(* 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_Space imports Complex_Main "~~/src/HOL/Library/Countable_Set" "~~/src/HOL/Library/Glbs" "~~/src/HOL/Library/FuncSet" Linear_Algebra Norm_Arith begin lemma dist_0_norm: fixes x :: "'a::real_normed_vector" shows "dist 0 x = norm x" unfolding dist_norm by simp lemma dist_double: "dist x y < d / 2 \ dist x z < d / 2 \ dist y z < d" using dist_triangle[of y z x] by (simp add: dist_commute) (* LEGACY *) lemma lim_subseq: "subseq r \ s ----> l \ (s o r) ----> l" by (rule LIMSEQ_subseq_LIMSEQ) 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) subsection {* Topological Basis *} context topological_space begin definition "topological_basis B = ((\b\B. open b) \ (\x. open x \ (\B'. B' \ B \ \B' = x)))" lemma topological_basis: "topological_basis B = (\x. open x \ (\B'. B' \ B \ \B' = x))" unfolding topological_basis_def apply safe apply fastforce apply fastforce apply (erule_tac x="x" in allE) apply simp apply (rule_tac x="{x}" in exI) apply auto done lemma topological_basis_iff: assumes "\B'. B' \ B \ open B'" shows "topological_basis B \ (\O'. open O' \ (\x\O'. \B'\B. x \ B' \ B' \ O'))" (is "_ \ ?rhs") proof safe fix O' and x::'a assume H: "topological_basis B" "open O'" "x \ O'" hence "(\B'\B. \B' = O')" by (simp add: topological_basis_def) then obtain B' where "B' \ B" "O' = \B'" by auto thus "\B'\B. x \ B' \ B' \ O'" using H by auto next assume H: ?rhs show "topological_basis B" using assms unfolding topological_basis_def proof safe fix O'::"'a set" assume "open O'" with H obtain f where "\x\O'. f x \ B \ x \ f x \ f x \ O'" by (force intro: bchoice simp: Bex_def) thus "\B'\B. \B' = O'" by (auto intro: exI[where x="{f x |x. x \ O'}"]) qed qed lemma topological_basisI: assumes "\B'. B' \ B \ open B'" assumes "\O' x. open O' \ x \ O' \ \B'\B. x \ B' \ B' \ O'" shows "topological_basis B" using assms by (subst topological_basis_iff) auto lemma topological_basisE: fixes O' assumes "topological_basis B" assumes "open O'" assumes "x \ O'" obtains B' where "B' \ B" "x \ B'" "B' \ O'" proof atomize_elim from assms have "\B'. B'\B \ open B'" by (simp add: topological_basis_def) with topological_basis_iff assms show "\B'. B' \ B \ x \ B' \ B' \ O'" using assms by (simp add: Bex_def) qed lemma topological_basis_open: assumes "topological_basis B" assumes "X \ B" shows "open X" using assms by (simp add: topological_basis_def) lemma topological_basis_imp_subbasis: assumes B: "topological_basis B" shows "open = generate_topology B" proof (intro ext iffI) fix S :: "'a set" assume "open S" with B obtain B' where "B' \ B" "S = \B'" unfolding topological_basis_def by blast then show "generate_topology B S" by (auto intro: generate_topology.intros dest: topological_basis_open) next fix S :: "'a set" assume "generate_topology B S" then show "open S" by induct (auto dest: topological_basis_open[OF B]) qed lemma basis_dense: fixes B::"'a set set" and f::"'a set \ 'a" assumes "topological_basis B" assumes 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" "X \ {}" from topological_basisE[OF `topological_basis B` `open X` choosefrom_basis[OF `X \ {}`]] guess B' . note B' = this thus "\B'\B. f B' \ X" by (auto intro!: choosefrom_basis) qed end lemma topological_basis_prod: assumes A: "topological_basis A" and B: "topological_basis B" shows "topological_basis ((\(a, b). a \ b) ` (A \ B))" unfolding topological_basis_def proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric]) fix S :: "('a \ 'b) set" assume "open S" then show "\X\A \ B. (\(a,b)\X. a \ b) = S" proof (safe intro!: exI[of _ "{x\A \ B. fst x \ snd x \ S}"]) fix x y assume "(x, y) \ S" from open_prod_elim[OF `open S` this] obtain a b where a: "open a""x \ a" and b: "open b" "y \ b" and "a \ b \ S" by (metis mem_Sigma_iff) moreover from topological_basisE[OF A a] guess A0 . moreover from topological_basisE[OF B b] guess B0 . ultimately show "(x, y) \ (\(a, b)\{X \ A \ B. fst X \ snd X \ S}. a \ b)" by (intro UN_I[of "(A0, B0)"]) auto qed auto qed (metis A B topological_basis_open open_Times) subsection {* Countable Basis *} locale countable_basis = fixes B::"'a::topological_space set set" assumes is_basis: "topological_basis B" assumes countable_basis: "countable B" begin lemma open_countable_basis_ex: assumes "open X" shows "\B' \ B. X = Union B'" using assms countable_basis is_basis unfolding topological_basis_def by blast lemma open_countable_basisE: assumes "open X" obtains B' where "B' \ B" "X = Union B'" using assms open_countable_basis_ex by (atomize_elim) simp lemma countable_dense_exists: shows "\D::'a set. countable D \ (\X. open X \ X \ {} \ (\d \ D. d \ X))" proof - let ?f = "(\B'. SOME x. x \ B')" have "countable (?f ` B)" using countable_basis by simp with basis_dense[OF is_basis, of ?f] show ?thesis by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI) qed lemma countable_dense_setE: obtains D :: "'a set" where "countable D" "\X. open X \ X \ {} \ \d \ D. d \ X" using countable_dense_exists by blast end class first_countable_topology = topological_space + assumes first_countable_basis: "\A. countable A \ (\a\A. x \ a \ open a) \ (\S. open S \ x \ S \ (\a\A. a \ S))" lemma (in first_countable_topology) countable_basis_at_decseq: obtains A :: "nat \ 'a set" where "\i. open (A i)" "\i. x \ (A i)" "\S. open S \ x \ S \ eventually (\i. A i \ S) sequentially" proof atomize_elim from first_countable_basis[of x] obtain A where "countable A" and nhds: "\a. a \ A \ open a" "\a. a \ A \ x \ a" and incl: "\S. open S \ x \ S \ \a\A. a \ S" by auto then have "A \ {}" by auto with `countable A` have r: "A = range (from_nat_into A)" by auto def F \ "\n. \i\n. from_nat_into A i" show "\A. (\i. open (A i)) \ (\i. x \ A i) \ (\S. open S \ x \ S \ eventually (\i. A i \ S) sequentially)" proof (safe intro!: exI[of _ F]) fix i show "open (F i)" using nhds(1) r by (auto simp: F_def intro!: open_INT) show "x \ F i" using nhds(2) r by (auto simp: F_def) next fix S assume "open S" "x \ S" from incl[OF this] obtain i where "F i \ S" by (subst (asm) r) (auto simp: F_def) moreover have "\j. i \ j \ F j \ F i" by (auto simp: F_def) ultimately show "eventually (\i. F i \ S) sequentially" by (auto simp: eventually_sequentially) qed qed lemma (in first_countable_topology) first_countable_basisE: obtains A where "countable A" "\a. a \ A \ x \ a" "\a. a \ A \ open a" "\S. open S \ x \ S \ (\a\A. a \ S)" using first_countable_basis[of x] by atomize_elim auto lemma (in first_countable_topology) first_countable_basis_Int_stableE: obtains A where "countable A" "\a. a \ A \ x \ a" "\a. a \ A \ open a" "\S. open S \ x \ S \ (\a\A. a \ S)" "\a b. a \ A \ b \ A \ a \ b \ A" proof atomize_elim from first_countable_basisE[of x] guess A' . note A' = this def A \ "(\N. \((\n. from_nat_into A' n) ` N)) ` (Collect finite::nat set set)" thus "\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" thus "x \ a" "open a" using A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into) next let ?int = "\N. \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) thus "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 thus "\a\A. a \ S" using a A' by (intro bexI[where x=a]) (auto simp: A_def intro: image_eqI[where x="{to_nat_on A' a}"]) qed qed instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology proof 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::('a\'b) set set. countable A \ (\a\A. x \ a \ open a) \ (\S. open S \ x \ S \ (\a\A. a \ S))" proof (intro exI[of _ "(\(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" "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' . moreover with 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\(\(a, b). a \ b) ` (A \ B). a \ S" by (auto intro!: bexI[of _ "a \ b"] bexI[of _ a] bexI[of _ b]) qed (simp add: A B) qed instance metric_space \ first_countable_topology proof fix x :: 'a show "\A. countable A \ (\a\A. x \ a \ open a) \ (\S. open S \ x \ S \ (\a\A. a \ S))" proof (intro exI, safe) fix S assume "open S" "x \ S" then obtain r where "0 < r" "{y. dist x y < r} \ S" by (auto simp: open_dist dist_commute subset_eq) moreover from reals_Archimedean[OF `0 < r`] guess n .. moreover then have "{y. dist x y < inverse (Suc n)} \ {y. dist x y < r}" by (auto simp: inverse_eq_divide) ultimately show "\a\range (\n. {y. dist x y < inverse (Suc n)}). a \ S" by auto qed (auto intro: open_ball) qed class second_countable_topology = topological_space + assumes ex_countable_subbasis: "\B::'a::topological_space set set. countable B \ open = generate_topology B" begin lemma ex_countable_basis: "\B::'a set set. countable B \ topological_basis B" proof - from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B" by blast let ?B = "Inter ` {b. finite b \ b \ B }" show ?thesis proof (intro exI conjI) show "countable ?B" by (intro countable_image countable_Collect_finite_subset B) { fix S assume "open S" then have "\B'\{b. finite b \ b \ B}. (\b\B'. \b) = S" unfolding B proof induct case UNIV show ?case by (intro exI[of _ "{{}}"]) simp next case (Int a b) then obtain x y where x: "a = UNION x Inter" "\i. i \ x \ finite i \ i \ B" and y: "b = UNION y Inter" "\i. i \ y \ finite i \ i \ B" by blast show ?case unfolding x y Int_UN_distrib2 by (intro exI[of _ "{i \ j| i j. i \ x \ j \ y}"]) (auto dest: x(2) y(2)) next case (UN K) then have "\k\K. \B'\{b. finite b \ b \ B}. UNION B' Inter = k" by auto then guess k unfolding bchoice_iff .. then show "\B'\{b. finite b \ b \ B}. UNION B' Inter = \K" by (intro exI[of _ "UNION K k"]) auto next case (Basis S) then show ?case by (intro exI[of _ "{{S}}"]) auto qed then have "(\B'\Inter ` {b. finite b \ b \ B}. \B' = S)" unfolding subset_image_iff by blast } then show "topological_basis ?B" unfolding topological_space_class.topological_basis_def by (safe intro!: topological_space_class.open_Inter) (simp_all add: B generate_topology.Basis subset_eq) qed qed end sublocale second_countable_topology < countable_basis "SOME B. countable B \ topological_basis B" using someI_ex[OF ex_countable_basis] by unfold_locales safe instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology proof obtain A :: "'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto moreover obtain B :: "'b set set" where "countable B" "topological_basis B" using ex_countable_basis by auto ultimately show "\B::('a \ 'b) set set. countable B \ open = generate_topology B" by (auto intro!: exI[of _ "(\(a, b). a \ b) ` (A \ B)"] topological_basis_prod topological_basis_imp_subbasis) qed instance second_countable_topology \ first_countable_topology proof fix x :: 'a def B \ "SOME B::'a set set. countable B \ topological_basis B" then have B: "countable B" "topological_basis B" using countable_basis is_basis by (auto simp: countable_basis is_basis) then show "\A. countable A \ (\a\A. x \ a \ open a) \ (\S. open S \ x \ S \ (\a\A. a \ S))" by (intro exI[of _ "{b\B. x \ b}"]) (fastforce simp: topological_space_class.topological_basis_def) qed subsection {* Polish spaces *} text {* Textbooks define Polish spaces as completely metrizable. We assume the topology to be complete for a given metric. *} class polish_space = complete_space + second_countable_topology subsection {* General notion of a topology as a value *} definition "istopology L \ L {} \ (\S T. L S \ L T \ L (S \ T)) \ (\K. Ball K L \ L (\ K))" typedef 'a topology = "{L::('a set) \ bool. istopology L}" morphisms "openin" "topology" unfolding istopology_def by blast lemma istopology_open_in[intro]: "istopology(openin U)" using openin[of U] by blast lemma topology_inverse': "istopology U \ openin (topology U) = U" using topology_inverse[unfolded mem_Collect_eq] . lemma topology_inverse_iff: "istopology U \ openin (topology U) = U" using topology_inverse[of U] istopology_open_in[of "topology U"] by auto lemma topology_eq: "T1 = T2 \ (\S. openin T1 S \ openin T2 S)" proof- { assume "T1=T2" hence "\S. openin T1 S \ openin T2 S" by simp } moreover { assume H: "\S. openin T1 S \ openin T2 S" hence "openin T1 = openin T2" by (simp add: fun_eq_iff) hence "topology (openin T1) = topology (openin T2)" by simp hence "T1 = T2" unfolding openin_inverse . } ultimately show ?thesis by blast qed text{* Infer the "universe" from union of all sets in the topology. *} definition "topspace T = \{S. openin T S}" subsubsection {* Main properties of open sets *} 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 (\K)" using openin[of U] unfolding istopology_def mem_Collect_eq by fast+ lemma openin_subset[intro]: "openin U S \ S \ topspace U" unfolding topspace_def by blast lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses) lemma openin_Int[intro]: "openin U S \ openin U T \ openin U (S \ T)" using openin_clauses by simp lemma openin_Union[intro]: "(\S \K. openin U S) \ openin U (\ K)" using openin_clauses by simp lemma openin_Un[intro]: "openin U S \ openin U T \ openin U (S \ T)" using openin_Union[of "{S,T}" U] by auto lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def) lemma openin_subopen: "openin U S \ (\x \ S. \T. openin U T \ x \ T \ T \ S)" (is "?lhs \ ?rhs") proof assume ?lhs then show ?rhs by auto next assume H: ?rhs let ?t = "\{T. openin U T \ T \ S}" have "openin U ?t" by (simp add: openin_Union) also have "?t = S" using H by auto finally show "openin U S" . qed subsubsection {* Closed sets *} definition "closedin U S \ S \ topspace U \ openin U (topspace U - S)" lemma closedin_subset: "closedin U S \ S \ topspace U" by (metis closedin_def) lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def) lemma closedin_topspace[intro,simp]: "closedin U (topspace U)" by (simp add: closedin_def) lemma closedin_Un[intro]: "closedin U S \ closedin U T \ closedin U (S \ T)" by (auto simp add: Diff_Un closedin_def) lemma Diff_Inter[intro]: "A - \S = \ {A - s|s. s\S}" by auto lemma closedin_Inter[intro]: assumes Ke: "K \ {}" and Kc: "\S \K. closedin U S" shows "closedin U (\ K)" using Ke Kc unfolding closedin_def Diff_Inter by auto lemma closedin_Int[intro]: "closedin U S \ closedin U T \ closedin U (S \ T)" using closedin_Inter[of "{S,T}" U] by auto lemma Diff_Diff_Int: "A - (A - B) = A \ B" by blast lemma openin_closedin_eq: "openin U S \ S \ topspace U \ closedin U (topspace U - S)" apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2) apply (metis openin_subset subset_eq) done lemma openin_closedin: "S \ topspace U \ (openin U S \ closedin U (topspace U - S))" by (simp add: openin_closedin_eq) lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)" proof- have "S - T = S \ (topspace U - T)" using openin_subset[of U S] oS cT by (auto simp add: topspace_def openin_subset) then show ?thesis using oS cT by (auto simp add: closedin_def) qed lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)" proof- have "S - T = S \ (topspace U - T)" using closedin_subset[of U S] oS cT by (auto simp add: topspace_def ) then show ?thesis using oS cT by (auto simp add: openin_closedin_eq) qed subsubsection {* Subspace topology *} definition "subtopology U V = topology (\T. \S. T = S \ V \ openin U S)" lemma istopology_subtopology: "istopology (\T. \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 = (\S. S \ V) ` Collect (openin U)" apply (rule set_eqI) apply (simp add: Ball_def image_iff) by metis from K[unfolded th0 subset_image_iff] obtain Sk where Sk: "Sk \ Collect (openin U)" "K = (\S. S \ V) ` Sk" by blast have "\K = (\Sk) \ V" using Sk by auto moreover have "openin U (\ Sk)" using Sk by (auto simp add: subset_eq) ultimately have "?L (\K)" by blast} ultimately show ?thesis unfolding subset_eq mem_Collect_eq istopology_def by blast qed lemma openin_subtopology: "openin (subtopology U V) S \ (\ T. (openin U T) \ (S = T \ V))" unfolding subtopology_def topology_inverse'[OF istopology_subtopology] by auto lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \ V" by (auto simp add: topspace_def openin_subtopology) lemma closedin_subtopology: "closedin (subtopology U V) S \ (\T. closedin U T \ S = T \ V)" unfolding closedin_def topspace_subtopology apply (simp add: openin_subtopology) apply (rule iffI) apply clarify apply (rule_tac x="topspace U - T" in exI) by auto lemma 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 done lemma subtopology_superset: assumes UV: "topspace U \ V" shows "subtopology U V = U" proof- {fix S {fix T assume T: "openin U T" "S = T \ V" from T openin_subset[OF T(1)] UV have eq: "S = T" by blast have "openin U S" unfolding eq using T by blast} moreover {assume S: "openin U S" hence "\T. openin U T \ S = T \ V" using openin_subset[OF S] UV by auto} ultimately have "(\T. openin U T \ S = T \ V) \ openin U S" by blast} then show ?thesis unfolding topology_eq openin_subtopology by blast qed lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U" by (simp add: subtopology_superset) lemma subtopology_UNIV[simp]: "subtopology U UNIV = U" by (simp add: subtopology_superset) subsubsection {* The standard Euclidean topology *} definition euclidean :: "'a::topological_space topology" where "euclidean = topology open" lemma open_openin: "open S \ openin euclidean S" unfolding euclidean_def apply (rule cong[where x=S and y=S]) apply (rule topology_inverse[symmetric]) apply (auto simp add: istopology_def) done lemma topspace_euclidean: "topspace euclidean = UNIV" apply (simp add: topspace_def) apply (rule set_eqI) by (auto simp add: open_openin[symmetric]) lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S" by (simp add: topspace_euclidean topspace_subtopology) lemma closed_closedin: "closed S \ closedin euclidean S" by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV) lemma open_subopen: "open S \ (\x\S. \T. open T \ x \ T \ T \ S)" by (simp add: open_openin openin_subopen[symmetric]) text {* Basic "localization" results are handy for connectedness. *} lemma openin_open: "openin (subtopology euclidean U) S \ (\T. open T \ (S = U \ T))" by (auto simp add: openin_subtopology open_openin[symmetric]) lemma openin_open_Int[intro]: "open S \ openin (subtopology euclidean U) (U \ S)" by (auto simp add: openin_open) lemma open_openin_trans[trans]: "open S \ open T \ T \ S \ openin (subtopology euclidean S) T" by (metis Int_absorb1 openin_open_Int) lemma open_subset: "S \ T \ open S \ openin (subtopology euclidean T) S" by (auto simp add: openin_open) lemma closedin_closed: "closedin (subtopology euclidean U) S \ (\T. closed T \ S = U \ T)" by (simp add: closedin_subtopology closed_closedin Int_ac) lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \ S)" by (metis closedin_closed) lemma closed_closedin_trans: "closed S \ closed T \ T \ S \ closedin (subtopology euclidean S) T" apply (subgoal_tac "S \ T = T" ) apply auto apply (frule closedin_closed_Int[of T S]) by simp lemma closed_subset: "S \ T \ closed S \ closedin (subtopology euclidean T) S" by (auto simp add: closedin_closed) lemma openin_euclidean_subtopology_iff: fixes S U :: "'a::metric_space set" shows "openin (subtopology euclidean U) S \ S \ U \ (\x\S. \e>0. \x'\U. dist x' x < e \ x'\ S)" (is "?lhs \ ?rhs") proof assume ?lhs thus ?rhs unfolding openin_open open_dist by blast next def T \ "{x. \a\S. \d>0. (\y\U. dist y a < d \ y \ S) \ dist x a < d}" have 1: "\x\T. \e>0. \y. dist y x < e \ y \ T" unfolding T_def apply clarsimp apply (rule_tac x="d - dist x a" in exI) apply (clarsimp simp add: less_diff_eq) apply (erule rev_bexI) apply (rule_tac x=d in exI, clarify) apply (erule le_less_trans [OF dist_triangle]) done assume ?rhs hence 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 fast qed text {* These "transitivity" results are handy too *} lemma openin_trans[trans]: "openin (subtopology euclidean T) S \ openin (subtopology euclidean U) T \ openin (subtopology euclidean U) S" unfolding open_openin openin_open by blast lemma openin_open_trans: "openin (subtopology euclidean T) S \ open T \ open S" by (auto simp add: openin_open intro: openin_trans) lemma closedin_trans[trans]: "closedin (subtopology euclidean T) S \ closedin (subtopology euclidean U) T ==> closedin (subtopology euclidean U) S" by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc) lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \ closed T \ closed S" by (auto simp add: closedin_closed intro: closedin_trans) subsection {* Open and closed balls *} definition ball :: "'a::metric_space \ real \ 'a set" where "ball x e = {y. dist x y < e}" definition cball :: "'a::metric_space \ real \ 'a set" where "cball x e = {y. dist x y \ e}" lemma mem_ball [simp]: "y \ ball x e \ dist x y < e" by (simp add: ball_def) lemma mem_cball [simp]: "y \ cball x e \ dist x y \ e" by (simp add: cball_def) lemma mem_ball_0: fixes x :: "'a::real_normed_vector" shows "x \ ball 0 e \ norm x < e" by (simp add: dist_norm) lemma mem_cball_0: fixes x :: "'a::real_normed_vector" shows "x \ cball 0 e \ norm x \ e" by (simp add: dist_norm) lemma centre_in_ball: "x \ ball x e \ 0 < e" by simp lemma centre_in_cball: "x \ cball x e \ 0 \ e" by simp lemma ball_subset_cball[simp,intro]: "ball x e \ cball x e" by (simp add: subset_eq) lemma subset_ball[intro]: "d <= e ==> ball x d \ ball x e" by (simp add: subset_eq) lemma subset_cball[intro]: "d <= e ==> cball x d \ cball x e" by (simp add: subset_eq) lemma ball_max_Un: "ball a (max r s) = ball a r \ ball a s" by (simp add: set_eq_iff) arith lemma ball_min_Int: "ball a (min r s) = ball a r \ ball a s" by (simp add: set_eq_iff) lemma diff_less_iff: "(a::real) - b > 0 \ a > b" "(a::real) - b < 0 \ a < b" "a - b < c \ a < c +b" "a - b > c \ a > c +b" by arith+ lemma diff_le_iff: "(a::real) - b \ 0 \ a \ b" "(a::real) - b \ 0 \ a \ b" "a - b \ c \ a \ c +b" "a - b \ c \ a \ c +b" by arith+ lemma open_ball[intro, simp]: "open (ball x e)" 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) by arith lemma open_contains_ball: "open S \ (\x\S. \e>0. ball x e \ S)" unfolding open_dist subset_eq mem_ball Ball_def dist_commute .. lemma openE[elim?]: assumes "open S" "x\S" obtains e where "e>0" "ball x e \ S" using assms unfolding open_contains_ball by auto lemma open_contains_ball_eq: "open S \ \x. x\S \ (\e>0. ball x e \ S)" by (metis open_contains_ball subset_eq centre_in_ball) lemma ball_eq_empty[simp]: "ball x e = {} \ e \ 0" unfolding mem_ball set_eq_iff apply (simp add: not_less) by (metis zero_le_dist order_trans dist_self) lemma ball_empty[intro]: "e \ 0 ==> ball x e = {}" by simp lemma euclidean_dist_l2: fixes x y :: "'a :: euclidean_space" shows "dist x y = setL2 (\i. 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 "0 < e" shows "\a b. (\i\Basis. a \ i \ \ \ b \ i \ \ ) \ x \ box a b \ box a b \ ball x e" proof - def e' \ "e / (2 * sqrt (real (DIM ('a))))" then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos simp: DIM_positive) have "\i. \y. y \ \ \ 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 \ \ \ 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 *\<^sub>R i" and ?b = "\i\Basis. b i *\<^sub>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))\)" 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))\ < (e/sqrt (real (DIM('a))))\" by (rule power_strict_mono) auto then show "(dist (x \ i) (y \ i))\ < e\ / real DIM('a)" by (simp add: power_divide) qed auto also have "\ = e" using `0 < e` by (simp add: real_eq_of_nat) finally show "y \ ball x e" by (auto simp: ball_def) qed (insert a b, auto simp: box_def) qed lemma open_UNION_box: fixes M :: "'a\euclidean_space set" assumes "open M" defines "a' \ \f :: 'a \ real \ real. (\(i::'a)\Basis. fst (f i) *\<^sub>R i)" defines "b' \ \f :: 'a \ real \ real. (\(i::'a)\Basis. snd (f i) *\<^sub>R i)" defines "I \ {f\Basis \\<^isub>E \ \ \. box (a' f) (b' f) \ M}" shows "M = (\f\I. box (a' f) (b' f))" proof safe 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 then obtain a b where ab: "x \ box a b" "\i \ Basis. a \ i \ \" "\i\Basis. b \ i \ \" "box a b \ ball x e" using rational_boxes[OF e(1)] by metis ultimately show "x \ (\f\I. box (a' f) (b' f))" by (intro UN_I[of "\i\Basis. (a \ i, b \ i)"]) (auto simp: euclidean_representation I_def a'_def b'_def) qed (auto simp: I_def) subsection{* Connectedness *} definition "connected S \ ~(\e1 e2. open e1 \ open e2 \ S \ (e1 \ e2) \ (e1 \ e2 \ S = {}) \ ~(e1 \ S = {}) \ ~(e2 \ S = {}))" lemma connected_local: "connected S \ ~(\e1 e2. openin (subtopology euclidean S) e1 \ openin (subtopology euclidean S) e2 \ S \ e1 \ e2 \ e1 \ e2 = {} \ ~(e1 = {}) \ ~(e2 = {}))" unfolding connected_def openin_open by (safe, blast+) lemma exists_diff: fixes P :: "'a set \ bool" shows "(\S. P(- S)) \ (\S. P S)" (is "?lhs \ ?rhs") proof- {assume "?lhs" hence ?rhs by blast } moreover {fix S assume H: "P S" have "S = - (- S)" by auto with H have "P (- (- S))" by metis } ultimately show ?thesis by metis qed lemma connected_clopen: "connected S \ (\T. openin (subtopology euclidean S) T \ closedin (subtopology euclidean S) T \ T = {} \ T = S)" (is "?lhs \ ?rhs") proof- have " \ connected S \ (\e1 e2. open e1 \ open (- e2) \ S \ e1 \ (- e2) \ e1 \ (- e2) \ S = {} \ e1 \ S \ {} \ (- e2) \ S \ {})" unfolding connected_def openin_open closedin_closed apply (subst exists_diff) by blast hence 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) by metis have th1: "?rhs \ \ (\t' t. closed t'\t = S\t' \ t\{} \ t\S \ (\t'. open t' \ t = S \ t'))" (is "_ \ \ (\t' t. ?Q t' t)") unfolding connected_def openin_open closedin_closed by auto {fix e2 {fix e1 have "?P e2 e1 \ (\t. closed e2 \ t = S\e2 \ open e1 \ t = S\e1 \ t\{} \ t\S)" by auto} then have "(\e1. ?P e2 e1) \ (\t. ?Q e2 t)" by metis} then have "\e2. (\e1. ?P e2 e1) \ (\t. ?Q e2 t)" by blast then show ?thesis unfolding th0 th1 by simp qed lemma connected_empty[simp, intro]: "connected {}" by (simp add: connected_def) subsection{* Limit points *} definition (in topological_space) islimpt:: "'a \ 'a set \ bool" (infixr "islimpt" 60) where "x islimpt S \ (\T. x\T \ open T \ (\y\S. y\T \ y\x))" lemma islimptI: assumes "\T. x \ T \ open T \ \y\S. y \ T \ y \ x" shows "x islimpt S" using assms unfolding islimpt_def by auto lemma islimptE: assumes "x islimpt S" and "x \ T" and "open T" obtains y where "y \ S" and "y \ T" and "y \ x" using assms unfolding islimpt_def by auto lemma islimpt_iff_eventually: "x islimpt S \ \ eventually (\y. y \ S) (at x)" unfolding islimpt_def eventually_at_topological by auto lemma islimpt_subset: "\x islimpt S; S \ T\ \ x islimpt T" unfolding islimpt_def by fast lemma islimpt_approachable: fixes x :: "'a::metric_space" shows "x islimpt S \ (\e>0. \x'\S. x' \ x \ dist x' x < e)" unfolding islimpt_iff_eventually eventually_at by fast lemma islimpt_approachable_le: fixes x :: "'a::metric_space" shows "x islimpt S \ (\e>0. \x'\ S. x' \ x \ dist x' x <= e)" unfolding islimpt_approachable using approachable_lt_le [where f="\y. dist y x" and P="\y. 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) text {* A perfect space has no isolated points. *} lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV" unfolding islimpt_UNIV_iff by (rule not_open_singleton) lemma perfect_choose_dist: fixes x :: "'a::{perfect_space, metric_space}" shows "0 < r \ \a. a \ x \ dist a x < r" using islimpt_UNIV [of x] by (simp add: islimpt_approachable) lemma closed_limpt: "closed S \ (\x. x islimpt S \ x \ S)" unfolding closed_def apply (subst open_subopen) apply (simp add: islimpt_def subset_eq) by (metis ComplE ComplI) lemma islimpt_EMPTY[simp]: "\ x islimpt {}" unfolding islimpt_def by auto lemma finite_set_avoid: fixes a :: "'a::metric_space" assumes fS: "finite S" shows "\d>0. \x\S. x \ a \ d <= dist a x" proof(induct rule: finite_induct[OF fS]) case 1 thus ?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 {assume "x = a" hence ?case using d by auto } moreover {assume xa: "x\a" let ?d = "min d (dist a x)" have dp: "?d > 0" using xa d(1) using dist_nz by auto from d have d': "\x\F. x\a \ ?d \ dist a x" by auto with dp xa have ?case by(auto intro!: exI[where x="?d"]) } ultimately show ?case by blast qed lemma islimpt_Un: "x islimpt (S \ T) \ x islimpt S \ x islimpt T" by (simp add: islimpt_iff_eventually eventually_conj_iff) lemma discrete_imp_closed: fixes S :: "'a::metric_space set" assumes e: "0 < e" and d: "\x \ S. \y \ S. dist y x < e \ y = x" shows "closed S" proof- {fix x assume C: "\e>0. \x'\S. x' \ x \ dist x' x < e" from e have e2: "e/2 > 0" by arith from C[rule_format, OF e2] obtain y where y: "y \ S" "y\x" "dist y x < e/2" by blast let ?m = "min (e/2) (dist x y) " from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym]) from C[rule_format, OF mp] obtain z where z: "z \ S" "z\x" "dist z x < ?m" by blast have th: "dist z y < e" using z y by (intro dist_triangle_lt [where z=x], simp) from d[rule_format, OF y(1) z(1) th] y z have False by (auto simp add: dist_commute)} then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a]) qed subsection {* Interior of a Set *} definition "interior S = \{T. open T \ T \ S}" lemma interiorI [intro?]: assumes "open T" and "x \ T" and "T \ S" shows "x \ interior S" using assms unfolding interior_def by fast lemma interiorE [elim?]: assumes "x \ interior S" obtains T where "open T" and "x \ T" and "T \ S" using assms unfolding interior_def by fast lemma open_interior [simp, intro]: "open (interior S)" by (simp add: interior_def open_Union) lemma interior_subset: "interior S \ S" by (auto simp add: interior_def) lemma interior_maximal: "T \ S \ open T \ T \ interior S" by (auto simp add: interior_def) lemma interior_open: "open S \ interior S = S" by (intro equalityI interior_subset interior_maximal subset_refl) lemma interior_eq: "interior S = S \ open S" by (metis open_interior interior_open) lemma open_subset_interior: "open S \ S \ interior T \ S \ T" by (metis interior_maximal interior_subset subset_trans) lemma interior_empty [simp]: "interior {} = {}" using open_empty by (rule interior_open) lemma interior_UNIV [simp]: "interior UNIV = UNIV" using open_UNIV by (rule interior_open) lemma interior_interior [simp]: "interior (interior S) = interior S" using open_interior by (rule interior_open) lemma interior_mono: "S \ T \ interior S \ interior T" by (auto simp add: interior_def) lemma interior_unique: assumes "T \ S" and "open T" assumes "\T'. T' \ S \ open T' \ T' \ T" shows "interior S = T" by (intro equalityI assms interior_subset open_interior interior_maximal) lemma interior_inter [simp]: "interior (S \ T) = interior S \ interior T" by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1 Int_lower2 interior_maximal interior_subset open_Int open_interior) lemma mem_interior: "x \ interior S \ (\e>0. ball x e \ S)" using open_contains_ball_eq [where S="interior S"] by (simp add: open_subset_interior) lemma interior_limit_point [intro]: fixes x :: "'a::perfect_space" assumes x: "x \ interior S" shows "x islimpt S" using x islimpt_UNIV [of x] unfolding interior_def islimpt_def apply (clarsimp, rename_tac T T') apply (drule_tac x="T \ T'" in spec) apply (auto simp add: open_Int) done lemma interior_closed_Un_empty_interior: assumes cS: "closed S" and iT: "interior T = {}" shows "interior (S \ T) = interior S" proof show "interior S \ interior (S \ T)" by (rule interior_mono, rule Un_upper1) next show "interior (S \ T) \ interior S" proof fix x assume "x \ interior (S \ T)" then obtain R where "open R" "x \ R" "R \ S \ T" .. show "x \ interior S" proof (rule ccontr) assume "x \ interior S" with `x \ R` `open R` obtain y where "y \ R - S" unfolding interior_def by fast from `open R` `closed S` have "open (R - S)" by (rule open_Diff) from `R \ S \ T` have "R - S \ T" by fast from `y \ R - S` `open (R - S)` `R - S \ T` `interior T = {}` show "False" unfolding interior_def by fast qed qed qed lemma interior_Times: "interior (A \ B) = interior A \ interior B" proof (rule interior_unique) show "interior A \ interior B \ A \ B" by (intro Sigma_mono interior_subset) show "open (interior A \ interior B)" by (intro open_Times open_interior) fix T assume "T \ A \ B" and "open T" thus "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 hence "open C" "open D" "C \ A" "D \ B" "x \ C" "y \ D" using `T \ A \ B` by auto thus "x \ interior A" and "y \ interior B" by (auto intro: interiorI) qed qed subsection {* Closure of a Set *} definition "closure S = S \ {x | x. x islimpt S}" lemma interior_closure: "interior S = - (closure (- S))" unfolding interior_def closure_def islimpt_def by auto lemma closure_interior: "closure S = - interior (- S)" unfolding interior_closure by simp lemma closed_closure[simp, intro]: "closed (closure S)" unfolding closure_interior by (simp add: closed_Compl) lemma closure_subset: "S \ closure S" unfolding closure_def by simp lemma closure_hull: "closure S = closed hull S" unfolding hull_def closure_interior interior_def by auto lemma closure_eq: "closure S = S \ closed S" unfolding closure_hull using closed_Inter by (rule hull_eq) lemma closure_closed [simp]: "closed S \ closure S = S" unfolding closure_eq . lemma closure_closure [simp]: "closure (closure S) = closure S" unfolding closure_hull by (rule hull_hull) lemma closure_mono: "S \ T \ closure S \ closure T" unfolding closure_hull by (rule hull_mono) lemma closure_minimal: "S \ T \ closed T \ closure S \ T" unfolding closure_hull by (rule hull_minimal) lemma closure_unique: assumes "S \ T" and "closed T" assumes "\T'. S \ T' \ closed T' \ T \ T'" shows "closure S = T" using assms unfolding closure_hull by (rule hull_unique) lemma closure_empty [simp]: "closure {} = {}" using closed_empty by (rule closure_closed) lemma closure_UNIV [simp]: "closure UNIV = UNIV" using closed_UNIV by (rule closure_closed) lemma closure_union [simp]: "closure (S \ T) = closure S \ closure T" unfolding closure_interior by simp lemma closure_eq_empty: "closure S = {} \ S = {}" using closure_empty closure_subset[of S] by blast lemma closure_subset_eq: "closure S \ S \ closed S" using closure_eq[of S] closure_subset[of S] by simp lemma open_inter_closure_eq_empty: "open S \ (S \ closure T) = {} \ S \ T = {}" using open_subset_interior[of S "- T"] using interior_subset[of "- T"] unfolding closure_interior by auto lemma open_inter_closure_subset: "open S \ (S \ (closure T)) \ closure(S \ T)" proof fix x assume as: "open S" "x \ S \ closure T" { assume *:"x islimpt T" have "x islimpt (S \ T)" proof (rule islimptI) fix A assume "x \ A" "open A" with as have "x \ A \ S" "open (A \ S)" by (simp_all add: open_Int) with * obtain y where "y \ T" "y \ A \ S" "y \ x" by (rule islimptE) hence "y \ S \ T" "y \ A \ y \ x" by simp_all thus "\y\(S \ T). y \ A \ y \ x" .. qed } then show "x \ closure (S \ T)" using as unfolding closure_def by blast qed lemma closure_complement: "closure (- S) = - interior S" unfolding closure_interior by simp lemma interior_complement: "interior (- S) = - closure S" unfolding closure_interior by simp lemma closure_Times: "closure (A \ B) = closure A \ closure B" proof (rule closure_unique) show "A \ B \ closure A \ closure B" by (intro Sigma_mono closure_subset) show "closed (closure A \ closure B)" by (intro closed_Times closed_closure) fix T assume "A \ B \ T" and "closed T" thus "closure A \ closure B \ T" apply (simp add: closed_def open_prod_def, clarify) apply (rule ccontr) apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D) apply (simp add: closure_interior interior_def) apply (drule_tac x=C in spec) apply (drule_tac x=D in spec) apply auto done qed subsection {* Frontier (aka boundary) *} definition "frontier S = closure S - interior S" lemma frontier_closed: "closed(frontier S)" by (simp add: frontier_def closed_Diff) lemma frontier_closures: "frontier S = (closure S) \ (closure(- S))" by (auto simp add: frontier_def interior_closure) lemma frontier_straddle: fixes a :: "'a::metric_space" shows "a \ frontier S \ (\e>0. (\x\S. dist a x < e) \ (\x. x \ S \ dist a x < e))" unfolding frontier_def closure_interior by (auto simp add: mem_interior subset_eq ball_def) lemma frontier_subset_closed: "closed S \ frontier S \ S" by (metis frontier_def closure_closed Diff_subset) lemma frontier_empty[simp]: "frontier {} = {}" by (simp add: frontier_def) lemma frontier_subset_eq: "frontier S \ S \ closed S" proof- { assume "frontier S \ S" hence "closure S \ S" using interior_subset unfolding frontier_def by auto hence "closed S" using closure_subset_eq by auto } thus ?thesis using frontier_subset_closed[of S] .. qed lemma frontier_complement: "frontier(- S) = frontier S" by (auto simp add: frontier_def closure_complement interior_complement) lemma frontier_disjoint_eq: "frontier S \ S = {} \ open S" using frontier_complement frontier_subset_eq[of "- S"] unfolding open_closed by auto subsection {* Filters and the ``eventually true'' quantifier *} definition indirection :: "'a::real_normed_vector \ 'a \ 'a filter" (infixr "indirection" 70) where "a indirection v = (at a) within {b. \c\0. b - a = scaleR c v}" text {* Identify Trivial limits, where we can't approach arbitrarily closely. *} lemma trivial_limit_within: shows "trivial_limit (at a within S) \ \ a islimpt S" proof assume "trivial_limit (at a within S)" thus "\ a islimpt S" unfolding trivial_limit_def unfolding eventually_within eventually_at_topological unfolding islimpt_def apply (clarsimp simp add: set_eq_iff) apply (rename_tac T, rule_tac x=T in exI) apply (clarsimp, drule_tac x=y in bspec, simp_all) done next assume "\ a islimpt S" thus "trivial_limit (at a within S)" unfolding trivial_limit_def unfolding eventually_within eventually_at_topological unfolding islimpt_def apply clarsimp apply (rule_tac x=T in exI) apply auto done qed lemma trivial_limit_at_iff: "trivial_limit (at a) \ \ a islimpt UNIV" using trivial_limit_within [of a UNIV] by simp lemma trivial_limit_at: fixes a :: "'a::perfect_space" shows "\ trivial_limit (at a)" by (rule at_neq_bot) lemma trivial_limit_at_infinity: "\ trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)" unfolding trivial_limit_def eventually_at_infinity apply clarsimp apply (subgoal_tac "\x::'a. x \ 0", clarify) apply (rule_tac x="scaleR (b / norm x) x" in exI, simp) apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def]) apply (drule_tac x=UNIV in spec, simp) done text {* Some property holds "sufficiently close" to the limit point. *} lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *) "eventually P (at a) \ (\d>0. \x. 0 < dist x a \ dist x a < d \ P x)" unfolding eventually_at dist_nz by auto lemma eventually_within: (* FIXME: this replaces Limits.eventually_within *) "eventually P (at a within S) \ (\d>0. \x\S. 0 < dist x a \ dist x a < d \ P x)" by (rule eventually_within_less) lemma eventually_happens: "eventually P net ==> trivial_limit net \ (\x. P x)" unfolding trivial_limit_def by (auto elim: eventually_rev_mp) lemma trivial_limit_eventually: "trivial_limit net \ eventually P net" by simp lemma trivial_limit_eq: "trivial_limit net \ (\P. eventually P net)" by (simp add: filter_eq_iff) text{* Combining theorems for "eventually" *} lemma eventually_rev_mono: "eventually P net \ (\x. P x \ Q x) \ eventually Q net" using eventually_mono [of P Q] by fast lemma not_eventually: "(\x. \ P x ) \ ~(trivial_limit net) ==> ~(eventually (\x. P x) net)" by (simp add: eventually_False) subsection {* Limits *} text{* Notation Lim to avoid collition with lim defined in analysis *} definition Lim :: "'a filter \ ('a \ 'b::t2_space) \ 'b" where "Lim A f = (THE l. (f ---> l) A)" lemma Lim: "(f ---> l) net \ trivial_limit net \ (\e>0. eventually (\x. dist (f x) l < e) net)" unfolding tendsto_iff trivial_limit_eq by auto text{* Show that they yield usual definitions in the various cases. *} lemma Lim_within_le: "(f ---> l)(at a within S) \ (\e>0. \d>0. \x\S. 0 < dist x a \ dist x a <= d \ dist (f x) l < e)" by (auto simp add: tendsto_iff eventually_within_le) 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_within) 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_at) 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 (\x. f x = l) net \ (f ---> l) net" by (rule topological_tendstoI, auto elim: eventually_rev_mono) text{* The expected monotonicity property. *} lemma Lim_within_empty: "(f ---> l) (net within {})" unfolding tendsto_def Limits.eventually_within by simp lemma Lim_within_subset: "(f ---> l) (net within S) \ T \ S \ (f ---> l) (net within T)" unfolding tendsto_def Limits.eventually_within by (auto elim!: eventually_elim1) lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)" shows "(f ---> l) (net within (S \ T))" using assms unfolding tendsto_def Limits.eventually_within apply clarify apply (drule spec, drule (1) mp, drule (1) mp) apply (drule spec, drule (1) mp, drule (1) mp) apply (auto elim: eventually_elim2) done lemma Lim_Un_univ: "(f ---> l) (net within S) \ (f ---> l) (net within T) \ S \ T = UNIV ==> (f ---> l) net" by (metis Lim_Un within_UNIV) text{* Interrelations between restricted and unrestricted limits. *} lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)" (* FIXME: rename *) unfolding tendsto_def Limits.eventually_within apply (clarify, drule spec, drule (1) mp, drule (1) mp) by (auto elim!: eventually_elim1) 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" "x \ A" "\y\A. y \ x \ y \ S \ P y" unfolding Limits.eventually_within Limits.eventually_at_topological by auto with T have "open (A \ T)" "x \ A \ T" "\y\(A \ T). y \ x \ P y" by auto then have "?rhs" unfolding Limits.eventually_at_topological by auto } moreover { assume "?rhs" hence "?lhs" unfolding Limits.eventually_within by (auto elim: eventually_elim1) } ultimately show "?thesis" .. qed lemma at_within_interior: "x \ interior S \ at x within S = at x" by (simp add: filter_eq_iff eventually_within_interior) lemma at_within_open: "\x \ S; open S\ \ at x within S = at x" by (simp only: at_within_interior interior_open) lemma Lim_within_open: fixes f :: "'a::topological_space \ 'b::topological_space" assumes"a \ S" "open S" shows "(f ---> l)(at a within S) \ (f ---> l)(at a)" using assms by (simp only: at_within_open) lemma Lim_within_LIMSEQ: fixes a :: "'a::metric_space" assumes "\S. (\n. S n \ a \ S n \ T) \ S ----> a \ (\n. 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 :: "real \ real" assumes mono: "\a b. a \ I \ b \ I \ x < a \ a \ b \ f a \ f b" assumes bnd: "\a. a \ I \ x < a \ K \ f a" shows "(f ---> Inf (f ` ({x<..} \ I))) (at x within ({x<..} \ I))" proof cases assume "{x<..} \ I = {}" then show ?thesis by (simp add: Lim_within_empty) next assume [simp]: "{x<..} \ I \ {}" show ?thesis proof (rule Lim_within_LIMSEQ, safe) fix S assume S: "\n. S n \ x \ S n \ {x <..} \ I" "S ----> x" show "(\n. f (S n)) ----> Inf (f ` ({x<..} \ I))" proof (rule LIMSEQ_I, rule ccontr) fix r :: real assume "0 < r" with Inf_close[of "f ` ({x<..} \ I)" r] obtain y where y: "x < y" "y \ I" "f y < Inf (f ` ({x <..} \ I)) + r" by auto from `x < y` have "0 < y - x" by auto from S(2)[THEN LIMSEQ_D, OF this] obtain N where N: "\n. N \ n \ \S n - x\ < y - x" by auto assume "\ (\N. \n\N. norm (f (S n) - Inf (f ` ({x<..} \ I))) < r)" moreover have "\n. Inf (f ` ({x<..} \ I)) \ f (S n)" using S bnd by (intro Inf_lower[where z=K]) auto ultimately obtain n where n: "N \ n" "r + Inf (f ` ({x<..} \ I)) \ f (S n)" by (auto simp: not_less field_simps) with N[OF n(1)] mono[OF _ `y \ I`, of "S n"] S(1)[THEN spec, of n] y show False by auto qed qed qed text{* Another limit point characterization. *} lemma islimpt_sequential: fixes x :: "'a::first_countable_topology" shows "x islimpt S \ (\f. (\n::nat. f n \ S - {x}) \ (f ---> x) sequentially)" (is "?lhs = ?rhs") proof assume ?lhs from countable_basis_at_decseq[of x] guess A . note A = this def f \ "\n. 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 "(\n. 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 (\x. f x \ S) sequentially" by (auto elim!: eventually_elim1) qed ultimately show ?rhs by fast next assume ?rhs then obtain f :: "nat \ 'a" where f: "\n. f n \ S - {x}" and lim: "f ----> x" by auto show ?lhs unfolding islimpt_def proof safe fix T assume "open T" "x \ T" from lim[THEN topological_tendstoD, OF this] f show "\y\S. y \ T \ y \ x" unfolding eventually_sequentially by auto qed qed lemma Lim_inv: (* TODO: delete *) fixes f :: "'a \ real" and A :: "'a filter" assumes "(f ---> l) A" and "l \ 0" shows "((inverse o f) ---> inverse l) A" unfolding o_def using assms by (rule tendsto_inverse) lemma Lim_null: fixes f :: "'a \ 'b::real_normed_vector" shows "(f ---> l) net \ ((\x. f(x) - l) ---> 0) net" by (simp add: Lim dist_norm) lemma Lim_null_comparison: fixes f :: "'a \ 'b::real_normed_vector" assumes "eventually (\x. norm (f x) \ g x) net" "(g ---> 0) net" shows "(f ---> 0) net" proof (rule metric_tendsto_imp_tendsto) show "(g ---> 0) net" by fact show "eventually (\x. dist (f x) 0 \ dist (g x) 0) net" using assms(1) by (rule eventually_elim1, simp add: dist_norm) qed lemma Lim_transform_bound: fixes f :: "'a \ 'b::real_normed_vector" fixes g :: "'a \ 'c::real_normed_vector" assumes "eventually (\n. norm(f n) <= norm(g n)) net" "(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" "eventually (\x. f(x) \ S) net" "\(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 (\x. f x \ - S) net" by (rule topological_tendstoD) with assms(2) have "eventually (\x. False) net" by (rule eventually_elim2) simp with assms(3) show "False" by (simp add: eventually_False) qed text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *} lemma Lim_dist_ubound: assumes "\(trivial_limit net)" "(f ---> l) net" "eventually (\x. 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 (\x. dist a (f x) \ {..e}) net" by (simp add: assms) show "\ trivial_limit net" by fact show "((\x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms) qed thus ?thesis by simp qed lemma Lim_norm_ubound: fixes f :: "'a \ 'b::real_normed_vector" assumes "\(trivial_limit net)" "(f ---> l) net" "eventually (\x. 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 (\x. norm (f x) \ {..e}) net" by (simp add: assms) show "\ trivial_limit net" by fact show "((\x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms) qed thus ?thesis by simp qed lemma Lim_norm_lbound: fixes f :: "'a \ 'b::real_normed_vector" assumes "\ (trivial_limit net)" "(f ---> l) net" "eventually (\x. 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 (\x. norm (f x) \ {e..}) net" by (simp add: assms) show "\ trivial_limit net" by fact show "((\x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms) qed thus ?thesis by simp qed text{* Uniqueness of the limit, when nontrivial. *} lemma tendsto_Lim: fixes f :: "'a \ 'b::t2_space" shows "~(trivial_limit net) \ (f ---> l) net ==> Lim net f = l" unfolding Lim_def using tendsto_unique[of net f] by auto text{* Limit under bilinear function *} lemma Lim_bilinear: assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h" shows "((\x. 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_within) 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" fixes l :: "'b::topological_space" shows "(f ---> l) (at a) \ ((\x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs") using LIM_offset_zero LIM_offset_zero_cancel .. text{* It's also sometimes useful to extract the limit point from the filter. *} definition netlimit :: "'a::t2_space filter \ 'a" where "netlimit net = (SOME a. ((\x. x) ---> a) net)" lemma netlimit_within: assumes "\ trivial_limit (at a within S)" shows "netlimit (at a within S) = a" unfolding netlimit_def apply (rule some_equality) apply (rule Lim_at_within) apply (rule tendsto_ident_at) apply (erule tendsto_unique [OF assms]) apply (rule Lim_at_within) apply (rule tendsto_ident_at) done lemma netlimit_at: fixes a :: "'a::{perfect_space,t2_space}" shows "netlimit (at a) = a" using netlimit_within [of a UNIV] by simp lemma lim_within_interior: "x \ interior S \ (f ---> l) (at x within S) \ (f ---> l) (at x)" by (simp add: 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 (simp add: at_within_interior netlimit_at) text{* Transformation of limit. *} lemma Lim_transform: fixes f g :: "'a::type \ 'b::real_normed_vector" assumes "((\x. f x - g x) ---> 0) net" "(f ---> l) net" shows "(g ---> l) net" using tendsto_diff [OF assms(2) assms(1)] by simp lemma Lim_transform_eventually: "eventually (\x. f x = g x) net \ (f ---> l) net \ (g ---> l) net" apply (rule topological_tendstoI) apply (drule (2) topological_tendstoD) apply (erule (1) eventually_elim2, simp) done lemma Lim_transform_within: assumes "0 < d" and "\x'\S. 0 < dist x' x \ dist x' x < d \ f x' = g x'" and "(f ---> l) (at x within S)" shows "(g ---> l) (at x within S)" proof (rule Lim_transform_eventually) show "eventually (\x. f x = g x) (at x within S)" unfolding eventually_within using assms(1,2) by auto show "(f ---> l) (at x within S)" by fact qed lemma Lim_transform_at: assumes "0 < d" and "\x'. 0 < dist x' x \ dist x' x < d \ f x' = g x'" and "(f ---> l) (at x)" shows "(g ---> l) (at x)" proof (rule Lim_transform_eventually) show "eventually (\x. f x = g x) (at x)" unfolding eventually_at using assms(1,2) by auto show "(f ---> l) (at x)" by fact qed text{* Common case assuming being away from some crucial point like 0. *} lemma Lim_transform_away_within: fixes a b :: "'a::t1_space" assumes "a \ b" and "\x\S. x \ a \ x \ b \ f x = g x" and "(f ---> l) (at a within S)" shows "(g ---> l) (at a within S)" proof (rule Lim_transform_eventually) show "(f ---> l) (at a within S)" by fact show "eventually (\x. f x = g x) (at a within S)" unfolding Limits.eventually_within eventually_at_topological by (rule exI [where x="- {b}"], simp add: open_Compl assms) qed lemma Lim_transform_away_at: fixes a b :: "'a::t1_space" assumes ab: "a\b" and fg: "\x. x \ a \ x \ b \ f x = g x" and fl: "(f ---> l) (at a)" shows "(g ---> l) (at a)" using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp text{* Alternatively, within an open set. *} lemma Lim_transform_within_open: assumes "open S" and "a \ S" and "\x\S. x \ a \ f x = g x" and "(f ---> l) (at a)" shows "(g ---> l) (at a)" proof (rule Lim_transform_eventually) show "eventually (\x. f x = g x) (at a)" unfolding eventually_at_topological using assms(1,2,3) by auto show "(f ---> l) (at a)" by fact qed text{* A congruence rule allowing us to transform limits assuming not at point. *} (* FIXME: Only one congruence rule for tendsto can be used at a time! *) lemma Lim_cong_within(*[cong add]*): assumes "a = b" "x = y" "S = T" assumes "\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 Limits.eventually_within eventually_at_topological using assms by simp lemma Lim_cong_at(*[cong add]*): assumes "a = b" "x = y" assumes "\x. x \ a \ f x = g x" shows "((\x. f x) ---> x) (at a) \ ((g ---> y) (at a))" unfolding tendsto_def eventually_at_topological using assms by simp text{* Useful lemmas on closure and set of possible sequential limits.*} lemma closure_sequential: fixes l :: "'a::first_countable_topology" shows "l \ closure S \ (\x. (\n. x n \ S) \ (x ---> l) sequentially)" (is "?lhs = ?rhs") proof assume "?lhs" moreover { assume "l \ S" hence "?rhs" using tendsto_const[of l sequentially] by auto } moreover { assume "l islimpt S" hence "?rhs" unfolding islimpt_sequential by auto } ultimately show "?rhs" unfolding closure_def by auto next assume "?rhs" thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto qed lemma closed_sequential_limits: fixes S :: "'a::first_countable_topology set" shows "closed S \ (\x l. (\n. x n \ S) \ (x ---> l) sequentially \ l \ S)" 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 metis lemma closure_approachable: fixes S :: "'a::metric_space set" shows "x \ closure S \ (\e>0. \y\S. dist y x < e)" apply (auto simp add: closure_def islimpt_approachable) by (metis dist_self) lemma closed_approachable: fixes S :: "'a::metric_space set" shows "closed S ==> (\e>0. \y\S. dist y x < e) \ x \ S" by (metis closure_closed closure_approachable) subsection {* 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: shows "0 \ infdist x A" using assms by (auto simp add: infdist_def) lemma infdist_le: assumes "a \ A" assumes "d = dist x a" shows "infdist x A \ d" using assms by (auto intro!: SupInf.Inf_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 auto qed lemma infdist_triangle: shows "infdist x A \ infdist y A + dist x y" proof cases assume "A = {}" thus ?thesis by (simp add: infdist_def) next assume "A \ {}" then obtain a where "a \ A" by auto have "infdist x A \ Inf {dist x y + dist y a |a. a \ A}" proof 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 Inf_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 thus "infdist x A \ d" by (rule infdist_le) qed qed also have "\ = dist x y + infdist y A" proof (rule Inf_eq, safe) fix a assume "a \ A" thus "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" hence "i - dist x y \ infdist y A" unfolding infdist_notempty[OF `A \ {}`] using `a \ A` by (intro Inf_greatest) (auto simp: field_simps) thus "i \ dist x y + infdist y A" by simp qed finally show ?thesis by simp qed lemma in_closure_iff_infdist_zero: assumes "A \ {}" shows "x \ closure A \ infdist x A = 0" proof assume "x \ closure A" show "infdist x A = 0" proof (rule ccontr) assume "infdist x A \ 0" with infdist_nonneg[of x A] have "infdist x A > 0" by auto hence "ball x (infdist x A) \ closure A = {}" apply auto by (metis `0 < infdist x A` `x \ closure A` closure_approachable dist_commute eucl_less_not_refl euclidean_trans(2) infdist_le) hence "x \ closure A" by (metis `0 < infdist x A` centre_in_ball disjoint_iff_not_equal) thus False using `x \ closure A` by simp qed next assume x: "infdist x A = 0" then obtain a where "a \ A" by atomize_elim (metis all_not_in_conv assms) show "x \ closure A" unfolding closure_approachable proof (safe, rule ccontr) fix e::real assume "0 < e" assume "\ (\y\A. dist y x < e)" hence "infdist x A \ e" using `a \ A` unfolding infdist_def by (force simp: dist_commute) with x `0 < e` show False by auto qed qed lemma in_closed_iff_infdist_zero: assumes "closed A" "A \ {}" shows "x \ A \ infdist x A = 0" proof - have "x \ closure A \ infdist x A = 0" by (rule in_closure_iff_infdist_zero) fact with assms show ?thesis by simp qed lemma tendsto_infdist [tendsto_intros]: assumes f: "(f ---> l) F" shows "((\x. infdist (f x) A) ---> infdist l A) F" proof (rule tendstoI) fix e ::real assume "0 < e" from tendstoD[OF f this] show "eventually (\x. dist (infdist (f x) A) (infdist l A) < e) F" proof (eventually_elim) fix x from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l] have "dist (infdist (f x) A) (infdist l A) \ dist (f x) l" by (simp add: dist_commute dist_real_def) also assume "dist (f x) l < e" finally show "dist (infdist (f x) A) (infdist l A) < e" . qed qed text{* Some other lemmas about sequences. *} lemma sequentially_offset: assumes "eventually (\i. P i) sequentially" shows "eventually (\i. P (i + k)) sequentially" using assms unfolding eventually_sequentially by (metis trans_le_add1) lemma seq_offset: assumes "(f ---> l) sequentially" shows "((\i. f (i + k)) ---> l) sequentially" using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *) lemma seq_offset_neg: "(f ---> l) sequentially ==> ((\i. f(i - k)) ---> l) sequentially" apply (rule topological_tendstoI) apply (drule (2) topological_tendstoD) apply (simp only: eventually_sequentially) apply (subgoal_tac "\N k (n::nat). N + k <= n ==> N <= n - k") apply metis by arith lemma seq_offset_rev: "((\i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially" by (rule LIMSEQ_offset) (* FIXME: redundant *) lemma seq_harmonic: "((\n. inverse (real n)) ---> 0) sequentially" using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc) 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 done lemma open_contains_cball: "open S \ (\x\S. \e>0. cball x e \ S)" proof- { fix x and e::real assume "x\S" "e>0" "ball x e \ S" hence "\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" hence "\d>0. ball x d \ S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto } ultimately show ?thesis unfolding open_contains_ball by auto qed lemma open_contains_cball_eq: "open S ==> (\x. x \ S \ (\e>0. cball x e \ S))" by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball) lemma mem_interior_cball: "x \ interior S \ (\e>0. cball x e \ S)" apply (simp add: interior_def, safe) apply (force simp add: open_contains_cball) apply (rule_tac x="ball x e" in exI) apply (simp add: subset_trans [OF ball_subset_cball]) done lemma islimpt_ball: fixes x y :: "'a::{real_normed_vector,perfect_space}" shows "y islimpt ball x e \ 0 < e \ y \ cball x e" (is "?lhs = ?rhs") proof assume "?lhs" { assume "e \ 0" hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto } hence "e > 0" by (metis not_less) moreover have "y \ cball x e" using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"] ball_subset_cball[of x e] `?lhs` unfolding closed_limpt by auto ultimately show "?rhs" by auto next assume "?rhs" hence "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 thus "\x'\ball x e. x' \ y \ dist x' y < d" proof(cases "x=y") case True hence False using `d \ dist x y` `d>0` by auto thus "\x'\ball x e. x' \ y \ dist x' y < d" by auto next case False have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))" unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto also have "\ = \- 1 + d / (2 * norm (x - y))\ * norm (x - y)" using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, 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)) *\<^sub>R (y - x) \ ball x e" using `d>0` by auto moreover have "(d / (2*dist y x)) *\<^sub>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)) *\<^sub>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 by auto ultimately show "\x'\ball x e. x' \ y \ dist x' y < d" by (rule_tac x="y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto qed next case False hence "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` ** by (rule_tac x=z in bexI, auto simp add: dist_commute) next case False thus "\x'\ball x e. x' \ y \ dist x' y < d" using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto) qed qed } thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto qed lemma closure_ball_lemma: fixes x y :: "'a::real_normed_vector" assumes "x \ y" shows "y islimpt ball x (dist x y)" proof (rule islimptI) fix T assume "y \ T" "open T" then obtain r where "0 < r" "\z. dist z y < r \ z \ T" unfolding open_dist by fast (* choose point between x and y, within distance r of y. *) def k \ "min 1 (r / (2 * dist x y))" def z \ "y + scaleR k (x - y)" have z_def2: "z = x + scaleR (1 - k) (y - x)" unfolding z_def by (simp add: algebra_simps) have "dist z y < r" unfolding z_def k_def using `0 < r` by (simp add: dist_norm min_def) hence "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 hence "z \ ball x (dist x y)" by simp have "z \ y" unfolding z_def k_def using `x \ y` `0 < r` by (simp add: min_def) show "\z\ball x (dist x y). z \ T \ z \ y" using `z \ ball x (dist x y)` `z \ T` `z \ y` by fast qed lemma closure_ball: fixes x :: "'a::real_normed_vector" shows "0 < e \ closure (ball x e) = cball x e" apply (rule equalityI) apply (rule closure_minimal) apply (rule ball_subset_cball) apply (rule closed_cball) apply (rule subsetI, rename_tac y) apply (simp add: le_less [where 'a=real]) apply (erule disjE) apply (rule subsetD [OF closure_subset], simp) apply (simp add: closure_def) apply clarify apply (rule closure_ball_lemma) apply (simp add: zero_less_dist_iff) done (* In a trivial vector space, this fails for e = 0. *) lemma interior_cball: fixes x :: "'a::{real_normed_vector, perfect_space}" shows "interior (cball x e) = ball x e" proof(cases "e\0") case False note cs = this from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover { fix y assume "y \ cball x e" hence False unfolding mem_cball using dist_nz[of x y] cs by auto } hence "cball x e = {}" by auto hence "interior (cball x e) = {}" using interior_empty by auto ultimately show ?thesis by blast next case True note cs = this have "ball x e \ cball x e" using ball_subset_cball by auto moreover { fix S y assume as: "S \ cball x e" "open S" "y\S" then obtain d where "d>0" and d:"\x'. dist x' y < d \ x' \ S" unfolding open_dist by blast then obtain xa where xa_y: "xa \ y" and xa: "dist xa y < d" using perfect_choose_dist [of d] by auto have "xa\S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute) hence xa_cball:"xa \ cball x e" using as(1) by auto hence "y \ ball x e" proof(cases "x = y") case True hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute) thus "y \ ball x e" using `x = y ` by simp next case False have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm using `d>0` norm_ge_zero[of "y - x"] `x \ y` by auto hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \ cball x e" using d as(1)[unfolded subset_eq] by blast have "y - x \ 0" using `x \ y` by auto hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym] using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x = norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)" by (auto simp add: dist_norm algebra_simps) also have "\ = norm ((1 + d / (2 * norm (y - x))) *\<^sub>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) thus "y \ ball x e" unfolding mem_ball using `d>0` by auto qed } hence "\S \ cball x e. open S \ S \ ball x e" by auto ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto qed lemma frontier_ball: fixes a :: "'a::real_normed_vector" shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}" apply (simp add: frontier_def closure_ball interior_open order_less_imp_le) apply (simp add: set_eq_iff) by arith lemma frontier_cball: fixes a :: "'a::{real_normed_vector, perfect_space}" shows "frontier(cball a e) = {x. dist a x = e}" apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le) apply (simp add: set_eq_iff) by arith lemma cball_eq_empty: "(cball x e = {}) \ e < 0" apply (simp add: set_eq_iff not_le) by (metis zero_le_dist dist_self order_less_le_trans) lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty) lemma cball_eq_sing: fixes x :: "'a::{metric_space,perfect_space}" shows "(cball x e = {x}) \ e = 0" proof (rule linorder_cases) assume e: "0 < e" obtain a where "a \ x" "dist a x < e" using perfect_choose_dist [OF e] by auto hence "a \ x" "dist x a \ e" by (auto simp add: dist_commute) with e show ?thesis by (auto simp add: set_eq_iff) qed auto lemma cball_sing: fixes x :: "'a::metric_space" shows "e = 0 ==> cball x e = {x}" by (auto simp add: set_eq_iff) subsection {* Boundedness *} (* FIXME: This has to be unified with BSEQ!! *) definition (in metric_space) bounded :: "'a set \ bool" where "bounded S \ (\x e. \y\S. dist x y \ e)" lemma bounded_subset_cball: "bounded S \ (\e x. S \ cball x e)" unfolding bounded_def subset_eq by auto lemma bounded_any_center: "bounded S \ (\e. \y\S. dist a y \ e)" unfolding bounded_def apply safe apply (rule_tac x="dist a x + e" in exI, clarify) apply (drule (1) bspec) apply (erule order_trans [OF dist_triangle add_left_mono]) apply auto done lemma bounded_iff: "bounded S \ (\a. \x\S. norm x \ a)" unfolding bounded_any_center [where a=0] by (simp add: dist_norm) lemma bounded_realI: assumes "\x\s. abs (x::real) \ B" shows "bounded s" unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI) using assms by auto lemma bounded_empty [simp]: "bounded {}" by (simp add: bounded_def) lemma bounded_subset: "bounded T \ S \ T ==> bounded S" by (metis bounded_def subset_eq) lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)" by (metis bounded_subset interior_subset) lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)" proof- from assms obtain x and a where a: "\y\S. dist x y \ a" unfolding bounded_def by auto { fix y assume "y \ closure S" then obtain f where f: "\n. f n \ S" "(f ---> y) sequentially" unfolding closure_sequential by auto have "\n. f n \ S \ dist x (f n) \ a" using a by simp hence "eventually (\n. 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 } thus ?thesis unfolding bounded_def by auto qed lemma bounded_cball[simp,intro]: "bounded (cball x e)" apply (simp add: bounded_def) apply (rule_tac x=x in exI) apply (rule_tac x=e in exI) apply auto done lemma bounded_ball[simp,intro]: "bounded(ball x e)" by (metis ball_subset_cball bounded_cball bounded_subset) lemma bounded_Un[simp]: "bounded (S \ T) \ bounded S \ bounded T" apply (auto simp add: bounded_def) 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, rename_tac z, safe) apply (drule (1) bspec, simp) apply (drule (1) bspec) apply (rule min_max.le_supI2) apply (erule order_trans [OF dist_triangle add_left_mono]) done lemma bounded_Union[intro]: "finite F \ (\S\F. bounded S) \ bounded(\F)" by (induct rule: finite_induct[of F], auto) lemma bounded_UN [intro]: "finite A \ \x\A. bounded (B x) \ bounded (\x\A. B x)" by (induct set: finite, auto) lemma bounded_insert [simp]: "bounded (insert x S) \ bounded S" proof - have "\y\{x}. dist x y \ 0" by simp hence "bounded {x}" unfolding bounded_def by fast thus ?thesis by (metis insert_is_Un bounded_Un) qed lemma finite_imp_bounded [intro]: "finite S \ bounded S" by (induct set: finite, simp_all) lemma bounded_pos: "bounded S \ (\b>0. \x\ S. norm x <= b)" apply (simp add: bounded_iff) apply (subgoal_tac "\x (y::real). 0 < 1 + abs y \ (x <= y \ x <= 1 + abs y)") by metis arith lemma Bseq_eq_bounded: "Bseq f \ bounded (range f)" unfolding Bseq_def bounded_pos by auto lemma bounded_Int[intro]: "bounded S \ bounded T \ bounded (S \ T)" by (metis Int_lower1 Int_lower2 bounded_subset) lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)" apply (metis Diff_subset bounded_subset) done lemma not_bounded_UNIV[simp, intro]: "\ bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)" proof(auto simp add: bounded_pos not_le) obtain x :: 'a where "x \ 0" using perfect_choose_dist [OF zero_less_one] by fast fix b::real assume b: "b >0" have b1: "b +1 \ 0" using b by simp with `x \ 0` have "b < norm (scaleR (b + 1) (sgn x))" by (simp add: norm_sgn) then show "\x::'a. b < norm x" .. qed lemma bounded_linear_image: assumes "bounded S" "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" hence "norm x \ b" using b by auto hence "norm (f x) \ B * b" using B(2) apply(erule_tac x=x in allE) by (metis B(1) B(2) order_trans mult_le_cancel_left_pos) } thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI) using b B mult_pos_pos [of b B] by (auto simp add: mult_commute) qed lemma bounded_scaling: fixes S :: "'a::real_normed_vector set" shows "bounded S \ bounded ((\x. c *\<^sub>R x) ` S)" apply (rule bounded_linear_image, assumption) apply (rule bounded_linear_scaleR_right) done lemma bounded_translation: fixes S :: "'a::real_normed_vector set" assumes "bounded S" shows "bounded ((\x. 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" hence "norm (a + x) \ b + norm a" using norm_triangle_ineq[of a x] b by auto } thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"] by (auto intro!: exI[of _ "b + norm a"]) qed text{* Some theorems on sups and infs using the notion "bounded". *} lemma bounded_real: 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" "S \ {}" shows "\x\S. x <= Sup S" and "\b. (\x\S. x <= b) \ Sup S <= b" proof fix x assume "x\S" thus "x \ Sup S" by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real) next show "\b. (\x\S. x \ b) \ Sup S \ b" using assms by (metis SupInf.Sup_least) qed lemma Sup_insert: fixes S :: "real set" shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))" by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal) lemma Sup_insert_finite: fixes S :: "real set" shows "finite S \ Sup(insert x S) = (if S = {} then x else max x (Sup S))" apply (rule Sup_insert) apply (rule finite_imp_bounded) by simp lemma bounded_has_Inf: fixes S :: "real set" assumes "bounded S" "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 thus "x \ Inf S" using `x\S` by (metis Inf_lower_EX abs_le_D2 minus_le_iff) next show "\b. (\x\S. x >= b) \ Inf S \ b" using assms by (metis SupInf.Inf_greatest) qed lemma Inf_insert: fixes S :: "real set" shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))" by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal) lemma Inf_insert_finite: fixes S :: "real set" shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))" by (rule Inf_insert, rule finite_imp_bounded, simp) subsection {* Compactness *} subsubsection{* Open-cover compactness *} definition compact :: "'a::topological_space set \ bool" where compact_eq_heine_borel: -- "This name is used for backwards compatibility" "compact S \ (\C. (\c\C. open c) \ S \ \C \ (\D\C. finite D \ S \ \D))" lemma compactI: assumes "\C. \t\C. open t \ s \ \ C \ \C'. C' \ C \ finite C' \ s \ \ C'" shows "compact s" unfolding compact_eq_heine_borel using assms by metis lemma compactE: assumes "compact s" and "\t\C. open t" and "s \ \C" obtains C' where "C' \ C" and "finite C'" and "s \ \C'" using assms unfolding compact_eq_heine_borel by metis lemma compactE_image: assumes "compact s" and "\t\C. open (f t)" and "s \ (\c\C. f c)" obtains C' where "C' \ C" and "finite C'" and "s \ (\c\C'. f c)" using assms unfolding ball_simps[symmetric] SUP_def by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s]) subsubsection {* Bolzano-Weierstrass property *} lemma heine_borel_imp_bolzano_weierstrass: assumes "compact s" "infinite t" "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 "\ x T. x \ T \ open T \ (\y\t. y \ T \ y = x)"] by auto obtain g where g:"g\{t. \x. x \ s \ t = f x}" "finite g" "s \ \g" using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \x. x\s \ t = f x}"]] using f by auto from g(1,3) have g':"\x\g. \xa \ s. x = f xa" by auto { fix x y assume "x\t" "y\t" "f x = f y" hence "x \ f x" "y \ f x \ y = x" using f[THEN bspec[where x=x]] and `t\s` by auto hence "x = y" using `f x = f y` and f[THEN bspec[where x=y]] and `y\t` and `t\s` by auto } hence "inj_on f t" unfolding inj_on_def by simp hence "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 hence "y = x" using f[THEN bspec[where x=y]] and `x\t` and `x\h`[unfolded `h = f y`] by auto hence False using `f x \ g` `h\g` unfolding `h = f y` by auto } hence "f ` t \ g" by auto ultimately show False using g(2) using finite_subset by auto qed lemma acc_point_range_imp_convergent_subsequence: fixes l :: "'a :: first_countable_topology" assumes l: "\U. l\U \ open U \ infinite (U \ range f)" shows "\r. subseq r \ (f \ r) ----> l" proof - from countable_basis_at_decseq[of l] guess A . note A = this def s \ "\n 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 "(\n. f (r n)) ----> l" proof (rule topological_tendstoI) fix S assume "open S" "l \ S" with A(3) have "eventually (\i. 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 (\i. f (r i) \ A i) sequentially" by (auto simp: eventually_sequentially) ultimately show "eventually (\i. f (r i) \ S) sequentially" by eventually_elim auto qed ultimately show "\r. subseq r \ (f \ r) ----> l" by (auto simp: convergent_def comp_def) qed lemma sequence_infinite_lemma: fixes f :: "nat \ 'a::t1_space" assumes "\n. f n \ l" and "(f ---> l) sequentially" shows "infinite (range f)" proof assume "finite (range f)" hence "closed (range f)" by (rule finite_imp_closed) hence "open (- range f)" by (rule open_Compl) from assms(1) have "l \ - range f" by auto from assms(2) have "eventually (\n. f n \ - range f) sequentially" using `open (- range f)` `l \ - range f` by (rule topological_tendstoD) thus False unfolding eventually_sequentially by auto qed lemma closure_insert: fixes x :: "'a::t1_space" shows "closure (insert x s) = insert x (closure s)" apply (rule closure_unique) apply (rule insert_mono [OF closure_subset]) apply (rule closed_insert [OF closed_closure]) apply (simp add: closure_minimal) done lemma islimpt_insert: fixes x :: "'a::t1_space" shows "x islimpt (insert a s) \ x islimpt s" proof assume *: "x islimpt (insert a s)" show "x islimpt s" proof (rule islimptI) fix t assume t: "x \ t" "open t" show "\y\s. y \ t \ y \ x" proof (cases "x = a") case True obtain y where "y \ insert a s" "y \ t" "y \ x" using * t by (rule islimptE) with `x = a` show ?thesis by auto next case False with t have t': "x \ t - {a}" "open (t - {a})" by (simp_all add: open_Diff) obtain y where "y \ insert a s" "y \ t - {a}" "y \ x" using * t' by (rule islimptE) thus ?thesis by auto qed qed next assume "x islimpt s" thus "x islimpt (insert a s)" by (rule islimpt_subset) auto qed lemma islimpt_finite: fixes x :: "'a::t1_space" shows "finite s \ \ x islimpt s" by (induct set: finite, simp_all add: islimpt_insert) lemma islimpt_union_finite: fixes x :: "'a::t1_space" shows "finite s \ x islimpt (s \ t) \ x islimpt t" by (simp add: islimpt_Un islimpt_finite) lemma islimpt_eq_acc_point: fixes l :: "'a :: t1_space" shows "l islimpt S \ (\U. l\U \ open U \ infinite (U \ S))" proof (safe intro!: islimptI) fix U assume "l islimpt S" "l \ U" "open U" "finite (U \ S)" then have "l islimpt S" "l \ (U - (U \ S - {l}))" "open (U - (U \ S - {l}))" by (auto intro: finite_imp_closed) then show False by (rule islimptE) auto next fix T assume *: "\U. l\U \ open U \ infinite (U \ S)" "l \ T" "open T" then have "infinite (T \ S - {l})" by auto then have "\x. x \ (T \ S - {l})" unfolding ex_in_conv by (intro notI) simp then show "\y\S. y \ T \ y \ l" by auto qed lemma islimpt_range_imp_convergent_subsequence: fixes l :: "'a :: {t1_space, first_countable_topology}" assumes l: "l islimpt (range f)" shows "\r. subseq r \ (f \ 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 (\n. 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..}" by auto hence "l' islimpt (f ` ({.. {N..}))" using assms(2) by simp hence "l' islimpt (f ` {.. f ` {N..})" by (simp add: image_Un) hence "l' islimpt (f ` {N..})" by (simp add: islimpt_union_finite) then obtain y where "y \ f ` {N..}" "y \ s" "y \ l'" using `l' \ s` `open s` by (rule islimptE) then obtain n where "N \ n" "f n \ s" "f n \ l'" by auto with `\n\N. f n \ t` have "f n \ s \ t" by simp with `s \ t = {}` show False by simp qed lemma bolzano_weierstrass_imp_closed: fixes s :: "'a::{first_countable_topology, t2_space} set" assumes "\t. infinite t \ t \ s --> (\x \ s. x islimpt t)" shows "closed s" proof- { fix x l assume as: "\n::nat. x n \ s" "(x ---> l) sequentially" hence "l \ s" proof(cases "\n. x n \ l") case False thus "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 thus "l\s" using sequence_unique_limpt[of x l l'] using as cas by auto qed } thus ?thesis unfolding closed_sequential_limits by fast qed lemma compact_imp_closed: fixes s :: "'a::t2_space set" assumes "compact s" shows "closed s" unfolding closed_def proof (rule openI) fix y assume "y \ - s" let ?C = "\x\s. {u. open u \ x \ u \ eventually (\y. y \ u) (nhds y)}" note `compact s` moreover have "\u\?C. open u" by simp moreover have "s \ \?C" proof fix x assume "x \ s" with `y \ - s` have "x \ y" by clarsimp hence "\u v. open u \ open v \ x \ u \ y \ v \ u \ v = {}" by (rule hausdorff) with `x \ s` show "x \ \?C" unfolding eventually_nhds by auto qed ultimately obtain D where "D \ ?C" and "finite D" and "s \ \D" by (rule compactE) from `D \ ?C` have "\x\D. eventually (\y. y \ x) (nhds y)" by auto with `finite D` have "eventually (\y. y \ \D) (nhds y)" by (simp add: eventually_Ball_finite) with `s \ \D` have "eventually (\y. y \ s) (nhds y)" by (auto elim!: eventually_mono [rotated]) thus "\t. open t \ y \ t \ t \ - s" by (simp add: eventually_nhds subset_eq) qed lemma compact_imp_bounded: assumes "compact U" shows "bounded U" proof - have "compact U" "\x\U. open (ball x 1)" "U \ (\x\U. ball x 1)" using assms by auto then obtain D where D: "D \ U" "finite D" "U \ (\x\D. ball x 1)" by (elim compactE_image) from `finite D` have "bounded (\x\D. ball x 1)" by (simp add: bounded_UN) thus "bounded U" using `U \ (\x\D. ball x 1)` by (rule bounded_subset) qed text{* In particular, some common special cases. *} lemma compact_empty[simp]: "compact {}" unfolding compact_eq_heine_borel by auto lemma compact_union [intro]: assumes "compact s" "compact t" shows " compact (s \ t)" proof (rule compactI) fix f assume *: "Ball f open" "s \ t \ \f" from * `compact s` obtain s' where "s' \ f \ finite s' \ s \ \s'" unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis moreover from * `compact t` obtain t' where "t' \ f \ finite t' \ t \ \t'" unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis ultimately show "\f'\f. finite f' \ s \ t \ \f'" by (auto intro!: exI[of _ "s' \ t'"]) qed lemma compact_Union [intro]: "finite S \ (\T. T \ S \ compact T) \ compact (\S)" by (induct set: finite) auto lemma compact_UN [intro]: "finite A \ (\x. x \ A \ compact (B x)) \ compact (\x\A. B x)" unfolding SUP_def by (rule compact_Union) auto lemma compact_inter_closed [intro]: assumes "compact s" and "closed t" shows "compact (s \ t)" proof (rule compactI) fix C assume C: "\c\C. open c" and cover: "s \ t \ \C" from C `closed t` have "\c\C \ {-t}. open c" by auto moreover from cover have "s \ \(C \ {-t})" by auto ultimately have "\D\C \ {-t}. finite D \ s \ \D" using `compact s` unfolding compact_eq_heine_borel by auto then guess D .. then show "\D\C. finite D \ s \ t \ \D" by (intro exI[of _ "D - {-t}"]) auto qed lemma closed_inter_compact [intro]: assumes "closed s" and "compact t" shows "compact (s \ t)" using compact_inter_closed [of t s] assms by (simp add: Int_commute) lemma compact_inter [intro]: fixes s t :: "'a :: t2_space set" assumes "compact s" and "compact t" shows "compact (s \ t)" using assms by (intro compact_inter_closed compact_imp_closed) lemma compact_sing [simp]: "compact {a}" unfolding compact_eq_heine_borel by auto lemma compact_insert [simp]: assumes "compact s" shows "compact (insert x s)" proof - have "compact ({x} \ s)" using compact_sing assms by (rule compact_union) thus ?thesis by simp qed lemma finite_imp_compact: shows "finite s \ compact s" by (induct set: finite) simp_all lemma open_delete: fixes s :: "'a::t1_space set" shows "open s \ open (s - {x})" by (simp add: open_Diff) text{* 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 \ \B \ {}) \ U \ \A \ {})" (is "_ \ ?R") proof (safe intro!: compact_eq_heine_borel[THEN iffD2]) fix A assume "compact U" and A: "\a\A. closed a" "U \ \A = {}" and fi: "\B \ A. finite B \ U \ \B \ {}" from A have "(\a\uminus`A. open a) \ U \ \uminus`A" by auto with `compact U` obtain B where "B \ A" "finite (uminus`B)" "U \ \(uminus`B)" unfolding compact_eq_heine_borel by (metis subset_image_iff) with fi[THEN spec, of B] show False by (auto dest: finite_imageD intro: inj_setminus) next fix A assume ?R and cover: "\a\A. open a" "U \ \A" from cover have "U \ \(uminus`A) = {}" "\a\uminus`A. closed a" by auto with `?R` obtain B where "B \ A" "finite (uminus`B)" "U \ \uminus`B = {}" by (metis subset_image_iff) then show "\T\A. finite T \ U \ \T" by (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD) qed lemma compact_imp_fip: "compact s \ \t \ f. closed t \ \f'. finite f' \ f' \ f \ (s \ (\ f') \ {}) \ s \ (\ f) \ {}" unfolding compact_fip by auto text{*Compactness expressed with filters*} definition "filter_from_subbase B = Abs_filter (\P. \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_def proof (rule eventually_Abs_filter is_filter.intro)+ show "?R (\x. 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 (\x. P x \ Q x)" by (intro exI[of _ "X \ Y"]) auto next 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]) auto qed lemma 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 auto lemma closure_iff_nhds_not_empty: "x \ closure X \ (\A. \S\A. open S \ x \ S \ X \ A \ {})" proof safe assume x: "x \ closure X" fix S A assume "open S" "x \ S" "X \ A = {}" "S \ A" then have "x \ closure (-S)" by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI) with x have "x \ closure X - closure (-S)" by auto also have "\ \ closure (X \ S)" using `open S` open_inter_closure_subset[of S X] by (simp add: closed_Compl ac_simps) finally have "X \ S \ {}" by auto then show False using `X \ A = {}` `S \ A` by auto next assume "\A S. S \ A \ open S \ x \ S \ X \ A \ {}" from this[THEN spec, of "- X", THEN spec, of "- closure X"] show "x \ closure X" by (simp add: closure_subset open_Compl) qed lemma compact_filter: "compact U \ (\F. F \ bot \ eventually (\x. 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" and F: "F \ bot" "eventually (\x. x \ U) F" from F have "U \ {}" by (auto simp: eventually_False) def Z \ "closure ` {A. eventually (\x. x \ A) F}" then have "\z\Z. closed z" by auto moreover have ev_Z: "\z. z \ Z \ eventually (\x. x \ z) F" unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset]) have "(\B \ Z. finite B \ U \ \B \ {})" proof (intro allI impI) fix B assume "finite B" "B \ Z" with `finite B` ev_Z have "eventually (\x. \b\B. x \ b) F" by (auto intro!: eventually_Ball_finite) with F(2) have "eventually (\x. x \ U \ (\B)) F" by eventually_elim auto with F show "U \ \B \ {}" by (intro notI) (simp add: eventually_False) qed ultimately have "U \ \Z \ {}" using `compact U` unfolding compact_fip by blast then obtain x where "x \ U" and x: "\z. z \ Z \ x \ z" by auto have "\P. eventually P (inf (nhds x) F) \ P \ bot" unfolding eventually_inf eventually_nhds proof safe fix P Q R S assume "eventually R F" "open S" "x \ S" with open_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 \ \B \ {}" "U \ \A = {}" def P' \ "(\a (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 \ \(B - {U}) \ {}" by auto with B show False by (auto simp: P'_def fun_eq_iff) qed moreover have "eventually (\x. x \ U) F" unfolding F_def by (rule eventually_filter_from_subbaseI) (auto simp: P'_def) moreover assume "\F. F \ bot \ eventually (\x. 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 (\x. 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 (\x. x \ A) (nhds x)" by (auto simp: eventually_nhds) with V have "eventually (\x. 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 \ \A" by auto with `U \ \A = {}` show False by auto qed definition "countably_compact U \ (\A. countable A \ (\a\A. open a) \ U \ \A \ (\T\A. finite T \ U \ \T))" lemma countably_compactE: assumes "countably_compact s" and "\t\C. open t" and "s \ \C" "countable C" obtains C' where "C' \ C" and "finite C'" and "s \ \C'" using assms unfolding countably_compact_def by metis lemma countably_compactI: assumes "\C. \t\C. open t \ s \ \C \ countable C \ (\C'\C. finite C' \ s \ \C')" shows "countably_compact s" using assms unfolding countably_compact_def by metis lemma compact_imp_countably_compact: "compact U \ countably_compact U" by (auto simp: compact_eq_heine_borel countably_compact_def) lemma countably_compact_imp_compact: assumes "countably_compact U" assumes ccover: "countable B" "\b\B. open b" assumes basis: "\T x. open T \ x \ T \ x \ U \ \b\B. x \ b \ b \ U \ T" shows "compact U" using `countably_compact U` unfolding compact_eq_heine_borel countably_compact_def proof safe fix A assume A: "\a\A. open a" "U \ \A" assume *: "\A. countable A \ (\a\A. open a) \ U \ \A \ (\T\A. finite T \ U \ \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 "\A \ U \ \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 \ \C" unfolding C_def by auto qed then have "U \ \C" using `U \ \A` by auto ultimately obtain T where "T\C" "finite T" "U \ \T" using * by metis moreover then have "\t\T. \a\A. t \ U \ a" by (auto simp: C_def) then guess f unfolding bchoice_iff Bex_def .. ultimately show "\T\A. finite T \ U \ \T" unfolding C_def by (intro exI[of _ "f`T"]) fastforce qed lemma countably_compact_imp_compact_second_countable: "countably_compact U \ compact (U :: 'a :: second_countable_topology set)" proof (rule countably_compact_imp_compact) fix T and x :: 'a assume "open T" "x \ T" from topological_basisE[OF is_basis this] guess b . then show "\b\SOME B. countable B \ topological_basis B. x \ b \ b \ U \ T" by auto qed (insert countable_basis topological_basis_open[OF is_basis], auto) lemma countably_compact_eq_compact: "countably_compact U \ compact (U :: 'a :: second_countable_topology set)" using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast subsubsection{* Sequential compactness *} definition seq_compact :: "'a::topological_space set \ bool" where "seq_compact S \ (\f. (\n. f n \ S) \ (\l\S. \r. subseq r \ ((f o r) ---> l) sequentially))" lemma seq_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 \ \A" "countable A" have subseq: "\X. range X \ U \ \r x. x \ U \ subseq r \ (X \ r) ----> x" using `seq_compact U` by (fastforce simp: seq_compact_def subset_eq) show "\T\A. finite T \ U \ \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 \ \T)" then have "\T. \x. T \ A \ finite T \ (x \ U - \T)" by auto then obtain X' where T: "\T. T \ A \ finite T \ X' T \ U - \T" by metis def X \ "\n. X' (from_nat_into A ` {.. n})" have X: "\n. X n \ U - (\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 \ r) ----> x" by auto from `x\U` `U \ \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 (\i. X (r i) \ from_nat_into A n) sequentially" unfolding tendsto_def by (auto simp: comp_def) then obtain N where "\i. N \ i \ X (r i) \ from_nat_into A n" by (auto simp: eventually_sequentially) moreover from X have "\i. n \ r i \ X (r i) \ from_nat_into A n" by auto moreover from `subseq r`[THEN seq_suble, of "max n N"] have "\i. n \ r i \ N \ i" by (auto intro!: exI[of _ "max n N"]) ultimately show False by auto qed qed qed lemma compact_imp_seq_compact: fixes U :: "'a :: first_countable_topology set" assumes "compact U" shows "seq_compact U" unfolding seq_compact_def proof safe fix X :: "nat \ 'a" assume "\n. X n \ U" then have "eventually (\x. 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 \ "\n 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 (\x. x \ A (Suc n)) (filtermap X sequentially)" by (auto simp: eventually_filtermap eventually_sequentially) moreover have "eventually (\x. x \ A (Suc n)) (nhds x)" using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds) ultimately have "eventually (\x. 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 "(\n. X (r n)) ----> x" proof (rule topological_tendstoI) fix S assume "open S" "x \ S" with A(3) have "eventually (\i. 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 (\i. X (r i) \ A i) sequentially" by (auto simp: eventually_sequentially) ultimately show "eventually (\i. X (r i) \ S) sequentially" by eventually_elim auto qed ultimately show "\x \ U. \r. subseq r \ (X \ r) ----> x" using `x \ U` by (auto simp: convergent_def comp_def) qed lemma seq_compactI: assumes "\f. \n. f n \ S \ \l\S. \r. subseq r \ ((f o r) ---> l) sequentially" shows "seq_compact S" unfolding seq_compact_def using assms by fast lemma seq_compactE: assumes "seq_compact S" "\n. f n \ S" obtains l r where "l \ S" "subseq r" "((f \ r) ---> l) sequentially" using assms unfolding seq_compact_def by fast lemma countably_compact_imp_acc_point: assumes "countably_compact s" "countable t" "infinite t" "t \ s" shows "\x\s. \U. x\U \ open U \ infinite (U \ t)" proof (rule ccontr) def C \ "(\F. 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 \ \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 \ (\F\E. interior (F \ (- t)))" by (metis (lifting) Union_image_eq finite_subset_image C_def) from s `t \ s` have "t \ \E" using interior_subset by blast moreover have "finite (\E)" using E by auto ultimately show False using `infinite t` by (auto simp: finite_subset) qed lemma countable_acc_point_imp_seq_compact: fixes s :: "'a::first_countable_topology set" assumes "\t. infinite t \ countable t \ t \ s --> (\x\s. \U. x\U \ open U \ infinite (U \ t))" shows "seq_compact s" proof - { fix f :: "nat \ 'a" assume f: "\n. f n \ s" have "\l\s. \r. subseq r \ ((f \ 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 hence "subseq r \ (f \ r) ----> f l" by (simp add: fr tendsto_const o_def) with f show "\l\s. \r. subseq r \ (f \ 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 \ 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 \ r) ---> l) sequentially" .. qed } thus ?thesis unfolding seq_compact_def by auto qed lemma seq_compact_eq_countably_compact: fixes U :: "'a :: first_countable_topology set" shows "seq_compact U \ countably_compact U" using countable_acc_point_imp_seq_compact countably_compact_imp_acc_point seq_compact_imp_countably_compact by metis lemma seq_compact_eq_acc_point: fixes s :: "'a :: first_countable_topology set" shows "seq_compact s \ (\t. infinite t \ countable t \ t \ s --> (\x\s. \U. x\U \ open U \ infinite (U \ t)))" using countable_acc_point_imp_seq_compact[of s] countably_compact_imp_acc_point[of s] seq_compact_imp_countably_compact[of s] by metis lemma seq_compact_eq_compact: fixes U :: "'a :: second_countable_topology set" shows "seq_compact U \ compact U" using seq_compact_eq_countably_compact countably_compact_eq_compact by blast lemma bolzano_weierstrass_imp_seq_compact: fixes s :: "'a::{t1_space, first_countable_topology} set" shows "\t. infinite t \ t \ s --> (\x \ s. x islimpt t) \ seq_compact s" by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point) subsubsection{* 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 blast fun helper_1::"('a::metric_space set) \ real \ nat \ 'a" where "helper_1 s e n = (SOME y::'a. y \ s \ (\m (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 \ (\((\x. ball x e) ` k))" proof(rule, rule, rule ccontr) fix e::real assume "e>0" and assm:"\ (\k. finite k \ k \ s \ s \ \(\x. ball x e) ` k)" def x \ "helper_1 s e" { fix n have "x n \ s \ (\m dist (x m) (x n) < e)" proof(induct_tac rule:nat_less_induct) fix n def Q \ "(\y. y \ s \ (\m dist (x m) y < e))" assume as:"\m s \ (\ma dist (x ma) (x m) < e)" have "\ s \ (\x\x ` {0..s" "z \ (\x\x ` {0.. s \ (\m dist (x m) (x n) < e)" unfolding Q_def by auto qed } hence "\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 \ r) ---> l) sequentially" using assms(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto from this(3) have "Cauchy (x \ r)" using LIMSEQ_imp_Cauchy by auto then obtain N::nat where N:"\m n. N \ m \ N \ n \ dist ((x \ r) m) ((x \ 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 auto qed subsubsection{* Heine-Borel theorem *} lemma seq_compact_imp_heine_borel: fixes s :: "'a :: metric_space set" assumes "seq_compact s" shows "compact s" proof - from seq_compact_imp_totally_bounded[OF `seq_compact s`] guess f unfolding choice_iff' .. note f = this def K \ "(\(x, r). ball x r) ` ((\e \ \ \ {0 <..}. f e) \ \)" 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 \ \" "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 \ \` `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 qed (rule `x \ ball k r`) qed qed lemma compact_eq_seq_compact_metric: "compact (s :: 'a::metric_space set) \ seq_compact s" using compact_imp_seq_compact seq_compact_imp_heine_borel by blast lemma compact_def: "compact (S :: 'a::metric_space set) \ (\f. (\n. f n \ S) \ (\l\S. \r. subseq r \ (f o r) ----> l))" unfolding compact_eq_seq_compact_metric seq_compact_def by auto subsubsection {* Complete the chain of compactness variants *} lemma compact_eq_bolzano_weierstrass: fixes s :: "'a::metric_space set" shows "compact s \ (\t. infinite t \ t \ s --> (\x \ s. x islimpt t))" (is "?lhs = ?rhs") proof assume ?lhs thus ?rhs using heine_borel_imp_bolzano_weierstrass[of s] by auto next assume ?rhs thus ?lhs unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact) qed lemma bolzano_weierstrass_imp_bounded: "\t. infinite t \ t \ s --> (\x \ s. x islimpt t) \ bounded s" using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass . 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 \ 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 \ r) ---> l) sequentially" using bounded_imp_convergent_subsequence [OF `bounded (range f)`] by auto from f have fr: "\n. (f \ 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 \ r) ---> l) sequentially" using `l \ s` r l by blast qed lemma compact_eq_bounded_closed: fixes s :: "'a::heine_borel set" shows "compact s \ bounded s \ closed s" (is "?lhs = ?rhs") proof assume ?lhs thus ?rhs using compact_imp_closed compact_imp_bounded by blast next assume ?rhs thus ?lhs using bounded_closed_imp_seq_compact[of s] unfolding compact_eq_seq_compact_metric by auto qed (* TODO: is this lemma necessary? *) lemma bounded_increasing_convergent: fixes s :: "nat \ real" shows "bounded {s n| n. True} \ \n. s n \ s (Suc n) \ \l. s ----> l" using Bseq_mono_convergent[of s] incseq_Suc_iff[of s] by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def) instance real :: heine_borel proof fix f :: "nat \ real" assume f: "bounded (range f)" obtain r where r: "subseq r" "monoseq (f \ r)" unfolding comp_def by (metis seq_monosub) moreover then have "Bseq (f \ r)" unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset) ultimately show "\l r. subseq r \ (f \ r) ----> l" using Bseq_monoseq_convergent[of "f \ r"] by (auto simp: convergent_def) qed lemma compact_lemma: fixes f :: "nat \ 'a::euclidean_space" assumes "bounded (range f)" shows "\d\Basis. \l::'a. \ r. subseq r \ (\e>0. eventually (\n. \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 (\n. \i\d. dist (f (r n) \ i) (l \ i) < e) sequentially)" proof(induct d) case empty thus ?case unfolding subseq_def by auto next case (insert k d) have k[intro]:"k\Basis" using insert by auto have s': "bounded ((\x. 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 (\n. \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 \ (\x. x \ k) ` range f" by simp have "bounded (range (\i. f (r1 i) \ k))" by (metis (lifting) bounded_subset f' image_subsetI s') then obtain l2 r2 where r2:"subseq r2" and lr2:"((\i. f (r1 (r2 i)) \ k) ---> l2) sequentially" using bounded_imp_convergent_subsequence[of "\i. f (r1 i) \ k"] by (auto simp: o_def) def r \ "r1 \ 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) *\<^sub>R i)::'a" { fix e::real assume "e>0" from lr1 `e>0` have N1:"eventually (\n. \i\d. dist (f (r1 n) \ i) (l1 \ i) < e) sequentially" by blast from lr2 `e>0` have N2:"eventually (\n. dist (f (r1 (r2 n)) \ k) l2 < e) sequentially" by (rule tendstoD) from r2 N1 have N1': "eventually (\n. \i\d. dist (f (r1 (r2 n)) \ i) (l1 \ i) < e) sequentially" by (rule eventually_subseq) have "eventually (\n. \i\(insert k d). dist (f (r n) \ i) (l \ i) < e) sequentially" using N1' N2 by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def) } ultimately show ?case by auto qed qed instance euclidean_space \ heine_borel proof fix f :: "nat \ 'a" assume f: "bounded (range f)" then obtain l::'a and r where r: "subseq r" and l: "\e>0. eventually (\n. \i\Basis. dist (f (r n) \ i) (l \ i) < e) sequentially" using compact_lemma [OF f] by blast { fix e::real assume "e>0" hence "0 < e / real_of_nat DIM('a)" by (auto intro!: divide_pos_pos DIM_positive) with l have "eventually (\n. \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 by (rule setL2_le_setsum) also have "\ < (\i\(Basis::'a set). e / (real_of_nat DIM('a)))" apply(rule setsum_strict_mono) using n by auto finally have "dist (f (r n)) l < e" by auto } ultimately have "eventually (\n. dist (f (r n)) l < e) sequentially" by (rule eventually_elim1) } hence *:"((f \ r) ---> l) sequentially" unfolding o_def tendsto_iff by simp with r show "\l r. subseq r \ ((f \ r) ---> l) sequentially" by auto qed lemma 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]) done lemma 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]) done instance prod :: (heine_borel, heine_borel) heine_borel proof fix f :: "nat \ 'a \ 'b" assume f: "bounded (range f)" from f have s1: "bounded (range (fst \ f))" unfolding image_comp by (rule bounded_fst) obtain l1 r1 where r1: "subseq r1" and l1: "(\n. fst (f (r1 n))) ----> l1" using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast from f have s2: "bounded (range (snd \ f \ r1))" by (auto simp add: image_comp intro: bounded_snd bounded_subset) obtain l2 r2 where r2: "subseq r2" and l2: "((\n. snd (f (r1 (r2 n)))) ---> l2) sequentially" using bounded_imp_convergent_subsequence [OF s2] unfolding o_def by fast have l1': "((\n. fst (f (r1 (r2 n)))) ---> l1) sequentially" using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def . have l: "((f \ (r1 \ r2)) ---> (l1, l2)) sequentially" using tendsto_Pair [OF l1' l2] unfolding o_def by simp have r: "subseq (r1 \ r2)" using r1 r2 unfolding subseq_def by simp show "\l r. subseq r \ ((f \ r) ---> l) sequentially" using l r by fast qed subsubsection{* Completeness *} definition complete :: "'a::metric_space set \ bool" where "complete s \ (\f. (\n. f n \ s) \ Cauchy f \ (\l\s. f ----> l))" lemma 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 \ 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) by auto from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] obtain M where M:"\n\M. dist ((f \ r) n) l < e/2" using `e>0` by auto { fix n::nat assume n:"n \ max N M" have "dist ((f \ r) n) l < e/2" using n M by auto moreover have "r n \ N" using lr'[of n] n by auto hence "dist (f n) ((f \ r) n) < e / 2" using N using 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) } hence "\N. \n\N. dist (f n) l < e" by blast } hence "\l\s. (f ---> l) sequentially" using `l\s` unfolding LIMSEQ_def by auto } thus ?thesis unfolding complete_def by auto qed lemma nat_approx_posE: fixes e::real assumes "0 < e" obtains n::nat where "1 / (Suc n) < e" proof atomize_elim have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))" by (rule divide_strict_left_mono) (auto 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" .. qed lemma compact_eq_totally_bounded: "compact s \ complete s \ (\e>0. \k. finite k \ s \ (\((\x. 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 \ (\x\k e. ball x e)" by (auto simp: choice_iff') show "compact s" proof cases assume "s = {}" thus "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 \ "\n. 1 / (2 * Suc n)" then have [simp]: "\n. 0 < e n" by auto def B \ "\n 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) (\n i. sel (Suc n) i)" have "subseq t" unfolding subseq_Suc_iff by (simp add: t_def sel) moreover have "\i. (f \ t) i \ s" using f by auto moreover { fix n have "(f \ t) n \ F n" by (cases n) (simp_all add: t_def sel) } note t = this have "Cauchy (f \ 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 \ t) n \ F (Suc N)" "(f \ 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 \ t) n) < e N" "dist x ((f \ t) m) < e N" by (auto simp: subset_eq) with dist_triangle[of "(f \ t) m" "(f \ t) n" x] `2 * e N < r` show "dist ((f \ t) m) ((f \ t) n) < r" by (simp add: dist_commute) qed ultimately show "\l\s. \r. subseq r \ (f \ r) ----> l" using assms unfolding complete_def by blast qed qed qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded) lemma cauchy: "Cauchy s \ (\e>0.\ N::nat. \n\N. dist(s n)(s N) < e)" (is "?lhs = ?rhs") proof- { assume ?rhs { fix e::real assume "e>0" with `?rhs` obtain N where N:"\n\N. dist (s n) (s N) < e/2" by (erule_tac x="e/2" in allE) auto { fix n m assume nm:"N \ m \ N \ n" hence "dist (s m) (s n) < e" using N using dist_triangle_half_l[of "s m" "s N" "e" "s n"] by blast } hence "\N. \m n. N \ m \ N \ n \ dist (s m) (s n) < e" by blast } hence ?lhs unfolding cauchy_def by blast } thus ?thesis unfolding cauchy_def using dist_triangle_half_l by blast qed lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)" proof- from assms obtain N::nat where "\m n. N \ m \ N \ n \ dist (s m) (s n) < 1" unfolding cauchy_def apply(erule_tac x= 1 in allE) by auto hence 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) by auto qed instance heine_borel < complete_space proof fix f :: "nat \ 'a" assume "Cauchy f" hence "bounded (range f)" by (rule cauchy_imp_bounded) hence "compact (closure (range f))" unfolding compact_eq_bounded_closed by auto hence "complete (closure (range f))" by (rule compact_imp_complete) moreover have "\n. f n \ closure (range f)" using closure_subset [of "range f"] by auto ultimately have "\l\closure (range f). (f ---> l) sequentially" using `Cauchy f` unfolding complete_def by auto then show "convergent f" unfolding convergent_def by auto qed instance euclidean_space \ banach .. lemma complete_univ: "complete (UNIV :: 'a::complete_space set)" proof(simp add: complete_def, rule, rule) fix f :: "nat \ 'a" assume "Cauchy f" hence "convergent f" by (rule Cauchy_convergent) thus "\l. f ----> l" unfolding convergent_def . qed lemma 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 hence "x \ s" using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto } thus "closed s" unfolding closed_limpt by auto qed lemma complete_eq_closed: fixes s :: "'a::complete_space set" shows "complete s \ closed s" (is "?lhs = ?rhs") proof assume ?lhs thus ?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 hence "\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 } thus ?lhs unfolding complete_def by auto qed lemma convergent_eq_cauchy: fixes s :: "nat \ 'a::complete_space" shows "(\l. (s ---> l) sequentially) \ Cauchy s" unfolding Cauchy_convergent_iff convergent_def .. lemma convergent_imp_bounded: fixes s :: "nat \ 'a::metric_space" shows "(s ---> l) sequentially \ bounded (range s)" by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy) lemma compact_cball[simp]: fixes x :: "'a::heine_borel" shows "compact(cball x e)" using compact_eq_bounded_closed bounded_cball closed_cball by blast lemma compact_frontier_bounded[intro]: fixes s :: "'a::heine_borel set" shows "bounded s ==> compact(frontier s)" unfolding frontier_def using compact_eq_bounded_closed by blast lemma compact_frontier[intro]: fixes s :: "'a::heine_borel set" shows "compact s ==> compact (frontier s)" using compact_eq_bounded_closed compact_frontier_bounded by blast lemma frontier_subset_compact: fixes s :: "'a::heine_borel set" shows "compact s ==> frontier s \ s" using frontier_subset_closed compact_eq_bounded_closed by blast subsection {* Bounded closed nest property (proof does not use Heine-Borel) *} lemma bounded_closed_nest: assumes "\n. closed(s n)" "\n. (s n \ {})" "(\m n. m \ n --> s n \ s m)" "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 "\n 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 \ r) ---> l) sequentially" unfolding seq_compact_def apply(erule_tac x=x in allE) using x using assms(3) by blast { fix n::nat { fix e::real assume "e>0" with lr(3) obtain N where N:"\m\N. dist ((x \ r) m) l < e" unfolding LIMSEQ_def by auto hence "dist ((x \ 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 hence "(x \ 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) by auto ultimately have "\y\s n. dist y l < e" by auto } hence "l \ s n" using closed_approachable[of "s n" l] assms(1) by blast } thus ?thesis by auto qed text {* 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 hence "\t. \n. t n \ s n" using choice[of "\ 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" hence "t m \ s N" "t n \ s N" using assms(3) t unfolding subset_eq t by blast+ hence "dist (t m) (t n) < e" using N by auto } hence "\N. \m n. N \ m \ N \ n \ dist (t m) (t n) < e" by auto } hence "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 by auto hence "\y\s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto } hence "l \ s n" using closed_approachable[of "s n" l] assms(1) by auto } then show ?thesis by auto qed text {* Strengthen it to the intersection actually being a singleton. *} lemma decreasing_closed_nest_sing: fixes s :: "nat \ 'a::complete_space set" assumes "\n. closed(s n)" "\n. s n \ {}" "\m n. m \ n --> s n \ s m" "\e>0. \n. \x \ (s n). \ y\(s n). dist x y < e" shows "\a. \(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 \ \(range s)" { fix e::real assume "e>0" hence "dist a b < e" using assms(4 )using b using a by blast } hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le) } with a have "\(range s) = {a}" unfolding image_def by auto thus ?thesis .. qed text{* Cauchy-type criteria for uniform convergence. *} lemma uniformly_convergent_eq_cauchy: fixes s::"nat \ 'b \ 'a::complete_space" shows "(\l. \e>0. \N. \n x. N \ n \ P x --> dist(s n x)(l x) < e) \ (\e>0. \N. \m n x. N \ m \ N \ n \ P x --> dist (s m x) (s n x) < e)" (is "?lhs = ?rhs") proof(rule) 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" hence "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 } hence "\N. \m n x. N \ m \ N \ n \ P x --> dist (s m x) (s n x) < e" by auto } thus ?rhs by auto next assume ?rhs hence "\x. P x \ Cauchy (\n. s n x)" unfolding cauchy_def apply auto by (erule_tac x=e in allE)auto then obtain l where l:"\x. P x \ ((\n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym] using choice[of "\x l. P x \ ((\n. 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] using `e>0` by(auto elim!: allE[where x="e/2"]) fix n::nat assume "n\N" hence "dist(s n x)(l x) < e" using `P x`and N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]] using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"] by (auto simp add: dist_commute) } hence "\N. \n x. N \ n \ P x \ dist(s n x)(l x) < e" by auto } thus ?lhs by auto qed lemma uniformly_cauchy_imp_uniformly_convergent: fixes s :: "nat \ 'a \ 'b::complete_space" assumes "\e>0.\N. \m (n::nat) x. N \ m \ N \ n \ P x --> dist(s m x)(s n x) < e" "\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[THEN sym] by auto moreover { fix x assume "P x" hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\n. s n x" "l x" "l' x"] using l and assms(2) unfolding LIMSEQ_def by blast } ultimately show ?thesis by auto qed subsection {* Continuity *} text {* Define continuity over a net to take in restrictions of the set. *} definition continuous :: "'a::t2_space filter \ ('a \ 'b::topological_space) \ bool" where "continuous net f \ (f ---> f(netlimit net)) net" lemma continuous_trivial_limit: "trivial_limit net ==> continuous net f" unfolding continuous_def tendsto_def trivial_limit_eq by auto lemma continuous_within: "continuous (at x within s) f \ (f ---> f(x)) (at x within s)" unfolding continuous_def unfolding tendsto_def using netlimit_within[of x s] by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually) lemma continuous_at: "continuous (at x) f \ (f ---> f(x)) (at x)" using continuous_within [of x UNIV f] by simp lemma continuous_isCont: "isCont f x = continuous (at x) f" unfolding isCont_def LIM_def unfolding continuous_at Lim_at unfolding dist_nz by auto lemma continuous_at_within: assumes "continuous (at x) f" shows "continuous (at x within s) f" using assms unfolding continuous_at continuous_within by (rule Lim_at_within) 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[THEN sym] apply(auto del: allE elim!:allE) apply(rule_tac x=d in exI) by auto lemma continuous_at_eps_delta: "continuous (at x) f \ (\e>0. \d>0. \x'. dist x' x < d --> dist(f x')(f x) < e)" using continuous_within_eps_delta [of x UNIV f] by simp text{* Versions in terms of open balls. *} lemma continuous_within_ball: "continuous (at x within s) f \ (\e>0. \d>0. f ` (ball x d \ s) \ ball (f x) e)" (is "?lhs = ?rhs") proof assume ?lhs { fix e::real assume "e>0" then obtain d where d: "d>0" "\xa\s. 0 < dist xa x \ dist xa x < d \ dist (f xa) (f x) < e" using `?lhs`[unfolded continuous_within Lim_within] by auto { fix y assume "y\f ` (ball x d \ s)" hence "y \ ball (f x) e" using d(2) unfolding dist_nz[THEN sym] apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto } hence "\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) } thus ?rhs by auto next assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto qed lemma continuous_at_ball: "continuous (at x) f \ (\e>0. \d>0. f ` (ball x d) \ ball (f x) e)" (is "?lhs = ?rhs") proof assume ?lhs thus ?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[THEN sym] by auto next assume ?rhs thus ?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) by (auto simp add: dist_commute dist_nz) qed text{* Define setwise continuity in terms of limits within the set. *} definition continuous_on :: "'a set \ ('a::topological_space \ 'b::topological_space) \ bool" where "continuous_on s f \ (\x\s. (f ---> f x) (at x within s))" lemma continuous_on_topological: "continuous_on s f \ (\x\s. \B. open B \ f x \ B \ (\A. open A \ x \ A \ (\y\s. y \ A \ f y \ B)))" unfolding continuous_on_def tendsto_def unfolding Limits.eventually_within eventually_at_topological by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto 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", simp) apply (simp add: dist_nz) done definition uniformly_continuous_on :: "'a set \ ('a::metric_space \ 'b::metric_space) \ bool" where "uniformly_continuous_on s f \ (\e>0. \d>0. \x\s. \x'\s. dist x' x < d \ dist (f x') (f x) < e)" text{* Some simple consequential lemmas. *} lemma uniformly_continuous_imp_continuous: " uniformly_continuous_on s f ==> continuous_on s f" unfolding uniformly_continuous_on_def continuous_on_iff by blast lemma continuous_at_imp_continuous_within: "continuous (at x) f ==> continuous (at x within s) f" unfolding continuous_within continuous_at using Lim_at_within by auto lemma Lim_trivial_limit: "trivial_limit net \ (f ---> l) net" unfolding tendsto_def by (simp add: trivial_limit_eq) lemma continuous_at_imp_continuous_on: assumes "\x\s. continuous (at x) f" shows "continuous_on s f" unfolding continuous_on_def proof fix x assume "x \ s" with assms have *: "(f ---> f (netlimit (at x))) (at x)" unfolding continuous_def by simp have "(f ---> f x) (at x)" proof (cases "trivial_limit (at x)") case True thus ?thesis by (rule Lim_trivial_limit) next case False hence 1: "netlimit (at x) = x" using netlimit_within [of x UNIV] by simp with * show ?thesis by simp qed thus "(f ---> f x) (at x within s)" by (rule Lim_at_within) qed lemma continuous_on_eq_continuous_within: "continuous_on s f \ (\x \ s. continuous (at x within s) f)" unfolding continuous_on_def continuous_def apply (rule ball_cong [OF refl]) apply (case_tac "trivial_limit (at x within s)") apply (simp add: Lim_trivial_limit) apply (simp add: netlimit_within) done lemmas continuous_on = continuous_on_def -- "legacy theorem name" lemma continuous_on_eq_continuous_at: shows "open s ==> (continuous_on s f \ (\x \ s. continuous (at x) f))" by (auto simp add: continuous_on continuous_at Lim_within_open) lemma continuous_within_subset: "continuous (at x within s) f \ t \ s ==> continuous (at x within t) f" unfolding continuous_within by(metis Lim_within_subset) lemma continuous_on_subset: shows "continuous_on s f \ t \ s ==> continuous_on t f" unfolding continuous_on by (metis subset_eq Lim_within_subset) lemma continuous_on_interior: shows "continuous_on s f \ x \ interior s \ continuous (at x) f" by (erule interiorE, drule (1) continuous_on_subset, simp add: continuous_on_eq_continuous_at) lemma continuous_on_eq: "(\x \ s. f x = g x) \ continuous_on s f \ continuous_on s g" unfolding continuous_on_def tendsto_def Limits.eventually_within by simp text {* Characterization of various kinds of continuity in terms of sequences. *} lemma continuous_within_sequentially: fixes f :: "'a::metric_space \ 'b::topological_space" shows "continuous (at a within s) f \ (\x. (\n::nat. x n \ s) \ (x ---> a) sequentially --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs") proof assume ?lhs { fix x::"nat \ 'a" assume x:"\n. x n \ s" "\e>0. eventually (\n. 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_within by auto have "eventually (\n. dist (x n) a < d) sequentially" using x(2) `d>0` by simp hence "eventually (\n. (f \ x) n \ T) sequentially" proof eventually_elim case (elim n) thus ?case using d x(1) `f a \ T` unfolding dist_nz[THEN sym] by auto qed } thus ?rhs unfolding tendsto_iff unfolding tendsto_def by simp next assume ?rhs thus ?lhs unfolding continuous_within tendsto_def [where l="f a"] by (simp add: sequentially_imp_eventually_within) qed lemma continuous_at_sequentially: fixes f :: "'a::metric_space \ 'b::topological_space" shows "continuous (at a) f \ (\x. (x ---> a) sequentially --> ((f o x) ---> f a) sequentially)" using continuous_within_sequentially[of a UNIV f] by simp lemma continuous_on_sequentially: fixes f :: "'a::metric_space \ 'b::topological_space" shows "continuous_on s f \ (\x. \a \ s. (\n. x(n) \ s) \ (x ---> a) sequentially --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs") proof assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto next assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto qed lemma uniformly_continuous_on_sequentially: "uniformly_continuous_on s f \ (\x y. (\n. x n \ s) \ (\n. y n \ s) \ ((\n. dist (x n) (y n)) ---> 0) sequentially \ ((\n. 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:"((\n. 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" hence "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 } hence "\N. \n\N. dist (f (x n)) (f (y n)) < e" by auto } hence "((\n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding LIMSEQ_def and dist_real_def by auto } thus ?rhs by auto next assume ?rhs { assume "\ ?lhs" then obtain e where "e>0" "\d>0. \x\s. \x'\s. dist x' x < d \ \ dist (f x') (f x) < e" unfolding uniformly_continuous_on_def by auto then obtain fa where fa:"\x. 0 < x \ fst (fa x) \ s \ snd (fa x) \ s \ dist (fst (fa x)) (snd (fa x)) < x \ \ dist (f (fst (fa x))) (f (snd (fa x))) < e" using choice[of "\d 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 \ "\n::nat. fst (fa (inverse (real n + 1)))" def y \ "\n::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" hence "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 hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto } hence "\N. \n\N. dist (x n) (y n) < e" by auto } hence "\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 hence False using fxy and `e>0` by auto } thus ?lhs unfolding uniformly_continuous_on_def by blast qed text{* The usual transformation theorems. *} lemma continuous_transform_within: fixes f g :: "'a::metric_space \ 'b::topological_space" assumes "0 < d" "x \ s" "\x' \ s. dist x' x < d --> f x' = g x'" "continuous (at x within s) f" shows "continuous (at x within s) g" unfolding continuous_within proof (rule Lim_transform_within) show "0 < d" by fact show "\x'\s. 0 < dist x' x \ dist x' x < d \ f x' = g x'" using assms(3) by auto have "f x = g x" using assms(1,2,3) by auto thus "(f ---> g x) (at x within s)" using assms(4) unfolding continuous_within by simp qed lemma continuous_transform_at: fixes f g :: "'a::metric_space \ 'b::topological_space" assumes "0 < d" "\x'. dist x' x < d --> f x' = g x'" "continuous (at x) f" shows "continuous (at x) g" using continuous_transform_within [of d x UNIV f g] assms by simp subsubsection {* Structural rules for pointwise continuity *} lemma continuous_within_id: "continuous (at a within s) (\x. x)" unfolding continuous_within by (rule tendsto_ident_at_within) lemma continuous_at_id: "continuous (at a) (\x. x)" unfolding continuous_at by (rule tendsto_ident_at) lemma continuous_const: "continuous F (\x. c)" unfolding continuous_def by (rule tendsto_const) lemma continuous_fst: "continuous F f \ continuous F (\x. fst (f x))" unfolding continuous_def by (rule tendsto_fst) lemma continuous_snd: "continuous F f \ continuous F (\x. snd (f x))" unfolding continuous_def by (rule tendsto_snd) lemma continuous_Pair: "continuous F f \ continuous F g \ continuous F (\x. (f x, g x))" unfolding continuous_def by (rule tendsto_Pair) lemma continuous_dist: assumes "continuous F f" and "continuous F g" shows "continuous F (\x. dist (f x) (g x))" using assms unfolding continuous_def by (rule tendsto_dist) lemma continuous_infdist: assumes "continuous F f" shows "continuous F (\x. infdist (f x) A)" using assms unfolding continuous_def by (rule tendsto_infdist) lemma continuous_norm: shows "continuous F f \ continuous F (\x. norm (f x))" unfolding continuous_def by (rule tendsto_norm) lemma continuous_infnorm: shows "continuous F f \ continuous F (\x. infnorm (f x))" unfolding continuous_def by (rule tendsto_infnorm) lemma continuous_add: fixes f g :: "'a::t2_space \ 'b::real_normed_vector" shows "\continuous F f; continuous F g\ \ continuous F (\x. f x + g x)" unfolding continuous_def by (rule tendsto_add) lemma continuous_minus: fixes f :: "'a::t2_space \ 'b::real_normed_vector" shows "continuous F f \ continuous F (\x. - f x)" unfolding continuous_def by (rule tendsto_minus) lemma continuous_diff: fixes f g :: "'a::t2_space \ 'b::real_normed_vector" shows "\continuous F f; continuous F g\ \ continuous F (\x. f x - g x)" unfolding continuous_def by (rule tendsto_diff) lemma continuous_scaleR: fixes g :: "'a::t2_space \ 'b::real_normed_vector" shows "\continuous F f; continuous F g\ \ continuous F (\x. f x *\<^sub>R g x)" unfolding continuous_def by (rule tendsto_scaleR) lemma continuous_mult: fixes f g :: "'a::t2_space \ 'b::real_normed_algebra" shows "\continuous F f; continuous F g\ \ continuous F (\x. f x * g x)" unfolding continuous_def by (rule tendsto_mult) lemma continuous_inner: assumes "continuous F f" and "continuous F g" shows "continuous F (\x. inner (f x) (g x))" using assms unfolding continuous_def by (rule tendsto_inner) lemma continuous_inverse: fixes f :: "'a::t2_space \ 'b::real_normed_div_algebra" assumes "continuous F f" and "f (netlimit F) \ 0" shows "continuous F (\x. inverse (f x))" using assms unfolding continuous_def by (rule tendsto_inverse) lemma continuous_at_within_inverse: fixes f :: "'a::t2_space \ 'b::real_normed_div_algebra" assumes "continuous (at a within s) f" and "f a \ 0" shows "continuous (at a within s) (\x. inverse (f x))" using assms unfolding continuous_within by (rule tendsto_inverse) lemma continuous_at_inverse: fixes f :: "'a::t2_space \ 'b::real_normed_div_algebra" assumes "continuous (at a) f" and "f a \ 0" shows "continuous (at a) (\x. inverse (f x))" using assms unfolding continuous_at by (rule tendsto_inverse) lemmas continuous_intros = continuous_at_id continuous_within_id continuous_const continuous_dist continuous_norm continuous_infnorm continuous_add continuous_minus continuous_diff continuous_scaleR continuous_mult continuous_inner continuous_at_inverse continuous_at_within_inverse subsubsection {* Structural rules for setwise continuity *} lemma continuous_on_id: "continuous_on s (\x. x)" unfolding continuous_on_def by (fast intro: tendsto_ident_at_within) lemma continuous_on_const: "continuous_on s (\x. c)" unfolding continuous_on_def by (auto intro: tendsto_intros) lemma continuous_on_norm: shows "continuous_on s f \ continuous_on s (\x. norm (f x))" unfolding continuous_on_def by (fast intro: tendsto_norm) lemma continuous_on_infnorm: shows "continuous_on s f \ continuous_on s (\x. infnorm (f x))" unfolding continuous_on by (fast intro: tendsto_infnorm) lemma continuous_on_minus: fixes f :: "'a::topological_space \ 'b::real_normed_vector" shows "continuous_on s f \ continuous_on s (\x. - f x)" unfolding continuous_on_def by (auto intro: tendsto_intros) lemma continuous_on_add: fixes f g :: "'a::topological_space \ 'b::real_normed_vector" shows "continuous_on s f \ continuous_on s g \ continuous_on s (\x. f x + g x)" unfolding continuous_on_def by (auto intro: tendsto_intros) lemma continuous_on_diff: fixes f g :: "'a::topological_space \ 'b::real_normed_vector" shows "continuous_on s f \ continuous_on s g \ continuous_on s (\x. f x - g x)" unfolding continuous_on_def by (auto intro: tendsto_intros) lemma (in bounded_linear) continuous_on: "continuous_on s g \ continuous_on s (\x. f (g x))" unfolding continuous_on_def by (fast intro: tendsto) lemma (in bounded_bilinear) continuous_on: "\continuous_on s f; continuous_on s g\ \ continuous_on s (\x. f x ** g x)" unfolding continuous_on_def by (fast intro: tendsto) lemma continuous_on_scaleR: fixes g :: "'a::topological_space \ 'b::real_normed_vector" assumes "continuous_on s f" and "continuous_on s g" shows "continuous_on s (\x. f x *\<^sub>R g x)" using bounded_bilinear_scaleR assms by (rule bounded_bilinear.continuous_on) lemma continuous_on_mult: fixes g :: "'a::topological_space \ 'b::real_normed_algebra" assumes "continuous_on s f" and "continuous_on s g" shows "continuous_on s (\x. f x * g x)" using bounded_bilinear_mult assms by (rule bounded_bilinear.continuous_on) lemma continuous_on_inner: fixes g :: "'a::topological_space \ 'b::real_inner" assumes "continuous_on s f" and "continuous_on s g" shows "continuous_on s (\x. inner (f x) (g x))" using bounded_bilinear_inner assms by (rule bounded_bilinear.continuous_on) lemma continuous_on_inverse: fixes f :: "'a::topological_space \ 'b::real_normed_div_algebra" assumes "continuous_on s f" and "\x\s. f x \ 0" shows "continuous_on s (\x. inverse (f x))" using assms unfolding continuous_on by (fast intro: tendsto_inverse) subsubsection {* Structural rules for uniform continuity *} lemma uniformly_continuous_on_id: shows "uniformly_continuous_on s (\x. x)" unfolding uniformly_continuous_on_def by auto lemma uniformly_continuous_on_const: shows "uniformly_continuous_on s (\x. c)" unfolding uniformly_continuous_on_def by simp lemma uniformly_continuous_on_dist: fixes f g :: "'a::metric_space \ 'b::metric_space" assumes "uniformly_continuous_on s f" assumes "uniformly_continuous_on s g" shows "uniformly_continuous_on s (\x. 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: "(\n. dist (f (x n)) (f (y n))) ----> 0" assume g: "(\n. dist (g (x n)) (g (y n))) ----> 0" have "(\n. \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) } thus ?thesis using assms unfolding uniformly_continuous_on_sequentially unfolding dist_real_def by simp qed lemma uniformly_continuous_on_norm: assumes "uniformly_continuous_on s f" shows "uniformly_continuous_on s (\x. 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: assumes "uniformly_continuous_on s g" shows "uniformly_continuous_on s (\x. 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: fixes f :: "'a::metric_space \ 'b::real_normed_vector" assumes "uniformly_continuous_on s f" shows "uniformly_continuous_on s (\x. c *\<^sub>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: fixes f :: "'a::metric_space \ 'b::real_normed_vector" shows "uniformly_continuous_on s f \ uniformly_continuous_on s (\x. - f x)" unfolding uniformly_continuous_on_def dist_minus . lemma uniformly_continuous_on_add: fixes f g :: "'a::metric_space \ 'b::real_normed_vector" assumes "uniformly_continuous_on s f" assumes "uniformly_continuous_on s g" shows "uniformly_continuous_on s (\x. 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: 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 (\x. 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. *} lemma continuous_within_topological: "continuous (at x within s) f \ (\B. open B \ f x \ B \ (\A. open A \ x \ A \ (\y\s. y \ A \ f y \ B)))" unfolding continuous_within unfolding tendsto_def Limits.eventually_within eventually_at_topological by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto lemma continuous_within_compose: assumes "continuous (at x within s) f" assumes "continuous (at (f x) within f ` s) g" shows "continuous (at x within s) (g o f)" using assms unfolding continuous_within_topological by simp metis lemma continuous_at_compose: assumes "continuous (at x) f" and "continuous (at (f x)) g" shows "continuous (at x) (g o f)" proof- have "continuous (at (f x) within range f) g" using assms(2) using continuous_within_subset[of "f x" UNIV g "range f"] by simp thus ?thesis using assms(1) using continuous_within_compose[of x UNIV f g] by simp qed lemma continuous_on_compose: "continuous_on s f \ continuous_on (f ` s) g \ continuous_on s (g o f)" unfolding continuous_on_topological by simp metis lemma uniformly_continuous_on_compose: 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 hence "\d>0. \x\s. \x'\s. dist x' x < d \ dist ((g \ f) x') ((g \ f) x) < e" using `d>0` using d by auto } thus ?thesis using assms unfolding uniformly_continuous_on_def by auto qed lemmas continuous_on_intros = continuous_on_id continuous_on_const continuous_on_compose continuous_on_norm continuous_on_infnorm continuous_on_add continuous_on_minus continuous_on_diff continuous_on_scaleR continuous_on_mult continuous_on_inverse continuous_on_inner uniformly_continuous_on_id uniformly_continuous_on_const uniformly_continuous_on_dist uniformly_continuous_on_norm uniformly_continuous_on_compose uniformly_continuous_on_add uniformly_continuous_on_minus uniformly_continuous_on_diff uniformly_continuous_on_cmul text{* Continuity in terms of open preimages. *} lemma continuous_at_open: shows "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, unfolded within_UNIV] unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto lemma continuous_on_open: shows "continuous_on s f \ (\t. openin (subtopology euclidean (f ` s)) t --> openin (subtopology euclidean s) {x \ s. f x \ t})" (is "?lhs = ?rhs") proof (safe) fix t :: "'b set" assume 1: "continuous_on s f" assume 2: "openin (subtopology euclidean (f ` s)) t" from 2 obtain B where B: "open B" and t: "t = f ` s \ B" unfolding openin_open by auto def U == "\{A. open A \ (\x\s. x \ A \ f x \ B)}" have "open U" unfolding U_def by (simp add: open_Union) moreover have "\x\s. x \ U \ f x \ t" proof (intro ballI iffI) fix x assume "x \ s" and "x \ U" thus "f x \ t" unfolding U_def t by auto next fix x assume "x \ s" and "f x \ t" hence "x \ s" and "f x \ B" unfolding t by auto with 1 B obtain A where "open A" "x \ A" "\y\s. y \ A \ f y \ B" unfolding t continuous_on_topological by metis then show "x \ U" unfolding U_def by auto qed ultimately have "open U \ {x \ s. f x \ t} = s \ U" by auto then show "openin (subtopology euclidean s) {x \ s. f x \ t}" unfolding openin_open by fast next assume "?rhs" show "continuous_on s f" unfolding continuous_on_topological proof (clarify) fix x and B assume "x \ s" and "open B" and "f x \ B" have "openin (subtopology euclidean (f ` s)) (f ` s \ B)" unfolding openin_open using `open B` by auto then have "openin (subtopology euclidean s) {x \ s. f x \ f ` s \ B}" using `?rhs` by fast then show "\A. open A \ x \ A \ (\y\s. y \ A \ f y \ B)" unfolding openin_open using `x \ s` and `f x \ B` by auto qed qed text {* Similarly in terms of closed sets. *} lemma continuous_on_closed: shows "continuous_on s f \ (\t. closedin (subtopology euclidean (f ` s)) t --> closedin (subtopology euclidean s) {x \ s. f x \ t})" (is "?lhs = ?rhs") proof assume ?lhs { fix t have *:"s - {x \ s. f x \ f ` s - t} = {x \ s. f x \ t}" by auto have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto assume as:"closedin (subtopology euclidean (f ` s)) t" hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto hence "closedin (subtopology euclidean s) {x \ s. f x \ t}" using `?lhs`[unfolded continuous_on_open, THEN spec[where x="(f ` s) - t"]] unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto } thus ?rhs by auto next assume ?rhs { fix t have *:"s - {x \ s. f x \ f ` s - t} = {x \ s. f x \ t}" by auto assume as:"openin (subtopology euclidean (f ` s)) t" hence "openin (subtopology euclidean s) {x \ s. f x \ t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]] unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto } thus ?lhs unfolding continuous_on_open by auto qed 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 thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \ f ` s"]] using * by auto qed lemma continuous_closed_in_preimage: assumes "continuous_on s f" "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 thus ?thesis using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \ f ` s"]] using * by auto qed lemma continuous_open_preimage: assumes "continuous_on s f" "open s" "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 thus ?thesis using open_Int[of s T, OF assms(2)] by auto qed lemma continuous_closed_preimage: assumes "continuous_on s f" "closed s" "closed t" shows "closed {x \ s. f x \ t}" proof- obtain T where T: "closed T" "{x \ s. f x \ t} = s \ T" using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto thus ?thesis using closed_Int[of s T, OF assms(2)] by auto qed lemma continuous_open_preimage_univ: shows "\x. continuous (at x) f \ open s \ open {x. f x \ s}" using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto lemma continuous_closed_preimage_univ: shows "(\x. continuous (at x) f) \ closed s ==> closed {x. f x \ s}" using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto lemma continuous_open_vimage: shows "\x. continuous (at x) f \ open s \ open (f -` s)" unfolding vimage_def by (rule continuous_open_preimage_univ) lemma continuous_closed_vimage: shows "\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" "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" .. hence "x \ f ` s" by auto then obtain y where y: "y \ s" "x = f y" by auto have "open (vimage f T)" using assms(1) `open T` by (rule continuous_open_vimage) moreover have "y \ vimage f T" using `x = f y` `x \ T` by simp moreover have "vimage f T \ s" using `T \ image f s` `inj f` unfolding inj_on_def subset_eq by auto ultimately have "y \ interior s" .. with `x = f y` show "x \ f ` interior s" .. qed text {* Equality of continuous functions on closure and related results. *} lemma continuous_closed_in_preimage_constant: fixes f :: "_ \ 'b::t1_space" shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \ s. f x = a}" using continuous_closed_in_preimage[of s f "{a}"] by auto lemma continuous_closed_preimage_constant: fixes f :: "_ \ 'b::t1_space" shows "continuous_on s f \ closed s ==> closed {x \ s. f x = a}" using continuous_closed_preimage[of s f "{a}"] by auto lemma continuous_constant_on_closure: fixes f :: "_ \ 'b::t1_space" assumes "continuous_on (closure s) f" "\x \ s. f x = a" shows "\x \ (closure s). f x = a" using continuous_closed_preimage_constant[of "closure s" f a] assms closure_minimal[of s "{x \ closure s. f x = a}"] closure_subset unfolding subset_eq by auto lemma image_closure_subset: assumes "continuous_on (closure s) f" "closed t" "(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 thus ?thesis by auto qed lemma continuous_on_closure_norm_le: fixes f :: "'a::metric_space \ 'b::real_normed_vector" assumes "continuous_on (closure s) f" "\y \ s. norm(f y) \ b" "x \ (closure s)" shows "norm(f x) \ b" proof- have *:"f ` s \ cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] 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) by (auto simp add: dist_norm) qed text {* Making a continuous function avoid some value in a neighbourhood. *} lemma continuous_within_avoid: fixes f :: "'a::metric_space \ 'b::t1_space" assumes "continuous (at x within s) f" and "f x \ a" shows "\e>0. \y \ s. dist x y < e --> f y \ a" proof- obtain U where "open U" and "f x \ U" and "a \ U" using t1_space [OF `f x \ a`] by fast have "(f ---> f x) (at x within s)" using assms(1) by (simp add: continuous_within) hence "eventually (\y. f y \ U) (at x within s)" using `open U` and `f x \ U` unfolding tendsto_def by fast hence "eventually (\y. f y \ a) (at x within s)" using `a \ U` by (fast elim: eventually_mono [rotated]) thus ?thesis unfolding Limits.eventually_within Limits.eventually_at by (rule ex_forward, cut_tac `f x \ a`, auto simp: dist_commute) qed lemma continuous_at_avoid: fixes f :: "'a::metric_space \ 'b::t1_space" assumes "continuous (at x) f" and "f x \ a" shows "\e>0. \y. dist x y < e \ f y \ a" using assms continuous_within_avoid[of x UNIV f a] by simp lemma continuous_on_avoid: fixes f :: "'a::metric_space \ 'b::t1_space" assumes "continuous_on s f" "x \ s" "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] assms(3) by auto lemma continuous_on_open_avoid: fixes f :: "'a::metric_space \ 'b::t1_space" assumes "continuous_on s f" "open s" "x \ s" "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)] continuous_at_avoid[of x f a] assms(4) by auto text {* Proving a function is constant by proving open-ness of level set. *} lemma continuous_levelset_open_in_cases: fixes f :: "_ \ 'b::t1_space" shows "connected s \ continuous_on s f \ openin (subtopology euclidean s) {x \ s. f x = a} ==> (\x \ s. f x \ a) \ (\x \ s. f x = a)" unfolding connected_clopen using continuous_closed_in_preimage_constant by auto lemma continuous_levelset_open_in: fixes f :: "_ \ 'b::t1_space" shows "connected s \ continuous_on s f \ openin (subtopology euclidean s) {x \ s. f x = a} \ (\x \ s. f x = a) ==> (\x \ s. f x = a)" using continuous_levelset_open_in_cases[of s f ] by meson lemma continuous_levelset_open: fixes f :: "_ \ 'b::t1_space" assumes "connected s" "continuous_on s f" "open {x \ s. f x = a}" "\x \ s. f x = a" shows "\x \ s. f x = a" using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast text {* Some arithmetical combinations (more to prove). *} lemma open_scaling[intro]: fixes s :: "'a::real_normed_vector set" assumes "c \ 0" "open s" shows "open((\x. c *\<^sub>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[THEN sym]] using mult_pos_pos[OF `e>0`] by auto moreover { fix y assume "dist y (c *\<^sub>R x) < e * \c\" hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1) assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff) hence "y \ op *\<^sub>R c ` s" using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"] e[THEN spec[where x="(1 / c) *\<^sub>R y"]] assms(1) unfolding dist_norm scaleR_scaleR by auto } ultimately have "\e>0. \x'. dist x' (c *\<^sub>R x) < e \ x' \ op *\<^sub>R c ` s" apply(rule_tac x="e * abs c" in exI) by auto } thus ?thesis unfolding open_dist by auto qed lemma minus_image_eq_vimage: fixes A :: "'a::ab_group_add set" shows "(\x. - x) ` A = (\x. - x) -` A" by (auto intro!: image_eqI [where f="\x. - x"]) lemma open_negations: fixes s :: "'a::real_normed_vector set" shows "open s ==> open ((\ 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((\x. a + x) ` s)" proof- { fix x have "continuous (at x) (\x. 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 "\x. x - a" s] using assms by auto qed lemma open_affinity: fixes s :: "'a::real_normed_vector set" assumes "open s" "c \ 0" shows "open ((\x. a + c *\<^sub>R x) ` s)" proof- have *:"(\x. a + c *\<^sub>R x) = (\x. a + x) \ (\x. c *\<^sub>R x)" unfolding o_def .. have "op + a ` op *\<^sub>R c ` s = (op + a \ op *\<^sub>R c) ` s" by auto thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto qed lemma interior_translation: fixes s :: "'a::real_normed_vector set" shows "interior ((\x. a + x) ` s) = (\x. 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 hence "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[THEN sym] by auto thus "x \ op + a ` interior s" unfolding image_iff apply(rule_tac x="x - a" in bexI) unfolding mem_interior using `e > 0` by auto next fix x assume "x \ op + a ` interior s" then obtain y e where "e>0" and e:"ball y e \ s" and y:"x = a + y" unfolding image_iff Bex_def mem_interior by auto { fix z have *:"a + y - z = y + a - z" by auto assume "z\ball x e" hence "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 hence "z \ op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"]) } hence "ball x e \ op + a ` s" unfolding subset_eq by auto thus "x \ interior (op + a ` s)" unfolding mem_interior using `e>0` by auto qed text {* Topological properties of linear functions. *} lemma linear_lim_0: assumes "bounded_linear f" shows "(f ---> 0) (at (0))" proof- interpret f: bounded_linear f by fact have "(f ---> f 0) (at 0)" using tendsto_ident_at by (rule f.tendsto) thus ?thesis unfolding f.zero . qed lemma linear_continuous_at: assumes "bounded_linear f" shows "continuous (at a) f" unfolding continuous_at using assms apply (rule bounded_linear.tendsto) apply (rule tendsto_ident_at) done lemma linear_continuous_within: shows "bounded_linear f ==> continuous (at x within s) f" using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto lemma linear_continuous_on: shows "bounded_linear f ==> continuous_on s f" using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto text {* Also bilinear functions, in composition form. *} lemma bilinear_continuous_at_compose: shows "continuous (at x) f \ continuous (at x) g \ bounded_bilinear h ==> continuous (at x) (\x. h (f x) (g x))" unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto lemma bilinear_continuous_within_compose: shows "continuous (at x within s) f \ continuous (at x within s) g \ bounded_bilinear h ==> continuous (at x within s) (\x. h (f x) (g x))" unfolding continuous_within using Lim_bilinear[of f "f x"] by auto lemma bilinear_continuous_on_compose: shows "continuous_on s f \ continuous_on s g \ bounded_bilinear h ==> continuous_on s (\x. 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 \ \C \ (\D\C. finite D \ S \ \D))" proof safe fix C assume "compact S" and "\c\C. openin (subtopology euclidean S) c" and "S \ \C" hence "\c\{T. open T \ S \ T \ C}. open c" and "S \ \{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 \ \D" by (rule compactE) hence "image (\T. S \ T) D \ C \ finite (image (\T. S \ T) D) \ S \ \(image (\T. S \ T) D)" by auto thus "\D\C. finite D \ S \ \D" .. next assume 1: "\C. (\c\C. openin (subtopology euclidean S) c) \ S \ \C \ (\D\C. finite D \ S \ \D)" show "compact S" proof (rule compactI) fix C let ?C = "image (\T. S \ T) C" assume "\t\C. open t" and "S \ \C" hence "(\c\?C. openin (subtopology euclidean S) c) \ S \ \?C" unfolding openin_open by auto with 1 obtain D where "D \ ?C" and "finite D" and "S \ \D" by metis let ?D = "inv_into C (\T. S \ T) ` D" have "?D \ C \ finite ?D \ S \ \?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 \ \D` show "S \ \?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 thus "\D\C. finite D \ S \ \D" .. qed qed lemma compact_continuous_image: assumes "continuous_on s f" and "compact s" shows "compact (f ` s)" using assms (* FIXME: long unstructured proof *) unfolding continuous_on_open unfolding compact_eq_openin_cover apply clarify apply (drule_tac x="image (\t. {x \ s. f x \ t}) C" in spec) apply (drule mp) apply (rule conjI) apply simp apply clarsimp apply (drule subsetD) apply (erule imageI) apply fast apply (erule thin_rl) apply clarify apply (rule_tac x="image (inv_into C (\t. {x \ s. f x \ t})) D" in exI) apply (intro conjI) apply clarify apply (rule inv_into_into) apply (erule (1) subsetD) apply (erule finite_imageI) apply (clarsimp, rename_tac x) apply (drule (1) subsetD, clarify) apply (drule (1) subsetD, clarify) apply (rule rev_bexI) apply assumption apply (subgoal_tac "{x \ s. f x \ t} \ (\t. {x \ s. f x \ t}) ` C") apply (drule f_inv_into_f) apply fast apply (erule imageI) done lemma connected_continuous_image: assumes "continuous_on s f" "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 hence False using as(1,2) using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto } thus ?thesis unfolding connected_clopen by auto qed text {* Continuity implies uniform continuity on a compact domain. *} lemma compact_uniformly_continuous: assumes f: "continuous_on s f" and s: "compact s" shows "uniformly_continuous_on s f" unfolding uniformly_continuous_on_def proof (cases, safe) fix e :: real assume "0 < e" "s \ {}" def [simp]: R \ "{(y, d). y \ s \ 0 < d \ ball y d \ s \ {x \ s. f x \ ball (f y) (e/2) } }" let ?b = "(\(y, d). ball y (d/2))" have "(\r\R. open (?b r))" "s \ (\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 \ (\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 \ (\(y, d)\D. ball y (d/2))" by (rule compactE_image) with `s \ {}` have [simp]: "\x. x < Min (snd ` D) \ (\(y, d)\D. x < d)" by (subst Min_gr_iff) auto show "\d>0. \x\s. \x'\s. dist x' x < d \ dist (f x') (f x) < e" proof (rule, safe) fix x x' assume in_s: "x' \ s" "x \ s" with D obtain y d where x: "x \ ball y (d/2)" "(y, d) \ D" by blast moreover assume "dist x x' < Min (snd`D) / 2" ultimately have "dist y x' < d" by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute) with D x in_s show "dist (f x) (f x') < e" by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq) qed (insert D, auto) qed auto text{* Continuity of inverse function on compact domain. *} lemma continuous_on_inv: fixes f :: "'a::topological_space \ 'b::t2_space" assumes "continuous_on s f" "compact s" "\x \ s. g (f x) = x" shows "continuous_on (f ` s) g" unfolding continuous_on_topological proof (clarsimp simp add: assms(3)) fix x :: 'a and B :: "'a set" assume "x \ s" and "open B" and "x \ B" have 1: "\x\s. f x \ f ` (s - B) \ x \ s - B" using assms(3) by (auto, metis) have "continuous_on (s - B) f" using `continuous_on s f` Diff_subset by (rule continuous_on_subset) moreover have "compact (s - B)" using `open B` and `compact s` unfolding Diff_eq by (intro compact_inter_closed closed_Compl) ultimately have "compact (f ` (s - B))" by (rule compact_continuous_image) hence "closed (f ` (s - B))" by (rule compact_imp_closed) hence "open (- f ` (s - B))" by (rule open_Compl) moreover have "f x \ - f ` (s - B)" using `x \ s` and `x \ B` by (simp add: 1) moreover have "\y\s. f y \ - f ` (s - B) \ y \ B" by (simp add: 1) ultimately show "\A. open A \ f x \ A \ (\y\s. f y \ A \ y \ B)" by fast qed text {* A uniformly convergent limit of continuous functions is continuous. *} lemma continuous_uniform_limit: fixes f :: "'a \ 'b::metric_space \ 'c::metric_space" assumes "\ trivial_limit F" assumes "eventually (\n. continuous_on s (f n)) F" assumes "\e>0. eventually (\n. \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 (\n. \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" hence "dist (f n y) (f n x) < e / 3" by (rule d [rule_format]) hence "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 hence "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 } hence "\d>0. \x'\s. dist x' x < d \ dist (g x') (g x) < e" using `d>0` by auto } thus ?thesis unfolding continuous_on_iff by auto qed subsection {* Topological stuff lifted from and dropped to R *} lemma open_real: fixes s :: "real set" shows "open s \ (\x \ s. \e>0. \x'. abs(x' - x) < e --> x' \ s)" (is "?lhs = ?rhs") unfolding open_dist dist_norm by simp lemma islimpt_approachable_real: fixes s :: "real set" shows "x islimpt s \ (\e>0. \x'\ s. x' \ x \ abs(x' - x) < e)" unfolding islimpt_approachable dist_norm by simp lemma closed_real: fixes s :: "real set" shows "closed s \ (\x. (\e>0. \x' \ s. x' \ x \ abs(x' - x) < e) --> x \ s)" unfolding closed_limpt islimpt_approachable dist_norm by simp lemma continuous_at_real_range: fixes f :: "'a::real_normed_vector \ real" shows "continuous (at x) f \ (\e>0. \d>0. \x'. norm(x' - x) < d --> abs(f x' - f x) < e)" unfolding continuous_at unfolding Lim_at unfolding dist_nz[THEN sym] 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) by auto lemma continuous_on_real_range: fixes f :: "'a::real_normed_vector \ real" shows "continuous_on s f \ (\x \ s. \e>0. \d>0. (\x' \ s. norm(x' - x) < d --> abs(f x' - f x) < e))" unfolding continuous_on_iff dist_norm by simp text {* Hence some handy theorems on distance, diameter etc. of/from a set. *} lemma compact_attains_sup: fixes s :: "real set" assumes "compact s" "s \ {}" shows "\x \ s. \y \ s. y \ x" proof- from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto { fix e::real assume as: "\x\s. x \ Sup s" "Sup s \ s" "0 < e" "\x'\s. x' = Sup s \ \ Sup s - x' < e" have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using `e>0` by auto } thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]] apply(rule_tac x="Sup s" in bexI) by auto qed lemma Inf: fixes S :: "real set" shows "S \ {} ==> (\b. b <=* S) ==> isGlb UNIV S (Inf S)" by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def) lemma compact_attains_inf: fixes s :: "real set" assumes "compact s" "s \ {}" shows "\x \ s. \y \ s. x \ y" proof- from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto { fix e::real assume as: "\x\s. x \ Inf s" "Inf s \ s" "0 < e" "\x'\s. x' = Inf s \ \ abs (x' - Inf s) < e" have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto moreover { fix x assume "x \ s" hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto have "Inf s + e \ x" using as(4)[THEN bspec[where x=x]] using as(2) `x\s` unfolding * by auto } hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using `e>0` by auto } thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]] apply(rule_tac x="Inf s" in bexI) by auto qed lemma continuous_attains_sup: fixes f :: "'a::topological_space \ real" shows "compact s \ s \ {} \ continuous_on s f ==> (\x \ s. \y \ s. f y \ f x)" using compact_attains_sup[of "f ` s"] using compact_continuous_image[of s f] by auto lemma continuous_attains_inf: fixes f :: "'a::topological_space \ real" shows "compact s \ s \ {} \ continuous_on s f \ (\x \ s. \y \ s. f x \ f y)" using compact_attains_inf[of "f ` s"] using compact_continuous_image[of s f] by auto 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 Lim_at_within tendsto_ident_at) } thus "continuous_on s (dist a)" unfolding continuous_on .. qed text {* For \emph{minimal} distance, we only need closure, not compactness. *} lemma distance_attains_inf: fixes a :: "'a::heine_borel" assumes "closed s" "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 = "cball a (dist b a) \ s" have "b \ ?B" using `b\s` by (simp add: dist_commute) hence "?B \ {}" by auto moreover { fix x assume "x\?B" fix e::real assume "e>0" { fix x' assume "x'\?B" and as:"dist x' x < e" from as have "\dist a x' - dist a x\ < e" unfolding abs_less_iff minus_diff_eq using dist_triangle2 [of a x' x] using dist_triangle [of a x x'] by arith } hence "\d>0. \x'\?B. dist x' x < d \ \dist a x' - dist a x\ < e" using `e>0` by auto } hence "continuous_on (cball a (dist b a) \ s) (dist a)" unfolding continuous_on Lim_within dist_norm real_norm_def by fast moreover have "compact ?B" using compact_cball[of a "dist b a"] unfolding compact_eq_bounded_closed using bounded_Int and closed_Int and assms(1) by auto ultimately obtain x where "x\cball a (dist b a) \ s" "\y\cball a (dist b a) \ s. dist a x \ dist a y" using continuous_attains_inf[of ?B "dist a"] by fastforce thus ?thesis by fastforce qed subsection {* Pasted sets *} lemma bounded_Times: assumes "bounded s" "bounded t" shows "bounded (s \ t)" proof- obtain x y a b where "\z\s. dist x z \ a" "\z\t. dist y z \ b" using assms [unfolded bounded_def] by auto then have "\z\s \ t. dist (x, y) z \ sqrt (a\ + b\)" by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono) thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto qed lemma mem_Times_iff: "x \ A \ B \ fst x \ A \ snd x \ B" by (induct x) simp lemma seq_compact_Times: "seq_compact s \ seq_compact t \ seq_compact (s \ t)" unfolding seq_compact_def apply clarify apply (drule_tac x="fst \ f" in spec) apply (drule mp, simp add: mem_Times_iff) apply (clarify, rename_tac l1 r1) apply (drule_tac x="snd \ f \ 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 \ r2" in exI) apply (rule conjI, simp add: subseq_def) apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption) apply (drule (1) tendsto_Pair) back apply (simp add: o_def) done text {* Generalize to @{class topological_space} *} lemma compact_Times: fixes s :: "'a::metric_space set" and t :: "'b::metric_space set" shows "compact s \ compact t \ compact (s \ t)" unfolding compact_eq_seq_compact_metric by (rule seq_compact_Times) text{* Hence some useful properties follow quite easily. *} lemma compact_scaling: fixes s :: "'a::real_normed_vector set" assumes "compact s" shows "compact ((\x. c *\<^sub>R x) ` s)" proof- let ?f = "\x. scaleR c x" have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right) show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f] using linear_continuous_at[OF *] assms by auto qed lemma compact_negations: fixes s :: "'a::real_normed_vector set" assumes "compact s" shows "compact ((\x. -x) ` s)" using compact_scaling [OF assms, of "- 1"] by auto lemma compact_sums: fixes s t :: "'a::real_normed_vector set" assumes "compact s" "compact t" shows "compact {x + y | x y. x \ s \ y \ t}" proof- have *:"{x + y | x y. x \ s \ y \ t} = (\z. fst z + snd z) ` (s \ t)" apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto have "continuous_on (s \ t) (\z. fst z + snd z)" unfolding continuous_on by (rule ballI) (intro tendsto_intros) thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto qed lemma compact_differences: fixes s t :: "'a::real_normed_vector set" assumes "compact s" "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) by auto thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto qed lemma compact_translation: fixes s :: "'a::real_normed_vector set" assumes "compact s" shows "compact ((\x. a + x) ` s)" proof- have "{x + y |x y. x \ s \ y \ {a}} = (\x. a + x) ` s" by auto thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto qed lemma compact_affinity: fixes s :: "'a::real_normed_vector set" assumes "compact s" shows "compact ((\x. a + c *\<^sub>R x) ` s)" proof- have "op + a ` op *\<^sub>R c ` s = (\x. a + c *\<^sub>R x) ` s" by auto thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto qed text {* Hence we get the following. *} lemma compact_sup_maxdistance: fixes s :: "'a::metric_space set" assumes "compact s" "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) (\x. dist (fst x) (snd x))" by (intro continuous_at_imp_continuous_on ballI continuous_dist continuous_isCont[THEN iffD1] isCont_fst isCont_snd isCont_ident) ultimately show ?thesis using continuous_attains_sup[of "s \ s" "\x. dist (fst x) (snd x)"] by auto qed text {* 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 Sup_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) qed lemma diameter_lower_bounded: fixes s :: "'a :: metric_space set" assumes s: "bounded s" and d: "0 < d" "d < diameter s" shows "\x\s. \y\s. d < dist x y" proof (rule ccontr) 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 Sup_least) (auto simp: not_less) with `d < diameter s` `s \ {}` show False by (auto simp: diameter_def) qed lemma diameter_bounded: assumes "bounded s" shows "\x\s. \y\s. dist x y \ diameter s" "\d>0. d < diameter s \ (\x\s. \y\s. dist x y > d)" using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms by auto lemma diameter_compact_attained: assumes "compact s" "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 hence "diameter s \ dist x y" unfolding diameter_def by clarsimp (rule Sup_least, fast+) thus ?thesis by (metis b diameter_bounded_bound order_antisym xys) qed text {* Related results with closure as the conclusion. *} lemma closed_scaling: fixes s :: "'a::real_normed_vector set" assumes "closed s" shows "closed ((\x. c *\<^sub>R x) ` s)" proof(cases "s={}") case True thus ?thesis by auto next case False show ?thesis proof(cases "c=0") have *:"(\x. 0) ` s = {0}" using `s\{}` by auto case True thus ?thesis apply auto unfolding * by auto next case False { fix x l assume as:"\n::nat. x n \ scaleR c ` s" "(x ---> l) sequentially" { fix n::nat have "scaleR (1 / c) (x n) \ s" using as(1)[THEN spec[where x=n]] using `c\0` by auto } moreover { fix e::real assume "e>0" hence "0 < e *\c\" using `c\0` mult_pos_pos[of e "abs c"] by auto then obtain N where "\n\N. dist (x n) l < e * \c\" using as(2)[unfolded LIMSEQ_def, THEN spec[where x="e * abs c"]] by auto hence "\N. \n\N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e" unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym] using mult_imp_div_pos_less[of "abs c" _ e] `c\0` by auto } hence "((\n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding LIMSEQ_def by auto ultimately have "l \ scaleR c ` s" using assms[unfolded closed_sequential_limits, THEN spec[where x="\n. scaleR (1/c) (x n)"], THEN spec[where x="scaleR (1/c) l"]] unfolding image_iff using `c\0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto } thus ?thesis unfolding closed_sequential_limits by fast qed qed lemma closed_negations: fixes s :: "'a::real_normed_vector set" assumes "closed s" shows "closed ((\x. -x) ` s)" using closed_scaling[OF assms, of "- 1"] by simp lemma compact_closed_sums: fixes s :: "'a::real_normed_vector set" assumes "compact s" "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 "\n 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:"(((\n. fst (f n)) \ r) ---> l') sequentially" using assms(1)[unfolded compact_def, THEN spec[where x="\ n. fst (f n)"]] using f(2) by auto have "((\n. 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 hence "l - l' \ t" using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\ n. snd (f (r n))"], THEN spec[where x="l - l'"]] using f(3) by auto hence "l \ ?S" using `l' \ s` apply auto apply(rule_tac x=l' in exI) apply(rule_tac x="l - l'" in exI) by auto } thus ?thesis unfolding closed_sequential_limits by fast qed lemma closed_compact_sums: fixes s t :: "'a::real_normed_vector set" assumes "closed s" "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) by auto thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp qed lemma compact_closed_differences: fixes s t :: "'a::real_normed_vector set" assumes "compact s" "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) by auto thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto qed lemma closed_compact_differences: fixes s t :: "'a::real_normed_vector set" assumes "closed s" "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) by auto thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp qed lemma closed_translation: fixes a :: "'a::real_normed_vector" assumes "closed s" shows "closed ((\x. a + x) ` s)" proof- have "{a + y |y. y \ s} = (op + a ` s)" by auto thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto qed lemma translation_Compl: fixes a :: "'a::ab_group_add" shows "(\x. a + x) ` (- t) = - ((\x. a + x) ` t)" apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto lemma translation_UNIV: fixes a :: "'a::ab_group_add" shows "range (\x. a + x) = UNIV" apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto lemma translation_diff: fixes a :: "'a::ab_group_add" shows "(\x. a + x) ` (s - t) = ((\x. a + x) ` s) - ((\x. a + x) ` t)" by auto lemma closure_translation: fixes a :: "'a::real_normed_vector" shows "closure ((\x. a + x) ` s) = (\x. 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) by auto show ?thesis unfolding closure_interior translation_Compl using interior_translation[of a "- s"] unfolding * by auto qed lemma frontier_translation: fixes a :: "'a::real_normed_vector" shows "frontier((\x. a + x) ` s) = (\x. a + x) ` (frontier s)" unfolding frontier_def translation_diff interior_translation closure_translation by auto subsection {* Separation between points and sets *} lemma separate_point_closed: fixes s :: "'a::heine_borel set" shows "closed s \ a \ s ==> (\d>0. \x\s. d \ dist a x)" proof(cases "s = {}") case True thus ?thesis by(auto intro!: exI[where x=1]) next case False assume "closed s" "a \ s" then obtain x where "x\s" "\y\s. dist a x \ dist a y" using `s \ {}` distance_attains_inf [of s a] by blast with `x\s` show ?thesis using dist_pos_lt[of a x] and`a \ s` by blast qed lemma separate_compact_closed: fixes s t :: "'a::heine_borel set" assumes "compact s" and "closed t" and "s \ t = {}" shows "\d>0. \x\s. \y\t. d \ dist x y" proof - (* FIXME: long proof *) let ?T = "\x\s. { ball x (d / 2) | d. 0 < d \ (\y\t. d \ dist x y) }" note `compact s` moreover have "\t\?T. open t" by auto moreover have "s \ \?T" apply auto apply (rule rev_bexI, assumption) apply (subgoal_tac "x \ t") apply (drule separate_point_closed [OF `closed t`]) apply clarify apply (rule_tac x="ball x (d / 2)" in exI) apply simp apply fast apply (cut_tac assms(3)) apply auto done ultimately obtain U where "U \ ?T" and "finite U" and "s \ \U" by (rule compactE) from `finite U` and `U \ ?T` have "\d>0. \x\\U. \y\t. d \ dist x y" apply (induct set: finite) apply simp apply (rule exI) apply (rule zero_less_one) apply clarsimp apply (rename_tac y e) apply (rule_tac x="min d (e / 2)" in exI) apply simp apply (subst ball_Un) apply (rule conjI) apply (intro ballI, rename_tac z) apply (rule min_max.le_infI2) apply (simp only: mem_ball) apply (drule (1) bspec) apply (cut_tac x=y and y=x and z=z in dist_triangle, arith) apply simp apply (intro ballI) apply (rule min_max.le_infI1) apply simp done with `s \ \U` show ?thesis by fast qed 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) by (auto simp add: dist_commute) qed subsection {* 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<.. (\i\Basis. a\i < x\i \ x\i < b\i)" "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}" hence "a \ i < x \ i \ x \ i < b \ i" unfolding mem_interval by auto hence "a\i < b\i" by auto hence False using as by auto } moreover { assume as:"\i\Basis. \ (b\i \ a\i)" let ?x = "(1/2) *\<^sub>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 hence "a\i < ((1/2) *\<^sub>R (a+b)) \ i" "((1/2) *\<^sub>R (a+b)) \ i < b\i" by (auto simp: inner_add_left) } hence "{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}" hence "a \ i \ x \ i \ x \ i \ b \ i" unfolding mem_interval by auto hence "a\i \ b\i" by auto hence False using as by auto } moreover { assume as:"\i\Basis. \ (b\i < a\i)" let ?x = "(1/2) *\<^sub>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 hence "a\i \ ((1/2) *\<^sub>R (a+b)) \ i" "((1/2) *\<^sub>R (a+b)) \ i \ b\i" by (auto simp: inner_add_left) } hence "{a .. b} \ {}" using mem_interval(2)[of "?x" a b] by auto } ultimately show ?th2 by blast qed lemma 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<..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<..i\Basis. a\i \ c\i \ d\i \ b\i) \ {c<.. {a .. b}" and "(\i\Basis. a\i \ c\i \ d\i \ b\i) \ {c<.. {a<.. {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<.. (\i\Basis. c\i \ d\i) --> (\i\Basis. a\i < c\i \ d\i < b\i)" (is ?th2) and "{c<.. {a .. b} \ (\i\Basis. c\i < d\i) --> (\i\Basis. a\i \ c\i \ d\i \ b\i)" (is ?th3) and "{c<.. {a<.. (\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<.. {a .. b}" "\i\Basis. c\i < d\i" hence "{c<.. {}" 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) *\<^sub>R j)::'a" assume as2: "a\i > c\i" { fix j :: 'a assume j:"j\Basis" hence "c \ j < ?x \ j \ ?x \ j < d \ j" apply(cases "j=i") using as(2)[THEN bspec[where x=j]] i by (auto simp add: as2) } hence "?x\{c<..{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 by auto ultimately have False using as by auto } hence "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) *\<^sub>R j)::'a" assume as2: "b\i < d\i" { fix j :: 'a assume "j\Basis" hence "d \ j > ?x \ j \ ?x \ j > c \ j" apply(cases "j=i") using as(2)[THEN bspec[where x=j]] by (auto simp add: as2) } hence "?x\{c<..{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 by auto ultimately have False using as by auto } hence "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 by(erule_tac x=xa in allE,erule_tac x=xa in allE,fastforce)+ { assume as:"{c<.. {a<..i\Basis. c\i < d\i" fix i :: 'a assume i:"i\Basis" from as(1) have "{c<.. {a..b}" using interval_open_subset_closed[of a b] by auto hence "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 by(erule_tac x=xa in allE, simp)+ qed lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows "{a .. b} \ {c .. d} = {(\i\Basis. max (a\i) (c\i) *\<^sub>R i) .. (\i\Basis. min (b\i) (d\i) *\<^sub>R i)}" unfolding set_eq_iff and Int_iff and mem_interval by auto lemma 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<.. (\i\Basis. (b\i < a\i \ d\i \ c\i \ b\i \ c\i \ d\i \ a\i))" (is ?th2) and "{a<.. {c .. d} = {} \ (\i\Basis. (b\i \ a\i \ d\i < c\i \ b\i \ c\i \ d\i \ a\i))" (is ?th3) and "{a<.. {c<.. (\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) *\<^sub>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 **) auto qed (* Moved interval_open_subset_closed a bit upwards *) lemma open_interval[intro]: fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..i\Basis. (\x. x\i) -` {a\i<..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 "(\i\Basis. (\x. x\i) -` {a\i<..i}) = {a<..i\Basis. (\x. 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 "(\i\Basis. (\x. x\i) -` {a\i .. b\i}) = {a .. b}" by (auto simp add: eucl_le [where 'a='a]) finally show "closed {a .. b}" . qed lemma interior_closed_interval [intro]: fixes a b :: "'a::ordered_euclidean_space" shows "interior {a..b} = {a<.. ?L" using interval_open_subset_closed open_interval by (rule interior_maximal) next { 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) *\<^sub>R i) x < e" "dist (x + (e / 2) *\<^sub>R i) x < e" unfolding dist_norm apply auto unfolding norm_minus_cancel using norm_Basis[OF i] `e>0` by auto hence "a \ i \ (x - (e / 2) *\<^sub>R i) \ i" "(x + (e / 2) *\<^sub>R i) \ i \ b \ i" using e[THEN spec[where x="x - (e/2) *\<^sub>R i"]] and e[THEN spec[where x="x + (e/2) *\<^sub>R i"]] unfolding mem_interval using i by blast+ hence "a \ i < x \ i" and "x \ i < b \ i" using `e>0` i by (auto simp: inner_diff_left inner_Basis inner_add_left) } hence "x \ {a<.. ?R" .. qed lemma 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" hence "\x\i\ \ \a\i\ + \b\i\" using x[THEN bspec[where x=i]] by auto } hence "(\i\Basis. \x \ i\) \ ?b" apply-apply(rule setsum_mono) by auto hence "norm x \ ?b" using norm_le_l1[of x] by auto } thus ?thesis unfolding interval and bounded_iff by auto qed lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b} \ bounded {a<.. UNIV) \ ({a<.. UNIV)" using bounded_interval[of a b] by auto lemma 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<.. {}" shows "((1/2) *\<^sub>R (a + b)) \ {a<..Basis" hence "a \ i < ((1 / 2) *\<^sub>R (a + b)) \ i \ ((1 / 2) *\<^sub>R (a + b)) \ i < b \ i" using assms[unfolded interval_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left) } thus ?thesis unfolding mem_interval by auto qed lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space" assumes x:"x \ {a<.. {a .. b}" and e:"0 < e" "e \ 1" shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \ {a<..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 *\<^sub>R x + (1 - e) *\<^sub>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 *\<^sub>R x + (1 - e) *\<^sub>R y) \ i < b \ i" unfolding inner_simps by auto } ultimately have "a \ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \ i \ (e *\<^sub>R x + (1 - e) *\<^sub>R y) \ i < b \ i" by auto } thus ?thesis unfolding mem_interval by auto qed lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space" assumes "{a<.. {}" shows "closure {a<.. {a .. b}" def f == "\n::nat. x + (inverse (real n + 1)) *\<^sub>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)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x = x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)" by (auto simp add: algebra_simps) hence "f n < b" and "a < f n" using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *] unfolding f_def by auto hence False using fn unfolding f_def using xc by auto } moreover { assume "\ (f ---> x) sequentially" { fix e::real assume "e>0" hence "\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) by auto then obtain N::nat where "inverse (real (N + 1)) < e" by auto hence "\n\N. inverse (real n + 1) < e" by (auto, 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) hence "\N::nat. \n\N. inverse (real n + 1) < e" by auto } hence "((\n. inverse (real n + 1)) ---> 0) sequentially" unfolding LIMSEQ_def by(auto simp add: dist_norm) hence "(f ---> x) sequentially" unfolding f_def using tendsto_add[OF tendsto_const, of "\n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x] using tendsto_scaleR [OF _ tendsto_const, of "\n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto } ultimately have "x \ closure {a<..a. s \ {-a<..0" and b:"\x\s. norm x \ b" using assms[unfolded bounded_pos] by auto def a \ "(\i\Basis. (b + 1) *\<^sub>R i)::'a" { fix x assume "x\s" fix i :: 'a assume i:"i\Basis" hence "(-a)\i < x\i" and "x\i < a\i" using b[THEN bspec[where x=x], OF `x\s`] and Basis_le_norm[OF i, of x] unfolding inner_simps and a_def by auto } thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a]) qed lemma bounded_subset_open_interval: fixes s :: "('a::ordered_euclidean_space) set" shows "bounded s ==> (\a b. s \ {a<..a. s \ {-a .. a}" proof- obtain a where "s \ {- a<.. (\a b. s \ {a .. b})" using bounded_subset_closed_interval_symmetric[of s] by auto lemma frontier_closed_interval: fixes a b :: "'a::ordered_euclidean_space" shows "frontier {a .. b} = {a .. b} - {a<.. {}" shows "{a<.. {c .. d} = {} \ {a<.. {c<..i\Basis. x\i \ b\i}" proof- { fix i :: 'a assume i:"i\Basis" fix x::"'a" assume x:"\e>0. \x'\{x. \i\Basis. x \ i \ b \ i}. x' \ x \ dist x' x < e" { assume "x\i > b\i" then obtain y where "y \ i \ b \ i" "y \ x" "dist y x < x\i - b\i" using x[THEN spec[where x="x\i - b\i"]] using i by auto hence False using Basis_le_norm[OF i, of "y - x"] unfolding dist_norm inner_simps using i by auto } hence "x\i \ b\i" by(rule ccontr)auto } thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast qed lemma closed_interval_right: fixes a::"'a::euclidean_space" shows "closed {x::'a. \i\Basis. a\i \ x\i}" proof- { fix i :: 'a assume i:"i\Basis" fix x::"'a" assume x:"\e>0. \x'\{x. \i\Basis. a \ i \ x \ i}. x' \ x \ dist x' x < e" { assume "a\i > x\i" then obtain y where "a \ i \ y \ i" "y \ x" "dist y x < a\i - x\i" using x[THEN spec[where x="a\i - x\i"]] i by auto hence False using Basis_le_norm[OF i, of "y - x"] unfolding dist_norm inner_simps by auto } hence "a\i \ x\i" by(rule ccontr)auto } thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast qed lemma open_box: "open (box a b)" proof - have "open (\i\Basis. (op \ i) -` {a \ i <..< b \ i})" by (auto intro!: continuous_open_vimage continuous_inner continuous_at_id continuous_const) also have "(\i\Basis. (op \ i) -` {a \ i <..< b \ i}) = box a b" by (auto simp add: box_def inner_commute) finally show ?thesis . qed instance euclidean_space \ second_countable_topology proof def a \ "\f :: 'a \ (real \ real). \i\Basis. fst (f i) *\<^sub>R i" then have a: "\f. (\i\Basis. fst (f i) *\<^sub>R i) = a f" by simp def b \ "\f :: 'a \ (real \ real). \i\Basis. snd (f i) *\<^sub>R i" then have b: "\f. (\i\Basis. snd (f i) *\<^sub>R i) = b f" by simp def B \ "(\f. box (a f) (b f)) ` (Basis \\<^isub>E (\ \ \))" have "Ball B open" by (simp add: B_def open_box) moreover have "(\A. open A \ (\B'\B. \B' = A))" proof safe fix A::"'a set" assume "open A" show "\B'\B. \B' = A" apply (rule exI[of _ "{b\B. b \ A}"]) apply (subst (3) open_UNION_box[OF `open A`]) apply (auto simp add: a b B_def) done qed ultimately have "topological_basis B" unfolding topological_basis_def by blast moreover have "countable B" unfolding B_def by (intro countable_image countable_PiE finite_Basis countable_SIGMA countable_rat) ultimately show "\B::'a set set. countable B \ open = generate_topology B" by (blast intro: topological_basis_imp_subbasis) qed instance euclidean_space \ polish_space .. 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<..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 within_UNIV by simp hence "open {x \ UNIV. f x \ s}" using open_UNIV `open s` by (rule continuous_open_preimage) thus "open (f -` s)" by (simp add: vimage_def) qed lemma isCont_closed_vimage: assumes "\x. isCont f x" and "closed s" shows "closed (f -` s)" using assms unfolding closed_def vimage_Compl [symmetric] by (rule isCont_open_vimage) lemma open_Collect_less: fixes f g :: "'a::topological_space \ real" assumes f: "\x. isCont f x" assumes g: "\x. isCont g x" shows "open {x. f x < g x}" proof - have "open ((\x. g x - f x) -` {0<..})" using isCont_diff [OF g f] open_real_greaterThan by (rule isCont_open_vimage) also have "((\x. g x - f x) -` {0<..}) = {x. f x < g x}" by auto finally show ?thesis . qed lemma closed_Collect_le: fixes f g :: "'a::topological_space \ real" assumes f: "\x. isCont f x" assumes g: "\x. isCont g x" shows "closed {x. f x \ g x}" proof - have "closed ((\x. g x - f x) -` {0..})" using isCont_diff [OF g f] closed_real_atLeast by (rule isCont_closed_vimage) also have "((\x. g x - f x) -` {0..}) = {x. f x \ g x}" by auto finally show ?thesis . qed lemma closed_Collect_eq: fixes f g :: "'a::topological_space \ 'b::t2_space" assumes f: "\x. isCont f x" assumes 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) hence "closed {(x::'b, y::'b). x = y}" unfolding closed_def split_def Collect_neg_eq . with isCont_Pair [OF f g] have "closed ((\x. (f x, g x)) -` {(x, y). x = y})" by (rule isCont_closed_vimage) also have "\ = {x. f x = g x}" by auto finally show ?thesis . qed lemma continuous_at_inner: "continuous (at x) (inner a)" unfolding continuous_at by (intro tendsto_intros) lemma closed_halfspace_le: "closed {x. inner a x \ b}" by (simp add: closed_Collect_le) lemma closed_halfspace_ge: "closed {x. inner a x \ b}" by (simp add: closed_Collect_le) lemma closed_hyperplane: "closed {x. inner a x = b}" by (simp add: closed_Collect_eq) lemma closed_halfspace_component_le: shows "closed {x::'a::euclidean_space. x\i \ a}" by (simp add: closed_Collect_le) lemma closed_halfspace_component_ge: shows "closed {x::'a::euclidean_space. x\i \ a}" by (simp add: 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: shows "open {x::'a::euclidean_space. x\i < a}" by (simp add: open_Collect_less) lemma open_halfspace_component_gt: shows "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 "{..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<..} = (\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} = (\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 ..} = (\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" "\ (trivial_limit net)" "eventually (\x. 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" "\ (trivial_limit net)" "eventually (\x. 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 (\x. 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] and Lim_component_le[OF net, of i b] by auto text{* Limits relative to a union. *} lemma eventually_within_Un: "eventually P (net within (s \ t)) \ eventually P (net within s) \ eventually P (net within t)" unfolding Limits.eventually_within by (auto elim!: eventually_rev_mp) lemma Lim_within_union: "(f ---> l) (net within (s \ t)) \ (f ---> l) (net within s) \ (f ---> l) (net 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 (\x. f x \ S) net)" unfolding tendsto_def trivial_limit_eq by auto lemma continuous_on_union: assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f" shows "continuous_on (s \ t) f" using assms unfolding continuous_on Lim_within_union unfolding Lim_topological trivial_limit_within closed_limpt by auto lemma continuous_on_cases: assumes "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" shows "continuous_on (s \ t) (\x. if P x then f x else g x)" proof- let ?h = "(\x. if P x then f x else g x)" have "\x\s. f x = (if P x then f x else g x)" using assms(5) by auto hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto moreover have "\x\t. g x = (if P x then f x else g x)" using assms(5) by auto hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto qed text{* Some more convenient intermediate-value theorem formulations. *} lemma connected_ivt_hyperplane: assumes "connected s" "x \ s" "y \ s" "inner a x \ b" "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"]] and assms(2-5) by auto qed lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows "connected s \ x \ s \ y \ s \ x\k \ a \ a \ y\k \ (\z\s. z\k = a)" using connected_ivt_hyperplane[of s x y "k::'a" a] by (auto simp: inner_commute) subsection {* 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 homeomorphic_def: "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 = "(\x. x)" in exI) apply(rule_tac x = "(\x. x)" in exI) by blast lemma homeomorphic_sym: "s homeomorphic t \ t homeomorphic s" unfolding homeomorphic_def unfolding homeomorphism_def by blast lemma homeomorphic_trans: assumes "s homeomorphic t" "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" hence "(g1 \ g2) ((f2 \ 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 \ f1) ` s = u" using fg1(2) fg2(2) by auto moreover have "continuous_on s (f2 \ f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto moreover { fix y assume "y\u" hence "(f2 \ f1) ((g1 \ 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 \ g2) ` u = s" using fg1(5) fg2(5) by auto moreover have "continuous_on u (g1 \ 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 \ f1" in exI) apply(rule_tac x="g1 \ g2" in exI) by auto qed lemma homeomorphic_minimal: "s homeomorphic t \ (\f g. (\x\s. f(x) \ t \ (g(f(x)) = x)) \ (\y\t. g(y) \ s \ (f(g(y)) = y)) \ continuous_on s f \ continuous_on t g)" unfolding homeomorphic_def homeomorphism_def apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto unfolding image_iff apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE) apply auto apply(rule_tac x="g x" in bexI) apply auto apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE) apply auto apply(rule_tac x="f x" in bexI) by auto text {* Relatively weak hypotheses if a set is compact. *} lemma homeomorphism_compact: fixes f :: "'a::topological_space \ 'b::t2_space" assumes "compact s" "continuous_on s f" "f ` s = t" "inj_on f s" shows "\g. homeomorphism s t f g" proof- def g \ "\x. 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 hence "g (f x) = x" using g by auto hence "f (g y) = y" unfolding x(1)[THEN sym] by auto } hence 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 hence "x \ s" unfolding g_def using someI2[of "\b. b\s \ f b = y" x' "\x. x\s"] unfolding y(2)[THEN sym] and g_def by auto } ultimately have "x\s \ x \ g ` t" .. } hence "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) by auto qed lemma homeomorphic_compact: fixes f :: "'a::topological_space \ 'b::t2_space" shows "compact s \ continuous_on s f \ (f ` s = t) \ inj_on f s \ s homeomorphic t" unfolding homeomorphic_def by (metis homeomorphism_compact) text{* Preservation of topological properties. *} lemma homeomorphic_compactness: "s homeomorphic t ==> (compact s \ compact t)" unfolding homeomorphic_def homeomorphism_def by (metis compact_continuous_image) text{* Results on translation, scaling etc. *} lemma homeomorphic_scaling: fixes s :: "'a::real_normed_vector set" assumes "c \ 0" shows "s homeomorphic ((\x. c *\<^sub>R x) ` s)" unfolding homeomorphic_minimal apply(rule_tac x="\x. c *\<^sub>R x" in exI) apply(rule_tac x="\x. (1 / c) *\<^sub>R x" in exI) using assms by (auto simp add: continuous_on_intros) lemma homeomorphic_translation: fixes s :: "'a::real_normed_vector set" shows "s homeomorphic ((\x. a + x) ` s)" unfolding homeomorphic_minimal apply(rule_tac x="\x. a + x" in exI) apply(rule_tac x="\x. -a + x" in exI) using continuous_on_add[OF continuous_on_const continuous_on_id] by auto lemma homeomorphic_affinity: fixes s :: "'a::real_normed_vector set" assumes "c \ 0" shows "s homeomorphic ((\x. a + c *\<^sub>R x) ` s)" proof- have *:"op + a ` op *\<^sub>R c ` s = (\x. a + c *\<^sub>R x) ` s" by auto show ?thesis using homeomorphic_trans using homeomorphic_scaling[OF assms, of s] using homeomorphic_translation[of "(\x. c *\<^sub>R x) ` s" a] unfolding * by auto qed lemma homeomorphic_balls: fixes a b ::"'a::real_normed_vector" assumes "0 < d" "0 < e" shows "(ball a d) homeomorphic (ball b e)" (is ?th) "(cball a d) homeomorphic (cball b e)" (is ?cth) proof- show ?th unfolding homeomorphic_minimal apply(rule_tac x="\x. b + (e/d) *\<^sub>R (x - a)" in exI) apply(rule_tac x="\x. a + (d/e) *\<^sub>R (x - b)" in exI) using assms apply (auto simp add: dist_commute) unfolding dist_norm apply (auto simp add: pos_divide_less_eq mult_strict_left_mono) unfolding continuous_on by (intro ballI tendsto_intros, simp)+ next show ?cth unfolding homeomorphic_minimal apply(rule_tac x="\x. b + (e/d) *\<^sub>R (x - a)" in exI) apply(rule_tac x="\x. a + (d/e) *\<^sub>R (x - b)" in exI) using assms apply (auto simp add: dist_commute) unfolding dist_norm apply (auto simp add: pos_divide_le_eq) unfolding continuous_on by (intro ballI tendsto_intros, simp)+ qed text{* "Isometry" (up to constant bounds) of injective linear map etc. *} lemma cauchy_isometric: fixes x :: "nat \ 'a::euclidean_space" assumes e:"0 < e" and s:"subspace s" and f:"bounded_linear f" and normf:"\x\s. norm(f x) \ e * norm(x)" and xs:"\n::nat. 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[THEN sym] by auto finally have "norm (x n - x N) < d" using `e>0` by simp } hence "\N. \n\N. norm (x n - x N) < d" by auto } thus ?thesis unfolding cauchy and dist_norm by auto qed lemma 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 "\ n xa. xa \ s \ g n = f xa"] by auto hence x:"\n. x n \ s" "\n. g n = f (x n)" by auto hence "f \ 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 `0l\f ` s. (g ---> l) sequentially" using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l] unfolding `f \ x = g` by auto } thus ?thesis unfolding complete_def by auto qed lemma injective_imp_isometric: fixes f::"'a::euclidean_space \ 'b::euclidean_space" assumes s:"closed s" "subspace s" and f:"bounded_linear f" "\x\s. (f x = 0) \ (x = 0)" shows "\e>0. \x\s. norm (f x) \ e * norm(x)" proof(cases "s \ {0::'a}") case True { fix x assume "x \ s" hence "x = 0" using True by auto hence "norm x \ norm (f x)" by auto } thus ?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 hence "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 hence "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 *[THEN sym] 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" hence "norm (f b) / norm b * norm x \ norm (f x)" proof(cases "x=0") case True thus "norm (f b) / norm b * norm x \ norm (f x)" by auto next case False hence *:"0 < norm a / norm x" using `a\0` unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos) have "\c. \x\s. c *\<^sub>R x \ s" using s[unfolded subspace_def] by auto hence "(norm a / norm x) *\<^sub>R x \ {x \ s. norm x = norm a}" using `x\s` and `x\0` by auto thus "norm (f b) / norm b * norm x \ norm (f x)" using b[THEN bspec[where x="(norm a / norm x) *\<^sub>R x"]] unfolding f.scaleR and ba using `x\0` `a\0` by (auto simp add: mult_commute pos_le_divide_eq pos_divide_le_eq) qed } ultimately show ?thesis by auto qed lemma closed_injective_image_subspace: fixes f :: "'a::euclidean_space \ 'b::euclidean_space" assumes "subspace s" "bounded_linear f" "\x\s. f x = 0 --> x = 0" "closed s" shows "closed(f ` s)" proof- obtain e where "e>0" and e:"\x\s. e * norm x \ norm (f x)" using injective_imp_isometric[OF assms(4,1,2,3)] by auto show ?thesis using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4) unfolding complete_eq_closed[THEN sym] by auto qed subsection {* Some properties of a canonical subspace *} lemma subspace_substandard: "subspace {x::'a::euclidean_space. (\i\Basis. P i \ x\i = 0)}" unfolding subspace_def by (auto simp: inner_add_left) lemma closed_substandard: "closed {x::'a::euclidean_space. \i\Basis. P i --> x\i = 0}" (is "closed ?A") proof- let ?D = "{i\Basis. P i}" have "closed (\i\?D. {x::'a. x\i = 0})" by (simp add: closed_INT closed_Collect_eq) also have "(\i\?D. {x::'a. x\i = 0}) = ?A" by auto finally show "closed ?A" . qed lemma dim_substandard: assumes d: "d \ Basis" shows "dim {x::'a::euclidean_space. \i\Basis. i \ d \ x\i = 0} = card d" (is "dim ?A = _") proof- let ?D = "Basis :: 'a set" have "d \ ?A" using d by (auto simp: inner_Basis) moreover { fix x::"'a" assume "x \ ?A" hence "finite d" "x \ ?A" using assms by(auto intro: finite_subset[OF _ finite_Basis]) from this d have "x \ span d" proof(induct d arbitrary: x) case empty hence "x=0" apply(rule_tac euclidean_eqI) by auto thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto next case (insert k F) hence *:"\i\Basis. i \ insert k F \ x \ i = 0" by auto have **:"F \ insert k F" by auto def y \ "x - (x\k) *\<^sub>R k" have y:"x = y + (x\k) *\<^sub>R k" unfolding y_def by auto { fix i assume i': "i \ F" "i \ Basis" hence "y \ i = 0" unfolding y_def using *[THEN bspec[where x=i]] insert by (auto simp: inner_simps inner_Basis) } hence "y \ span F" using insert by auto hence "y \ span (insert k F)" using span_mono[of F "insert k F"] using assms by auto moreover have "k \ span (insert k F)" by(rule span_superset, auto) hence "(x\k) *\<^sub>R k \ span (insert k F)" using span_mul by auto ultimately have "y + (x\k) *\<^sub>R k \ span (insert k F)" using span_add by auto thus ?case using y by auto qed } hence "?A \ span d" by auto moreover { fix x assume "x \ d" hence "x \ ?D" using assms by auto } hence "independent d" using independent_mono[OF independent_Basis, of d] and assms by auto moreover have "d \ ?D" unfolding subset_eq using assms by auto ultimately show ?thesis using dim_unique[of d ?A] by auto qed text{* Hence closure and completeness of all subspaces. *} lemma ex_card: assumes "n \ card A" shows "\S\A. card S = n" proof cases assume "finite A" from ex_bij_betw_nat_finite[OF this] guess f .. moreover with `n \ card A` have "{..< n} \ {..< card A}" "inj_on f {..< n}" by (auto simp: bij_betw_def intro: subset_inj_on) ultimately have "f ` {..< n} \ A" "card (f ` {..< n}) = n" by (auto simp: bij_betw_def card_image) then show ?thesis by blast next assume "\ finite A" with `n \ card A` show ?thesis by force qed lemma closed_subspace: fixes s::"('a::euclidean_space) set" assumes "subspace s" shows "closed s" proof- have "dim s \ card (Basis :: 'a set)" using dim_subset_UNIV by auto with ex_card[OF this] obtain d :: "'a set" where t: "card d = dim s" and d: "d \ Basis" by auto let ?t = "{x::'a. \i\Basis. i \ d \ x\i = 0}" have "\f. linear f \ f ` {x::'a. \i\Basis. i \ d \ x \ i = 0} = s \ inj_on f {x::'a. \i\Basis. i \ d \ x \ i = 0}" using dim_substandard[of d] t d assms by (intro subspace_isomorphism[OF subspace_substandard[of "\i. 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 auto qed lemma complete_subspace: fixes s :: "('a::euclidean_space) set" shows "subspace s ==> complete s" using complete_eq_closed closed_subspace by auto lemma dim_closure: fixes s :: "('a::euclidean_space) set" shows "dim(closure s) = dim s" (is "?dc = ?d") proof- have "?dc \ ?d" using closure_minimal[OF span_inc, of s] using closed_subspace[OF subspace_span, of s] using dim_subset[of "closure s" "span s"] unfolding dim_span by auto thus ?thesis using dim_subset[OF closure_subset, of s] by auto qed subsection {* Affine transformations of intervals *} lemma real_affinity_le: "0 < (m::'a::linordered_field) ==> (m * x + c \ y \ x \ inverse(m) * y + -(c / m))" by (simp add: field_simps 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 "(\x. m *\<^sub>R x + c) ` {a .. b} = (if {a .. b} = {} then {} else (if 0 \ m then {m *\<^sub>R a + c .. m *\<^sub>R b + c} else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))" proof(cases "m=0") { fix x assume "x \ c" "c \ x" hence "x=c" unfolding eucl_le[where 'a='a] apply- apply(subst euclidean_eq_iff) by (auto intro: order_antisym) } moreover case True moreover have "c \ {m *\<^sub>R a + c..m *\<^sub>R b + c}" unfolding True by(auto simp add: eucl_le[where 'a='a]) ultimately show ?thesis by auto next case False { fix y assume "a \ y" "y \ b" "m > 0" hence "m *\<^sub>R a + c \ m *\<^sub>R y + c" "m *\<^sub>R y + c \ m *\<^sub>R b + c" unfolding eucl_le[where 'a='a] by (auto simp: inner_simps) } moreover { fix y assume "a \ y" "y \ b" "m < 0" hence "m *\<^sub>R b + c \ m *\<^sub>R y + c" "m *\<^sub>R y + c \ m *\<^sub>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" "m *\<^sub>R a + c \ y" "y \ m *\<^sub>R b + c" hence "y \ (\x. m *\<^sub>R x + c) ` {a..b}" unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a] apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"]) by(auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff inner_simps) } moreover { fix y assume "m *\<^sub>R b + c \ y" "y \ m *\<^sub>R a + c" "m < 0" hence "y \ (\x. m *\<^sub>R x + c) ` {a..b}" unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a] apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"]) by(auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff inner_simps) } ultimately show ?thesis using False by auto qed lemma image_smult_interval:"(\x. m *\<^sub>R (x::_::ordered_euclidean_space)) ` {a..b} = (if {a..b} = {} then {} else if 0 \ m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})" using image_affinity_interval[of m 0 a b] by auto subsection {* Banach fixed point theorem (not really topological...) *} lemma banach_fix: assumes s:"complete s" "s \ {}" and c:"0 \ c" "c < 1" and f:"(f ` s) \ s" and lipschitz:"\x\s. \y\s. dist (f x) (f y) \ c * dist x y" shows "\! x\s. (f x = x)" proof- have "1 - c > 0" using c by auto from s(2) obtain z0 where "z0 \ s" by auto def z \ "\n. (f ^^ n) z0" { fix n::nat have "z n \ s" unfolding z_def proof(induct n) case 0 thus ?case using `z0 \s` by auto next case Suc thus ?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 thus ?case unfolding d_def by auto next case (Suc m) hence "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 thus ?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" hence "\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: *) thus ?thesis using `e>0` by auto next case False hence "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"] by (metis False d_def less_le) hence "0 < e * (1 - c) / d" using `e>0` and `1-c>0` 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 hence **:"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" hence "dist (z n) (z m) < e" proof(cases "n = m") case True thus ?thesis using `e>0` by auto next case False thus ?thesis using as and *[of n m] *[of m n] unfolding nat_neq_iff by (auto simp add: dist_commute) qed } thus ?thesis by auto qed } hence "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" hence "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 hence 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 hence "f x = x" unfolding e_def by auto moreover { fix y assume "f y = y" "y\s" hence "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 hence "dist x y = 0" unfolding mult_le_cancel_right1 using c and zero_le_dist[of x y] by auto hence "y = x" by auto } ultimately show ?thesis using `x\s` by blast+ qed subsection {* Edelstein fixed point theorem *} lemma edelstein_fix: fixes s :: "'a::metric_space set" assumes s:"compact s" "s \ {}" and gs:"(g ` s) \ s" and dist:"\x\s. \y\s. x \ y \ dist (g x) (g y) < dist x y" shows "\! x\s. g x = x" proof(cases "\x\s. g x \ x") obtain x where "x\s" using s(2) by auto case False hence g:"\x\s. g x = x" by auto { fix y assume "y\s" hence "x = y" using `x\s` and dist[THEN bspec[where x=x], THEN bspec[where x=y]] unfolding g[THEN bspec[where x=x], OF `x\s`] unfolding g[THEN bspec[where x=y], OF `y\s`] by auto } thus ?thesis using `x\s` and g by blast+ next case True then obtain x where [simp]:"x\s" and "g x \ x" by auto { fix x y assume "x \ s" "y \ s" hence "dist (g x) (g y) \ dist x y" using dist[THEN bspec[where x=x], THEN bspec[where x=y]] by auto } note dist' = this def y \ "g x" have [simp]:"y\s" unfolding y_def using gs[unfolded image_subset_iff] and `x\s` by blast def f \ "\n. g ^^ n" have [simp]:"\n z. g (f n z) = f (Suc n) z" unfolding f_def by auto have [simp]:"\z. f 0 z = z" unfolding f_def by auto { fix n::nat and z assume "z\s" have "f n z \ s" unfolding f_def proof(induct n) case 0 thus ?case using `z\s` by simp next case (Suc n) thus ?case using gs[unfolded image_subset_iff] by auto qed } note fs = this { fix m n ::nat assume "m\n" fix w z assume "w\s" "z\s" have "dist (f n w) (f n z) \ dist (f m w) (f m z)" using `m\n` proof(induct n) case 0 thus ?case by auto next case (Suc n) thus ?case proof(cases "m\n") case True thus ?thesis using Suc(1) using dist'[OF fs fs, OF `w\s` `z\s`, of n n] by auto next case False hence mn:"m = Suc n" using Suc(2) by simp show ?thesis unfolding mn by auto qed qed } note distf = this def h \ "\n. (f n x, f n y)" let ?s2 = "s \ s" obtain l r where "l\?s2" and r:"subseq r" and lr:"((h \ r) ---> l) sequentially" using compact_Times [OF s(1) s(1), unfolded compact_def, THEN spec[where x=h]] unfolding h_def using fs[OF `x\s`] and fs[OF `y\s`] by blast def a \ "fst l" def b \ "snd l" have lab:"l = (a, b)" unfolding a_def b_def by simp have [simp]:"a\s" "b\s" unfolding a_def b_def using `l\?s2` by auto have lima:"((fst \ (h \ r)) ---> a) sequentially" and limb:"((snd \ (h \ r)) ---> b) sequentially" using lr unfolding o_def a_def b_def by (rule tendsto_intros)+ { fix n::nat have *:"\fx fy (x::'a) y. dist fx fy \ dist x y \ \ (\dist fx fy - dist a b\ < dist a b - dist x y)" by auto { assume as:"dist a b > dist (f n x) (f n y)" then obtain Na Nb where "\m\Na. dist (f (r m) x) a < (dist a b - dist (f n x) (f n y)) / 2" and "\m\Nb. dist (f (r m) y) b < (dist a b - dist (f n x) (f n y)) / 2" using lima limb unfolding h_def LIMSEQ_def by (fastforce simp del: less_divide_eq_numeral1) hence "\dist (f (r (Na + Nb + n)) x) (f (r (Na + Nb + n)) y) - dist a b\ < dist a b - dist (f n x) (f n y)" apply - apply (drule_tac x="Na+Nb+n" in spec, drule mp, simp) apply (drule_tac x="Na+Nb+n" in spec, drule mp, simp) apply (drule (1) add_strict_mono, simp only: real_sum_of_halves) apply (erule le_less_trans [rotated]) apply (erule thin_rl) apply (rule abs_leI) apply (simp add: diff_le_iff add_assoc) apply (rule order_trans [OF dist_triangle add_left_mono]) apply (subst add_commute, rule dist_triangle2) apply (simp add: diff_le_iff add_assoc) apply (rule order_trans [OF dist_triangle3 add_left_mono]) apply (subst add_commute, rule dist_triangle) done moreover have "\dist (f (r (Na + Nb + n)) x) (f (r (Na + Nb + n)) y) - dist a b\ \ dist a b - dist (f n x) (f n y)" using distf[of n "r (Na+Nb+n)", OF _ `x\s` `y\s`] using seq_suble[OF r, of "Na+Nb+n"] using *[of "f (r (Na + Nb + n)) x" "f (r (Na + Nb + n)) y" "f n x" "f n y"] by auto ultimately have False by simp } hence "dist a b \ dist (f n x) (f n y)" by(rule ccontr)auto } note ab_fn = this have [simp]:"a = b" proof(rule ccontr) def e \ "dist a b - dist (g a) (g b)" assume "a\b" hence "e > 0" unfolding e_def using dist by fastforce hence "\n. dist (f n x) a < e/2 \ dist (f n y) b < e/2" using lima limb unfolding LIMSEQ_def apply (auto elim!: allE[where x="e/2"]) apply(rename_tac N N', rule_tac x="r (max N N')" in exI) unfolding h_def by fastforce then obtain n where n:"dist (f n x) a < e/2 \ dist (f n y) b < e/2" by auto have "dist (f (Suc n) x) (g a) \ dist (f n x) a" using dist[THEN bspec[where x="f n x"], THEN bspec[where x="a"]] and fs by auto moreover have "dist (f (Suc n) y) (g b) \ dist (f n y) b" using dist[THEN bspec[where x="f n y"], THEN bspec[where x="b"]] and fs by auto ultimately have "dist (f (Suc n) x) (g a) + dist (f (Suc n) y) (g b) < e" using n by auto thus False unfolding e_def using ab_fn[of "Suc n"] using dist_triangle2 [of "f (Suc n) y" "g a" "g b"] using dist_triangle2 [of "f (Suc n) x" "f (Suc n) y" "g a"] by auto qed have [simp]:"\n. f (Suc n) x = f n y" unfolding f_def y_def by(induct_tac n)auto { fix x y assume "x\s" "y\s" moreover fix e::real assume "e>0" ultimately have "dist y x < e \ dist (g y) (g x) < e" using dist by fastforce } hence "continuous_on s g" unfolding continuous_on_iff by auto hence "((snd \ h \ r) ---> g a) sequentially" unfolding continuous_on_sequentially apply (rule allE[where x="\n. (fst \ h \ r) n"]) apply (erule ballE[where x=a]) using lima unfolding h_def o_def using fs[OF `x\s`] by (auto simp add: y_def) hence "g a = a" using tendsto_unique[OF trivial_limit_sequentially limb, of "g a"] unfolding `a=b` and o_assoc by auto moreover { fix x assume "x\s" "g x = x" "x\a" hence "False" using dist[THEN bspec[where x=a], THEN bspec[where x=x]] using `g a = a` and `a\s` by auto } ultimately show "\!x\s. g x = x" using `a\s` by blast qed declare tendsto_const [intro] (* FIXME: move *) end