (simp add: eventually_False) qed ultimately have "U ∩ \<Inter>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_inter_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 ∩ \<Inter>B ≠ {}" "U ∩ \<Inter>A = {}" def F ≡ "INF a:insert U A. principal a" have "F ≠ bot" unfolding F_def proof (rule INF_filter_not_bot) fix X assume "X ⊆ insert U A" "finite X" moreover with A(2)[THEN spec, of "X - {U}"] have "U ∩ \<Inter>(X - {U}) ≠ {}" by auto ultimately show "(INF a:X. principal a) ≠ bot" by (auto simp add: 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" "x ∈ S" "S ⊆ A" then have "eventually (λx. x ∈ A) (nhds x)" by (auto simp: eventually_nhds) with V have "eventually (λx. x ∈ V ∩ A) (inf (nhds x) F)" by (auto simp: eventually_inf) with x show "V ∩ A ≠ {}" by (auto simp del: Int_iff simp add: trivial_limit_def) qed then have "x ∈ V" using `V ∈ A` A(1) by simp } with `x∈U` have "x ∈ U ∩ \<Inter>A" by auto with `U ∩ \<Inter>A = {}` show False by auto qed definition "countably_compact U <-> (∀A. countable A --> (∀a∈A. open a) --> U ⊆ \<Union>A --> (∃T⊆A. finite T ∧ U ⊆ \<Union>T))" lemma countably_compactE: assumes "countably_compact s" and "∀t∈C. open t" and "s ⊆ \<Union>C" "countable C" obtains C' where "C' ⊆ C" and "finite C'" and "s ⊆ \<Union>C'" using assms unfolding countably_compact_def by metis lemma countably_compactI: assumes "!!C. ∀t∈C. open t ==> s ⊆ \<Union>C ==> countable C ==> (∃C'⊆C. finite C' ∧ s ⊆ \<Union>C')" shows "countably_compact s" using assms unfolding countably_compact_def by metis lemma compact_imp_countably_compact: "compact U ==> countably_compact U" by (auto simp: compact_eq_heine_borel countably_compact_def) lemma countably_compact_imp_compact: assumes "countably_compact U" and ccover: "countable B" "∀b∈B. open b" and basis: "!!T x. open T ==> x ∈ T ==> x ∈ U ==> ∃b∈B. x ∈ b ∧ b ∩ U ⊆ T" shows "compact U" using `countably_compact U` unfolding compact_eq_heine_borel countably_compact_def proof safe fix A assume A: "∀a∈A. open a" "U ⊆ \<Union>A" assume *: "∀A. countable A --> (∀a∈A. open a) --> U ⊆ \<Union>A --> (∃T⊆A. finite T ∧ U ⊆ \<Union>T)" moreover def C ≡ "{b∈B. ∃a∈A. b ∩ U ⊆ a}" ultimately have "countable C" "∀a∈C. open a" unfolding C_def using ccover by auto moreover have "\<Union>A ∩ U ⊆ \<Union>C" proof safe fix x a assume "x ∈ U" "x ∈ a" "a ∈ A" with basis[of a x] A obtain b where "b ∈ B" "x ∈ b" "b ∩ U ⊆ a" by blast with `a ∈ A` show "x ∈ \<Union>C" unfolding C_def by auto qed then have "U ⊆ \<Union>C" using `U ⊆ \<Union>A` by auto ultimately obtain T where T: "T⊆C" "finite T" "U ⊆ \<Union>T" using * by metis then have "∀t∈T. ∃a∈A. t ∩ U ⊆ a" by (auto simp: C_def) then obtain f where "∀t∈T. f t ∈ A ∧ t ∩ U ⊆ f t" unfolding bchoice_iff Bex_def .. with T show "∃T⊆A. finite T ∧ U ⊆ \<Union>T" unfolding C_def by (intro exI[of _ "f`T"]) fastforce qed lemma countably_compact_imp_compact_second_countable: "countably_compact U ==> compact (U :: 'a :: second_countable_topology set)" proof (rule countably_compact_imp_compact) fix T and x :: 'a assume "open T" "x ∈ T" from topological_basisE[OF is_basis this] obtain b where "b ∈ (SOME B. countable B ∧ topological_basis B)" "x ∈ b" "b ⊆ T" . then show "∃b∈SOME B. countable B ∧ topological_basis B. x ∈ b ∧ b ∩ U ⊆ T" by blast qed (insert countable_basis topological_basis_open[OF is_basis], auto) lemma countably_compact_eq_compact: "countably_compact U <-> compact (U :: 'a :: second_countable_topology set)" using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast subsubsection{* Sequential compactness *} definition seq_compact :: "'a::topological_space set => bool" where "seq_compact S <-> (∀f. (∀n. f n ∈ S) --> (∃l∈S. ∃r. subseq r ∧ ((f o r) ---> l) sequentially))" lemma seq_compactI: assumes "!!f. ∀n. f n ∈ S ==> ∃l∈S. ∃r. subseq r ∧ ((f o r) ---> l) sequentially" shows "seq_compact S" unfolding seq_compact_def using assms by fast lemma seq_compactE: assumes "seq_compact S" "∀n. f n ∈ S" obtains l r where "l ∈ S" "subseq r" "((f o r) ---> l) sequentially" using assms unfolding seq_compact_def by fast lemma closed_sequentially: (* TODO: move upwards *) assumes "closed s" and "∀n. f n ∈ s" and "f ----> l" shows "l ∈ s" proof (rule ccontr) assume "l ∉ s" with `closed s` and `f ----> l` have "eventually (λn. f n ∈ - s) sequentially" by (fast intro: topological_tendstoD) with `∀n. f n ∈ s` show "False" by simp qed lemma seq_compact_inter_closed: assumes "seq_compact s" and "closed t" shows "seq_compact (s ∩ t)" proof (rule seq_compactI) fix f assume "∀n::nat. f n ∈ s ∩ t" hence "∀n. f n ∈ s" and "∀n. f n ∈ t" by simp_all from `seq_compact s` and `∀n. f n ∈ s` obtain l r where "l ∈ s" and r: "subseq r" and l: "(f o r) ----> l" by (rule seq_compactE) from `∀n. f n ∈ t` have "∀n. (f o r) n ∈ t" by simp from `closed t` and this and l have "l ∈ t" by (rule closed_sequentially) with `l ∈ s` and r and l show "∃l∈s ∩ t. ∃r. subseq r ∧ (f o r) ----> l" by fast qed lemma seq_compact_closed_subset: assumes "closed s" and "s ⊆ t" and "seq_compact t" shows "seq_compact s" using assms seq_compact_inter_closed [of t s] by (simp add: Int_absorb1) lemma seq_compact_imp_countably_compact: fixes U :: "'a :: first_countable_topology set" assumes "seq_compact U" shows "countably_compact U" proof (safe intro!: countably_compactI) fix A assume A: "∀a∈A. open a" "U ⊆ \<Union>A" "countable A" have subseq: "!!X. range X ⊆ U ==> ∃r x. x ∈ U ∧ subseq r ∧ (X o r) ----> x" using `seq_compact U` by (fastforce simp: seq_compact_def subset_eq) show "∃T⊆A. finite T ∧ U ⊆ \<Union>T" proof cases assume "finite A" with A show ?thesis by auto next assume "infinite A" then have "A ≠ {}" by auto show ?thesis proof (rule ccontr) assume "¬ (∃T⊆A. finite T ∧ U ⊆ \<Union>T)" then have "∀T. ∃x. T ⊆ A ∧ finite T --> (x ∈ U - \<Union>T)" by auto then obtain X' where T: "!!T. T ⊆ A ==> finite T ==> X' T ∈ U - \<Union>T" by metis def X ≡ "λn. X' (from_nat_into A ` {.. n})" have X: "!!n. X n ∈ U - (\<Union>i≤n. from_nat_into A i)" using `A ≠ {}` unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into) then have "range X ⊆ U" by auto with subseq[of X] obtain r x where "x ∈ U" and r: "subseq r" "(X o r) ----> x" by auto from `x∈U` `U ⊆ \<Union>A` from_nat_into_surj[OF `countable A`] obtain n where "x ∈ from_nat_into A n" by auto with r(2) A(1) from_nat_into[OF `A ≠ {}`, of n] have "eventually (λi. X (r i) ∈ from_nat_into A n) sequentially" unfolding tendsto_def by (auto simp: comp_def) then obtain N where "!!i. N ≤ i ==> X (r i) ∈ from_nat_into A n" by (auto simp: eventually_sequentially) moreover from X have "!!i. n ≤ r i ==> X (r i) ∉ from_nat_into A n" by auto moreover from `subseq r`[THEN seq_suble, of "max n N"] have "∃i. n ≤ r i ∧ N ≤ i" by (auto intro!: exI[of _ "max n N"]) ultimately show False by auto qed qed qed lemma compact_imp_seq_compact: fixes U :: "'a :: first_countable_topology set" assumes "compact U" shows "seq_compact U" unfolding seq_compact_def proof safe fix X :: "nat => 'a" assume "∀n. X n ∈ U" then have "eventually (λx. x ∈ U) (filtermap X sequentially)" by (auto simp: eventually_filtermap) moreover have "filtermap X sequentially ≠ bot" by (simp add: trivial_limit_def eventually_filtermap) ultimately obtain x where "x ∈ U" and x: "inf (nhds x) (filtermap X sequentially) ≠ bot" (is "?F ≠ _") using `compact U` by (auto simp: compact_filter) 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 def s ≡ "λn i. SOME j. i < j ∧ X j ∈ A (Suc n)" { fix n i have "∃a. i < a ∧ X a ∈ A (Suc n)" proof (rule ccontr) assume "¬ (∃a>i. X a ∈ A (Suc n))" then have "!!a. Suc i ≤ a ==> X a ∉ A (Suc n)" by auto then have "eventually (λx. x ∉ A (Suc n)) (filtermap X sequentially)" by (auto simp: eventually_filtermap eventually_sequentially) moreover have "eventually (λx. x ∈ A (Suc n)) (nhds x)" using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds) ultimately have "eventually (λx. False) ?F" by (auto simp add: eventually_inf) with x show False by (simp add: eventually_False) qed then have "i < s n i" "X (s n i) ∈ A (Suc n)" unfolding s_def by (auto intro: someI2_ex) } note s = this def r ≡ "rec_nat (s 0 0) s" have "subseq r" by (auto simp: r_def s subseq_Suc_iff) moreover have "(λn. X (r n)) ----> x" proof (rule topological_tendstoI) fix S assume "open S" "x ∈ S" with A(3) have "eventually (λi. A i ⊆ S) sequentially" by auto moreover { fix i assume "Suc 0 ≤ i" then have "X (r i) ∈ A i" by (cases i) (simp_all add: r_def s) } then have "eventually (λi. X (r i) ∈ A i) sequentially" by (auto simp: eventually_sequentially) ultimately show "eventually (λi. X (r i) ∈ S) sequentially" by eventually_elim auto qed ultimately show "∃x ∈ U. ∃r. subseq r ∧ (X o r) ----> x" using `x ∈ U` by (auto simp: convergent_def comp_def) qed lemma countably_compact_imp_acc_point: assumes "countably_compact s" and "countable t" and "infinite t" and "t ⊆ s" shows "∃x∈s. ∀U. x∈U ∧ open U --> infinite (U ∩ t)" proof (rule ccontr) def C ≡ "(λF. interior (F ∪ (- t))) ` {F. finite F ∧ F ⊆ t }" note `countably_compact s` moreover have "∀t∈C. open t" by (auto simp: C_def) moreover assume "¬ (∃x∈s. ∀U. x∈U ∧ open U --> infinite (U ∩ t))" then have s: "!!x. x ∈ s ==> ∃U. x∈U ∧ open U ∧ finite (U ∩ t)" by metis have "s ⊆ \<Union>C" using `t ⊆ s` unfolding C_def Union_image_eq apply (safe dest!: s) apply (rule_tac a="U ∩ t" in UN_I) apply (auto intro!: interiorI simp add: finite_subset) done moreover from `countable t` have "countable C" unfolding C_def by (auto intro: countable_Collect_finite_subset) ultimately obtain D where "D ⊆ C" "finite D" "s ⊆ \<Union>D" by (rule countably_compactE) then obtain E where E: "E ⊆ {F. finite F ∧ F ⊆ t }" "finite E" and s: "s ⊆ (\<Union>F∈E. interior (F ∪ (- t)))" by (metis (lifting) Union_image_eq finite_subset_image C_def) from s `t ⊆ s` have "t ⊆ \<Union>E" using interior_subset by blast moreover have "finite (\<Union>E)" using E by auto ultimately show False using `infinite t` by (auto simp: finite_subset) qed lemma countable_acc_point_imp_seq_compact: fixes s :: "'a::first_countable_topology set" assumes "∀t. infinite t ∧ countable t ∧ t ⊆ s --> (∃x∈s. ∀U. x∈U ∧ open U --> infinite (U ∩ t))" shows "seq_compact s" proof - { fix f :: "nat => 'a" assume f: "∀n. f n ∈ s" have "∃l∈s. ∃r. subseq r ∧ ((f o r) ---> l) sequentially" proof (cases "finite (range f)") case True obtain l where "infinite {n. f n = f l}" using pigeonhole_infinite[OF _ True] by auto then obtain r where "subseq r" and fr: "∀n. f (r n) = f l" using infinite_enumerate by blast then have "subseq r ∧ (f o r) ----> f l" by (simp add: fr o_def) with f show "∃l∈s. ∃r. subseq r ∧ (f o r) ----> l" by auto next case False with f assms have "∃x∈s. ∀U. x∈U ∧ open U --> infinite (U ∩ range f)" by auto then obtain l where "l ∈ s" "∀U. l∈U ∧ open U --> infinite (U ∩ range f)" .. from this(2) have "∃r. subseq r ∧ ((f o r) ---> l) sequentially" using acc_point_range_imp_convergent_subsequence[of l f] by auto with `l ∈ s` show "∃l∈s. ∃r. subseq r ∧ ((f o r) ---> l) sequentially" .. qed } then show ?thesis unfolding seq_compact_def by auto qed lemma seq_compact_eq_countably_compact: fixes U :: "'a :: first_countable_topology set" shows "seq_compact U <-> countably_compact U" using countable_acc_point_imp_seq_compact countably_compact_imp_acc_point seq_compact_imp_countably_compact by metis lemma seq_compact_eq_acc_point: fixes s :: "'a :: first_countable_topology set" shows "seq_compact s <-> (∀t. infinite t ∧ countable t ∧ t ⊆ s --> (∃x∈s. ∀U. x∈U ∧ open U --> infinite (U ∩ t)))" using countable_acc_point_imp_seq_compact[of s] countably_compact_imp_acc_point[of s] seq_compact_imp_countably_compact[of s] by metis lemma seq_compact_eq_compact: fixes U :: "'a :: second_countable_topology set" shows "seq_compact U <-> compact U" using seq_compact_eq_countably_compact countably_compact_eq_compact by blast lemma bolzano_weierstrass_imp_seq_compact: fixes s :: "'a::{t1_space, first_countable_topology} set" shows "∀t. infinite t ∧ t ⊆ s --> (∃x ∈ s. x islimpt t) ==> seq_compact s" by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point) subsubsection{* Totally bounded *} lemma cauchy_def: "Cauchy s <-> (∀e>0. ∃N. ∀m n. m ≥ N ∧ n ≥ N --> dist(s m)(s n) < e)" unfolding Cauchy_def by metis lemma seq_compact_imp_totally_bounded: assumes "seq_compact s" shows "∀e>0. ∃k. finite k ∧ k ⊆ s ∧ s ⊆ (\<Union>x∈k. ball x e)" proof - { fix e::real assume "e > 0" assume *: "!!k. finite k ==> k ⊆ s ==> ¬ s ⊆ (\<Union>x∈k. ball x e)" let ?Q = "λx n r. r ∈ s ∧ (∀m < (n::nat). ¬ (dist (x m) r < e))" have "∃x. ∀n::nat. ?Q x n (x n)" proof (rule dependent_wellorder_choice) fix n x assume "!!y. y < n ==> ?Q x y (x y)" then have "¬ s ⊆ (\<Union>x∈x ` {0..<n}. ball x e)" using *[of "x ` {0 ..< n}"] by (auto simp: subset_eq) then obtain z where z:"z∈s" "z ∉ (\<Union>x∈x ` {0..<n}. ball x e)" unfolding subset_eq by auto show "∃r. ?Q x n r" using z by auto qed simp then obtain x where "∀n::nat. x n ∈ s" and x:"!!n m. m < n ==> ¬ (dist (x m) (x n) < e)" by blast then obtain l r where "l ∈ s" and r:"subseq r" and "((x o r) ---> l) sequentially" using assms by (metis seq_compact_def) from this(3) have "Cauchy (x o r)" using LIMSEQ_imp_Cauchy by auto then obtain N::nat where "!!m n. N ≤ m ==> N ≤ n ==> dist ((x o r) m) ((x o r) n) < e" unfolding cauchy_def using `e > 0` by blast then have False using x[of "r N" "r (N+1)"] r by (auto simp: subseq_def) } then show ?thesis by metis qed subsubsection{* Heine-Borel theorem *} lemma seq_compact_imp_heine_borel: fixes s :: "'a :: metric_space set" assumes "seq_compact s" shows "compact s" proof - from seq_compact_imp_totally_bounded[OF `seq_compact s`] obtain f where f: "∀e>0. finite (f e) ∧ f e ⊆ s ∧ s ⊆ (\<Union>x∈f e. ball x e)" unfolding choice_iff' .. def K ≡ "(λ(x, r). ball x r) ` ((\<Union>e ∈ \<rat> ∩ {0 <..}. f e) × \<rat>)" have "countably_compact s" using `seq_compact s` by (rule seq_compact_imp_countably_compact) then show "compact s" proof (rule countably_compact_imp_compact) show "countable K" unfolding K_def using f by (auto intro: countable_finite countable_subset countable_rat intro!: countable_image countable_SIGMA countable_UN) show "∀b∈K. open b" by (auto simp: K_def) next fix T x assume T: "open T" "x ∈ T" and x: "x ∈ s" from openE[OF T] obtain e where "0 < e" "ball x e ⊆ T" by auto then have "0 < e / 2" "ball x (e / 2) ⊆ T" by auto from Rats_dense_in_real[OF `0 < e / 2`] obtain r where "r ∈ \<rat>" "0 < r" "r < e / 2" by auto from f[rule_format, of r] `0 < r` `x ∈ s` obtain k where "k ∈ f r" "x ∈ ball k r" unfolding Union_image_eq by auto from `r ∈ \<rat>` `0 < r` `k ∈ f r` have "ball k r ∈ K" by (auto simp: K_def) then show "∃b∈K. x ∈ b ∧ b ∩ s ⊆ T" proof (rule bexI[rotated], safe) fix y assume "y ∈ ball k r" with `r < e / 2` `x ∈ ball k r` have "dist x y < e" by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute) with `ball x e ⊆ T` show "y ∈ T" by auto next show "x ∈ ball k r" by fact qed qed qed lemma compact_eq_seq_compact_metric: "compact (s :: 'a::metric_space set) <-> seq_compact s" using compact_imp_seq_compact seq_compact_imp_heine_borel by blast lemma compact_def: "compact (S :: 'a::metric_space set) <-> (∀f. (∀n. f n ∈ S) --> (∃l∈S. ∃r. subseq r ∧ (f o r) ----> l))" unfolding compact_eq_seq_compact_metric seq_compact_def by auto subsubsection {* Complete the chain of compactness variants *} lemma compact_eq_bolzano_weierstrass: fixes s :: "'a::metric_space set" shows "compact s <-> (∀t. infinite t ∧ t ⊆ s --> (∃x ∈ s. x islimpt t))" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs using heine_borel_imp_bolzano_weierstrass[of s] by auto next assume ?rhs then show ?lhs unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact) qed lemma bolzano_weierstrass_imp_bounded: "∀t. infinite t ∧ t ⊆ s --> (∃x ∈ s. x islimpt t) ==> bounded s" using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass . subsection {* Metric spaces with the Heine-Borel property *} text {* A metric space (or topological vector space) is said to have the Heine-Borel property if every closed and bounded subset is compact. *} class heine_borel = metric_space + assumes bounded_imp_convergent_subsequence: "bounded (range f) ==> ∃l r. subseq r ∧ ((f o r) ---> l) sequentially" lemma bounded_closed_imp_seq_compact: fixes s::"'a::heine_borel set" assumes "bounded s" and "closed s" shows "seq_compact s" proof (unfold seq_compact_def, clarify) fix f :: "nat => 'a" assume f: "∀n. f n ∈ s" with `bounded s` have "bounded (range f)" by (auto intro: bounded_subset) obtain l r where r: "subseq r" and l: "((f o r) ---> l) sequentially" using bounded_imp_convergent_subsequence [OF `bounded (range f)`] by auto from f have fr: "∀n. (f o r) n ∈ s" by simp have "l ∈ s" using `closed s` fr l by (rule closed_sequentially) show "∃l∈s. ∃r. subseq r ∧ ((f o r) ---> l) sequentially" using `l ∈ s` r l by blast qed lemma compact_eq_bounded_closed: fixes s :: "'a::heine_borel set" shows "compact s <-> bounded s ∧ closed s" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs using compact_imp_closed compact_imp_bounded by blast next assume ?rhs then show ?lhs using bounded_closed_imp_seq_compact[of s] unfolding compact_eq_seq_compact_metric by auto qed (* TODO: is this lemma necessary? *) lemma bounded_increasing_convergent: fixes s :: "nat => real" shows "bounded {s n| n. True} ==> ∀n. s n ≤ s (Suc n) ==> ∃l. s ----> l" using Bseq_mono_convergent[of s] incseq_Suc_iff[of s] by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def) instance real :: heine_borel proof fix f :: "nat => real" assume f: "bounded (range f)" obtain r where r: "subseq r" "monoseq (f o r)" unfolding comp_def by (metis seq_monosub) then have "Bseq (f o r)" unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset) with r show "∃l r. subseq r ∧ (f o r) ----> l" using Bseq_monoseq_convergent[of "f o r"] by (auto simp: convergent_def) qed lemma compact_lemma: fixes f :: "nat => 'a::euclidean_space" assumes "bounded (range f)" shows "∀d⊆Basis. ∃l::'a. ∃ r. subseq r ∧ (∀e>0. eventually (λn. ∀i∈d. dist (f (r n) • i) (l • i) < e) sequentially)" proof safe fix d :: "'a set" assume d: "d ⊆ Basis" with finite_Basis have "finite d" by (blast intro: finite_subset) from this d show "∃l::'a. ∃r. subseq r ∧ (∀e>0. eventually (λn. ∀i∈d. dist (f (r n) • i) (l • i) < e) sequentially)" proof (induct d) case empty then show ?case unfolding subseq_def by auto next case (insert k d) have k[intro]: "k ∈ Basis" using insert by auto have s': "bounded ((λx. x • k) ` range f)" using `bounded (range f)` by (auto intro!: bounded_linear_image bounded_linear_inner_left) obtain l1::"'a" and r1 where r1: "subseq r1" and lr1: "∀e > 0. eventually (λn. ∀i∈d. dist (f (r1 n) • i) (l1 • i) < e) sequentially" using insert(3) using insert(4) by auto have f': "∀n. f (r1 n) • k ∈ (λx. x • k) ` range f" by simp have "bounded (range (λi. f (r1 i) • k))" by (metis (lifting) bounded_subset f' image_subsetI s') then obtain l2 r2 where r2:"subseq r2" and lr2:"((λi. f (r1 (r2 i)) • k) ---> l2) sequentially" using bounded_imp_convergent_subsequence[of "λi. f (r1 i) • k"] by (auto simp: o_def) def r ≡ "r1 o r2" have r:"subseq r" using r1 and r2 unfolding r_def o_def subseq_def by auto moreover def l ≡ "(∑i∈Basis. (if i = k then l2 else l1•i) *⇩_{R}i)::'a" { fix e::real assume "e > 0" from lr1 `e > 0` have N1: "eventually (λn. ∀i∈d. dist (f (r1 n) • i) (l1 • i) < e) sequentially" by blast from lr2 `e > 0` have N2:"eventually (λn. dist (f (r1 (r2 n)) • k) l2 < e) sequentially" by (rule tendstoD) from r2 N1 have N1': "eventually (λn. ∀i∈d. dist (f (r1 (r2 n)) • i) (l1 • i) < e) sequentially" by (rule eventually_subseq) have "eventually (λn. ∀i∈(insert k d). dist (f (r n) • i) (l • i) < e) sequentially" using N1' N2 by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def) } ultimately show ?case by auto qed qed instance euclidean_space ⊆ heine_borel proof fix f :: "nat => 'a" assume f: "bounded (range f)" then obtain l::'a and r where r: "subseq r" and l: "∀e>0. eventually (λn. ∀i∈Basis. dist (f (r n) • i) (l • i) < e) sequentially" using compact_lemma [OF f] by blast { fix e::real assume "e > 0" hence "e / real_of_nat DIM('a) > 0" by (simp add: DIM_positive) with l have "eventually (λn. ∀i∈Basis. dist (f (r n) • i) (l • i) < e / (real_of_nat DIM('a))) sequentially" by simp moreover { fix n assume n: "∀i∈Basis. dist (f (r n) • i) (l • i) < e / (real_of_nat DIM('a))" have "dist (f (r n)) l ≤ (∑i∈Basis. dist (f (r n) • i) (l • i))" apply (subst euclidean_dist_l2) using zero_le_dist apply (rule setL2_le_setsum) done also have "… < (∑i∈(Basis::'a set). e / (real_of_nat DIM('a)))" apply (rule setsum_strict_mono) using n apply auto done finally have "dist (f (r n)) l < e" by auto } ultimately have "eventually (λn. dist (f (r n)) l < e) sequentially" by (rule eventually_elim1) } then have *: "((f o r) ---> l) sequentially" unfolding o_def tendsto_iff by simp with r show "∃l r. subseq r ∧ ((f o r) ---> l) sequentially" by auto qed lemma bounded_fst: "bounded s ==> bounded (fst ` s)" unfolding bounded_def by (metis (erased, hide_lams) dist_fst_le image_iff order_trans) lemma bounded_snd: "bounded s ==> bounded (snd ` s)" unfolding bounded_def by (metis (no_types, hide_lams) dist_snd_le image_iff order.trans) instance prod :: (heine_borel, heine_borel) heine_borel proof fix f :: "nat => 'a × 'b" assume f: "bounded (range f)" then have "bounded (fst ` range f)" by (rule bounded_fst) then have s1: "bounded (range (fst o f))" by (simp add: image_comp) obtain l1 r1 where r1: "subseq r1" and l1: "(λn. fst (f (r1 n))) ----> l1" using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast from f have s2: "bounded (range (snd o f o r1))" by (auto simp add: image_comp intro: bounded_snd bounded_subset) obtain l2 r2 where r2: "subseq r2" and l2: "((λn. snd (f (r1 (r2 n)))) ---> l2) sequentially" using bounded_imp_convergent_subsequence [OF s2] unfolding o_def by fast have l1': "((λn. fst (f (r1 (r2 n)))) ---> l1) sequentially" using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def . have l: "((f o (r1 o r2)) ---> (l1, l2)) sequentially" using tendsto_Pair [OF l1' l2] unfolding o_def by simp have r: "subseq (r1 o r2)" using r1 r2 unfolding subseq_def by simp show "∃l r. subseq r ∧ ((f o r) ---> l) sequentially" using l r by fast qed subsubsection {* Completeness *} definition complete :: "'a::metric_space set => bool" where "complete s <-> (∀f. (∀n. f n ∈ s) ∧ Cauchy f --> (∃l∈s. f ----> l))" lemma completeI: assumes "!!f. ∀n. f n ∈ s ==> Cauchy f ==> ∃l∈s. f ----> l" shows "complete s" using assms unfolding complete_def by fast lemma completeE: assumes "complete s" and "∀n. f n ∈ s" and "Cauchy f" obtains l where "l ∈ s" and "f ----> l" using assms unfolding complete_def by fast lemma compact_imp_complete: assumes "compact s" shows "complete s" proof - { fix f assume as: "(∀n::nat. f n ∈ s)" "Cauchy f" from as(1) obtain l r where lr: "l∈s" "subseq r" "(f o r) ----> l" using assms unfolding compact_def by blast note lr' = seq_suble [OF lr(2)] { fix e :: real assume "e > 0" from as(2) obtain N where N:"∀m n. N ≤ m ∧ N ≤ n --> dist (f m) (f n) < e/2" unfolding cauchy_def using `e > 0` apply (erule_tac x="e/2" in allE) apply auto done from lr(3)[unfolded lim_sequentially, THEN spec[where x="e/2"]] obtain M where M:"∀n≥M. dist ((f o r) n) l < e/2" using `e > 0` by auto { fix n :: nat assume n: "n ≥ max N M" have "dist ((f o r) n) l < e/2" using n M by auto moreover have "r n ≥ N" using lr'[of n] n by auto then have "dist (f n) ((f o r) n) < e / 2" using N and n by auto ultimately have "dist (f n) l < e" using dist_triangle_half_r[of "f (r n)" "f n" e l] by (auto simp add: dist_commute) } then have "∃N. ∀n≥N. dist (f n) l < e" by blast } then have "∃l∈s. (f ---> l) sequentially" using `l∈s` unfolding lim_sequentially by auto } then show ?thesis unfolding complete_def by auto qed lemma nat_approx_posE: fixes e::real assumes "0 < e" obtains n :: nat where "1 / (Suc n) < e" proof atomize_elim have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))" by (rule divide_strict_left_mono) (auto simp: `0 < e`) also have "1 / (ceiling (1/e)) ≤ 1 / (1/e)" by (rule divide_left_mono) (auto simp: `0 < e`) also have "… = e" by simp finally show "∃n. 1 / real (Suc n) < e" .. qed lemma compact_eq_totally_bounded: "compact s <-> complete s ∧ (∀e>0. ∃k. finite k ∧ s ⊆ (\<Union>x∈k. ball x e))" (is "_ <-> ?rhs") proof assume assms: "?rhs" then obtain k where k: "!!e. 0 < e ==> finite (k e)" "!!e. 0 < e ==> s ⊆ (\<Union>x∈k e. ball x e)" by (auto simp: choice_iff') show "compact s" proof cases assume "s = {}" then show "compact s" by (simp add: compact_def) next assume "s ≠ {}" show ?thesis unfolding compact_def proof safe fix f :: "nat => 'a" assume f: "∀n. f n ∈ s" def e ≡ "λn. 1 / (2 * Suc n)" then have [simp]: "!!n. 0 < e n" by auto def B ≡ "λn U. SOME b. infinite {n. f n ∈ b} ∧ (∃x. b ⊆ ball x (e n) ∩ U)" { fix n U assume "infinite {n. f n ∈ U}" then have "∃b∈k (e n). infinite {i∈{n. f n ∈ U}. f i ∈ ball b (e n)}" using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq) then obtain a where "a ∈ k (e n)" "infinite {i ∈ {n. f n ∈ U}. f i ∈ ball a (e n)}" .. then have "∃b. infinite {i. f i ∈ b} ∧ (∃x. b ⊆ ball x (e n) ∩ U)" by (intro exI[of _ "ball a (e n) ∩ U"] exI[of _ a]) (auto simp: ac_simps) from someI_ex[OF this] have "infinite {i. f i ∈ B n U}" "∃x. B n U ⊆ ball x (e n) ∩ U" unfolding B_def by auto } note B = this def F ≡ "rec_nat (B 0 UNIV) B" { fix n have "infinite {i. f i ∈ F n}" by (induct n) (auto simp: F_def B) } then have F: "!!n. ∃x. F (Suc n) ⊆ ball x (e n) ∩ F n" using B by (simp add: F_def) then have F_dec: "!!m n. m ≤ n ==> F n ⊆ F m" using decseq_SucI[of F] by (auto simp: decseq_def) obtain sel where sel: "!!k i. i < sel k i" "!!k i. f (sel k i) ∈ F k" proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI) fix k i have "infinite ({n. f n ∈ F k} - {.. i})" using `infinite {n. f n ∈ F k}` by auto from infinite_imp_nonempty[OF this] show "∃x>i. f x ∈ F k" by (simp add: set_eq_iff not_le conj_commute) qed def t ≡ "rec_nat (sel 0 0) (λn i. sel (Suc n) i)" have "subseq t" unfolding subseq_Suc_iff by (simp add: t_def sel) moreover have "∀i. (f o t) i ∈ s" using f by auto moreover { fix n have "(f o t) n ∈ F n" by (cases n) (simp_all add: t_def sel) } note t = this have "Cauchy (f o t)" proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE) fix r :: real and N n m assume "1 / Suc N < r" "Suc N ≤ n" "Suc N ≤ m" then have "(f o t) n ∈ F (Suc N)" "(f o t) m ∈ F (Suc N)" "2 * e N < r" using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc) with F[of N] obtain x where "dist x ((f o t) n) < e N" "dist x ((f o t) m) < e N" by (auto simp: subset_eq) with dist_triangle[of "(f o t) m" "(f o t) n" x] `2 * e N < r` show "dist ((f o t) m) ((f o t) n) < r" by (simp add: dist_commute) qed ultimately show "∃l∈s. ∃r. subseq r ∧ (f o r) ----> l" using assms unfolding complete_def by blast qed qed qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded) lemma cauchy: "Cauchy s <-> (∀e>0.∃ N::nat. ∀n≥N. dist(s n)(s N) < e)" (is "?lhs = ?rhs") proof - { assume ?rhs { fix e::real assume "e>0" with `?rhs` obtain N where N:"∀n≥N. dist (s n) (s N) < e/2" by (erule_tac x="e/2" in allE) auto { fix n m assume nm:"N ≤ m ∧ N ≤ n" then have "dist (s m) (s n) < e" using N using dist_triangle_half_l[of "s m" "s N" "e" "s n"] by blast } then have "∃N. ∀m n. N ≤ m ∧ N ≤ n --> dist (s m) (s n) < e" by blast } then have ?lhs unfolding cauchy_def by blast } then show ?thesis unfolding cauchy_def using dist_triangle_half_l by blast qed lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)" proof - from assms obtain N :: nat where "∀m n. N ≤ m ∧ N ≤ n --> dist (s m) (s n) < 1" unfolding cauchy_def apply (erule_tac x= 1 in allE) apply auto done then have N:"∀n. N ≤ n --> dist (s N) (s n) < 1" by auto moreover have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto then obtain a where a:"∀x∈s ` {0..N}. dist (s N) x ≤ a" unfolding bounded_any_center [where a="s N"] by auto ultimately show "?thesis" unfolding bounded_any_center [where a="s N"] apply (rule_tac x="max a 1" in exI) apply auto apply (erule_tac x=y in allE) apply (erule_tac x=y in ballE) apply auto done qed instance heine_borel < complete_space proof fix f :: "nat => 'a" assume "Cauchy f" then have "bounded (range f)" by (rule cauchy_imp_bounded) then have "compact (closure (range f))" unfolding compact_eq_bounded_closed by auto then have "complete (closure (range f))" by (rule compact_imp_complete) moreover have "∀n. f n ∈ closure (range f)" using closure_subset [of "range f"] by auto ultimately have "∃l∈closure (range f). (f ---> l) sequentially" using `Cauchy f` unfolding complete_def by auto then show "convergent f" unfolding convergent_def by auto qed instance euclidean_space ⊆ banach .. lemma complete_UNIV: "complete (UNIV :: ('a::complete_space) set)" proof (rule completeI) fix f :: "nat => 'a" assume "Cauchy f" then have "convergent f" by (rule Cauchy_convergent) then show "∃l∈UNIV. f ----> l" unfolding convergent_def by simp qed lemma complete_imp_closed: assumes "complete s" shows "closed s" proof (unfold closed_sequential_limits, clarify) fix f x assume "∀n. f n ∈ s" and "f ----> x" from `f ----> x` have "Cauchy f" by (rule LIMSEQ_imp_Cauchy) with `complete s` and `∀n. f n ∈ s` obtain l where "l ∈ s" and "f ----> l" by (rule completeE) from `f ----> x` and `f ----> l` have "x = l" by (rule LIMSEQ_unique) with `l ∈ s` show "x ∈ s" by simp qed lemma complete_inter_closed: assumes "complete s" and "closed t" shows "complete (s ∩ t)" proof (rule completeI) fix f assume "∀n. f n ∈ s ∩ t" and "Cauchy f" then have "∀n. f n ∈ s" and "∀n. f n ∈ t" by simp_all from `complete s` obtain l where "l ∈ s" and "f ----> l" using `∀n. f n ∈ s` and `Cauchy f` by (rule completeE) from `closed t` and `∀n. f n ∈ t` and `f ----> l` have "l ∈ t" by (rule closed_sequentially) with `l ∈ s` and `f ----> l` show "∃l∈s ∩ t. f ----> l" by fast qed lemma complete_closed_subset: assumes "closed s" and "s ⊆ t" and "complete t" shows "complete s" using assms complete_inter_closed [of t s] by (simp add: Int_absorb1) lemma complete_eq_closed: fixes s :: "('a::complete_space) set" shows "complete s <-> closed s" proof assume "closed s" then show "complete s" using subset_UNIV complete_UNIV by (rule complete_closed_subset) next assume "complete s" then show "closed s" by (rule complete_imp_closed) qed lemma convergent_eq_cauchy: fixes s :: "nat => 'a::complete_space" shows "(∃l. (s ---> l) sequentially) <-> Cauchy s" unfolding Cauchy_convergent_iff convergent_def .. lemma convergent_imp_bounded: fixes s :: "nat => 'a::metric_space" shows "(s ---> l) sequentially ==> bounded (range s)" by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy) lemma compact_cball[simp]: fixes x :: "'a::heine_borel" shows "compact (cball x e)" using compact_eq_bounded_closed bounded_cball closed_cball by blast lemma compact_frontier_bounded[intro]: fixes s :: "'a::heine_borel set" shows "bounded s ==> compact (frontier s)" unfolding frontier_def using compact_eq_bounded_closed by blast lemma compact_frontier[intro]: fixes s :: "'a::heine_borel set" shows "compact s ==> compact (frontier s)" using compact_eq_bounded_closed compact_frontier_bounded by blast lemma frontier_subset_compact: fixes s :: "'a::heine_borel set" shows "compact s ==> frontier s ⊆ s" using frontier_subset_closed compact_eq_bounded_closed by blast subsection {* Bounded closed nest property (proof does not use Heine-Borel) *} lemma bounded_closed_nest: fixes s :: "nat => ('a::heine_borel) set" assumes "∀n. closed (s n)" and "∀n. s n ≠ {}" and "∀m n. m ≤ n --> s n ⊆ s m" and "bounded (s 0)" shows "∃a. ∀n. a ∈ s n" proof - from assms(2) obtain x where x: "∀n. x n ∈ s n" using choice[of "λn x. x ∈ s n"] by auto from assms(4,1) have "seq_compact (s 0)" by (simp add: bounded_closed_imp_seq_compact) then obtain l r where lr: "l ∈ s 0" "subseq r" "(x o r) ----> l" using x and assms(3) unfolding seq_compact_def by blast have "∀n. l ∈ s n" proof fix n :: nat have "closed (s n)" using assms(1) by simp moreover have "∀i. (x o r) i ∈ s i" using x and assms(3) and lr(2) [THEN seq_suble] by auto then have "∀i. (x o r) (i + n) ∈ s n" using assms(3) by (fast intro!: le_add2) moreover have "(λi. (x o r) (i + n)) ----> l" using lr(3) by (rule LIMSEQ_ignore_initial_segment) ultimately show "l ∈ s n" by (rule closed_sequentially) qed then show ?thesis .. qed text {* Decreasing case does not even need compactness, just completeness. *} lemma decreasing_closed_nest: fixes s :: "nat => ('a::complete_space) set" assumes "∀n. closed (s n)" "∀n. s n ≠ {}" "∀m n. m ≤ n --> s n ⊆ s m" "∀e>0. ∃n. ∀x∈s n. ∀y∈s n. dist x y < e" shows "∃a. ∀n. a ∈ s n" proof - have "∀n. ∃x. x ∈ s n" using assms(2) by auto then have "∃t. ∀n. t n ∈ s n" using choice[of "λn x. x ∈ s n"] by auto then obtain t where t: "∀n. t n ∈ s n" by auto { fix e :: real assume "e > 0" then obtain N where N:"∀x∈s N. ∀y∈s N. dist x y < e" using assms(4) by auto { fix m n :: nat assume "N ≤ m ∧ N ≤ n" then have "t m ∈ s N" "t n ∈ s N" using assms(3) t unfolding subset_eq t by blast+ then have "dist (t m) (t n) < e" using N by auto } then have "∃N. ∀m n. N ≤ m ∧ N ≤ n --> dist (t m) (t n) < e" by auto } then have "Cauchy t" unfolding cauchy_def by auto then obtain l where l:"(t ---> l) sequentially" using complete_UNIV unfolding complete_def by auto { fix n :: nat { fix e :: real assume "e > 0" then obtain N :: nat where N: "∀n≥N. dist (t n) l < e" using l[unfolded lim_sequentially] by auto have "t (max n N) ∈ s n" using assms(3) unfolding subset_eq apply (erule_tac x=n in allE) apply (erule_tac x="max n N" in allE) using t apply auto done then have "∃y∈s n. dist y l < e" apply (rule_tac x="t (max n N)" in bexI) using N apply auto done } then have "l ∈ s n" using closed_approachable[of "s n" l] assms(1) by auto } then show ?thesis by auto qed text {* Strengthen it to the intersection actually being a singleton. *} lemma decreasing_closed_nest_sing: fixes s :: "nat => 'a::complete_space set" assumes "∀n. closed(s n)" "∀n. s n ≠ {}" "∀m n. m ≤ n --> s n ⊆ s m" "∀e>0. ∃n. ∀x ∈ (s n). ∀ y∈(s n). dist x y < e" shows "∃a. \<Inter>(range s) = {a}" proof - obtain a where a: "∀n. a ∈ s n" using decreasing_closed_nest[of s] using assms by auto { fix b assume b: "b ∈ \<Inter>(range s)" { fix e :: real assume "e > 0" then have "dist a b < e" using assms(4) and b and a by blast } then have "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le) } with a have "\<Inter>(range s) = {a}" unfolding image_def by auto then show ?thesis .. qed text{* Cauchy-type criteria for uniform convergence. *} lemma uniformly_convergent_eq_cauchy: fixes s::"nat => 'b => 'a::complete_space" shows "(∃l. ∀e>0. ∃N. ∀n x. N ≤ n ∧ P x --> dist(s n x)(l x) < e) <-> (∀e>0. ∃N. ∀m n x. N ≤ m ∧ N ≤ n ∧ P x --> dist (s m x) (s n x) < e)" (is "?lhs = ?rhs") proof assume ?lhs then obtain l where l:"∀e>0. ∃N. ∀n x. N ≤ n ∧ P x --> dist (s n x) (l x) < e" by auto { fix e :: real assume "e > 0" then obtain N :: nat where N: "∀n x. N ≤ n ∧ P x --> dist (s n x) (l x) < e / 2" using l[THEN spec[where x="e/2"]] by auto { fix n m :: nat and x :: "'b" assume "N ≤ m ∧ N ≤ n ∧ P x" then have "dist (s m x) (s n x) < e" using N[THEN spec[where x=m], THEN spec[where x=x]] using N[THEN spec[where x=n], THEN spec[where x=x]] using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto } then have "∃N. ∀m n x. N ≤ m ∧ N ≤ n ∧ P x --> dist (s m x) (s n x) < e" by auto } then show ?rhs by auto next assume ?rhs then have "∀x. P x --> Cauchy (λn. s n x)" unfolding cauchy_def apply auto apply (erule_tac x=e in allE) apply auto done then obtain l where l: "∀x. P x --> ((λn. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[symmetric] using choice[of "λx l. P x --> ((λn. s n x) ---> l) sequentially"] by auto { fix e :: real assume "e > 0" then obtain N where N:"∀m n x. N ≤ m ∧ N ≤ n ∧ P x --> dist (s m x) (s n x) < e/2" using `?rhs`[THEN spec[where x="e/2"]] by auto { fix x assume "P x" then obtain M where M:"∀n≥M. dist (s n x) (l x) < e/2" using l[THEN spec[where x=x], unfolded lim_sequentially] and `e > 0` by (auto elim!: allE[where x="e/2"]) fix n :: nat assume "n ≥ N" then have "dist(s n x)(l x) < e" using `P x`and N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]] using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"] by (auto simp add: dist_commute) } then have "∃N. ∀n x. N ≤ n ∧ P x --> dist(s n x)(l x) < e" by auto } then show ?lhs by auto qed lemma uniformly_cauchy_imp_uniformly_convergent: fixes s :: "nat => 'a => 'b::complete_space" assumes "∀e>0.∃N. ∀m (n::nat) x. N ≤ m ∧ N ≤ n ∧ P x --> dist(s m x)(s n x) < e" and "∀x. P x --> (∀e>0. ∃N. ∀n. N ≤ n --> dist(s n x)(l x) < e)" shows "∀e>0. ∃N. ∀n x. N ≤ n ∧ P x --> dist(s n x)(l x) < e" proof - obtain l' where l:"∀e>0. ∃N. ∀n x. N ≤ n ∧ P x --> dist (s n x) (l' x) < e" using assms(1) unfolding uniformly_convergent_eq_cauchy[symmetric] by auto moreover { fix x assume "P x" then have "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "λn. s n x" "l x" "l' x"] using l and assms(2) unfolding lim_sequentially by blast } ultimately show ?thesis by auto qed subsection {* Continuity *} text{* Derive the epsilon-delta forms, which we often use as "definitions" *} lemma continuous_within_eps_delta: "continuous (at x within s) f <-> (∀e>0. ∃d>0. ∀x'∈ s. dist x' x < d --> dist (f x') (f x) < e)" unfolding continuous_within and Lim_within apply auto apply (metis dist_nz dist_self) apply blast done lemma continuous_at_eps_delta: "continuous (at x) f <-> (∀e > 0. ∃d > 0. ∀x'. dist x' x < d --> dist (f x') (f x) < e)" using continuous_within_eps_delta [of x UNIV f] by simp lemma continuous_at_right_real_increasing: fixes f :: "real => real" assumes nondecF: "!!x y. x ≤ y ==> f x ≤ f y" shows "continuous (at_right a) f <-> (∀e>0. ∃d>0. f (a + d) - f a < e)" apply (simp add: greaterThan_def dist_real_def continuous_within Lim_within_le) apply (intro all_cong ex_cong) apply safe apply (erule_tac x="a + d" in allE) apply simp apply (simp add: nondecF field_simps) apply (drule nondecF) apply simp done lemma continuous_at_left_real_increasing: assumes nondecF: "!! x y. x ≤ y ==> f x ≤ ((f y) :: real)" shows "(continuous (at_left (a :: real)) f) = (∀e > 0. ∃delta > 0. f a - f (a - delta) < e)" apply (simp add: lessThan_def dist_real_def continuous_within Lim_within_le) apply (intro all_cong ex_cong) apply safe apply (erule_tac x="a - d" in allE) apply simp apply (simp add: nondecF field_simps) apply (cut_tac x="a - d" and y="x" in nondecF) apply simp_all done text{* Versions in terms of open balls. *} lemma continuous_within_ball: "continuous (at x within s) f <-> (∀e > 0. ∃d > 0. f ` (ball x d ∩ s) ⊆ ball (f x) e)" (is "?lhs = ?rhs") proof assume ?lhs { fix e :: real assume "e > 0" then obtain d where d: "d>0" "∀xa∈s. 0 < dist xa x ∧ dist xa x < d --> dist (f xa) (f x) < e" using `?lhs`[unfolded continuous_within Lim_within] by auto { fix y assume "y ∈ f ` (ball x d ∩ s)" then have "y ∈ ball (f x) e" using d(2) unfolding dist_nz[symmetric] apply (auto simp add: dist_commute) apply (erule_tac x=xa in ballE) apply auto using `e > 0` apply auto done } then have "∃d>0. f ` (ball x d ∩ s) ⊆ ball (f x) e" using `d > 0` unfolding subset_eq ball_def by (auto simp add: dist_commute) } then show ?rhs by auto next assume ?rhs then show ?lhs unfolding continuous_within Lim_within ball_def subset_eq apply (auto simp add: dist_commute) apply (erule_tac x=e in allE) apply auto done qed lemma continuous_at_ball: "continuous (at x) f <-> (∀e>0. ∃d>0. f ` (ball x d) ⊆ ball (f x) e)" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball apply auto apply (erule_tac x=e in allE) apply auto apply (rule_tac x=d in exI) apply auto apply (erule_tac x=xa in allE) apply (auto simp add: dist_commute dist_nz) unfolding dist_nz[symmetric] apply auto done next assume ?rhs then show ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball apply auto apply (erule_tac x=e in allE) apply auto apply (rule_tac x=d in exI) apply auto apply (erule_tac x="f xa" in allE) apply (auto simp add: dist_commute dist_nz) done qed text{* Define setwise continuity in terms of limits within the set. *} lemma continuous_on_iff: "continuous_on s f <-> (∀x∈s. ∀e>0. ∃d>0. ∀x'∈s. dist x' x < d --> dist (f x') (f x) < e)" unfolding continuous_on_def Lim_within by (metis dist_pos_lt dist_self) definition uniformly_continuous_on :: "'a set => ('a::metric_space => 'b::metric_space) => bool" where "uniformly_continuous_on s f <-> (∀e>0. ∃d>0. ∀x∈s. ∀x'∈s. dist x' x < d --> dist (f x') (f x) < e)" text{* Some simple consequential lemmas. *} lemma uniformly_continuous_imp_continuous: "uniformly_continuous_on s f ==> continuous_on s f" unfolding uniformly_continuous_on_def continuous_on_iff by blast lemma continuous_at_imp_continuous_within: "continuous (at x) f ==> continuous (at x within s) f" unfolding continuous_within continuous_at using Lim_at_within by auto lemma Lim_trivial_limit: "trivial_limit net ==> (f ---> l) net" by simp lemmas continuous_on = continuous_on_def -- "legacy theorem name" lemma continuous_within_subset: "continuous (at x within s) f ==> t ⊆ s ==> continuous (at x within t) f" unfolding continuous_within by(metis tendsto_within_subset) lemma continuous_on_interior: "continuous_on s f ==> x ∈ interior s ==> continuous (at x) f" by (metis continuous_on_eq_continuous_at continuous_on_subset interiorE) lemma continuous_on_eq: "(∀x ∈ s. f x = g x) ==> continuous_on s f ==> continuous_on s g" unfolding continuous_on_def tendsto_def eventually_at_topological by simp text {* Characterization of various kinds of continuity in terms of sequences. *} lemma continuous_within_sequentially: fixes f :: "'a::metric_space => 'b::topological_space" shows "continuous (at a within s) f <-> (∀x. (∀n::nat. x n ∈ s) ∧ (x ---> a) sequentially --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs") proof assume ?lhs { fix x :: "nat => 'a" assume x: "∀n. x n ∈ s" "∀e>0. eventually (λn. dist (x n) a < e) sequentially" fix T :: "'b set" assume "open T" and "f a ∈ T" with `?lhs` obtain d where "d>0" and d:"∀x∈s. 0 < dist x a ∧ dist x a < d --> f x ∈ T" unfolding continuous_within tendsto_def eventually_at by (auto simp: dist_nz) have "eventually (λn. dist (x n) a < d) sequentially" using x(2) `d>0` by simp then have "eventually (λn. (f o x) n ∈ T) sequentially" proof eventually_elim case (elim n) then show ?case using d x(1) `f a ∈ T` unfolding dist_nz[symmetric] by auto qed } then show ?rhs unfolding tendsto_iff tendsto_def by simp next assume ?rhs then show ?lhs unfolding continuous_within tendsto_def [where l="f a"] by (simp add: sequentially_imp_eventually_within) qed lemma continuous_at_sequentially: fixes f :: "'a::metric_space => 'b::topological_space" shows "continuous (at a) f <-> (∀x. (x ---> a) sequentially --> ((f o x) ---> f a) sequentially)" using continuous_within_sequentially[of a UNIV f] by simp lemma continuous_on_sequentially: fixes f :: "'a::metric_space => 'b::topological_space" shows "continuous_on s f <-> (∀x. ∀a ∈ s. (∀n. x(n) ∈ s) ∧ (x ---> a) sequentially --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs") proof assume ?rhs then show ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto next assume ?lhs then show ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto qed lemma uniformly_continuous_on_sequentially: "uniformly_continuous_on s f <-> (∀x y. (∀n. x n ∈ s) ∧ (∀n. y n ∈ s) ∧ ((λn. dist (x n) (y n)) ---> 0) sequentially --> ((λn. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs") proof assume ?lhs { fix x y assume x: "∀n. x n ∈ s" and y: "∀n. y n ∈ s" and xy: "((λn. dist (x n) (y n)) ---> 0) sequentially" { fix e :: real assume "e > 0" then obtain d where "d > 0" and d: "∀x∈s. ∀x'∈s. dist x' x < d --> dist (f x') (f x) < e" using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto obtain N where N: "∀n≥N. dist (x n) (y n) < d" using xy[unfolded lim_sequentially dist_norm] and `d>0` by auto { fix n assume "n≥N" then have "dist (f (x n)) (f (y n)) < e" using N[THEN spec[where x=n]] using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]] using x and y unfolding dist_commute by simp } then have "∃N. ∀n≥N. dist (f (x n)) (f (y n)) < e" by auto } then have "((λn. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding lim_sequentially and dist_real_def by auto } then show ?rhs by auto next assume ?rhs { assume "¬ ?lhs" then obtain e where "e > 0" "∀d>0. ∃x∈s. ∃x'∈s. dist x' x < d ∧ ¬ dist (f x') (f x) < e" unfolding uniformly_continuous_on_def by auto then obtain fa where fa: "∀x. 0 < x --> fst (fa x) ∈ s ∧ snd (fa x) ∈ s ∧ dist (fst (fa x)) (snd (fa x)) < x ∧ ¬ dist (f (fst (fa x))) (f (snd (fa x))) < e" using choice[of "λd x. d>0 --> fst x ∈ s ∧ snd x ∈ s ∧ dist (snd x) (fst x) < d ∧ ¬ dist (f (snd x)) (f (fst x)) < e"] unfolding Bex_def by (auto simp add: dist_commute) def x ≡ "λn::nat. fst (fa (inverse (real n + 1)))" def y ≡ "λn::nat. snd (fa (inverse (real n + 1)))" have xyn: "∀n. x n ∈ s ∧ y n ∈ s" and xy0: "∀n. dist (x n) (y n) < inverse (real n + 1)" and fxy:"∀n. ¬ dist (f (x n)) (f (y n)) < e" unfolding x_def and y_def using fa by auto { fix e :: real assume "e > 0" then obtain N :: nat where "N ≠ 0" and N: "0 < inverse (real N) ∧ inverse (real N) < e" unfolding real_arch_inv[of e] by auto { fix n :: nat assume "n ≥ N" then have "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N≠0` by auto also have "… < e" using N by auto finally have "inverse (real n + 1) < e" by auto then have "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto } then have "∃N. ∀n≥N. dist (x n) (y n) < e" by auto } then have "∀e>0. ∃N. ∀n≥N. dist (f (x n)) (f (y n)) < e" using `?rhs`[THEN spec[where x=x], THEN spec[where x=y]] and xyn unfolding lim_sequentially dist_real_def by auto then have False using fxy and `e>0` by auto } then show ?lhs unfolding uniformly_continuous_on_def by blast qed text{* The usual transformation theorems. *} lemma continuous_transform_within: fixes f g :: "'a::metric_space => 'b::topological_space" assumes "0 < d" and "x ∈ s" and "∀x' ∈ s. dist x' x < d --> f x' = g x'" and "continuous (at x within s) f" shows "continuous (at x within s) g" unfolding continuous_within proof (rule Lim_transform_within) show "0 < d" by fact show "∀x'∈s. 0 < dist x' x ∧ dist x' x < d --> f x' = g x'" using assms(3) by auto have "f x = g x" using assms(1,2,3) by auto then show "(f ---> g x) (at x within s)" using assms(4) unfolding continuous_within by simp qed lemma continuous_transform_at: fixes f g :: "'a::metric_space => 'b::topological_space" assumes "0 < d" and "∀x'. dist x' x < d --> f x' = g x'" and "continuous (at x) f" shows "continuous (at x) g" using continuous_transform_within [of d x UNIV f g] assms by simp subsubsection {* Structural rules for pointwise continuity *} lemmas continuous_within_id = continuous_ident lemmas continuous_at_id = isCont_ident lemma continuous_infdist[continuous_intros]: assumes "continuous F f" shows "continuous F (λx. infdist (f x) A)" using assms unfolding continuous_def by (rule tendsto_infdist) lemma continuous_infnorm[continuous_intros]: "continuous F f ==> continuous F (λx. infnorm (f x))" unfolding continuous_def by (rule tendsto_infnorm) lemma continuous_inner[continuous_intros]: assumes "continuous F f" and "continuous F g" shows "continuous F (λx. inner (f x) (g x))" using assms unfolding continuous_def by (rule tendsto_inner) lemmas continuous_at_inverse = isCont_inverse subsubsection {* Structural rules for setwise continuity *} lemma continuous_on_infnorm[continuous_intros]: "continuous_on s f ==> continuous_on s (λx. infnorm (f x))" unfolding continuous_on by (fast intro: tendsto_infnorm) lemma continuous_on_inner[continuous_intros]: fixes g :: "'a::topological_space => 'b::real_inner" assumes "continuous_on s f" and "continuous_on s g" shows "continuous_on s (λx. inner (f x) (g x))" using bounded_bilinear_inner assms by (rule bounded_bilinear.continuous_on) subsubsection {* Structural rules for uniform continuity *} lemma uniformly_continuous_on_id[continuous_intros]: "uniformly_continuous_on s (λx. x)" unfolding uniformly_continuous_on_def by auto lemma uniformly_continuous_on_const[continuous_intros]: "uniformly_continuous_on s (λx. c)" unfolding uniformly_continuous_on_def by simp lemma uniformly_continuous_on_dist[continuous_intros]: fixes f g :: "'a::metric_space => 'b::metric_space" assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g" shows "uniformly_continuous_on s (λx. dist (f x) (g x))" proof - { fix a b c d :: 'b have "¦dist a b - dist c d¦ ≤ dist a c + dist b d" using dist_triangle2 [of a b c] dist_triangle2 [of b c d] using dist_triangle3 [of c d a] dist_triangle [of a d b] by arith } note le = this { fix x y assume f: "(λn. dist (f (x n)) (f (y n))) ----> 0" assume g: "(λn. dist (g (x n)) (g (y n))) ----> 0" have "(λn. ¦dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))¦) ----> 0" by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]], simp add: le) } then show ?thesis using assms unfolding uniformly_continuous_on_sequentially unfolding dist_real_def by simp qed lemma uniformly_continuous_on_norm[continuous_intros]: assumes "uniformly_continuous_on s f" shows "uniformly_continuous_on s (λx. norm (f x))" unfolding norm_conv_dist using assms by (intro uniformly_continuous_on_dist uniformly_continuous_on_const) lemma (in bounded_linear) uniformly_continuous_on[continuous_intros]: assumes "uniformly_continuous_on s g" shows "uniformly_continuous_on s (λx. f (g x))" using assms unfolding uniformly_continuous_on_sequentially unfolding dist_norm tendsto_norm_zero_iff diff[symmetric] by (auto intro: tendsto_zero) lemma uniformly_continuous_on_cmul[continuous_intros]: fixes f :: "'a::metric_space => 'b::real_normed_vector" assumes "uniformly_continuous_on s f" shows "uniformly_continuous_on s (λx. c *⇩_{R}f(x))" using bounded_linear_scaleR_right assms by (rule bounded_linear.uniformly_continuous_on) lemma dist_minus: fixes x y :: "'a::real_normed_vector" shows "dist (- x) (- y) = dist x y" unfolding dist_norm minus_diff_minus norm_minus_cancel .. lemma uniformly_continuous_on_minus[continuous_intros]: fixes f :: "'a::metric_space => 'b::real_normed_vector" shows "uniformly_continuous_on s f ==> uniformly_continuous_on s (λx. - f x)" unfolding uniformly_continuous_on_def dist_minus . lemma uniformly_continuous_on_add[continuous_intros]: fixes f g :: "'a::metric_space => 'b::real_normed_vector" assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g" shows "uniformly_continuous_on s (λx. f x + g x)" using assms unfolding uniformly_continuous_on_sequentially unfolding dist_norm tendsto_norm_zero_iff add_diff_add by (auto intro: tendsto_add_zero) lemma uniformly_continuous_on_diff[continuous_intros]: fixes f :: "'a::metric_space => 'b::real_normed_vector" assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g" shows "uniformly_continuous_on s (λx. f x - g x)" using assms uniformly_continuous_on_add [of s f "- g"] by (simp add: fun_Compl_def uniformly_continuous_on_minus) text{* Continuity of all kinds is preserved under composition. *} lemmas continuous_at_compose = isCont_o lemma uniformly_continuous_on_compose[continuous_intros]: assumes "uniformly_continuous_on s f" "uniformly_continuous_on (f ` s) g" shows "uniformly_continuous_on s (g o f)" proof - { fix e :: real assume "e > 0" then obtain d where "d > 0" and d: "∀x∈f ` s. ∀x'∈f ` s. dist x' x < d --> dist (g x') (g x) < e" using assms(2) unfolding uniformly_continuous_on_def by auto obtain d' where "d'>0" "∀x∈s. ∀x'∈s. dist x' x < d' --> dist (f x') (f x) < d" using `d > 0` using assms(1) unfolding uniformly_continuous_on_def by auto then have "∃d>0. ∀x∈s. ∀x'∈s. dist x' x < d --> dist ((g o f) x') ((g o f) x) < e" using `d>0` using d by auto } then show ?thesis using assms unfolding uniformly_continuous_on_def by auto qed text{* Continuity in terms of open preimages. *} lemma continuous_at_open: "continuous (at x) f <-> (∀t. open t ∧ f x ∈ t --> (∃s. open s ∧ x ∈ s ∧ (∀x' ∈ s. (f x') ∈ t)))" unfolding continuous_within_topological [of x UNIV f] unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto lemma continuous_imp_tendsto: assumes "continuous (at x0) f" and "x ----> x0" shows "(f o x) ----> (f x0)" proof (rule topological_tendstoI) fix S assume "open S" "f x0 ∈ S" then obtain T where T_def: "open T" "x0 ∈ T" "∀x∈T. f x ∈ S" using assms continuous_at_open by metis then have "eventually (λn. x n ∈ T) sequentially" using assms T_def by (auto simp: tendsto_def) then show "eventually (λn. (f o x) n ∈ S) sequentially" using T_def by (auto elim!: eventually_elim1) qed lemma continuous_on_open: "continuous_on s f <-> (∀t. openin (subtopology euclidean (f ` s)) t --> openin (subtopology euclidean s) {x ∈ s. f x ∈ t})" unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong) text {* Similarly in terms of closed sets. *} lemma continuous_on_closed: "continuous_on s f <-> (∀t. closedin (subtopology euclidean (f ` s)) t --> closedin (subtopology euclidean s) {x ∈ s. f x ∈ t})" unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong) text {* Half-global and completely global cases. *} lemma continuous_open_in_preimage: assumes "continuous_on s f" "open t" shows "openin (subtopology euclidean s) {x ∈ s. f x ∈ t}" proof - have *: "∀x. x ∈ s ∧ f x ∈ t <-> x ∈ s ∧ f x ∈ (t ∩ f ` s)" by auto have "openin (subtopology euclidean (f ` s)) (t ∩ f ` s)" using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto then show ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t ∩ f ` s"]] using * by auto qed lemma continuous_closed_in_preimage: assumes "continuous_on s f" and "closed t" shows "closedin (subtopology euclidean s) {x ∈ s. f x ∈ t}" proof - have *: "∀x. x ∈ s ∧ f x ∈ t <-> x ∈ s ∧ f x ∈ (t ∩ f ` s)" by auto have "closedin (subtopology euclidean (f ` s)) (t ∩ f ` s)" using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto then show ?thesis using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t ∩ f ` s"]] using * by auto qed lemma continuous_open_preimage: assumes "continuous_on s f" and "open s" and "open t" shows "open {x ∈ s. f x ∈ t}" proof- obtain T where T: "open T" "{x ∈ s. f x ∈ t} = s ∩ T" using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto then show ?thesis using open_Int[of s T, OF assms(2)] by auto qed lemma continuous_closed_preimage: assumes "continuous_on s f" and "closed s" and "closed t" shows "closed {x ∈ s. f x ∈ t}" proof- obtain T where "closed T" "{x ∈ s. f x ∈ t} = s ∩ T" using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto then show ?thesis using closed_Int[of s T, OF assms(2)] by auto qed lemma continuous_open_preimage_univ: "∀x. continuous (at x) f ==> open s ==> open {x. f x ∈ s}" using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto lemma continuous_closed_preimage_univ: "(∀x. continuous (at x) f) ==> closed s ==> closed {x. f x ∈ s}" using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto lemma continuous_open_vimage: "∀x. continuous (at x) f ==> open s ==> open (f -` s)" unfolding vimage_def by (rule continuous_open_preimage_univ) lemma continuous_closed_vimage: "∀x. continuous (at x) f ==> closed s ==> closed (f -` s)" unfolding vimage_def by (rule continuous_closed_preimage_univ) lemma interior_image_subset: assumes "∀x. continuous (at x) f" and "inj f" shows "interior (f ` s) ⊆ f ` (interior s)" proof fix x assume "x ∈ interior (f ` s)" then obtain T where as: "open T" "x ∈ T" "T ⊆ f ` s" .. then have "x ∈ f ` s" by auto then obtain y where y: "y ∈ s" "x = f y" by auto have "open (vimage f T)" using assms(1) `open T` by (rule continuous_open_vimage) moreover have "y ∈ vimage f T" using `x = f y` `x ∈ T` by simp moreover have "vimage f T ⊆ s" using `T ⊆ image f s` `inj f` unfolding inj_on_def subset_eq by auto ultimately have "y ∈ interior s" .. with `x = f y` show "x ∈ f ` interior s" .. qed text {* Equality of continuous functions on closure and related results. *} lemma continuous_closed_in_preimage_constant: fixes f :: "_ => 'b::t1_space" shows "continuous_on s f ==> closedin (subtopology euclidean s) {x ∈ s. f x = a}" using continuous_closed_in_preimage[of s f "{a}"] by auto lemma continuous_closed_preimage_constant: fixes f :: "_ => 'b::t1_space" shows "continuous_on s f ==> closed s ==> closed {x ∈ s. f x = a}" using continuous_closed_preimage[of s f "{a}"] by auto lemma continuous_constant_on_closure: fixes f :: "_ => 'b::t1_space" assumes "continuous_on (closure s) f" and "∀x ∈ s. f x = a" shows "∀x ∈ (closure s). f x = a" using continuous_closed_preimage_constant[of "closure s" f a] assms closure_minimal[of s "{x ∈ closure s. f x = a}"] closure_subset unfolding subset_eq by auto lemma image_closure_subset: assumes "continuous_on (closure s) f" and "closed t" and "(f ` s) ⊆ t" shows "f ` (closure s) ⊆ t" proof - have "s ⊆ {x ∈ closure s. f x ∈ t}" using assms(3) closure_subset by auto moreover have "closed {x ∈ closure s. f x ∈ t}" using continuous_closed_preimage[OF assms(1)] and assms(2) by auto ultimately have "closure s = {x ∈ closure s . f x ∈ t}" using closure_minimal[of s "{x ∈ closure s. f x ∈ t}"] by auto then show ?thesis by auto qed lemma continuous_on_closure_norm_le: fixes f :: "'a::metric_space => 'b::real_normed_vector" assumes "continuous_on (closure s) f" and "∀y ∈ s. norm(f y) ≤ b" and "x ∈ (closure s)" shows "norm (f x) ≤ b" proof - have *: "f ` s ⊆ cball 0 b" using assms(2)[unfolded mem_cball_0[symmetric]] by auto show ?thesis using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3) unfolding subset_eq apply (erule_tac x="f x" in ballE) apply (auto simp add: dist_norm) done qed text {* Making a continuous function avoid some value in a neighbourhood. *} lemma continuous_within_avoid: fixes f :: "'a::metric_space => 'b::t1_space" assumes "continuous (at x within s) f" and "f x ≠ a" shows "∃e>0. ∀y ∈ s. dist x y < e --> f y ≠ a" proof - obtain U where "open U" and "f x ∈ U" and "a ∉ U" using t1_space [OF `f x ≠ a`] by fast have "(f ---> f x) (at x within s)" using assms(1) by (simp add: continuous_within) then have "eventually (λy. f y ∈ U) (at x within s)" using `open U` and `f x ∈ U` unfolding tendsto_def by fast then have "eventually (λy. f y ≠ a) (at x within s)" using `a ∉ U` by (fast elim: eventually_mono [rotated]) then show ?thesis using `f x ≠ a` by (auto simp: dist_commute zero_less_dist_iff eventually_at) qed lemma continuous_at_avoid: fixes f :: "'a::metric_space => 'b::t1_space" assumes "continuous (at x) f" and "f x ≠ a" shows "∃e>0. ∀y. dist x y < e --> f y ≠ a" using assms continuous_within_avoid[of x UNIV f a] by simp lemma continuous_on_avoid: fixes f :: "'a::metric_space => 'b::t1_space" assumes "continuous_on s f" and "x ∈ s" and "f x ≠ a" shows "∃e>0. ∀y ∈ s. dist x y < e --> f y ≠ a" using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x], OF assms(2)] continuous_within_avoid[of x s f a] using assms(3) by auto lemma continuous_on_open_avoid: fixes f :: "'a::metric_space => 'b::t1_space" assumes "continuous_on s f" and "open s" and "x ∈ s" and "f x ≠ a" shows "∃e>0. ∀y. dist x y < e --> f y ≠ a" using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)] using continuous_at_avoid[of x f a] assms(4) by auto text {* Proving a function is constant by proving open-ness of level set. *} lemma continuous_levelset_open_in_cases: fixes f :: "_ => 'b::t1_space" shows "connected s ==> continuous_on s f ==> openin (subtopology euclidean s) {x ∈ s. f x = a} ==> (∀x ∈ s. f x ≠ a) ∨ (∀x ∈ s. f x = a)" unfolding connected_clopen using continuous_closed_in_preimage_constant by auto lemma continuous_levelset_open_in: fixes f :: "_ => 'b::t1_space" shows "connected s ==> continuous_on s f ==> openin (subtopology euclidean s) {x ∈ s. f x = a} ==> (∃x ∈ s. f x = a) ==> (∀x ∈ s. f x = a)" using continuous_levelset_open_in_cases[of s f ] by meson lemma continuous_levelset_open: fixes f :: "_ => 'b::t1_space" assumes "connected s" and "continuous_on s f" and "open {x ∈ s. f x = a}" and "∃x ∈ s. f x = a" shows "∀x ∈ s. f x = a" using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast text {* Some arithmetical combinations (more to prove). *} lemma open_scaling[intro]: fixes s :: "'a::real_normed_vector set" assumes "c ≠ 0" and "open s" shows "open((λx. c *⇩_{R}x) ` s)" proof - { fix x assume "x ∈ s" then obtain e where "e>0" and e:"∀x'. dist x' x < e --> x' ∈ s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]] by auto have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[symmetric]] `e>0` by auto moreover { fix y assume "dist y (c *⇩_{R}x) < e * ¦c¦" then have "norm ((1 / c) *⇩_{R}y - x) < e" unfolding dist_norm using norm_scaleR[of c "(1 / c) *⇩_{R}y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1) assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff) then have "y ∈ op *⇩_{R}c ` s" using rev_image_eqI[of "(1 / c) *⇩_{R}y" s y "op *⇩_{R}c"] using e[THEN spec[where x="(1 / c) *⇩_{R}y"]] using assms(1) unfolding dist_norm scaleR_scaleR by auto } ultimately have "∃e>0. ∀x'. dist x' (c *⇩_{R}x) < e --> x' ∈ op *⇩_{R}c ` s" apply (rule_tac x="e * abs c" in exI) apply auto done } then show ?thesis unfolding open_dist by auto qed lemma minus_image_eq_vimage: fixes A :: "'a::ab_group_add set" shows "(λx. - x) ` A = (λx. - x) -` A" by (auto intro!: image_eqI [where f="λx. - x"]) lemma open_negations: fixes s :: "'a::real_normed_vector set" shows "open s ==> open ((λx. - x) ` s)" using open_scaling [of "- 1" s] by simp lemma open_translation: fixes s :: "'a::real_normed_vector set" assumes "open s" shows "open((λx. a + x) ` s)" proof - { fix x have "continuous (at x) (λx. x - a)" by (intro continuous_diff continuous_at_id continuous_const) } moreover have "{x. x - a ∈ s} = op + a ` s" by force ultimately show ?thesis using continuous_open_preimage_univ[of "λx. x - a" s] using assms by auto qed lemma open_affinity: fixes s :: "'a::real_normed_vector set" assumes "open s" "c ≠ 0" shows "open ((λx. a + c *⇩_{R}x) ` s)" proof - have *: "(λx. a + c *⇩_{R}x) = (λx. a + x) o (λx. c *⇩_{R}x)" unfolding o_def .. have "op + a ` op *⇩_{R}c ` s = (op + a o op *⇩_{R}c) ` s" by auto then show ?thesis using assms open_translation[of "op *⇩_{R}c ` s" a] unfolding * by auto qed lemma interior_translation: fixes s :: "'a::real_normed_vector set" shows "interior ((λx. a + x) ` s) = (λx. a + x) ` (interior s)" proof (rule set_eqI, rule) fix x assume "x ∈ interior (op + a ` s)" then obtain e where "e > 0" and e: "ball x e ⊆ op + a ` s" unfolding mem_interior by auto then have "ball (x - a) e ⊆ s" unfolding subset_eq Ball_def mem_ball dist_norm by (auto simp add: diff_diff_eq) then show "x ∈ op + a ` interior s" unfolding image_iff apply (rule_tac x="x - a" in bexI) unfolding mem_interior using `e > 0` apply auto done next fix x assume "x ∈ op + a ` interior s" then obtain y e where "e > 0" and e: "ball y e ⊆ s" and y: "x = a + y" unfolding image_iff Bex_def mem_interior by auto { fix z have *: "a + y - z = y + a - z" by auto assume "z ∈ ball x e" then have "z - a ∈ s" using e[unfolded subset_eq, THEN bspec[where x="z - a"]] unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 * by auto then have "z ∈ op + a ` s" unfolding image_iff by (auto intro!: bexI[where x="z - a"]) } then have "ball x e ⊆ op + a ` s" unfolding subset_eq by auto then show "x ∈ interior (op + a ` s)" unfolding mem_interior using `e > 0` by auto qed text {* Topological properties of linear functions. *} lemma linear_lim_0: assumes "bounded_linear f" shows "(f ---> 0) (at (0))" proof - interpret f: bounded_linear f by fact have "(f ---> f 0) (at 0)" using tendsto_ident_at by (rule f.tendsto) then show ?thesis unfolding f.zero . qed lemma linear_continuous_at: assumes "bounded_linear f" shows "continuous (at a) f" unfolding continuous_at using assms apply (rule bounded_linear.tendsto) apply (rule tendsto_ident_at) done lemma linear_continuous_within: "bounded_linear f ==> continuous (at x within s) f" using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto lemma linear_continuous_on: "bounded_linear f ==> continuous_on s f" using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto text {* Also bilinear functions, in composition form. *} lemma bilinear_continuous_at_compose: "continuous (at x) f ==> continuous (at x) g ==> bounded_bilinear h ==> continuous (at x) (λx. h (f x) (g x))" unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto lemma bilinear_continuous_within_compose: "continuous (at x within s) f ==> continuous (at x within s) g ==> bounded_bilinear h ==> continuous (at x within s) (λx. h (f x) (g x))" unfolding continuous_within using Lim_bilinear[of f "f x"] by auto lemma bilinear_continuous_on_compose: "continuous_on s f ==> continuous_on s g ==> bounded_bilinear h ==> continuous_on s (λx. h (f x) (g x))" unfolding continuous_on_def by (fast elim: bounded_bilinear.tendsto) text {* Preservation of compactness and connectedness under continuous function. *} lemma compact_eq_openin_cover: "compact S <-> (∀C. (∀c∈C. openin (subtopology euclidean S) c) ∧ S ⊆ \<Union>C --> (∃D⊆C. finite D ∧ S ⊆ \<Union>D))" proof safe fix C assume "compact S" and "∀c∈C. openin (subtopology euclidean S) c" and "S ⊆ \<Union>C" then have "∀c∈{T. open T ∧ S ∩ T ∈ C}. open c" and "S ⊆ \<Union>{T. open T ∧ S ∩ T ∈ C}" unfolding openin_open by force+ with `compact S` obtain D where "D ⊆ {T. open T ∧ S ∩ T ∈ C}" and "finite D" and "S ⊆ \<Union>D" by (rule compactE) then have "image (λT. S ∩ T) D ⊆ C ∧ finite (image (λT. S ∩ T) D) ∧ S ⊆ \<Union>(image (λT. S ∩ T) D)" by auto then show "∃D⊆C. finite D ∧ S ⊆ \<Union>D" .. next assume 1: "∀C. (∀c∈C. openin (subtopology euclidean S) c) ∧ S ⊆ \<Union>C --> (∃D⊆C. finite D ∧ S ⊆ \<Union>D)" show "compact S" proof (rule compactI) fix C let ?C = "image (λT. S ∩ T) C" assume "∀t∈C. open t" and "S ⊆ \<Union>C" then have "(∀c∈?C. openin (subtopology euclidean S) c) ∧ S ⊆ \<Union>?C" unfolding openin_open by auto with 1 obtain D where "D ⊆ ?C" and "finite D" and "S ⊆ \<Union>D" by metis let ?D = "inv_into C (λT. S ∩ T) ` D" have "?D ⊆ C ∧ finite ?D ∧ S ⊆ \<Union>?D" proof (intro conjI) from `D ⊆ ?C` show "?D ⊆ C" by (fast intro: inv_into_into) from `finite D` show "finite ?D" by (rule finite_imageI) from `S ⊆ \<Union>D` show "S ⊆ \<Union>?D" apply (rule subset_trans) apply clarsimp apply (frule subsetD [OF `D ⊆ ?C`, THEN f_inv_into_f]) apply (erule rev_bexI, fast) done qed then show "∃D⊆C. finite D ∧ S ⊆ \<Union>D" .. qed qed lemma connected_continuous_image: assumes "continuous_on s f" and "connected s" shows "connected(f ` s)" proof - { fix T assume as: "T ≠ {}" "T ≠ f ` s" "openin (subtopology euclidean (f ` s)) T" "closedin (subtopology euclidean (f ` s)) T" have "{x ∈ s. f x ∈ T} = {} ∨ {x ∈ s. f x ∈ T} = s" using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]] using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]] using assms(2)[unfolded connected_clopen, THEN spec[where x="{x ∈ s. f x ∈ T}"]] as(3,4) by auto then have False using as(1,2) using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto } then show ?thesis unfolding connected_clopen by auto qed text {* Continuity implies uniform continuity on a compact domain. *} lemma compact_uniformly_continuous: assumes f: "continuous_on s f" and s: "compact s" shows "uniformly_continuous_on s f" unfolding uniformly_continuous_on_def proof (cases, safe) fix e :: real assume "0 < e" "s ≠ {}" def [simp]: R ≡ "{(y, d). y ∈ s ∧ 0 < d ∧ ball y d ∩ s ⊆ {x ∈ s. f x ∈ ball (f y) (e/2) } }" let ?b = "(λ(y, d). ball y (d/2))" have "(∀r∈R. open (?b r))" "s ⊆ (\<Union>r∈R. ?b r)" proof safe fix y assume "y ∈ s" from continuous_open_in_preimage[OF f open_ball] obtain T where "open T" and T: "{x ∈ s. f x ∈ ball (f y) (e/2)} = T ∩ s" unfolding openin_subtopology open_openin by metis then obtain d where "ball y d ⊆ T" "0 < d" using `0 < e` `y ∈ s` by (auto elim!: openE) with T `y ∈ s` show "y ∈ (\<Union>r∈R. ?b r)" by (intro UN_I[of "(y, d)"]) auto qed auto with s obtain D where D: "finite D" "D ⊆ R" "s ⊆ (\<Union>(y, d)∈D. ball y (d/2))" by (rule compactE_image) with `s ≠ {}` have [simp]: "!!x. x < Min (snd ` D) <-> (∀(y, d)∈D. x < d)" by (subst Min_gr_iff) auto show "∃d>0. ∀x∈s. ∀x'∈s. dist x' x < d --> dist (f x') (f x) < e" proof (rule, safe) fix x x' assume in_s: "x' ∈ s" "x ∈ s" with D obtain y d where x: "x ∈ ball y (d/2)" "(y, d) ∈ D" by blast moreover assume "dist x x' < Min (snd`D) / 2" ultimately have "dist y x' < d" by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute) with D x in_s show "dist (f x) (f x') < e" by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq) qed (insert D, auto) qed auto text {* A uniformly convergent limit of continuous functions is continuous. *} lemma continuous_uniform_limit: fixes f :: "'a => 'b::metric_space => 'c::metric_space" assumes "¬ trivial_limit F" and "eventually (λn. continuous_on s (f n)) F" and "∀e>0. eventually (λn. ∀x∈s. dist (f n x) (g x) < e) F" shows "continuous_on s g" proof - { fix x and e :: real assume "x∈s" "e>0" have "eventually (λn. ∀x∈s. dist (f n x) (g x) < e / 3) F" using `e>0` assms(3)[THEN spec[where x="e/3"]] by auto from eventually_happens [OF eventually_conj [OF this assms(2)]] obtain n where n:"∀x∈s. dist (f n x) (g x) < e / 3" "continuous_on s (f n)" using assms(1) by blast have "e / 3 > 0" using `e>0` by auto then obtain d where "d>0" and d:"∀x'∈s. dist x' x < d --> dist (f n x') (f n x) < e / 3" using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x∈s`, THEN spec[where x="e/3"]] by blast { fix y assume "y ∈ s" and "dist y x < d" then have "dist (f n y) (f n x) < e / 3" by (rule d [rule_format]) then have "dist (f n y) (g x) < 2 * e / 3" using dist_triangle [of "f n y" "g x" "f n x"] using n(1)[THEN bspec[where x=x], OF `x∈s`] by auto then have "dist (g y) (g x) < e" using n(1)[THEN bspec[where x=y], OF `y∈s`] using dist_triangle3 [of "g y" "g x" "f n y"] by auto } then have "∃d>0. ∀x'∈s. dist x' x < d --> dist (g x') (g x) < e" using `d>0` by auto } then show ?thesis unfolding continuous_on_iff by auto qed subsection {* Topological stuff lifted from and dropped to R *} lemma open_real: fixes s :: "real set" shows "open s <-> (∀x ∈ s. ∃e>0. ∀x'. abs(x' - x) < e --> x' ∈ s)" unfolding open_dist dist_norm by simp lemma islimpt_approachable_real: fixes s :: "real set" shows "x islimpt s <-> (∀e>0. ∃x'∈ s. x' ≠ x ∧ abs(x' - x) < e)" unfolding islimpt_approachable dist_norm by simp lemma closed_real: fixes s :: "real set" shows "closed s <-> (∀x. (∀e>0. ∃x' ∈ s. x' ≠ x ∧ abs(x' - x) < e) --> x ∈ s)" unfolding closed_limpt islimpt_approachable dist_norm by simp lemma continuous_at_real_range: fixes f :: "'a::real_normed_vector => real" shows "continuous (at x) f <-> (∀e>0. ∃d>0. ∀x'. norm(x' - x) < d --> abs(f x' - f x) < e)" unfolding continuous_at unfolding Lim_at unfolding dist_nz[symmetric] unfolding dist_norm apply auto apply (erule_tac x=e in allE) apply auto apply (rule_tac x=d in exI) apply auto apply (erule_tac x=x' in allE) apply auto apply (erule_tac x=e in allE) apply auto done lemma continuous_on_real_range: fixes f :: "'a::real_normed_vector => real" shows "continuous_on s f <-> (∀x ∈ s. ∀e>0. ∃d>0. (∀x' ∈ s. norm(x' - x) < d --> abs(f x' - f x) < e))" unfolding continuous_on_iff dist_norm by simp text {* Hence some handy theorems on distance, diameter etc. of/from a set. *} lemma distance_attains_sup: assumes "compact s" "s ≠ {}" shows "∃x∈s. ∀y∈s. dist a y ≤ dist a x" proof (rule continuous_attains_sup [OF assms]) { fix x assume "x∈s" have "(dist a ---> dist a x) (at x within s)" by (intro tendsto_dist tendsto_const tendsto_ident_at) } then show "continuous_on s (dist a)" unfolding continuous_on .. qed text {* For \emph{minimal} distance, we only need closure, not compactness. *} lemma distance_attains_inf: fixes a :: "'a::heine_borel" assumes "closed s" and "s ≠ {}" shows "∃x∈s. ∀y∈s. dist a x ≤ dist a y" proof - from assms(2) obtain b where "b ∈ s" by auto let ?B = "s ∩ cball a (dist b a)" have "?B ≠ {}" using `b ∈ s` by (auto simp add: dist_commute) moreover have "continuous_on ?B (dist a)" by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_at_id continuous_const) moreover have "compact ?B" by (intro closed_inter_compact `closed s` compact_cball) ultimately obtain x where "x ∈ ?B" "∀y∈?B. dist a x ≤ dist a y" by (metis continuous_attains_inf) then show ?thesis by fastforce qed subsection {* Pasted sets *} lemma bounded_Times: assumes "bounded s" "bounded t" shows "bounded (s × t)" proof - obtain x y a b where "∀z∈s. dist x z ≤ a" "∀z∈t. dist y z ≤ b" using assms [unfolded bounded_def] by auto then have "∀z∈s × t. dist (x, y) z ≤ sqrt (a⇧^{2}+ b⇧^{2})" by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono) then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto qed lemma mem_Times_iff: "x ∈ A × B <-> fst x ∈ A ∧ snd x ∈ B" by (induct x) simp lemma seq_compact_Times: "seq_compact s ==> seq_compact t ==> seq_compact (s × t)" unfolding seq_compact_def apply clarify apply (drule_tac x="fst o f" in spec) apply (drule mp, simp add: mem_Times_iff) apply (clarify, rename_tac l1 r1) apply (drule_tac x="snd o f o r1" in spec) apply (drule mp, simp add: mem_Times_iff) apply (clarify, rename_tac l2 r2) apply (rule_tac x="(l1, l2)" in rev_bexI, simp) apply (rule_tac x="r1 o r2" in exI) apply (rule conjI, simp add: subseq_def) apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption) apply (drule (1) tendsto_Pair) back apply (simp add: o_def) done lemma compact_Times: assumes "compact s" "compact t" shows "compact (s × t)" proof (rule compactI) fix C assume C: "∀t∈C. open t" "s × t ⊆ \<Union>C" have "∀x∈s. ∃a. open a ∧ x ∈ a ∧ (∃d⊆C. finite d ∧ a × t ⊆ \<Union>d)" proof fix x assume "x ∈ s" have "∀y∈t. ∃a b c. c ∈ C ∧ open a ∧ open b ∧ x ∈ a ∧ y ∈ b ∧ a × b ⊆ c" (is "∀y∈t. ?P y") proof fix y assume "y ∈ t" with `x ∈ s` C obtain c where "c ∈ C" "(x, y) ∈ c" "open c" by auto then show "?P y" by (auto elim!: open_prod_elim) qed then obtain a b c where b: "!!y. y ∈ t ==> open (b y)" and c: "!!y. y ∈ t ==> c y ∈ C ∧ open (a y) ∧ open (b y) ∧ x ∈ a y ∧ y ∈ b y ∧ a y × b y ⊆ c y" by metis then have "∀y∈t. open (b y)" "t ⊆ (\<Union>y∈t. b y)" by auto from compactE_image[OF `compact t` this] obtain D where D: "D ⊆ t" "finite D" "t ⊆ (\<Union>y∈D. b y)" by auto moreover from D c have "(\<Inter>y∈D. a y) × t ⊆ (\<Union>y∈D. c y)" by (fastforce simp: subset_eq) ultimately show "∃a. open a ∧ x ∈ a ∧ (∃d⊆C. finite d ∧ a × t ⊆ \<Union>d)" using c by (intro exI[of _ "c`D"] exI[of _ "\<Inter>(a`D)"] conjI) (auto intro!: open_INT) qed then obtain a d where a: "∀x∈s. open (a x)" "s ⊆ (\<Union>x∈s. a x)" and d: "!!x. x ∈ s ==> d x ⊆ C ∧ finite (d x) ∧ a x × t ⊆ \<Union>d x" unfolding subset_eq UN_iff by metis moreover from compactE_image[OF `compact s` a] obtain e where e: "e ⊆ s" "finite e" and s: "s ⊆ (\<Union>x∈e. a x)" by auto moreover { from s have "s × t ⊆ (\<Union>x∈e. a x × t)" by auto also have "… ⊆ (\<Union>x∈e. \<Union>d x)" using d `e ⊆ s` by (intro UN_mono) auto finally have "s × t ⊆ (\<Union>x∈e. \<Union>d x)" . } ultimately show "∃C'⊆C. finite C' ∧ s × t ⊆ \<Union>C'" by (intro exI[of _ "(\<Union>x∈e. d x)"]) (auto simp add: subset_eq) qed text{* Hence some useful properties follow quite easily. *} lemma compact_scaling: fixes s :: "'a::real_normed_vector set" assumes "compact s" shows "compact ((λx. c *⇩_{R}x) ` s)" proof - let ?f = "λx. scaleR c x" have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right) show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f] using linear_continuous_at[OF *] assms by auto qed lemma compact_negations: fixes s :: "'a::real_normed_vector set" assumes "compact s" shows "compact ((λx. - x) ` s)" using compact_scaling [OF assms, of "- 1"] by auto lemma compact_sums: fixes s t :: "'a::real_normed_vector set" assumes "compact s" and "compact t" shows "compact {x + y | x y. x ∈ s ∧ y ∈ t}" proof - have *: "{x + y | x y. x ∈ s ∧ y ∈ t} = (λz. fst z + snd z) ` (s × t)" apply auto unfolding image_iff apply (rule_tac x="(xa, y)" in bexI) apply auto done have "continuous_on (s × t) (λz. fst z + snd z)" unfolding continuous_on by (rule ballI) (intro tendsto_intros) then show ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto qed lemma compact_differences: fixes s t :: "'a::real_normed_vector set" assumes "compact s" and "compact t" shows "compact {x - y | x y. x ∈ s ∧ y ∈ t}" proof- have "{x - y | x y. x∈s ∧ y ∈ t} = {x + y | x y. x ∈ s ∧ y ∈ (uminus ` t)}" apply auto apply (rule_tac x= xa in exI) apply auto done then show ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto qed lemma compact_translation: fixes s :: "'a::real_normed_vector set" assumes "compact s" shows "compact ((λx. a + x) ` s)" proof - have "{x + y |x y. x ∈ s ∧ y ∈ {a}} = (λx. a + x) ` s" by auto then show ?thesis using compact_sums[OF assms compact_sing[of a]] by auto qed lemma compact_affinity: fixes s :: "'a::real_normed_vector set" assumes "compact s" shows "compact ((λx. a + c *⇩_{R}x) ` s)" proof - have "op + a ` op *⇩_{R}c ` s = (λx. a + c *⇩_{R}x) ` s" by auto then show ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto qed text {* Hence we get the following. *} lemma compact_sup_maxdistance: fixes s :: "'a::metric_space set" assumes "compact s" and "s ≠ {}" shows "∃x∈s. ∃y∈s. ∀u∈s. ∀v∈s. dist u v ≤ dist x y" proof - have "compact (s × s)" using `compact s` by (intro compact_Times) moreover have "s × s ≠ {}" using `s ≠ {}` by auto moreover have "continuous_on (s × s) (λx. dist (fst x) (snd x))" by (intro continuous_at_imp_continuous_on ballI continuous_intros) ultimately show ?thesis using continuous_attains_sup[of "s × s" "λx. dist (fst x) (snd x)"] by auto qed text {* We can state this in terms of diameter of a set. *} definition diameter :: "'a::metric_space set => real" where "diameter S = (if S = {} then 0 else SUP (x,y):S×S. dist x y)" lemma diameter_bounded_bound: fixes s :: "'a :: metric_space set" assumes s: "bounded s" "x ∈ s" "y ∈ s" shows "dist x y ≤ diameter s" proof - from s obtain z d where z: "!!x. x ∈ s ==> dist z x ≤ d" unfolding bounded_def by auto have "bdd_above (split dist ` (s×s))" proof (intro bdd_aboveI, safe) fix a b assume "a ∈ s" "b ∈ s" with z[of a] z[of b] dist_triangle[of a b z] show "dist a b ≤ 2 * d" by (simp add: dist_commute) qed moreover have "(x,y) ∈ s×s" using s by auto ultimately have "dist x y ≤ (SUP (x,y):s×s. dist x y)" by (rule cSUP_upper2) simp with `x ∈ s` show ?thesis by (auto simp add: diameter_def) qed lemma diameter_lower_bounded: fixes s :: "'a :: metric_space set" assumes s: "bounded s" and d: "0 < d" "d < diameter s" shows "∃x∈s. ∃y∈s. d < dist x y" proof (rule ccontr) assume contr: "¬ ?thesis" moreover have "s ≠ {}" using d by (auto simp add: diameter_def) ultimately have "diameter s ≤ d" by (auto simp: not_less diameter_def intro!: cSUP_least) with `d < diameter s` show False by auto qed lemma diameter_bounded: assumes "bounded s" shows "∀x∈s. ∀y∈s. dist x y ≤ diameter s" and "∀d>0. d < diameter s --> (∃x∈s. ∃y∈s. dist x y > d)" using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms by auto lemma diameter_compact_attained: assumes "compact s" and "s ≠ {}" shows "∃x∈s. ∃y∈s. dist x y = diameter s" proof - have b: "bounded s" using assms(1) by (rule compact_imp_bounded) then obtain x y where xys: "x∈s" "y∈s" and xy: "∀u∈s. ∀v∈s. dist u v ≤ dist x y" using compact_sup_maxdistance[OF assms] by auto then have "diameter s ≤ dist x y" unfolding diameter_def apply clarsimp apply (rule cSUP_least) apply fast+ done then show ?thesis by (metis b diameter_bounded_bound order_antisym xys) qed text {* Related results with closure as the conclusion. *} lemma closed_scaling: fixes s :: "'a::real_normed_vector set" assumes "closed s" shows "closed ((λx. c *⇩_{R}x) ` s)" proof (cases "c = 0") case True then show ?thesis by (auto simp add: image_constant_conv) next case False from assms have "closed ((λx. inverse c *⇩_{R}x) -` s)" by (simp add: continuous_closed_vimage) also have "(λx. inverse c *⇩_{R}x) -` s = (λx. c *⇩_{R}x) ` s" using `c ≠ 0` by (auto elim: image_eqI [rotated]) finally show ?thesis . qed lemma closed_negations: fixes s :: "'a::real_normed_vector set" assumes "closed s" shows "closed ((λx. -x) ` s)" using closed_scaling[OF assms, of "- 1"] by simp lemma compact_closed_sums: fixes s :: "'a::real_normed_vector set" assumes "compact s" and "closed t" shows "closed {x + y | x y. x ∈ s ∧ y ∈ t}" proof - let ?S = "{x + y |x y. x ∈ s ∧ y ∈ t}" { fix x l assume as: "∀n. x n ∈ ?S" "(x ---> l) sequentially" from as(1) obtain f where f: "∀n. x n = fst (f n) + snd (f n)" "∀n. fst (f n) ∈ s" "∀n. snd (f n) ∈ t" using choice[of "λn y. x n = (fst y) + (snd y) ∧ fst y ∈ s ∧ snd y ∈ t"] by auto obtain l' r where "l'∈s" and r: "subseq r" and lr: "(((λn. fst (f n)) o r) ---> l') sequentially" using assms(1)[unfolded compact_def, THEN spec[where x="λ n. fst (f n)"]] using f(2) by auto have "((λn. snd (f (r n))) ---> l - l') sequentially" using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1) unfolding o_def by auto then have "l - l' ∈ t" using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="λ n. snd (f (r n))"], THEN spec[where x="l - l'"]] using f(3) by auto then have "l ∈ ?S" using `l' ∈ s` apply auto apply (rule_tac x=l' in exI) apply (rule_tac x="l - l'" in exI) apply auto done } then show ?thesis unfolding closed_sequential_limits by fast qed lemma closed_compact_sums: fixes s t :: "'a::real_normed_vector set" assumes "closed s" and "compact t" shows "closed {x + y | x y. x ∈ s ∧ y ∈ t}" proof - have "{x + y |x y. x ∈ t ∧ y ∈ s} = {x + y |x y. x ∈ s ∧ y ∈ t}" apply auto apply (rule_tac x=y in exI) apply auto apply (rule_tac x=y in exI) apply auto done then show ?thesis using compact_closed_sums[OF assms(2,1)] by simp qed lemma compact_closed_differences: fixes s t :: "'a::real_normed_vector set" assumes "compact s" and "closed t" shows "closed {x - y | x y. x ∈ s ∧ y ∈ t}" proof - have "{x + y |x y. x ∈ s ∧ y ∈ uminus ` t} = {x - y |x y. x ∈ s ∧ y ∈ t}" apply auto apply (rule_tac x=xa in exI) apply auto apply (rule_tac x=xa in exI) apply auto done then show ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto qed lemma closed_compact_differences: fixes s t :: "'a::real_normed_vector set" assumes "closed s" and "compact t" shows "closed {x - y | x y. x ∈ s ∧ y ∈ t}" proof - have "{x + y |x y. x ∈ s ∧ y ∈ uminus ` t} = {x - y |x y. x ∈ s ∧ y ∈ t}" apply auto apply (rule_tac x=xa in exI) apply auto apply (rule_tac x=xa in exI) apply auto done then show ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp qed lemma closed_translation: fixes a :: "'a::real_normed_vector" assumes "closed s" shows "closed ((λx. a + x) ` s)" proof - have "{a + y |y. y ∈ s} = (op + a ` s)" by auto then show ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto qed lemma translation_Compl: fixes a :: "'a::ab_group_add" shows "(λx. a + x) ` (- t) = - ((λx. a + x) ` t)" apply (auto simp add: image_iff) apply (rule_tac x="x - a" in bexI) apply auto done lemma translation_UNIV: fixes a :: "'a::ab_group_add" shows "range (λx. a + x) = UNIV" apply (auto simp add: image_iff) apply (rule_tac x="x - a" in exI) apply auto done lemma translation_diff: fixes a :: "'a::ab_group_add" shows "(λx. a + x) ` (s - t) = ((λx. a + x) ` s) - ((λx. a + x) ` t)" by auto lemma closure_translation: fixes a :: "'a::real_normed_vector" shows "closure ((λx. a + x) ` s) = (λx. a + x) ` (closure s)" proof - have *: "op + a ` (- s) = - op + a ` s" apply auto unfolding image_iff apply (rule_tac x="x - a" in bexI) apply auto done show ?thesis unfolding closure_interior translation_Compl using interior_translation[of a "- s"] unfolding * by auto qed lemma frontier_translation: fixes a :: "'a::real_normed_vector" shows "frontier((λx. a + x) ` s) = (λx. a + x) ` (frontier s)" unfolding frontier_def translation_diff interior_translation closure_translation by auto subsection {* Separation between points and sets *} lemma separate_point_closed: fixes s :: "'a::heine_borel set" assumes "closed s" and "a ∉ s" shows "∃d>0. ∀x∈s. d ≤ dist a x" proof (cases "s = {}") case True then show ?thesis by(auto intro!: exI[where x=1]) next case False from assms obtain x where "x∈s" "∀y∈s. dist a x ≤ dist a y" using `s ≠ {}` distance_attains_inf [of s a] by blast with `x∈s` show ?thesis using dist_pos_lt[of a x] and`a ∉ s` by blast qed lemma separate_compact_closed: fixes s t :: "'a::heine_borel set" assumes "compact s" and t: "closed t" "s ∩ t = {}" shows "∃d>0. ∀x∈s. ∀y∈t. d ≤ dist x y" proof cases assume "s ≠ {} ∧ t ≠ {}" then have "s ≠ {}" "t ≠ {}" by auto let ?inf = "λx. infdist x t" have "continuous_on s ?inf" by (auto intro!: continuous_at_imp_continuous_on continuous_infdist continuous_at_id) then obtain x where x: "x ∈ s" "∀y∈s. ?inf x ≤ ?inf y" using continuous_attains_inf[OF `compact s` `s ≠ {}`] by auto then have "0 < ?inf x" using t `t ≠ {}` in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg) moreover have "∀x'∈s. ∀y∈t. ?inf x ≤ dist x' y" using x by (auto intro: order_trans infdist_le) ultimately show ?thesis by auto qed (auto intro!: exI[of _ 1]) lemma separate_closed_compact: fixes s t :: "'a::heine_borel set" assumes "closed s" and "compact t" and "s ∩ t = {}" shows "∃d>0. ∀x∈s. ∀y∈t. d ≤ dist x y" proof - have *: "t ∩ s = {}" using assms(3) by auto show ?thesis using separate_compact_closed[OF assms(2,1) *] apply auto apply (rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE) apply (auto simp add: dist_commute) done qed subsection {* Closure of halfspaces and hyperplanes *} lemma isCont_open_vimage: assumes "!!x. isCont f x" and "open s" shows "open (f -` s)" proof - from assms(1) have "continuous_on UNIV f" unfolding isCont_def continuous_on_def by simp then have "open {x ∈ UNIV. f x ∈ s}" using open_UNIV `open s` by (rule continuous_open_preimage) then show "open (f -` s)" by (simp add: vimage_def) qed lemma isCont_closed_vimage: assumes "!!x. isCont f x" and "closed s" shows "closed (f -` s)" using assms unfolding closed_def vimage_Compl [symmetric] by (rule isCont_open_vimage) lemma open_Collect_less: fixes f g :: "'a::t2_space => real" assumes f: "!!x. isCont f x" and g: "!!x. isCont g x" shows "open {x. f x < g x}" proof - have "open ((λx. g x - f x) -` {0<..})" using isCont_diff [OF g f] open_real_greaterThan by (rule isCont_open_vimage) also have "((λx. g x - f x) -` {0<..}) = {x. f x < g x}" by auto finally show ?thesis . qed lemma closed_Collect_le: fixes f g :: "'a::t2_space => real" assumes f: "!!x. isCont f x" and g: "!!x. isCont g x" shows "closed {x. f x ≤ g x}" proof - have "closed ((λx. g x - f x) -` {0..})" using isCont_diff [OF g f] closed_real_atLeast by (rule isCont_closed_vimage) also have "((λx. g x - f x) -` {0..}) = {x. f x ≤ g x}" by auto finally show ?thesis . qed lemma closed_Collect_eq: fixes f g :: "'a::t2_space => 'b::t2_space" assumes f: "!!x. isCont f x" and g: "!!x. isCont g x" shows "closed {x. f x = g x}" proof - have "open {(x::'b, y::'b). x ≠ y}" unfolding open_prod_def by (auto dest!: hausdorff) then have "closed {(x::'b, y::'b). x = y}" unfolding closed_def split_def Collect_neg_eq . with isCont_Pair [OF f g] have "closed ((λx. (f x, g x)) -` {(x, y). x = y})" by (rule isCont_closed_vimage) also have "… = {x. f x = g x}" by auto finally show ?thesis . qed lemma continuous_at_inner: "continuous (at x) (inner a)" unfolding continuous_at by (intro tendsto_intros) lemma closed_halfspace_le: "closed {x. inner a x ≤ b}" by (simp add: closed_Collect_le) lemma closed_halfspace_ge: "closed {x. inner a x ≥ b}" by (simp add: closed_Collect_le) lemma closed_hyperplane: "closed {x. inner a x = b}" by (simp add: closed_Collect_eq) lemma closed_halfspace_component_le: "closed {x::'a::euclidean_space. x•i ≤ a}" by (simp add: closed_Collect_le) lemma closed_halfspace_component_ge: "closed {x::'a::euclidean_space. x•i ≥ a}" by (simp add: closed_Collect_le) lemma closed_interval_left: fixes b :: "'a::euclidean_space" shows "closed {x::'a. ∀i∈Basis. x•i ≤ b•i}" by (simp add: Collect_ball_eq closed_INT closed_Collect_le) lemma closed_interval_right: fixes a :: "'a::euclidean_space" shows "closed {x::'a. ∀i∈Basis. a•i ≤ x•i}" by (simp add: Collect_ball_eq closed_INT closed_Collect_le) text {* Openness of halfspaces. *} lemma open_halfspace_lt: "open {x. inner a x < b}" by (simp add: open_Collect_less) lemma open_halfspace_gt: "open {x. inner a x > b}" by (simp add: open_Collect_less) lemma open_halfspace_component_lt: "open {x::'a::euclidean_space. x•i < a}" by (simp add: open_Collect_less) lemma open_halfspace_component_gt: "open {x::'a::euclidean_space. x•i > a}" by (simp add: open_Collect_less) text {* This gives a simple derivation of limit component bounds. *} lemma Lim_component_le: fixes f :: "'a => 'b::euclidean_space" assumes "(f ---> l) net" and "¬ (trivial_limit net)" and "eventually (λx. f(x)•i ≤ b) net" shows "l•i ≤ b" by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)]) lemma Lim_component_ge: fixes f :: "'a => 'b::euclidean_space" assumes "(f ---> l) net" and "¬ (trivial_limit net)" and "eventually (λx. b ≤ (f x)•i) net" shows "b ≤ l•i" by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)]) lemma Lim_component_eq: fixes f :: "'a => 'b::euclidean_space" assumes net: "(f ---> l) net" "¬ trivial_limit net" and ev:"eventually (λx. f(x)•i = b) net" shows "l•i = b" using ev[unfolded order_eq_iff eventually_conj_iff] using Lim_component_ge[OF net, of b i] using Lim_component_le[OF net, of i b] by auto text {* Limits relative to a union. *} lemma eventually_within_Un: "eventually P (at x within (s ∪ t)) <-> eventually P (at x within s) ∧ eventually P (at x within t)" unfolding eventually_at_filter by (auto elim!: eventually_rev_mp) lemma Lim_within_union: "(f ---> l) (at x within (s ∪ t)) <-> (f ---> l) (at x within s) ∧ (f ---> l) (at x within t)" unfolding tendsto_def by (auto simp add: eventually_within_Un) lemma Lim_topological: "(f ---> l) net <-> trivial_limit net ∨ (∀S. open S --> l ∈ S --> eventually (λx. f x ∈ S) net)" unfolding tendsto_def trivial_limit_eq by auto text{* Some more convenient intermediate-value theorem formulations. *} lemma connected_ivt_hyperplane: assumes "connected s" and "x ∈ s" and "y ∈ s" and "inner a x ≤ b" and "b ≤ inner a y" shows "∃z ∈ s. inner a z = b" proof (rule ccontr) assume as:"¬ (∃z∈s. inner a z = b)" let ?A = "{x. inner a x < b}" let ?B = "{x. inner a x > b}" have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto moreover have "?A ∩ ?B = {}" by auto moreover have "s ⊆ ?A ∪ ?B" using as by auto ultimately show False using assms(1)[unfolded connected_def not_ex, THEN spec[where x="?A"], THEN spec[where x="?B"]] using assms(2-5) by auto qed lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows "connected s ==> x ∈ s ==> y ∈ s ==> x•k ≤ a ==> a ≤ y•k ==> (∃z∈s. z•k = a)" using connected_ivt_hyperplane[of s x y "k::'a" a] by (auto simp: inner_commute) subsection {* Intervals *} lemma open_box[intro]: "open (box a b)" proof - have "open (\<Inter>i∈Basis. (op • i) -` {a • i <..< b • i})" by (auto intro!: continuous_open_vimage continuous_inner continuous_at_id continuous_const) also have "(\<Inter>i∈Basis. (op • i) -` {a • i <..< b • i}) = box a b" by (auto simp add: box_def inner_commute) finally show ?thesis . qed instance euclidean_space ⊆ second_countable_topology proof def a ≡ "λf :: 'a => (real × real). ∑i∈Basis. fst (f i) *⇩_{R}i" then have a: "!!f. (∑i∈Basis. fst (f i) *⇩_{R}i) = a f" by simp def b ≡ "λf :: 'a => (real × real). ∑i∈Basis. snd (f i) *⇩_{R}i" then have b: "!!f. (∑i∈Basis. snd (f i) *⇩_{R}i) = b f" by simp def B ≡ "(λf. box (a f) (b f)) ` (Basis ->⇩_{E}(\<rat> × \<rat>))" have "Ball B open" by (simp add: B_def open_box) moreover have "(∀A. open A --> (∃B'⊆B. \<Union>B' = A))" proof safe fix A::"'a set" assume "open A" show "∃B'⊆B. \<Union>B' = A" apply (rule exI[of _ "{b∈B. b ⊆ A}"]) apply (subst (3) open_UNION_box[OF `open A`]) apply (auto simp add: a b B_def) done qed ultimately have "topological_basis B" unfolding topological_basis_def by blast moreover have "countable B" unfolding B_def by (intro countable_image countable_PiE finite_Basis countable_SIGMA countable_rat) ultimately show "∃B::'a set set. countable B ∧ open = generate_topology B" by (blast intro: topological_basis_imp_subbasis) qed instance euclidean_space ⊆ polish_space .. lemma closed_cbox[intro]: fixes a b :: "'a::euclidean_space" shows "closed (cbox a b)" proof - have "closed (\<Inter>i∈Basis. (λx. x•i) -` {a•i .. b•i})" by (intro closed_INT ballI continuous_closed_vimage allI linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left) also have "(\<Inter>i∈Basis. (λx. x•i) -` {a•i .. b•i}) = cbox a b" by (auto simp add: cbox_def) finally show "closed (cbox a b)" . qed lemma interior_cbox [intro]: fixes a b :: "'a::euclidean_space" shows "interior (cbox a b) = box a b" (is "?L = ?R") proof(rule subset_antisym) show "?R ⊆ ?L" using box_subset_cbox open_box by (rule interior_maximal) { fix x assume "x ∈ interior (cbox a b)" then obtain s where s: "open s" "x ∈ s" "s ⊆ cbox a b" .. then obtain e where "e>0" and e:"∀x'. dist x' x < e --> x' ∈ cbox a b" unfolding open_dist and subset_eq by auto { fix i :: 'a assume i: "i ∈ Basis" have "dist (x - (e / 2) *⇩_{R}i) x < e" and "dist (x + (e / 2) *⇩_{R}i) x < e" unfolding dist_norm apply auto unfolding norm_minus_cancel using norm_Basis[OF i] `e>0` apply auto done then have "a • i ≤ (x - (e / 2) *⇩_{R}i) • i" and "(x + (e / 2) *⇩_{R}i) • i ≤ b • i" using e[THEN spec[where x="x - (e/2) *⇩_{R}i"]] and e[THEN spec[where x="x + (e/2) *⇩_{R}i"]] unfolding mem_box using i by blast+ then have "a • i < x • i" and "x • i < b • i" using `e>0` i by (auto simp: inner_diff_left inner_Basis inner_add_left) } then have "x ∈ box a b" unfolding mem_box by auto } then show "?L ⊆ ?R" .. qed lemma bounded_cbox: fixes a :: "'a::euclidean_space" shows "bounded (cbox a b)" proof - let ?b = "∑i∈Basis. ¦a•i¦ + ¦b•i¦" { fix x :: "'a" assume x: "∀i∈Basis. a • i ≤ x • i ∧ x • i ≤ b • i" { fix i :: 'a assume "i ∈ Basis" then have "¦x•i¦ ≤ ¦a•i¦ + ¦b•i¦" using x[THEN bspec[where x=i]] by auto } then have "(∑i∈Basis. ¦x • i¦) ≤ ?b" apply - apply (rule setsum_mono) apply auto done then have "norm x ≤ ?b" using norm_le_l1[of x] by auto } then show ?thesis unfolding cbox_def bounded_iff by auto qed lemma bounded_box: fixes a :: "'a::euclidean_space" shows "bounded (box a b)" using bounded_cbox[of a b] using box_subset_cbox[of a b] using bounded_subset[of "cbox a b" "box a b"] by simp lemma not_interval_univ: fixes a :: "'a::euclidean_space" shows "cbox a b ≠ UNIV" "box a b ≠ UNIV" using bounded_box[of a b] bounded_cbox[of a b] by auto lemma compact_cbox: fixes a :: "'a::euclidean_space" shows "compact (cbox a b)" using bounded_closed_imp_seq_compact[of "cbox a b"] using bounded_cbox[of a b] by (auto simp: compact_eq_seq_compact_metric) lemma box_midpoint: fixes a :: "'a::euclidean_space" assumes "box a b ≠ {}" shows "((1/2) *⇩_{R}(a + b)) ∈ box a b" proof - { fix i :: 'a assume "i ∈ Basis" then have "a • i < ((1 / 2) *⇩_{R}(a + b)) • i ∧ ((1 / 2) *⇩_{R}(a + b)) • i < b • i" using assms[unfolded box_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left) } then show ?thesis unfolding mem_box by auto qed lemma open_cbox_convex: fixes x :: "'a::euclidean_space" assumes x: "x ∈ box a b" and y: "y ∈ cbox a b" and e: "0 < e" "e ≤ 1" shows "(e *⇩_{R}x + (1 - e) *⇩_{R}y) ∈ box a b" proof - { fix i :: 'a assume i: "i ∈ Basis" have "a • i = e * (a • i) + (1 - e) * (a • i)" unfolding left_diff_distrib by simp also have "… < e * (x • i) + (1 - e) * (y • i)" apply (rule add_less_le_mono) using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all using x unfolding mem_box using i apply simp using y unfolding mem_box using i apply simp done finally have "a • i < (e *⇩_{R}x + (1 - e) *⇩_{R}y) • i" unfolding inner_simps by auto moreover { have "b • i = e * (b•i) + (1 - e) * (b•i)" unfolding left_diff_distrib by simp also have "… > e * (x • i) + (1 - e) * (y • i)" apply (rule add_less_le_mono) using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all using x unfolding mem_box using i apply simp using y unfolding mem_box using i apply simp done finally have "(e *⇩_{R}x + (1 - e) *⇩_{R}y) • i < b • i" unfolding inner_simps by auto } ultimately have "a • i < (e *⇩_{R}x + (1 - e) *⇩_{R}y) • i ∧ (e *⇩_{R}x + (1 - e) *⇩_{R}y) • i < b • i" by auto } then show ?thesis unfolding mem_box by auto qed lemma closure_box: fixes a :: "'a::euclidean_space" assumes "box a b ≠ {}" shows "closure (box a b) = cbox a b" proof - have ab: "a <e b" using assms by (simp add: eucl_less_def box_ne_empty) let ?c = "(1 / 2) *⇩_{R}(a + b)" { fix x assume as:"x ∈ cbox a b" def f ≡ "λn::nat. x + (inverse (real n + 1)) *⇩_{R}(?c - x)" { fix n assume fn: "f n <e b --> a <e f n --> f n = x" and xc: "x ≠ ?c" have *: "0 < inverse (real n + 1)" "inverse (real n + 1) ≤ 1" unfolding inverse_le_1_iff by auto have "(inverse (real n + 1)) *⇩_{R}((1 / 2) *⇩_{R}(a + b)) + (1 - inverse (real n + 1)) *⇩_{R}x = x + (inverse (real n + 1)) *⇩_{R}(((1 / 2) *⇩_{R}(a + b)) - x)" by (auto simp add: algebra_simps) then have "f n <e b" and "a <e f n" using open_cbox_convex[OF box_midpoint[OF assms] as *] unfolding f_def by (auto simp: box_def eucl_less_def) then have False using fn unfolding f_def using xc by auto } moreover { assume "¬ (f ---> x) sequentially" { fix e :: real assume "e > 0" then have "∃N::nat. inverse (real (N + 1)) < e" using real_arch_inv[of e] apply (auto simp add: Suc_pred') apply (rule_tac x="n - 1" in exI) apply auto done then obtain N :: nat where "inverse (real (N + 1)) < e" by auto then have "∀n≥N. inverse (real n + 1) < e" apply auto apply (metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero) done then have "∃N::nat. ∀n≥N. inverse (real n + 1) < e" by auto } then have "((λn. inverse (real n + 1)) ---> 0) sequentially" unfolding lim_sequentially by(auto simp add: dist_norm) then have "(f ---> x) sequentially" unfolding f_def using tendsto_add[OF tendsto_const, of "λn::nat. (inverse (real n + 1)) *⇩_{R}((1 / 2) *⇩_{R}(a + b) - x)" 0 sequentially x] using tendsto_scaleR [OF _ tendsto_const, of "λn::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *⇩_{R}(a + b) - x)"] by auto } ultimately have "x ∈ closure (box a b)" using as and box_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by (cases "x=?c") (auto simp: in_box_eucl_less) } then show ?thesis using closure_minimal[OF box_subset_cbox, of a b] by blast qed lemma bounded_subset_box_symmetric: fixes s::"('a::euclidean_space) set" assumes "bounded s" shows "∃a. s ⊆ box (-a) a" proof - obtain b where "b>0" and b: "∀x∈s. norm x ≤ b" using assms[unfolded bounded_pos] by auto def a ≡ "(∑i∈Basis. (b + 1) *⇩_{R}i)::'a" { fix x assume "x ∈ s" fix i :: 'a assume i: "i ∈ Basis" then have "(-a)•i < x•i" and "x•i < a•i" using b[THEN bspec[where x=x], OF `x∈s`] using Basis_le_norm[OF i, of x] unfolding inner_simps and a_def by auto } then show ?thesis by (auto intro: exI[where x=a] simp add: box_def) qed lemma bounded_subset_open_interval: fixes s :: "('a::euclidean_space) set" shows "bounded s ==> (∃a b. s ⊆ box a b)" by (auto dest!: bounded_subset_box_symmetric) lemma bounded_subset_cbox_symmetric: fixes s :: "('a::euclidean_space) set" assumes "bounded s" shows "∃a. s ⊆ cbox (-a) a" proof - obtain a where "s ⊆ box (-a) a" using bounded_subset_box_symmetric[OF assms] by auto then show ?thesis using box_subset_cbox[of "-a" a] by auto qed lemma bounded_subset_cbox: fixes s :: "('a::euclidean_space) set" shows "bounded s ==> ∃a b. s ⊆ cbox a b" using bounded_subset_cbox_symmetric[of s] by auto lemma frontier_cbox: fixes a b :: "'a::euclidean_space" shows "frontier (cbox a b) = cbox a b - box a b" unfolding frontier_def unfolding interior_cbox and closure_closed[OF closed_cbox] .. lemma frontier_box: fixes a b :: "'a::euclidean_space" shows "frontier (box a b) = (if box a b = {} then {} else cbox a b - box a b)" proof (cases "box a b = {}") case True then show ?thesis using frontier_empty by auto next case False then show ?thesis unfolding frontier_def and closure_box[OF False] and interior_open[OF open_box] by auto qed lemma inter_interval_mixed_eq_empty: fixes a :: "'a::euclidean_space" assumes "box c d ≠ {}" shows "box a b ∩ cbox c d = {} <-> box a b ∩ box c d = {}" unfolding closure_box[OF assms, symmetric] unfolding open_inter_closure_eq_empty[OF open_box] .. lemma diameter_cbox: fixes a b::"'a::euclidean_space" shows "(∀i ∈ Basis. a • i ≤ b • i) ==> diameter (cbox a b) = dist a b" by (force simp add: diameter_def SUP_def simp del: Sup_image_eq intro!: cSup_eq_maximum setL2_mono simp: euclidean_dist_l2[where 'a='a] cbox_def dist_norm) lemma eucl_less_eq_halfspaces: fixes a :: "'a::euclidean_space" shows "{x. x <e a} = (\<Inter>i∈Basis. {x. x • i < a • i})" "{x. a <e x} = (\<Inter>i∈Basis. {x. a • i < x • i})" by (auto simp: eucl_less_def) lemma eucl_le_eq_halfspaces: fixes a :: "'a::euclidean_space" shows "{x. ∀i∈Basis. x • i ≤ a • i} = (\<Inter>i∈Basis. {x. x • i ≤ a • i})" "{x. ∀i∈Basis. a • i ≤ x • i} = (\<Inter>i∈Basis. {x. a • i ≤ x • i})" by auto lemma open_Collect_eucl_less[simp, intro]: fixes a :: "'a::euclidean_space" shows "open {x. x <e a}" "open {x. a <e x}" by (auto simp: eucl_less_eq_halfspaces open_halfspace_component_lt open_halfspace_component_gt) lemma closed_Collect_eucl_le[simp, intro]: fixes a :: "'a::euclidean_space" shows "closed {x. ∀i∈Basis. a • i ≤ x • i}" "closed {x. ∀i∈Basis. x • i ≤ a • i}" unfolding eucl_le_eq_halfspaces by (simp_all add: closed_INT closed_Collect_le) lemma image_affinity_cbox: fixes m::real fixes a b c :: "'a::euclidean_space" shows "(λx. m *⇩_{R}x + c) ` cbox a b = (if cbox a b = {} then {} else (if 0 ≤ m then cbox (m *⇩_{R}a + c) (m *⇩_{R}b + c) else cbox (m *⇩_{R}b + c) (m *⇩_{R}a + c)))" proof (cases "m = 0") case True { fix x assume "∀i∈Basis. x • i ≤ c • i" "∀i∈Basis. c • i ≤ x • i" then have "x = c" apply - apply (subst euclidean_eq_iff) apply (auto intro: order_antisym) done } moreover have "c ∈ cbox (m *⇩_{R}a + c) (m *⇩_{R}b + c)" unfolding True by (auto simp add: cbox_sing) ultimately show ?thesis using True by (auto simp: cbox_def) next case False { fix y assume "∀i∈Basis. a • i ≤ y • i" "∀i∈Basis. y • i ≤ b • i" "m > 0" then have "∀i∈Basis. (m *⇩_{R}a + c) • i ≤ (m *⇩_{R}y + c) • i" and "∀i∈Basis. (m *⇩_{R}y + c) • i ≤ (m *⇩_{R}b + c) • i" by (auto simp: inner_distrib) } moreover { fix y assume "∀i∈Basis. a • i ≤ y • i" "∀i∈Basis. y • i ≤ b • i" "m < 0" then have "∀i∈Basis. (m *⇩_{R}b + c) • i ≤ (m *⇩_{R}y + c) • i" and "∀i∈Basis. (m *⇩_{R}y + c) • i ≤ (m *⇩_{R}a + c) • i" by (auto simp add: mult_left_mono_neg inner_distrib) } moreover { fix y assume "m > 0" and "∀i∈Basis. (m *⇩_{R}a + c) • i ≤ y • i" and "∀i∈Basis. y • i ≤ (m *⇩_{R}b + c) • i" then have "y ∈ (λx. m *⇩_{R}x + c) ` cbox a b" unfolding image_iff Bex_def mem_box apply (intro exI[where x="(1 / m) *⇩_{R}(y - c)"]) apply (auto simp add: pos_le_divide_eq pos_divide_le_eq mult.commute diff_le_iff inner_distrib inner_diff_left) done } moreover { fix y assume "∀i∈Basis. (m *⇩_{R}b + c) • i ≤ y • i" "∀i∈Basis. y • i ≤ (m *⇩_{R}a + c) • i" "m < 0" then have "y ∈ (λx. m *⇩_{R}x + c) ` cbox a b" unfolding image_iff Bex_def mem_box apply (intro exI[where x="(1 / m) *⇩_{R}(y - c)"]) apply (auto simp add: neg_le_divide_eq neg_divide_le_eq mult.commute diff_le_iff inner_distrib inner_diff_left) done } ultimately show ?thesis using False by (auto simp: cbox_def) qed lemma image_smult_cbox:"(λx. m *⇩_{R}(x::_::euclidean_space)) ` cbox a b = (if cbox a b = {} then {} else if 0 ≤ m then cbox (m *⇩_{R}a) (m *⇩_{R}b) else cbox (m *⇩_{R}b) (m *⇩_{R}a))" using image_affinity_cbox[of m 0 a b] by auto subsection {* Homeomorphisms *} definition "homeomorphism s t f g <-> (∀x∈s. (g(f x) = x)) ∧ (f ` s = t) ∧ continuous_on s f ∧ (∀y∈t. (f(g y) = y)) ∧ (g ` t = s) ∧ continuous_on t g" definition homeomorphic :: "'a::topological_space set => 'b::topological_space set => bool" (infixr "homeomorphic" 60) where "s homeomorphic t ≡ (∃f g. homeomorphism s t f g)" lemma homeomorphic_refl: "s homeomorphic s" unfolding homeomorphic_def unfolding homeomorphism_def using continuous_on_id apply (rule_tac x = "(λx. x)" in exI) apply (rule_tac x = "(λx. x)" in exI) apply blast done lemma homeomorphic_sym: "s homeomorphic t <-> t homeomorphic s" unfolding homeomorphic_def unfolding homeomorphism_def by blast lemma homeomorphic_trans: assumes "s homeomorphic t" and "t homeomorphic u" shows "s homeomorphic u" proof - obtain f1 g1 where fg1: "∀x∈s. g1 (f1 x) = x" "f1 ` s = t" "continuous_on s f1" "∀y∈t. f1 (g1 y) = y" "g1 ` t = s" "continuous_on t g1" using assms(1) unfolding homeomorphic_def homeomorphism_def by auto obtain f2 g2 where fg2: "∀x∈t. g2 (f2 x) = x" "f2 ` t = u" "continuous_on t f2" "∀y∈u. f2 (g2 y) = y" "g2 ` u = t" "continuous_on u g2" using assms(2) unfolding homeomorphic_def homeomorphism_def by auto { fix x assume "x∈s" then have "(g1 o g2) ((f2 o f1) x) = x" using fg1(1)[THEN bspec[where x=x]] and fg2(1)[THEN bspec[where x="f1 x"]] and fg1(2) by auto } moreover have "(f2 o f1) ` s = u" using fg1(2) fg2(2) by auto moreover have "continuous_on s (f2 o f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto moreover { fix y assume "y∈u" then have "(f2 o f1) ((g1 o g2) y) = y" using fg2(4)[THEN bspec[where x=y]] and fg1(4)[THEN bspec[where x="g2 y"]] and fg2(5) by auto } moreover have "(g1 o g2) ` u = s" using fg1(5) fg2(5) by auto moreover have "continuous_on u (g1 o g2)" using continuous_on_compose[OF fg2(6)] and fg1(6) unfolding fg2(5) by auto ultimately show ?thesis unfolding homeomorphic_def homeomorphism_def apply (rule_tac x="f2 o f1" in exI) apply (rule_tac x="g1 o g2" in exI) apply auto done qed lemma homeomorphic_minimal: "s homeomorphic t <-> (∃f g. (∀x∈s. f(x) ∈ t ∧ (g(f(x)) = x)) ∧ (∀y∈t. g(y) ∈ s ∧ (f(g(y)) = y)) ∧ continuous_on s f ∧ continuous_on t g)" unfolding homeomorphic_def homeomorphism_def apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto unfolding image_iff apply (erule_tac x="g x" in ballE) apply (erule_tac x="x" in ballE) apply auto apply (rule_tac x="g x" in bexI) apply auto apply (erule_tac x="f x" in ballE) apply (erule_tac x="x" in ballE) apply auto apply (rule_tac x="f x" in bexI) apply auto done text {* Relatively weak hypotheses if a set is compact. *} lemma homeomorphism_compact: fixes f :: "'a::topological_space => 'b::t2_space" assumes "compact s" "continuous_on s f" "f ` s = t" "inj_on f s" shows "∃g. homeomorphism s t f g" proof - def g ≡ "λx. SOME y. y∈s ∧ f y = x" have g: "∀x∈s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto { fix y assume "y ∈ t" then obtain x where x:"f x = y" "x∈s" using assms(3) by auto then have "g (f x) = x" using g by auto then have "f (g y) = y" unfolding x(1)[symmetric] by auto } then have g':"∀x∈t. f (g x) = x" by auto moreover { fix x have "x∈s ==> x ∈ g ` t" using g[THEN bspec[where x=x]] unfolding image_iff using assms(3) by (auto intro!: bexI[where x="f x"]) moreover { assume "x∈g ` t" then obtain y where y:"y∈t" "g y = x" by auto then obtain x' where x':"x'∈s" "f x' = y" using assms(3) by auto then have "x ∈ s" unfolding g_def using someI2[of "λb. b∈s ∧ f b = y" x' "λx. x∈s"] unfolding y(2)[symmetric] and g_def by auto } ultimately have "x∈s <-> x ∈ g ` t" .. } then have "g ` t = s" by auto ultimately show ?thesis unfolding homeomorphism_def homeomorphic_def apply (rule_tac x=g in exI) using g and assms(3) and continuous_on_inv[OF assms(2,1), of g, unfolded assms(3)] and assms(2) apply auto done qed lemma homeomorphic_compact: fixes f :: "'a::topological_space => 'b::t2_space" shows "compact s ==> continuous_on s f ==> (f ` s = t) ==> inj_on f s ==> s homeomorphic t" unfolding homeomorphic_def by (metis homeomorphism_compact) text{* Preservation of topological properties. *} lemma homeomorphic_compactness: "s homeomorphic t ==> (compact s <-> compact t)" unfolding homeomorphic_def homeomorphism_def by (metis compact_continuous_image) text{* Results on translation, scaling etc. *} lemma homeomorphic_scaling: fixes s :: "'a::real_normed_vector set" assumes "c ≠ 0" shows "s homeomorphic ((λx. c *⇩_{R}x) ` s)" unfolding homeomorphic_minimal apply (rule_tac x="λx. c *⇩_{R}x" in exI) apply (rule_tac x="λx. (1 / c) *⇩_{R}x" in exI) using assms apply (auto simp add: continuous_intros) done lemma homeomorphic_translation: fixes s :: "'a::real_normed_vector set" shows "s homeomorphic ((λx. a + x) ` s)" unfolding homeomorphic_minimal apply (rule_tac x="λx. a + x" in exI) apply (rule_tac x="λx. -a + x" in exI) using continuous_on_add [OF continuous_on_const continuous_on_id, of s a] continuous_on_add [OF continuous_on_const continuous_on_id, of "plus a ` s" "- a"] apply auto done lemma homeomorphic_affinity: fixes s :: "'a::real_normed_vector set" assumes "c ≠ 0" shows "s homeomorphic ((λx. a + c *⇩_{R}x) ` s)" proof - have *: "op + a ` op *⇩_{R}c ` s = (λx. a + c *⇩_{R}x) ` s" by auto show ?thesis using homeomorphic_trans using homeomorphic_scaling[OF assms, of s] using homeomorphic_translation[of "(λx. c *⇩_{R}x) ` s" a] unfolding * by auto qed lemma homeomorphic_balls: fixes a b ::"'a::real_normed_vector" assumes "0 < d" "0 < e" shows "(ball a d) homeomorphic (ball b e)" (is ?th) and "(cball a d) homeomorphic (cball b e)" (is ?cth) proof - show ?th unfolding homeomorphic_minimal apply(rule_tac x="λx. b + (e/d) *⇩_{R}(x - a)" in exI) apply(rule_tac x="λx. a + (d/e) *⇩_{R}(x - b)" in exI) using assms apply (auto intro!: continuous_intros simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono) done show ?cth unfolding homeomorphic_minimal apply(rule_tac x="λx. b + (e/d) *⇩_{R}(x - a)" in exI) apply(rule_tac x="λx. a + (d/e) *⇩_{R}(x - b)" in exI) using assms apply (auto intro!: continuous_intros simp: dist_commute dist_norm pos_divide_le_eq mult_strict_left_mono) done qed text{* "Isometry" (up to constant bounds) of injective linear map etc. *} lemma cauchy_isometric: assumes e: "e > 0" and s: "subspace s" and f: "bounded_linear f" and normf: "∀x∈s. norm (f x) ≥ e * norm x" and xs: "∀n. x n ∈ s" and cf: "Cauchy (f o x)" shows "Cauchy x" proof - interpret f: bounded_linear f by fact { fix d :: real assume "d > 0" then obtain N where N:"∀n≥N. norm (f (x n) - f (x N)) < e * d" using cf[unfolded cauchy o_def dist_norm, THEN spec[where x="e*d"]] e by auto { fix n assume "n≥N" have "e * norm (x n - x N) ≤ norm (f (x n - x N))" using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]] using normf[THEN bspec[where x="x n - x N"]] by auto also have "norm (f (x n - x N)) < e * d" using `N ≤ n` N unfolding f.diff[symmetric] by auto finally have "norm (x n - x N) < d" using `e>0` by simp } then have "∃N. ∀n≥N. norm (x n - x N) < d" by auto } then show ?thesis unfolding cauchy and dist_norm by auto qed lemma complete_isometric_image: assumes "0 < e" and s: "subspace s" and f: "bounded_linear f" and normf: "∀x∈s. norm(f x) ≥ e * norm(x)" and cs: "complete s" shows "complete (f ` s)" proof - { fix g assume as:"∀n::nat. g n ∈ f ` s" and cfg:"Cauchy g" then obtain x where "∀n. x n ∈ s ∧ g n = f (x n)" using choice[of "λ n xa. xa ∈ s ∧ g n = f xa"] by auto then have x:"∀n. x n ∈ s" "∀n. g n = f (x n)" by auto then have "f o x = g" unfolding fun_eq_iff by auto then obtain l where "l∈s" and l:"(x ---> l) sequentially" using cs[unfolded complete_def, THEN spec[where x="x"]] using cauchy_isometric[OF `0 < e` s f normf] and cfg and x(1) by auto then have "∃l∈f ` s. (g ---> l) sequentially" using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l] unfolding `f o x = g` by auto } then show ?thesis unfolding complete_def by auto qed lemma injective_imp_isometric: fixes f :: "'a::euclidean_space => 'b::euclidean_space" assumes s: "closed s" "subspace s" and f: "bounded_linear f" "∀x∈s. f x = 0 --> x = 0" shows "∃e>0. ∀x∈s. norm (f x) ≥ e * norm x" proof (cases "s ⊆ {0::'a}") case True { fix x assume "x ∈ s" then have "x = 0" using True by auto then have "norm x ≤ norm (f x)" by auto } then show ?thesis by (auto intro!: exI[where x=1]) next interpret f: bounded_linear f by fact case False then obtain a where a: "a ≠ 0" "a ∈ s" by auto from False have "s ≠ {}" by auto let ?S = "{f x| x. (x ∈ s ∧ norm x = norm a)}" let ?S' = "{x::'a. x∈s ∧ norm x = norm a}" let ?S'' = "{x::'a. norm x = norm a}" have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by auto then have "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto moreover have "?S' = s ∩ ?S''" by auto ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto moreover have *:"f ` ?S' = ?S" by auto ultimately have "compact ?S" using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto then have "closed ?S" using compact_imp_closed by auto moreover have "?S ≠ {}" using a by auto ultimately obtain b' where "b'∈?S" "∀y∈?S. norm b' ≤ norm y" using distance_attains_inf[of ?S 0] unfolding dist_0_norm by auto then obtain b where "b∈s" and ba: "norm b = norm a" and b: "∀x∈{x ∈ s. norm x = norm a}. norm (f b) ≤ norm (f x)" unfolding *[symmetric] unfolding image_iff by auto let ?e = "norm (f b) / norm b" have "norm b > 0" using ba and a and norm_ge_zero by auto moreover have "norm (f b) > 0" using f(2)[THEN bspec[where x=b], OF `b∈s`] using `norm b >0` unfolding zero_less_norm_iff by auto ultimately have "0 < norm (f b) / norm b" by simp moreover { fix x assume "x∈s" then have "norm (f b) / norm b * norm x ≤ norm (f x)" proof (cases "x=0") case True then show "norm (f b) / norm b * norm x ≤ norm (f x)" by auto next case False then have *: "0 < norm a / norm x" using `a≠0` unfolding zero_less_norm_iff[symmetric] by simp have "∀c. ∀x∈s. c *⇩_{R}x ∈ s" using s[unfolded subspace_def] by auto then have "(norm a / norm x) *⇩_{R}x ∈ {x ∈ s. norm x = norm a}" using `x∈s` and `x≠0` by auto then show "norm (f b) / norm b * norm x ≤ norm (f x)" using b[THEN bspec[where x="(norm a / norm x) *⇩_{R}x"]] unfolding f.scaleR and ba using `x≠0` `a≠0` by (auto simp add: mult.commute pos_le_divide_eq pos_divide_le_eq) qed } ultimately show ?thesis by auto qed lemma closed_injective_image_subspace: fixes f :: "'a::euclidean_space => 'b::euclidean_space" assumes "subspace s" "bounded_linear f" "∀x∈s. f x = 0 --> x = 0" "closed s" shows "closed(f ` s)" proof - obtain e where "e > 0" and e: "∀x∈s. e * norm x ≤ norm (f x)" using injective_imp_isometric[OF assms(4,1,2,3)] by auto show ?thesis using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4) unfolding complete_eq_closed[symmetric] by auto qed subsection {* Some properties of a canonical subspace *} lemma subspace_substandard: "subspace {x::'a::euclidean_space. (∀i∈Basis. P i --> x•i = 0)}" unfolding subspace_def by (auto simp: inner_add_left) lemma closed_substandard: "closed {x::'a::euclidean_space. ∀i∈Basis. P i --> x•i = 0}" (is "closed ?A") proof - let ?D = "{i∈Basis. P i}" have "closed (\<Inter>i∈?D. {x::'a. x•i = 0})" by (simp add: closed_INT closed_Collect_eq) also have "(\<Inter>i∈?D. {x::'a. x•i = 0}) = ?A" by auto finally show "closed ?A" . qed lemma dim_substandard: assumes d: "d ⊆ Basis" shows "dim {x::'a::euclidean_space. ∀i∈Basis. i ∉ d --> x•i = 0} = card d" (is "dim ?A = _") proof (rule dim_unique) show "d ⊆ ?A" using d by (auto simp: inner_Basis) show "independent d" using independent_mono [OF independent_Basis d] . show "?A ⊆ span d" proof (clarify) fix x assume x: "∀i∈Basis. i ∉ d --> x • i = 0" have "finite d" using finite_subset [OF d finite_Basis] . then have "(∑i∈d. (x • i) *⇩_{R}i) ∈ span d" by (simp add: span_setsum span_clauses) also have "(∑i∈d. (x • i) *⇩_{R}i) = (∑i∈Basis. (x • i) *⇩_{R}i)" by (rule setsum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp add: x) finally show "x ∈ span d" unfolding euclidean_representation . qed qed simp text{* Hence closure and completeness of all subspaces. *} lemma ex_card: assumes "n ≤ card A" shows "∃S⊆A. card S = n" proof cases assume "finite A" from ex_bij_betw_nat_finite[OF this] obtain f where f: "bij_betw f {0..<card A} A" .. moreover from f `n ≤ card A` have "{..< n} ⊆ {..< card A}" "inj_on f {..< n}" by (auto simp: bij_betw_def intro: subset_inj_on) ultimately have "f ` {..< n} ⊆ A" "card (f ` {..< n}) = n" by (auto simp: bij_betw_def card_image) then show ?thesis by blast next assume "¬ finite A" with `n ≤ card A` show ?thesis by force qed lemma closed_subspace: fixes s :: "'a::euclidean_space set" assumes "subspace s" shows "closed s" proof - have "dim s ≤ card (Basis :: 'a set)" using dim_subset_UNIV by auto with ex_card[OF this] obtain d :: "'a set" where t: "card d = dim s" and d: "d ⊆ Basis" by auto let ?t = "{x::'a. ∀i∈Basis. i ∉ d --> x•i = 0}" have "∃f. linear f ∧ f ` {x::'a. ∀i∈Basis. i ∉ d --> x • i = 0} = s ∧ inj_on f {x::'a. ∀i∈Basis. i ∉ d --> x • i = 0}" using dim_substandard[of d] t d assms by (intro subspace_isomorphism[OF subspace_substandard[of "λi. i ∉ d"]]) (auto simp: inner_Basis) then obtain f where f: "linear f" "f ` {x. ∀i∈Basis. i ∉ d --> x • i = 0} = s" "inj_on f {x. ∀i∈Basis. i ∉ d --> x • i = 0}" by blast interpret f: bounded_linear f using f unfolding linear_conv_bounded_linear by auto { fix x have "x∈?t ==> f x = 0 ==> x = 0" using f.zero d f(3)[THEN inj_onD, of x 0] by auto } moreover have "closed ?t" using closed_substandard . moreover have "subspace ?t" using subspace_substandard . ultimately show ?thesis using closed_injective_image_subspace[of ?t f] unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto qed lemma complete_subspace: fixes s :: "('a::euclidean_space) set" shows "subspace s ==> complete s" using complete_eq_closed closed_subspace by auto lemma dim_closure: fixes s :: "('a::euclidean_space) set" shows "dim(closure s) = dim s" (is "?dc = ?d") proof - have "?dc ≤ ?d" using closure_minimal[OF span_inc, of s] using closed_subspace[OF subspace_span, of s] using dim_subset[of "closure s" "span s"] unfolding dim_span by auto then show ?thesis using dim_subset[OF closure_subset, of s] by auto qed subsection {* Affine transformations of intervals *} lemma real_affinity_le: "0 < (m::'a::linordered_field) ==> (m * x + c ≤ y <-> x ≤ inverse(m) * y + -(c / m))" by (simp add: field_simps) lemma real_le_affinity: "0 < (m::'a::linordered_field) ==> (y ≤ m * x + c <-> inverse(m) * y + -(c / m) ≤ x)" by (simp add: field_simps) lemma real_affinity_lt: "0 < (m::'a::linordered_field) ==> (m * x + c < y <-> x < inverse(m) * y + -(c / m))" by (simp add: field_simps) lemma real_lt_affinity: "0 < (m::'a::linordered_field) ==> (y < m * x + c <-> inverse(m) * y + -(c / m) < x)" by (simp add: field_simps) lemma real_affinity_eq: "(m::'a::linordered_field) ≠ 0 ==> (m * x + c = y <-> x = inverse(m) * y + -(c / m))" by (simp add: field_simps) lemma real_eq_affinity: "(m::'a::linordered_field) ≠ 0 ==> (y = m * x + c <-> inverse(m) * y + -(c / m) = x)" by (simp add: field_simps) subsection {* Banach fixed point theorem (not really topological...) *} lemma banach_fix: assumes s: "complete s" "s ≠ {}" and c: "0 ≤ c" "c < 1" and f: "(f ` s) ⊆ s" and lipschitz: "∀x∈s. ∀y∈s. dist (f x) (f y) ≤ c * dist x y" shows "∃!x∈s. f x = x" proof - have "1 - c > 0" using c by auto from s(2) obtain z0 where "z0 ∈ s" by auto def z ≡ "λn. (f ^^ n) z0" { fix n :: nat have "z n ∈ s" unfolding z_def proof (induct n) case 0 then show ?case using `z0 ∈ s` by auto next case Suc then show ?case using f by auto qed } note z_in_s = this def d ≡ "dist (z 0) (z 1)" have fzn:"!!n. f (z n) = z (Suc n)" unfolding z_def by auto { fix n :: nat have "dist (z n) (z (Suc n)) ≤ (c ^ n) * d" proof (induct n) case 0 then show ?case unfolding d_def by auto next case (Suc m) then have "c * dist (z m) (z (Suc m)) ≤ c ^ Suc m * d" using `0 ≤ c` using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by auto then show ?case using lipschitz[THEN bspec[where x="z m"], OF z_in_s, THEN bspec[where x="z (Suc m)"], OF z_in_s] unfolding fzn and mult_le_cancel_left by auto qed } note cf_z = this { fix n m :: nat have "(1 - c) * dist (z m) (z (m+n)) ≤ (c ^ m) * d * (1 - c ^ n)" proof (induct n) case 0 show ?case by auto next case (Suc k) have "(1 - c) * dist (z m) (z (m + Suc k)) ≤ (1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))" using dist_triangle and c by (auto simp add: dist_triangle) also have "… ≤ (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)" using cf_z[of "m + k"] and c by auto also have "… ≤ c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d" using Suc by (auto simp add: field_simps) also have "… = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)" unfolding power_add by (auto simp add: field_simps) also have "… ≤ (c ^ m) * d * (1 - c ^ Suc k)" using c by (auto simp add: field_simps) finally show ?case by auto qed } note cf_z2 = this { fix e :: real assume "e > 0" then have "∃N. ∀m n. N ≤ m ∧ N ≤ n --> dist (z m) (z n) < e" proof (cases "d = 0") case True have *: "!!x. ((1 - c) * x ≤ 0) = (x ≤ 0)" using `1 - c > 0` by (metis mult_zero_left mult.commute real_mult_le_cancel_iff1) from True have "!!n. z n = z0" using cf_z2[of 0] and c unfolding z_def by (simp add: *) then show ?thesis using `e>0` by auto next case False then have "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"] by (metis False d_def less_le) hence "0 < e * (1 - c) / d" using `e>0` and `1-c>0` by auto then obtain N where N:"c ^ N < e * (1 - c) / d" using real_arch_pow_inv[of "e * (1 - c) / d" c] and c by auto { fix m n::nat assume "m>n" and as:"m≥N" "n≥N" have *:"c ^ n ≤ c ^ N" using `n≥N` and c using power_decreasing[OF `n≥N`, of c] by auto have "1 - c ^ (m - n) > 0" using c and power_strict_mono[of c 1 "m - n"] using `m>n` by auto hence **: "d * (1 - c ^ (m - n)) / (1 - c) > 0" using `d>0` `0 < 1 - c` by auto have "dist (z m) (z n) ≤ c ^ n * d * (1 - c ^ (m - n)) / (1 - c)" using cf_z2[of n "m - n"] and `m>n` unfolding pos_le_divide_eq[OF `1-c>0`] by (auto simp add: mult.commute dist_commute) also have "… ≤ c ^ N * d * (1 - c ^ (m - n)) / (1 - c)" using mult_right_mono[OF * order_less_imp_le[OF **]] unfolding mult.assoc by auto also have "… < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)" using mult_strict_right_mono[OF N **] unfolding mult.assoc by auto also have "… = e * (1 - c ^ (m - n))" using c and `d>0` and `1 - c > 0` by auto also have "… ≤ e" using c and `1 - c ^ (m - n) > 0` and `e>0` using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto finally have "dist (z m) (z n) < e" by auto } note * = this { fix m n :: nat assume as: "N ≤ m" "N ≤ n" then have "dist (z n) (z m) < e" proof (cases "n = m") case True then show ?thesis using `e>0` by auto next case False then show ?thesis using as and *[of n m] *[of m n] unfolding nat_neq_iff by (auto simp add: dist_commute) qed } then show ?thesis by auto qed } then have "Cauchy z" unfolding cauchy_def by auto then obtain x where "x∈s" and x:"(z ---> x) sequentially" using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto def e ≡ "dist (f x) x" have "e = 0" proof (rule ccontr) assume "e ≠ 0" then have "e > 0" unfolding e_def using zero_le_dist[of "f x" x] by (metis dist_eq_0_iff dist_nz e_def) then obtain N where N:"∀n≥N. dist (z n) x < e / 2" using x[unfolded lim_sequentially, THEN spec[where x="e/2"]] by auto then have N':"dist (z N) x < e / 2" by auto have *: "c * dist (z N) x ≤ dist (z N) x" unfolding mult_le_cancel_right2 using zero_le_dist[of "z N" x] and c by (metis dist_eq_0_iff dist_nz order_less_asym less_le) have "dist (f (z N)) (f x) ≤ c * dist (z N) x" using lipschitz[THEN bspec[where x="z N"], THEN bspec[where x=x]] using z_in_s[of N] `x∈s` using c by auto also have "… < e / 2" using N' and c using * by auto finally show False unfolding fzn using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x] unfolding e_def by auto qed then have "f x = x" unfolding e_def by auto moreover { fix y assume "f y = y" "y∈s" then have "dist x y ≤ c * dist x y" using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]] using `x∈s` and `f x = x` by auto then have "dist x y = 0" unfolding mult_le_cancel_right1 using c and zero_le_dist[of x y] by auto then have "y = x" by auto } ultimately show ?thesis using `x∈s` by blast+ qed subsection {* Edelstein fixed point theorem *} lemma edelstein_fix: fixes s :: "'a::metric_space set" assumes s: "compact s" "s ≠ {}" and gs: "(g ` s) ⊆ s" and dist: "∀x∈s. ∀y∈s. x ≠ y --> dist (g x) (g y) < dist x y" shows "∃!x∈s. g x = x" proof - let ?D = "(λx. (x, x)) ` s" have D: "compact ?D" "?D ≠ {}" by (rule compact_continuous_image) (auto intro!: s continuous_Pair continuous_within_id simp: continuous_on_eq_continuous_within) have "!!x y e. x ∈ s ==> y ∈ s ==> 0 < e ==> dist y x < e ==> dist (g y) (g x) < e" using dist by fastforce then have "continuous_on s g" unfolding continuous_on_iff by auto then have cont: "continuous_on ?D (λx. dist ((g o fst) x) (snd x))" unfolding continuous_on_eq_continuous_within by (intro continuous_dist ballI continuous_within_compose) (auto intro!: continuous_fst continuous_snd continuous_within_id simp: image_image) obtain a where "a ∈ s" and le: "!!x. x ∈ s ==> dist (g a) a ≤ dist (g x) x" using continuous_attains_inf[OF D cont] by auto have "g a = a" proof (rule ccontr) assume "g a ≠ a" with `a ∈ s` gs have "dist (g (g a)) (g a) < dist (g a) a" by (intro dist[rule_format]) auto moreover have "dist (g a) a ≤ dist (g (g a)) (g a)" using `a ∈ s` gs by (intro le) auto ultimately show False by auto qed moreover have "!!x. x ∈ s ==> g x = x ==> x = a" using dist[THEN bspec[where x=a]] `g a = a` and `a∈s` by auto ultimately show "∃!x∈s. g x = x" using `a ∈ s` by blast qed no_notation eucl_less (infix "<e" 50) end