Theory Topology_Euclidean_Space

theory Topology_Euclidean_Space
imports Countable_Set Glbs 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*)header {* Elementary topology in Euclidean space. *}theory Topology_Euclidean_Spaceimports  Complex_Main  "~~/src/HOL/Library/Countable_Set"  "~~/src/HOL/Library/Glbs"  "~~/src/HOL/Library/FuncSet"  Linear_Algebra  Norm_Arithbeginlemma dist_0_norm:  fixes x :: "'a::real_normed_vector"  shows "dist 0 x = norm x"unfolding dist_norm by simplemma 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)lemmas real_isGlb_unique = isGlb_unique[where 'a=real]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) autosubsection {* Topological Basis *}context topological_spacebegindefinition "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  donelemma 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 autonext  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'}"])  qedqedlemma 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) autolemma 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)qedlemma 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])qedlemma 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 ≠ {}]]  guess B' . note B' = this  then show "∃B'∈B. f B' ∈ X"    by (auto intro!: choosefrom_basis)qedendlemma 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_defproof (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 topological_basisE[OF A a] guess A0 .    moreover from topological_basisE[OF B b] guess B0 .    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 autoqed (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"beginlemma open_countable_basis_ex:  assumes "open X"  shows "∃B' ⊆ B. X = Union B'"  using assms countable_basis is_basis  unfolding topological_basis_def by blastlemma open_countable_basisE:  assumes "open X"  obtains B' where "B' ⊆ B" "X = Union B'"  using assms open_countable_basis_ex  by (atomize_elim) simplemma 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)qedlemma countable_dense_setE:  obtains D :: "'a set"  where "countable D" "!!X. open X ==> X ≠ {} ==> ∃d ∈ D. d ∈ X"  using countable_dense_exists by blastendlemma (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  donelemma (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  from first_countable_basisE[of x] guess A' . note A' = this  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}"])  qedqedlemma (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 autonext  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 autoqedinstance prod :: (first_countable_topology, first_countable_topology) first_countable_topologyproof  fix x :: "'a × 'b"  from first_countable_basisE[of "fst x"] guess A :: "'a set set" . note A = this  from first_countable_basisE[of "snd x"] guess B :: "'b set set" . note B = this  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"    from open_prod_elim[OF this] guess a' b' . note a'b' = this    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)qedclass second_countable_topology = topological_space +  assumes ex_countable_subbasis:    "∃B::'a::topological_space set set. countable B ∧ open = generate_topology B"beginlemma 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 guess k 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)  qedqedendsublocale second_countable_topology <  countable_basis "SOME B. countable B ∧ topological_basis B"  using someI_ex[OF ex_countable_basis]  by unfold_locales safeinstance prod :: (second_countable_topology, second_countable_topology) second_countable_topologyproof  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)qedinstance second_countable_topology ⊆ first_countable_topologyproof  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)+qedsubsection {* 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_topologysubsection {* 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 blastlemma istopology_open_in[intro]: "istopology(openin U)"  using openin[of U] by blastlemma 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 autolemma 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 simpnext  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 .qedtext{* 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 blastlemma 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 simplemma openin_Union[intro]: "(∀S ∈K. openin U S) ==> openin U (\<Union> K)"  using openin_clauses by simplemma openin_Un[intro]: "openin U S ==> openin U T ==> openin U (S ∪ T)"  using openin_Union[of "{S,T}" U] by autolemma 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 autonext  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" .qedsubsubsection {* 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 autolemma 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 autolemma closedin_Int[intro]: "closedin U S ==> closedin U T ==> closedin U (S ∩ T)"  using closedin_Inter[of "{S,T}" U] by autolemma Diff_Diff_Int: "A - (A - B) = A ∩ B"  by blastlemma 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)  donelemma 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)qedlemma 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)qedsubsubsection {* 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)"      apply (rule set_eqI)      apply (simp add: Ball_def image_iff)      apply metis      done    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 blastqedlemma openin_subtopology: "openin (subtopology U V) S <-> (∃T. openin U T ∧ S = T ∩ V)"  unfolding subtopology_def topology_inverse'[OF istopology_subtopology]  by autolemma 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  apply (simp add: openin_subtopology)  apply (rule iffI)  apply clarify  apply (rule_tac x="topspace U - T" in exI)  apply auto  donelemma openin_subtopology_refl: "openin (subtopology U V) V <-> V ⊆ topspace U"  unfolding openin_subtopology  apply (rule iffI, clarify)  apply (frule openin_subset[of U])  apply blast  apply (rule exI[where x="topspace U"])  apply auto  donelemma 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 blastqedlemma 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)  donelemma topspace_euclidean: "topspace euclidean = UNIV"  apply (simp add: topspace_def)  apply (rule set_eqI)  apply (auto simp add: open_openin[symmetric])  donelemma 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"  apply (subgoal_tac "S ∩ T = T" )  apply auto  apply (frule closedin_closed_Int[of T S])  apply simp  donelemma 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 blastnext  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)    apply (erule rev_bexI)    apply (rule_tac x=d in exI, clarify)    apply (erule le_less_trans [OF dist_triangle])    done  assume ?rhs then have 2: "S = U ∩ T"    unfolding T_def    apply auto    apply (drule (1) bspec, erule rev_bexI)    apply auto    done  from 1 2 show ?lhs    unfolding openin_open open_dist by fastqedtext {* 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 blastlemma 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 simplemma centre_in_cball: "x ∈ cball x e <-> 0 ≤ e"  by simplemma 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) arithlemma 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)"  unfolding open_dist ball_def mem_Collect_eq Ball_def  unfolding dist_commute  apply clarify  apply (rule_tac x="e - dist xa x" in exI)  using dist_triangle_alt[where z=x]  apply (clarsimp simp add: diff_less_iff)  apply atomize  apply (erule_tac x="y" in allE)  apply (erule_tac x="xa" in allE)  apply arith  donelemma 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 autolemma 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)  donelemma ball_empty[intro]: "e ≤ 0 ==> ball x e = {}" by simplemma 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)definition "box a b = {x. ∀i∈Basis. a • i < x • i ∧ x • i < b • i}"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 intro!: divide_pos_pos 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] guess a .. note a = this  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] guess b .. note b = this  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)qedlemma 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)qedsubsection{* 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  apply safe  apply blast+  donelemma 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 metisqedlemma 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    apply (subst exists_diff)    apply blast    done  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 simpqedsubsection{* 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 autolemma 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 autolemma islimpt_iff_eventually: "x islimpt S <-> ¬ eventually (λy. y ∉ S) (at x)"  unfolding islimpt_def eventually_at_topological by autolemma islimpt_subset: "x islimpt S ==> S ⊆ T ==> x islimpt T"  unfolding islimpt_def by fastlemma 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 fastlemma 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 blasttext {* 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)  donelemma islimpt_EMPTY[simp]: "¬ x islimpt {}"  unfolding islimpt_def by autolemma 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"])  qedqedlemma 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])qedsubsection {* 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 fastlemma interiorE [elim?]:  assumes "x ∈ interior S"  obtains T where "open T" and "x ∈ T" and "T ⊆ S"  using assms unfolding interior_def by fastlemma 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)  donelemma 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  qedqedlemma 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)  qedqedsubsection {* 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 autolemma closure_interior: "closure S = - interior (- S)"  unfolding interior_closure by simplemma 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 simplemma closure_hull: "closure S = closed hull S"  unfolding hull_def closure_interior interior_def by autolemma 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 simplemma closure_eq_empty: "closure S = {} <-> S = {}"  using closure_empty closure_subset[of S]  by blastlemma closure_subset_eq: "closure S ⊆ S <-> closed S"  using closure_eq[of S] closure_subset[of S]  by simplemma 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 autolemma 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 blastqedlemma closure_complement: "closure (- S) = - interior S"  unfolding closure_interior by simplemma interior_complement: "interior (- S) = - closure S"  unfolding closure_interior by simplemma 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    doneqedlemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"  unfolding closure_def using islimpt_punctured by blastsubsection {* 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] ..qedlemma 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 autosubsection {* 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)    donenext  assume "¬ a islimpt S"  then show "trivial_limit (at a within S)"    unfolding trivial_limit_def    unfolding eventually_at_topological    unfolding islimpt_def    apply clarsimp    apply (rule_tac x=T in exI)    apply auto    doneqedlemma trivial_limit_at_iff: "trivial_limit (at a) <-> ¬ a islimpt UNIV"  using trivial_limit_within [of a UNIV] by simplemma 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)  donelemma 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 autolemma 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 simplemma 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 fastlemma 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 autotext{* 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)  }qedlemma 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 simpnext  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)      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 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  qedqedtext{* 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] guess A . note A = this  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 fastnext  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  qedqedlemma 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)qedlemma 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)qedtext{* 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"proof -  have "dist a l ∈ {..e}"  proof (rule Lim_in_closed_set)    show "closed {..e}"      by simp    show "eventually (λx. dist a (f x) ∈ {..e}) net"      by (simp add: assms)    show "¬ trivial_limit net"      by fact    show "((λx. dist a (f x)) ---> dist a l) net"      by (intro tendsto_intros assms)  qed  then show ?thesis by simpqedlemma 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"proof -  have "norm l ∈ {..e}"  proof (rule Lim_in_closed_set)    show "closed {..e}"      by simp    show "eventually (λx. norm (f x) ∈ {..e}) net"      by (simp add: assms)    show "¬ trivial_limit net"      by fact    show "((λx. norm (f x)) ---> norm l) net"      by (intro tendsto_intros assms)  qed  then show ?thesis by simpqedlemma 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"proof -  have "norm l ∈ {e..}"  proof (rule Lim_in_closed_set)    show "closed {e..}"      by simp    show "eventually (λx. norm (f x) ∈ {e..}) net"      by (simp add: assms)    show "¬ trivial_limit net"      by fact    show "((λx. norm (f x)) ---> norm l) net"      by (intro tendsto_intros assms)  qed  then show ?thesis by simpqedtext{* 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 simplemma 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 simplemma 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)  donelemma 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 factqedlemma 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 autoqedtext{* 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)qedlemma 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 simptext{* 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 factqedtext{* 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 simplemma 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 simptext{* 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 autonext  assume "?rhs"  then show "?lhs" unfolding closure_def islimpt_sequential by autoqedlemma closed_sequential_limits:  fixes S :: "'a::first_countable_topology set"  shows "closed S <-> (∀x l. (∀n. x n ∈ S) ∧ (x ---> l) sequentially --> l ∈ S)"  unfolding closed_limpt  using closure_sequential [where 'a='a] closure_closed [where 'a='a]    closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]  by metislemma 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)  donelemma 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 ≠ {}" "∀x∈S. B ≤ x"  shows "Inf S ∈ closure S"proof -  have *: "∀x∈S. Inf S ≤ x"    using cInf_lower_EX[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[of _ B]) 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 autoqedlemma closed_contains_Inf:  fixes S :: "real set"  assumes "S ≠ {}" "∀x∈S. B ≤ x"    and "closed S"  shows "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 autoqedsubsection {* Infimum Distance *}definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a ∈ A})"lemma infdist_notempty: "A ≠ {} ==> infdist x A = Inf {dist x a|a. a ∈ 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:  assumes "a ∈ A"    and "d = dist x a"  shows "infdist x A ≤ d"  using assms by (auto intro!: cInf_lower[where z=0] simp add: infdist_def)lemma infdist_zero[simp]:  assumes "a ∈ A"  shows "infdist a A = 0"proof -  from infdist_le[OF assms, of "dist a a"] have "infdist a A ≤ 0"    by auto  with infdist_nonneg[of a A] assms show "infdist a A = 0"    by autoqedlemma 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 "dist x a ∈ {dist x a |a. a ∈ A}"        using a ∈ A by auto      show "dist x a ≤ d"        unfolding d by (rule dist_triangle)      fix d      assume "d ∈ {dist x a |a. a ∈ A}"      then obtain a where "a ∈ A" "d = dist x a"        by auto      then show "infdist x A ≤ d"        by (rule infdist_le)    qed  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 simpqedlemma 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 0 < infdist x A x ∈ closure A closure_approachable dist_commute        eucl_less_not_refl euclidean_trans(2) infdist_le)      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  qednext  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  qedqedlemma 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 simpqedlemma 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" .  qedqedtext{* 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  donelemma 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_cball: "closed (cball x e)"  unfolding cball_def closed_def  unfolding Collect_neg_eq [symmetric] not_le  apply (clarsimp simp add: open_dist, rename_tac y)  apply (rule_tac x="dist x y - e" in exI, clarsimp)  apply (rename_tac x')  apply (cut_tac x=x and y=x' and z=y in dist_triangle)  apply simp  donelemma 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 autoqedlemma 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])  donelemma 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 autonext  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 autoqedlemma 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 divide_pos_pos 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 fastqedlemma 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 blastnext  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      then have **:"d / (2 * norm (y - x)) > 0"        unfolding zero_less_norm_iff[symmetric]        using d>0 divide_pos_pos[of d "2*norm (y - x)"] 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 autoqedlemma 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  donelemma 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  donelemma 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)  donelemma 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 autolemma 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 autolemma bounded_any_center: "bounded S <-> (∃e. ∀y∈S. dist a y ≤ e)"  unfolding bounded_def  apply safe  apply (rule_tac x="dist a x + e" in exI)  apply clarify  apply (drule (1) bspec)  apply (erule order_trans [OF dist_triangle add_left_mono])  apply auto  donelemma 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  apply (rule_tac x=0 in exI)  using assms  apply auto  donelemma 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 autoqedlemma 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  donelemma 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)  apply (rename_tac x y r s)  apply (rule_tac x=x in exI)  apply (rule_tac x="max r (dist x y + s)" in exI)  apply (rule ballI)  apply safe  apply (drule (1) bspec)  apply simp  apply (drule (1) bspec)  apply (rule min_max.le_supI2)  apply (erule order_trans [OF dist_triangle add_left_mono])  donelemma bounded_Union[intro]: "finite F ==> ∀S∈F. bounded S ==> bounded (\<Union>F)"  by (induct rule: finite_induct[of F]) autolemma bounded_UN [intro]: "finite A ==> ∀x∈A. bounded (B x) ==> bounded (\<Union>x∈A. B x)"  by (induct set: finite) autolemma 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)qedlemma finite_imp_bounded [intro]: "finite S ==> bounded S"  by (induct set: finite) simp_alllemma 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  donelemma Bseq_eq_bounded:  fixes f :: "nat => 'a::real_normed_vector"  shows "Bseq f <-> bounded (range f)"  unfolding Bseq_def bounded_pos by autolemma 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" ..qedlemma 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: mult_ac)  {    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 mult_pos_pos [of b B]    apply (auto simp add: mult_commute)    doneqedlemma 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)  donelemma 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"])qedtext{* Some theorems on sups and infs using the notion "bounded". *}lemma bounded_real:  fixes S :: "real set"  shows "bounded S <-> (∃a. ∀x∈S. abs x ≤ a)"  by (simp add: bounded_iff)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  fix x  assume "x∈S"  then show "x ≤ Sup S"    by (metis cSup_upper abs_le_D1 assms(1) bounded_real)next  show "∀b. (∀x∈S. x ≤ b) --> Sup S ≤ b"    using assms by (metis cSup_least)qedlemma Sup_insert:  fixes S :: "real set"  shows "bounded S ==> Sup (insert x S) = (if S = {} then x else max x (Sup S))"  apply (subst cSup_insert_If)  apply (rule bounded_has_Sup(1)[of S, rule_format])  apply (auto simp: sup_max)  donelemma 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  donelemma 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  fix x  assume "x ∈ S"  from assms(1) obtain a where a: "∀x∈S. ¦x¦ ≤ a"    unfolding bounded_real by auto  then show "x ≥ Inf S" using x ∈ S    by (metis cInf_lower_EX abs_le_D2 minus_le_iff)next  show "∀b. (∀x∈S. x ≥ b) --> Inf S ≥ b"    using assms by (metis cInf_greatest)qedlemma Inf_insert:  fixes S :: "real set"  shows "bounded S ==> Inf (insert x S) = (if S = {} then x else min x (Inf S))"  apply (subst cInf_insert_if)  apply (rule bounded_has_Inf(1)[of S, rule_format])  apply (auto simp: inf_min)  donelemma 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  donesubsection {* 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 autoqedlemma 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] guess A . note A = this  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 ≡ "nat_rec (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)qedlemma 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 autoqedlemma 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)  donelemma 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  qednext  assume "x islimpt s"  then show "x islimpt (insert a s)"    by (rule islimpt_subset) autoqedlemma 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) autonext  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 autoqedlemma 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 simpqedlemma 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 fastqedlemma 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)qedtext{* 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]) metis  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]) metis  ultimately show "∃f'⊆f. finite f' ∧ s ∪ t ⊆ \<Union>f'"    by (auto intro!: exI[of _ "s' ∪ t'"])qedlemma compact_Union [intro]: "finite S ==> (!!T. T ∈ S ==> compact T) ==> compact (\<Union>S)"  by (induct set: finite) autolemma compact_UN [intro]:  "finite A ==> (!!x. x ∈ A ==> compact (B x)) ==> compact (\<Union>x∈A. B x)"  unfolding SUP_def by (rule compact_Union) autolemma 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 autolemma 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 simpqedlemma finite_imp_compact: "finite s ==> compact s"  by (induct set: finite) simp_alllemma open_delete:  fixes s :: "'a::t1_space set"  shows "open s ==> open (s - {x})"  by (simp add: open_Diff)text{* Finite intersection property *}lemma inj_setminus: "inj_on uminus (A::'a set set)"  by (auto simp: inj_on_def)lemma compact_fip:  "compact U <->    (∀A. (∀a∈A. closed a) --> (∀B ⊆ A. finite B --> U ∩ \<Inter>B ≠ {}) --> U ∩ \<Inter>A ≠ {})"  (is "_ <-> ?R")proof (safe intro!: compact_eq_heine_borel[THEN iffD2])  fix A  assume "compact U"    and A: "∀a∈A. closed a" "U ∩ \<Inter>A = {}"    and fi: "∀B ⊆ A. finite B --> U ∩ \<Inter>B ≠ {}"  from A have "(∀a∈uminusA. open a) ∧ U ⊆ \<Union>(uminusA)"    by auto  with compact U obtain B where "B ⊆ A" "finite (uminusB)" "U ⊆ \<Union>(uminusB)"    unfolding compact_eq_heine_borel by (metis subset_image_iff)  with fi[THEN spec, of B] show False    by (auto dest: finite_imageD intro: inj_setminus)next  fix A  assume ?R  assume "∀a∈A. open a" "U ⊆ \<Union>A"  then have "U ∩ \<Inter>(uminusA) = {}" "∀a∈uminusA. closed a"    by auto  with ?R obtain B where "B ⊆ A" "finite (uminusB)" "U ∩ \<Inter>(uminusB) = {}"    by (metis subset_image_iff)  then show "∃T⊆A. finite T ∧ U ⊆ \<Union>T"    by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)qedlemma compact_imp_fip:  "compact s ==> ∀t ∈ f. closed t ==> ∀f'. finite f' ∧ f' ⊆ f --> (s ∩ (\<Inter> f') ≠ {}) ==>    s ∩ (\<Inter> f) ≠ {}"  unfolding compact_fip by autotext{*Compactness expressed with filters*}definition "filter_from_subbase B = Abs_filter (λP. ∃X ⊆ B. finite X ∧ Inf X ≤ P)"lemma eventually_filter_from_subbase:  "eventually P (filter_from_subbase B) <-> (∃X ⊆ B. finite X ∧ Inf X ≤ P)"    (is "_ <-> ?R P")  unfolding filter_from_subbase_defproof (rule eventually_Abs_filter is_filter.intro)+  show "?R (λx. True)"    by (rule exI[of _ "{}"]) (simp add: le_fun_def)next  fix P Q assume "?R P" then guess X ..  moreover assume "?R Q" then guess Y ..  ultimately show "?R (λx. P x ∧ Q x)"    by (intro exI[of _ "X ∪ Y"]) autonext  fix P Q  assume "?R P" then guess X ..  moreover assume "∀x. P x --> Q x"  ultimately show "?R Q"    by (intro exI[of _ X]) autoqedlemma eventually_filter_from_subbaseI: "P ∈ B ==> eventually P (filter_from_subbase B)"  by (subst eventually_filter_from_subbase) (auto intro!: exI[of _ "{P}"])lemma filter_from_subbase_not_bot:  "∀X ⊆ B. finite X --> Inf X ≠ bot ==> filter_from_subbase B ≠ bot"  unfolding trivial_limit_def eventually_filter_from_subbase by autolemma 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 autonext  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)qedlemma 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 have "eventually (λx. ∀b∈B. x ∈ b) F"      by (auto intro!: eventually_Ball_finite)    with F(2) have "eventually (λx. x ∈ U ∩ (\<Inter>B)) F"      by eventually_elim auto    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 P' ≡ "(λa (x::'a). x ∈ a)"  then have inj_P': "!!A. inj_on P' A"    by (auto intro!: inj_onI simp: fun_eq_iff)  def F ≡ "filter_from_subbase (P'  insert U A)"  have "F ≠ bot"    unfolding F_def  proof (safe intro!: filter_from_subbase_not_bot)    fix X    assume "X ⊆ P'  insert U A" "finite X" "Inf X = bot"    then obtain B where "B ⊆ insert U A" "finite B" and B: "Inf (P'  B) = bot"      unfolding subset_image_iff by (auto intro: inj_P' finite_imageD)    with A(2)[THEN spec, of "B - {U}"] have "U ∩ \<Inter>(B - {U}) ≠ {}"      by auto    with B show False      by (auto simp: P'_def fun_eq_iff)  qed  moreover have "eventually (λx. x ∈ U) F"    unfolding F_def by (rule eventually_filter_from_subbaseI) (auto simp: P'_def)  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 V: "eventually (λx. x ∈ V) F"      by (auto simp add: F_def image_iff P'_def intro!: eventually_filter_from_subbaseI)    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 autoqeddefinition "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 metislemma 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 metislemma 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_defproof 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 guess f unfolding bchoice_iff Bex_def ..  with T show "∃T⊆A. finite T ∧ U ⊆ \<Union>T"    unfolding C_def by (intro exI[of _ "fT"]) fastforceqedlemma 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] guess b .  then show "∃b∈SOME B. countable B ∧ topological_basis B. x ∈ b ∧ b ∩ U ⊆ T"    by autoqed (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 blastsubsubsection{* 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_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  qedqedlemma compact_imp_seq_compact:  fixes U :: "'a :: first_countable_topology set"  assumes "compact U"  shows "seq_compact U"  unfolding seq_compact_defproof 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] guess A . note A = this  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 ≡ "nat_rec (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)qedlemma 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 fastlemma 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 fastlemma 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 guess 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)qedlemma 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 tendsto_const 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 autoqedlemma 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 metislemma 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 metislemma 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 blastlemma 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{* Total boundedness *}lemma cauchy_def: "Cauchy s <-> (∀e>0. ∃N. ∀m n. m ≥ N ∧ n ≥ N --> dist(s m)(s n) < e)"  unfolding Cauchy_def by metisfun helper_1 :: "('a::metric_space set) => real => nat => 'a"where  "helper_1 s e n = (SOME y::'a. y ∈ s ∧ (∀m<n. ¬ (dist (helper_1 s e m) y < e)))"declare helper_1.simps[simp del]lemma seq_compact_imp_totally_bounded:  assumes "seq_compact s"  shows "∀e>0. ∃k. finite k ∧ k ⊆ s ∧ s ⊆ (\<Union>((λx. ball x e)  k))"proof (rule, rule, rule ccontr)  fix e::real  assume "e > 0"  assume assm: "¬ (∃k. finite k ∧ k ⊆ s ∧ s ⊆ \<Union>((λx. ball x e)  k))"  def x ≡ "helper_1 s e"  {    fix n    have "x n ∈ s ∧ (∀m<n. ¬ dist (x m) (x n) < e)"    proof (induct n rule: nat_less_induct)      fix n      def Q ≡ "(λy. y ∈ s ∧ (∀m<n. ¬ dist (x m) y < e))"      assume as: "∀m<n. x m ∈ s ∧ (∀ma<m. ¬ dist (x ma) (x m) < e)"      have "¬ s ⊆ (\<Union>x∈x  {0..<n}. ball x e)"        using assm        apply simp        apply (erule_tac x="x  {0 ..< n}" in allE)        using as        apply auto        done      then obtain z where z:"z∈s" "z ∉ (\<Union>x∈x  {0..<n}. ball x e)"        unfolding subset_eq by auto      have "Q (x n)"        unfolding x_def and helper_1.simps[of s e n]        apply (rule someI2[where a=z])        unfolding x_def[symmetric] and Q_def        using z        apply auto        done      then show "x n ∈ s ∧ (∀m<n. ¬ dist (x m) (x n) < e)"        unfolding Q_def by auto    qed  }  then have "∀n::nat. x n ∈ s" and x:"∀n. ∀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(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto  from this(3) have "Cauchy (x o r)"    using LIMSEQ_imp_Cauchy by auto  then obtain N::nat where N:"∀m n. N ≤ m ∧ N ≤ n --> dist ((x o r) m) ((x o r) n) < e"    unfolding cauchy_def using e>0 by auto  show False    using N[THEN spec[where x=N], THEN spec[where x="N+1"]]    using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]    using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]]    by autoqedsubsubsection{* 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]  guess f unfolding choice_iff' .. note f = this  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  qedqedlemma 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 blastlemma 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 autosubsubsection {* 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 autonext  assume ?rhs  then show ?lhs    unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)qedlemma 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 .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    unfolding closed_sequential_limits by blast  show "∃l∈s. ∃r. subseq r ∧ ((f o r) ---> l) sequentially"    using l ∈ s r l by blastqedlemma 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 blastnext  assume ?rhs  then show ?lhs    using bounded_closed_imp_seq_compact[of s]    unfolding compact_eq_seq_compact_metric    by autoqed(* 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_borelproof  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)qedlemma 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  qedqedinstance euclidean_space ⊆ heine_borelproof  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"    then have "e / real_of_nat DIM('a) > 0"      by (auto intro!: divide_pos_pos 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 autoqedlemma bounded_fst: "bounded s ==> bounded (fst  s)"  unfolding bounded_def  apply clarify  apply (rule_tac x="a" in exI)  apply (rule_tac x="e" in exI)  apply clarsimp  apply (drule (1) bspec)  apply (simp add: dist_Pair_Pair)  apply (erule order_trans [OF real_sqrt_sum_squares_ge1])  donelemma bounded_snd: "bounded s ==> bounded (snd  s)"  unfolding bounded_def  apply clarify  apply (rule_tac x="b" in exI)  apply (rule_tac x="e" in exI)  apply clarsimp  apply (drule (1) bspec)  apply (simp add: dist_Pair_Pair)  apply (erule order_trans [OF real_sqrt_sum_squares_ge2])  doneinstance prod :: (heine_borel, heine_borel) heine_borelproof  fix f :: "nat => 'a × 'b"  assume f: "bounded (range f)"  from f have s1: "bounded (range (fst o f))"    unfolding image_comp by (rule bounded_fst)  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 fastqedsubsubsection{* Completeness *}definition complete :: "'a::metric_space set => bool"  where "complete s <-> (∀f. (∀n. f n ∈ s) ∧ Cauchy f --> (∃l∈s. f ----> l))"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 LIMSEQ_def, 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 LIMSEQ_def by auto  }  then show ?thesis unfolding complete_def by autoqedlemma 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 intro!: mult_pos_pos simp: 0 < e)  also have "1 / (ceiling (1/e)) ≤ 1 / (1/e)"    by (rule divide_left_mono) (auto intro!: divide_pos_pos simp: 0 < e)  also have "… = e" by simp  finally show  "∃n. 1 / real (Suc n) < e" ..qedlemma compact_eq_totally_bounded:  "compact s <-> complete s ∧ (∀e>0. ∃k. finite k ∧ s ⊆ (\<Union>((λx. ball x e)  k)))"    (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 guess a ..        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 ≡ "nat_rec (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 ≡ "nat_rec (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  qedqed (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 blastqedlemma 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    doneqedinstance heine_borel < complete_spaceproof  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 autoqedinstance euclidean_space ⊆ banach ..lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"proof (simp add: complete_def, rule, rule)  fix f :: "nat => 'a"  assume "Cauchy f"  then have "convergent f" by (rule Cauchy_convergent)  then show "∃l. f ----> l" unfolding convergent_def .qedlemma complete_imp_closed:  assumes "complete s"  shows "closed s"proof -  {    fix x    assume "x islimpt s"    then obtain f where f: "∀n. f n ∈ s - {x}" "(f ---> x) sequentially"      unfolding islimpt_sequential by auto    then obtain l where l: "l∈s" "(f ---> l) sequentially"      using complete s[unfolded complete_def] using LIMSEQ_imp_Cauchy[of f x] by auto    then have "x ∈ s"      using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto  }  then show "closed s" unfolding closed_limpt by autoqedlemma complete_eq_closed:  fixes s :: "'a::complete_space set"  shows "complete s <-> closed s" (is "?lhs = ?rhs")proof  assume ?lhs  then show ?rhs by (rule complete_imp_closed)next  assume ?rhs  {    fix f    assume as:"∀n::nat. f n ∈ s" "Cauchy f"    then obtain l where "(f ---> l) sequentially"      using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto    then have "∃l∈s. (f ---> l) sequentially"      using ?rhs[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]]      using as(1) by auto  }  then show ?lhs unfolding complete_def by autoqedlemma 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 blastlemma 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 blastlemma compact_frontier[intro]:  fixes s :: "'a::heine_borel set"  shows "compact s ==> compact (frontier s)"  using compact_eq_bounded_closed compact_frontier_bounded  by blastlemma frontier_subset_compact:  fixes s :: "'a::heine_borel set"  shows "compact s ==> frontier s ⊆ s"  using frontier_subset_closed compact_eq_bounded_closed  by blastsubsection {* Bounded closed nest property (proof does not use Heine-Borel) *}lemma bounded_closed_nest:  assumes "∀n. closed(s n)"    and "∀n. (s n ≠ {})"    and "(∀m n. m ≤ n --> s n ⊆ s m)"    and "bounded(s 0)"  shows "∃a::'a::heine_borel. ∀n::nat. a ∈ s(n)"proof -  from assms(2) obtain x where x:"∀n::nat. x n ∈ s n"    using choice[of "λn x. x∈ s n"] by auto  from assms(4,1) have *:"seq_compact (s 0)"    using bounded_closed_imp_seq_compact[of "s 0"] by auto  then obtain l r where lr:"l∈s 0" "subseq r" "((x o r) ---> l) sequentially"    unfolding seq_compact_def    apply (erule_tac x=x in allE)    using x using assms(3)    apply blast    done  {    fix n :: nat    {      fix e :: real      assume "e>0"      with lr(3) obtain N where N:"∀m≥N. dist ((x o r) m) l < e"        unfolding LIMSEQ_def by auto      then have "dist ((x o r) (max N n)) l < e" by auto      moreover      have "r (max N n) ≥ n" using lr(2) using seq_suble[of r "max N n"]        by auto      then have "(x o r) (max N n) ∈ s n"        using x        apply (erule_tac x=n in allE)        using x        apply (erule_tac x="r (max N n)" in allE)        using assms(3)        apply (erule_tac x=n in allE)        apply (erule_tac x="r (max N n)" in allE)        apply auto        done      ultimately have "∃y∈s n. dist y l < e"        by auto    }    then have "l ∈ s n"      using closed_approachable[of "s n" l] assms(1) by blast  }  then show ?thesis by autoqedtext {* Decreasing case does not even need compactness, just completeness. *}lemma decreasing_closed_nest:  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::'a::complete_space. ∀n::nat. 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 "λ n 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 LIMSEQ_def] 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 autoqedtext {* 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 ..qedtext{* 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 autonext  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 LIMSEQ_def] 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 xand 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 autoqedlemma 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 LIMSEQ_def by blast  }  ultimately show ?thesis by autoqedsubsection {* 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  unfolding dist_nz[symmetric]  apply (auto del: allE elim!:allE)  apply(rule_tac x=d in exI)  apply auto  donelemma 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 simptext{* 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 autonext  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    doneqedlemma 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    donenext  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)    doneqedtext{* 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  apply (intro ball_cong [OF refl] all_cong ex_cong)  apply (rename_tac y, case_tac "y = x")  apply simp  apply (simp add: dist_nz)  donedefinition 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 blastlemma 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 autolemma Lim_trivial_limit: "trivial_limit net ==> (f ---> l) net"  by simplemmas 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"  apply (erule interiorE)  apply (drule (1) continuous_on_subset)  apply (simp add: continuous_on_eq_continuous_at)  donelemma 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 simptext {* 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 simpnext  assume ?rhs  then show ?lhs    unfolding continuous_within tendsto_def [where l="f a"]    by (simp add: sequentially_imp_eventually_within)qedlemma 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 simplemma 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 autonext  assume ?lhs  then show ?rhs    unfolding continuous_on_eq_continuous_within    using continuous_within_sequentially[of _ s f]    by autoqedlemma 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 LIMSEQ_def 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 LIMSEQ_def and dist_real_def by auto  }  then show ?rhs by autonext  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 LIMSEQ_def dist_real_def by auto    then have False using fxy and e>0 by auto  }  then show ?lhs    unfolding uniformly_continuous_on_def by blastqedtext{* 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_withinproof (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 simpqedlemma 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 simpsubsubsection {* Structural rules for pointwise continuity *}lemmas continuous_within_id = continuous_identlemmas continuous_at_id = isCont_identlemma 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_inversesubsubsection {* Structural rules for setwise continuity *}lemma continuous_on_infnorm[continuous_on_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_on_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_on_intros]:  "uniformly_continuous_on s (λx. x)"  unfolding uniformly_continuous_on_def by autolemma uniformly_continuous_on_const[continuous_on_intros]:  "uniformly_continuous_on s (λx. c)"  unfolding uniformly_continuous_on_def by simplemma uniformly_continuous_on_dist[continuous_on_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 simpqedlemma uniformly_continuous_on_norm[continuous_on_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_on_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_on_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_on_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_on_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_on_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)"  unfolding ab_diff_minus using assms  by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)text{* Continuity of all kinds is preserved under composition. *}lemmas continuous_at_compose = isCont_olemma uniformly_continuous_on_compose[continuous_on_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 autoqedtext{* 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) autolemma 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)qedlemma 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 autoqedlemma 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 autoqedlemma 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 autoqedlemma 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 autoqedlemma 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 autolemma 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 autolemma 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" ..qedtext {* 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 autolemma 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 autolemma 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 autolemma 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 autoqedlemma 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)    doneqedtext {* 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)qedlemma 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 simplemma 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 autolemma 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 autotext {* 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 autolemma 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 mesonlemma 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 fasttext {* 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]]      using mult_pos_pos[OF 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 autoqedlemma 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 ((λ x. -x)  s)"  unfolding scaleR_minus1_left [symmetric]  by (rule open_scaling, auto)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 autoqedlemma 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 autoqedlemma 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    apply auto    apply (erule_tac x="a + xa" in allE)    unfolding ab_group_add_class.diff_diff_eq[symmetric]    apply auto    done  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    donenext  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 autoqedtext {* 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 .qedlemma 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)  donelemma 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 autolemma 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 autotext {* 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 autolemma 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 autolemma 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" ..  qedqedlemma 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 autoqedtext {* 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_defproof (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 (sndD) / 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 autotext {* 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 autoqedsubsection {* 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 simplemma 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 simplemma 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 simplemma 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  donelemma 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 simptext {* 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 ..qedtext {* 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 fastforceqedsubsection {* 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 autoqedlemma mem_Times_iff: "x ∈ A × B <-> fst x ∈ A ∧ snd x ∈ B"  by (induct x) simplemma 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)  donelemma 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 _ "cD"] exI[of _ "\<Inter>(aD)"] 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)qedtext{* 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 autoqedlemma 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 autolemma 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 autoqedlemma 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    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 autoqedlemma 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 autoqedlemma 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 autoqedtext {* 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 autoqedtext {* We can state this in terms of diameter of a set. *}definition "diameter s = (if s = {} then 0::real else Sup {dist x y | x y. x ∈ s ∧ y ∈ s})"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 -  let ?D = "{dist x y |x y. x ∈ s ∧ y ∈ s}"  from s obtain z d where z: "!!x. x ∈ s ==> dist z x ≤ d"    unfolding bounded_def by auto  have "dist x y ≤ Sup ?D"  proof (rule cSup_upper, 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 (insert s, auto)  with x ∈ s show ?thesis    by (auto simp add: diameter_def)qedlemma 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)  let ?D = "{dist x y |x y. x ∈ s ∧ y ∈ s}"  assume contr: "¬ ?thesis"  moreover  from d have "s ≠ {}"    by (auto simp: diameter_def)  then have "?D ≠ {}" by auto  ultimately have "Sup ?D ≤ d"    by (intro cSup_least) (auto simp: not_less)  with d < diameter s s ≠ {} show False    by (auto simp: diameter_def)qedlemma 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 autolemma 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)qedtext {* 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 .qedlemma 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 simplemma 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 fastqedlemma 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 simpqedlemma 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 autoqedlemma 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 simpqedlemma 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 autoqedlemma 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  donelemma 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  donelemma translation_diff:  fixes a :: "'a::ab_group_add"  shows "(λx. a + x)  (s - t) = ((λx. a + x)  s) - ((λx. a + x)  t)"  by autolemma 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 autoqedlemma 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 autosubsection {* 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] anda ∉ s    by blastqedlemma 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 autoqed (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)    doneqedsubsection {* Intervals *}lemma interval:  fixes a :: "'a::ordered_euclidean_space"  shows "{a <..< b} = {x::'a. ∀i∈Basis. a•i < x•i ∧ x•i < b•i}"    and "{a .. b} = {x::'a. ∀i∈Basis. a•i ≤ x•i ∧ x•i ≤ b•i}"  by (auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])lemma mem_interval:  fixes a :: "'a::ordered_euclidean_space"  shows "x ∈ {a<..<b} <-> (∀i∈Basis. a•i < x•i ∧ x•i < b•i)"    and "x ∈ {a .. b} <-> (∀i∈Basis. a•i ≤ x•i ∧ x•i ≤ b•i)"  using interval[of a b]  by (auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])lemma interval_eq_empty:  fixes a :: "'a::ordered_euclidean_space"  shows "({a <..< b} = {} <-> (∃i∈Basis. b•i ≤ a•i))" (is ?th1)    and "({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∈{a <..< b}"    then have "a • i < x • i ∧ x • i < b • i"      unfolding mem_interval 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 "{a <..< b} ≠ {}"      using mem_interval(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∈{a .. b}"    then have "a • i ≤ x • i ∧ x • i ≤ b • i"      unfolding mem_interval 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 "{a .. b} ≠ {}"      using mem_interval(2)[of "?x" a b] by auto  }  ultimately show ?th2 by blastqedlemma interval_ne_empty:  fixes a :: "'a::ordered_euclidean_space"  shows "{a  ..  b} ≠ {} <-> (∀i∈Basis. a•i ≤ b•i)"  and "{a <..< b} ≠ {} <-> (∀i∈Basis. a•i < b•i)"  unfolding interval_eq_empty[of a b] by fastforce+lemma interval_sing:  fixes a :: "'a::ordered_euclidean_space"  shows "{a .. a} = {a}"    and "{a<..<a} = {}"  unfolding set_eq_iff mem_interval eq_iff [symmetric]  by (auto intro: euclidean_eqI simp: ex_in_conv)lemma subset_interval_imp:  fixes a :: "'a::ordered_euclidean_space"  shows "(∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i) ==> {c .. d} ⊆ {a .. b}"    and "(∀i∈Basis. a•i < c•i ∧ d•i < b•i) ==> {c .. d} ⊆ {a<..<b}"    and "(∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i) ==> {c<..<d} ⊆ {a .. b}"    and "(∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i) ==> {c<..<d} ⊆ {a<..<b}"  unfolding subset_eq[unfolded Ball_def] unfolding mem_interval  by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+lemma interval_open_subset_closed:  fixes a :: "'a::ordered_euclidean_space"  shows "{a<..<b} ⊆ {a .. b}"  unfolding subset_eq [unfolded Ball_def] mem_interval  by (fast intro: less_imp_le)lemma subset_interval:  fixes a :: "'a::ordered_euclidean_space"  shows "{c .. d} ⊆ {a .. b} <-> (∀i∈Basis. c•i ≤ d•i) --> (∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i)" (is ?th1)    and "{c .. d} ⊆ {a<..<b} <-> (∀i∈Basis. c•i ≤ d•i) --> (∀i∈Basis. a•i < c•i ∧ d•i < b•i)" (is ?th2)    and "{c<..<d} ⊆ {a .. b} <-> (∀i∈Basis. c•i < d•i) --> (∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i)" (is ?th3)    and "{c<..<d} ⊆ {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_interval    by (auto intro: order_trans)  show ?th2    unfolding subset_eq and Ball_def and mem_interval    by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)  {    assume as: "{c<..<d} ⊆ {a .. b}" "∀i∈Basis. c•i < d•i"    then have "{c<..<d} ≠ {}"      unfolding interval_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∈{c<..<d}"        using i unfolding mem_interval by auto      moreover      have "?x ∉ {a .. b}"        unfolding mem_interval        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∈{c<..<d}"        unfolding mem_interval by auto      moreover      have "?x∉{a .. b}"        unfolding mem_interval        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_interval    apply (rule, rule, rule, rule)    apply (rule part1)    unfolding subset_eq and Ball_def and mem_interval    prefer 4    apply auto    apply (erule_tac x=xa in allE, erule_tac x=xa in allE, fastforce)+    done  {    assume as: "{c<..<d} ⊆ {a<..<b}" "∀i∈Basis. c•i < d•i"    fix i :: 'a    assume i:"i∈Basis"    from as(1) have "{c<..<d} ⊆ {a..b}"      using interval_open_subset_closed[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_interval    apply (rule, rule, rule, rule)    apply (rule *)    unfolding subset_eq and Ball_def and mem_interval    prefer 4    apply auto    apply (erule_tac x=xa in allE, simp)+    doneqedlemma inter_interval:  fixes a :: "'a::ordered_euclidean_space"  shows "{a .. b} ∩ {c .. d} =    {(∑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_interval  by autolemma disjoint_interval:  fixes a::"'a::ordered_euclidean_space"  shows "{a .. b} ∩ {c .. d} = {} <-> (∃i∈Basis. (b•i < a•i ∨ d•i < c•i ∨ b•i < c•i ∨ d•i < a•i))" (is ?th1)    and "{a .. b} ∩ {c<..<d} = {} <-> (∃i∈Basis. (b•i < a•i ∨ d•i ≤ c•i ∨ b•i ≤ c•i ∨ d•i ≤ a•i))" (is ?th2)    and "{a<..<b} ∩ {c .. d} = {} <-> (∃i∈Basis. (b•i ≤ a•i ∨ d•i < c•i ∨ b•i ≤ c•i ∨ d•i ≤ a•i))" (is ?th3)    and "{a<..<b} ∩ {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_interval 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 **) autoqed(* Moved interval_open_subset_closed a bit upwards *)lemma open_interval[intro]:  fixes a b :: "'a::ordered_euclidean_space"  shows "open {a<..<b}"proof -  have "open (\<Inter>i∈Basis. (λx. x•i) - {a•i<..<b•i})"    by (intro open_INT finite_lessThan ballI continuous_open_vimage allI      linear_continuous_at open_real_greaterThanLessThan finite_Basis bounded_linear_inner_left)  also have "(\<Inter>i∈Basis. (λx. x•i) - {a•i<..<b•i}) = {a<..<b}"    by (auto simp add: eucl_less [where 'a='a])  finally show "open {a<..<b}" .qedlemma closed_interval[intro]:  fixes a b :: "'a::ordered_euclidean_space"  shows "closed {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}) = {a .. b}"    by (auto simp add: eucl_le [where 'a='a])  finally show "closed {a .. b}" .qedlemma interior_closed_interval [intro]:  fixes a b :: "'a::ordered_euclidean_space"  shows "interior {a..b} = {a<..<b}" (is "?L = ?R")proof(rule subset_antisym)  show "?R ⊆ ?L"    using interval_open_subset_closed open_interval    by (rule interior_maximal)  {    fix x    assume "x ∈ interior {a..b}"    then obtain s where s: "open s" "x ∈ s" "s ⊆ {a..b}" ..    then obtain e where "e>0" and e:"∀x'. dist x' x < e --> x' ∈ {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_interval        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 ∈ {a<..<b}"      unfolding mem_interval by auto  }  then show "?L ⊆ ?R" ..qedlemma bounded_closed_interval:  fixes a :: "'a::ordered_euclidean_space"  shows "bounded {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 interval and bounded_iff by autoqedlemma bounded_interval:  fixes a :: "'a::ordered_euclidean_space"  shows "bounded {a .. b} ∧ bounded {a<..<b}"  using bounded_closed_interval[of a b]  using interval_open_subset_closed[of a b]  using bounded_subset[of "{a..b}" "{a<..<b}"]  by simplemma not_interval_univ:  fixes a :: "'a::ordered_euclidean_space"  shows "{a .. b} ≠ UNIV ∧ {a<..<b} ≠ UNIV"  using bounded_interval[of a b] by autolemma compact_interval:  fixes a :: "'a::ordered_euclidean_space"  shows "compact {a .. b}"  using bounded_closed_imp_seq_compact[of "{a..b}"] using bounded_interval[of a b]  by (auto simp: compact_eq_seq_compact_metric)lemma open_interval_midpoint:  fixes a :: "'a::ordered_euclidean_space"  assumes "{a<..<b} ≠ {}"  shows "((1/2) *⇩R (a + b)) ∈ {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 interval_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left)  }  then show ?thesis unfolding mem_interval by autoqedlemma open_closed_interval_convex:  fixes x :: "'a::ordered_euclidean_space"  assumes x: "x ∈ {a<..<b}"    and y: "y ∈ {a .. b}"    and e: "0 < e" "e ≤ 1"  shows "(e *⇩R x + (1 - e) *⇩R y) ∈ {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_interval using i      apply simp      using y unfolding mem_interval 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_interval        using i        apply simp        using y        unfolding mem_interval        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_interval by autoqedlemma closure_open_interval:  fixes a :: "'a::ordered_euclidean_space"  assumes "{a<..<b} ≠ {}"  shows "closure {a<..<b} = {a .. b}"proof -  have ab: "a < b"    using assms[unfolded interval_ne_empty]    apply (subst eucl_less)    apply auto    done  let ?c = "(1 / 2) *⇩R (a + b)"  {    fix x    assume as:"x ∈ {a .. b}"    def f ≡ "λn::nat. x + (inverse (real n + 1)) *⇩R (?c - x)"    {      fix n      assume fn: "f n < b --> a < 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 < b" and "a < f n"        using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *]        unfolding f_def by auto      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 LIMSEQ_def 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 {a<..<b}"      using as and open_interval_midpoint[OF assms]      unfolding closure_def      unfolding islimpt_sequential      by (cases "x=?c") auto  }  then show ?thesis    using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blastqedlemma bounded_subset_open_interval_symmetric:  fixes s::"('a::ordered_euclidean_space) set"  assumes "bounded s"  shows "∃a. s ⊆ {-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: eucl_less[where 'a='a])qedlemma bounded_subset_open_interval:  fixes s :: "('a::ordered_euclidean_space) set"  shows "bounded s ==> (∃a b. s ⊆ {a<..<b})"  by (auto dest!: bounded_subset_open_interval_symmetric)lemma bounded_subset_closed_interval_symmetric:  fixes s :: "('a::ordered_euclidean_space) set"  assumes "bounded s"  shows "∃a. s ⊆ {-a .. a}"proof -  obtain a where "s ⊆ {- a<..<a}"    using bounded_subset_open_interval_symmetric[OF assms] by auto  then show ?thesis    using interval_open_subset_closed[of "-a" a] by autoqedlemma bounded_subset_closed_interval:  fixes s :: "('a::ordered_euclidean_space) set"  shows "bounded s ==> ∃a b. s ⊆ {a .. b}"  using bounded_subset_closed_interval_symmetric[of s] by autolemma frontier_closed_interval:  fixes a b :: "'a::ordered_euclidean_space"  shows "frontier {a .. b} = {a .. b} - {a<..<b}"  unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..lemma frontier_open_interval:  fixes a b :: "'a::ordered_euclidean_space"  shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"proof (cases "{a<..<b} = {}")  case True  then show ?thesis    using frontier_empty by autonext  case False  then show ?thesis    unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval]    by autoqedlemma inter_interval_mixed_eq_empty:  fixes a :: "'a::ordered_euclidean_space"  assumes "{c<..<d} ≠ {}"  shows "{a<..<b} ∩ {c .. d} = {} <-> {a<..<b} ∩ {c<..<d} = {}"  unfolding closure_open_interval[OF assms, symmetric]  unfolding open_inter_closure_eq_empty[OF open_interval] ..lemma open_box: "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 .qedinstance euclidean_space ⊆ second_countable_topologyproof  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)qedinstance euclidean_space ⊆ polish_space ..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_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)  "is_interval {a<..<b}" (is ?th2) proof -  show ?th1 ?th2  unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff    by(meson order_trans le_less_trans less_le_trans less_trans)+ qedlemma is_interval_empty: "is_interval {}"  unfolding is_interval_def  by simplemma is_interval_univ: "is_interval UNIV"  unfolding is_interval_def  by simpsubsection {* 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)qedlemma 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 .qedlemma 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 .qedlemma 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 .qedlemma 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{* Instantiation for intervals on @{text ordered_euclidean_space} *}lemma eucl_lessThan_eq_halfspaces:  fixes a :: "'a::ordered_euclidean_space"  shows "{..<a} = (\<Inter>i∈Basis. {x. x • i < a • i})"  by (auto simp: eucl_less[where 'a='a])lemma eucl_greaterThan_eq_halfspaces:  fixes a :: "'a::ordered_euclidean_space"  shows "{a<..} = (\<Inter>i∈Basis. {x. a • i < x • i})"  by (auto simp: eucl_less[where 'a='a])lemma eucl_atMost_eq_halfspaces:  fixes a :: "'a::ordered_euclidean_space"  shows "{.. a} = (\<Inter>i∈Basis. {x. x • i ≤ a • i})"  by (auto simp: eucl_le[where 'a='a])lemma eucl_atLeast_eq_halfspaces:  fixes a :: "'a::ordered_euclidean_space"  shows "{a ..} = (\<Inter>i∈Basis. {x. a • i ≤ x • i})"  by (auto simp: eucl_le[where 'a='a])lemma open_eucl_lessThan[simp, intro]:  fixes a :: "'a::ordered_euclidean_space"  shows "open {..< a}"  by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)lemma open_eucl_greaterThan[simp, intro]:  fixes a :: "'a::ordered_euclidean_space"  shows "open {a <..}"  by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)lemma closed_eucl_atMost[simp, intro]:  fixes a :: "'a::ordered_euclidean_space"  shows "closed {.. a}"  unfolding eucl_atMost_eq_halfspaces  by (simp add: closed_INT closed_Collect_le)lemma closed_eucl_atLeast[simp, intro]:  fixes a :: "'a::ordered_euclidean_space"  shows "closed {a ..}"  unfolding eucl_atLeast_eq_halfspaces  by (simp add: closed_INT closed_Collect_le)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 autotext {* 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 autotext{* 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 autoqedlemma 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 {* 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  donelemma homeomorphic_sym: "s homeomorphic t <-> t homeomorphic s"  unfolding homeomorphic_def  unfolding homeomorphism_def  by blastlemma 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    doneqedlemma 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  donetext {* 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    doneqedlemma 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_on_intros)  donelemma 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]  apply auto  donelemma 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 autoqedlemma 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_on_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_on_intros      simp: dist_commute dist_norm pos_divide_le_eq mult_strict_left_mono)    doneqedtext{* "Isometry" (up to constant bounds) of injective linear map etc.           *}lemma cauchy_isometric:  fixes x :: "nat => 'a::euclidean_space"  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"]]        and e and mult_pos_pos[of e d]      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 autoqedlemma complete_isometric_image:  fixes f :: "'a::euclidean_space => 'b::euclidean_space"  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 autoqedlemma 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 only: divide_pos_pos)  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 only: divide_pos_pos)      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 autoqedlemma 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 autoqedsubsection {* 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" .qedlemma 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_zero_cong_left [OF finite_Basis d]) (auto simp add: x)    finally show "x ∈ span d"      unfolding euclidean_representation .  qedqed simptext{* 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] guess f .. note f = this  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 blastnext  assume "¬ finite A"  with n ≤ card A show ?thesis by forceqedlemma 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 guess f by (elim exE conjE) note f = this  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 autoqedlemma complete_subspace:  fixes s :: "('a::euclidean_space) set"  shows "subspace s ==> complete s"  using complete_eq_closed closed_subspace by autolemma 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 autoqedsubsection {* 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 inverse_eq_divide)lemma real_le_affinity: "0 < (m::'a::linordered_field) ==> (y ≤ m * x + c <-> inverse(m) * y + -(c / m) ≤ x)"  by (simp add: field_simps inverse_eq_divide)lemma real_affinity_lt: "0 < (m::'a::linordered_field) ==> (m * x + c < y <-> x < inverse(m) * y + -(c / m))"  by (simp add: field_simps inverse_eq_divide)lemma real_lt_affinity: "0 < (m::'a::linordered_field) ==> (y < m * x + c <-> inverse(m) * y + -(c / m) < x)"  by (simp add: field_simps inverse_eq_divide)lemma real_affinity_eq: "(m::'a::linordered_field) ≠ 0 ==> (m * x + c = y <-> x = inverse(m) * y + -(c / m))"  by (simp add: field_simps inverse_eq_divide)lemma real_eq_affinity: "(m::'a::linordered_field) ≠ 0 ==> (y = m * x + c  <-> inverse(m) * y + -(c / m) = x)"  by (simp add: field_simps inverse_eq_divide)lemma image_affinity_interval: fixes m::real  fixes a b c :: "'a::ordered_euclidean_space"  shows "(λx. m *⇩R x + c)  {a .. b} =    (if {a .. b} = {} then {}     else (if 0 ≤ m then {m *⇩R a + c .. m *⇩R b + c}     else {m *⇩R b + c .. m *⇩R a + c}))"proof (cases "m = 0")  case True  {    fix x    assume "x ≤ c" "c ≤ x"    then have "x = c"      unfolding eucl_le[where 'a='a]      apply -      apply (subst euclidean_eq_iff)      apply (auto intro: order_antisym)      done  }  moreover have "c ∈ {m *⇩R a + c..m *⇩R b + c}"    unfolding True by (auto simp add: eucl_le[where 'a='a])  ultimately show ?thesis using True by autonext  case False  {    fix y    assume "a ≤ y" "y ≤ b" "m > 0"    then have "m *⇩R a + c ≤ m *⇩R y + c" and "m *⇩R y + c ≤ m *⇩R b + c"      unfolding eucl_le[where 'a='a] by (auto simp: inner_simps)  }  moreover  {    fix y    assume "a ≤ y" "y ≤ b" "m < 0"    then have "m *⇩R b + c ≤ m *⇩R y + c" and "m *⇩R y + c ≤ m *⇩R a + c"      unfolding eucl_le[where 'a='a] by (auto simp add: mult_left_mono_neg inner_simps)  }  moreover  {    fix y    assume "m > 0" and "m *⇩R a + c ≤ y" and "y ≤ m *⇩R b + c"    then have "y ∈ (λx. m *⇩R x + c)  {a..b}"      unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]      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_simps)      done  }  moreover  {    fix y    assume "m *⇩R b + c ≤ y" "y ≤ m *⇩R a + c" "m < 0"    then have "y ∈ (λx. m *⇩R x + c)  {a..b}"      unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]      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_simps)      done  }  ultimately show ?thesis using False by autoqedlemma image_smult_interval:"(λx. m *⇩R (x::_::ordered_euclidean_space))  {a..b} =  (if {a..b} = {} then {} else if 0 ≤ m then {m *⇩R a..m *⇩R b} else {m *⇩R b..m *⇩R a})"  using image_affinity_interval[of m 0 a b] by autosubsection {* 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)      then have "0 < e * (1 - c) / d"        using e>0 and 1-c>0        using divide_pos_pos[of "e * (1 - c)" d] and mult_pos_pos[of e "1 - c"]        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        then have **: "d * (1 - c ^ (m - n)) / (1 - c) > 0"          using mult_pos_pos[OF d>0, of "1 - c ^ (m - n)"]          using divide_pos_pos[of "d * (1 - c ^ (m - n))" "1 - c"]          using 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 LIMSEQ_def, 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+qedsubsection {* 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 blastqeddeclare tendsto_const [intro] (* FIXME: move *)end