imports Countable_Set Glbs FuncSet Linear_Algebra Norm_Arith

(* title: HOL/Library/Topology_Euclidian_Space.thy

Author: Amine Chaieb, University of Cambridge

Author: Robert Himmelmann, TU Muenchen

Author: Brian Huffman, Portland State University

*)

header {* Elementary topology in Euclidean space. *}

theory Topology_Euclidean_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)

lemma Lim_within_open:

fixes f :: "'a::topological_space => 'b::topological_space"

shows "a ∈ S ==> open S ==> (f ---> l)(at a within S) <-> (f ---> l)(at a)"

by (fact tendsto_within_open)

lemma continuous_on_union:

"closed s ==> closed t ==> continuous_on s f ==> continuous_on t f ==> continuous_on (s ∪ t) f"

by (fact continuous_on_closed_Un)

lemma continuous_on_cases:

"closed s ==> closed t ==> continuous_on s f ==> continuous_on t g ==>

∀x. (x∈s ∧ ¬ P x) ∨ (x ∈ t ∧ P x) --> f x = g x ==>

continuous_on (s ∪ t) (λx. if P x then f x else g x)"

by (rule continuous_on_If) auto

subsection {* Topological Basis *}

context topological_space

begin

definition "topological_basis B <->

(∀b∈B. open b) ∧ (∀x. open x --> (∃B'. B' ⊆ B ∧ \<Union>B' = x))"

lemma topological_basis:

"topological_basis B <-> (∀x. open x <-> (∃B'. B' ⊆ B ∧ \<Union>B' = x))"

unfolding topological_basis_def

apply safe

apply fastforce

apply fastforce

apply (erule_tac x="x" in allE)

apply simp

apply (rule_tac x="{x}" in exI)

apply auto

done

lemma topological_basis_iff:

assumes "!!B'. B' ∈ B ==> open B'"

shows "topological_basis B <-> (∀O'. open O' --> (∀x∈O'. ∃B'∈B. x ∈ B' ∧ B' ⊆ O'))"

(is "_ <-> ?rhs")

proof safe

fix O' and x::'a

assume H: "topological_basis B" "open O'" "x ∈ O'"

then have "(∃B'⊆B. \<Union>B' = O')" by (simp add: topological_basis_def)

then obtain B' where "B' ⊆ B" "O' = \<Union>B'" by auto

then show "∃B'∈B. x ∈ B' ∧ B' ⊆ O'" using H by auto

next

assume H: ?rhs

show "topological_basis B"

using assms unfolding topological_basis_def

proof safe

fix O' :: "'a set"

assume "open O'"

with H obtain f where "∀x∈O'. f x ∈ B ∧ x ∈ f x ∧ f x ⊆ O'"

by (force intro: bchoice simp: Bex_def)

then show "∃B'⊆B. \<Union>B' = O'"

by (auto intro: exI[where x="{f x |x. x ∈ O'}"])

qed

qed

lemma topological_basisI:

assumes "!!B'. B' ∈ B ==> open B'"

and "!!O' x. open O' ==> x ∈ O' ==> ∃B'∈B. x ∈ B' ∧ B' ⊆ O'"

shows "topological_basis B"

using assms by (subst topological_basis_iff) auto

lemma topological_basisE:

fixes O'

assumes "topological_basis B"

and "open O'"

and "x ∈ O'"

obtains B' where "B' ∈ B" "x ∈ B'" "B' ⊆ O'"

proof atomize_elim

from assms have "!!B'. B'∈B ==> open B'"

by (simp add: topological_basis_def)

with topological_basis_iff assms

show "∃B'. B' ∈ B ∧ x ∈ B' ∧ B' ⊆ O'"

using assms by (simp add: Bex_def)

qed

lemma topological_basis_open:

assumes "topological_basis B"

and "X ∈ B"

shows "open X"

using assms by (simp add: topological_basis_def)

lemma topological_basis_imp_subbasis:

assumes B: "topological_basis B"

shows "open = generate_topology B"

proof (intro ext iffI)

fix S :: "'a set"

assume "open S"

with B obtain B' where "B' ⊆ B" "S = \<Union>B'"

unfolding topological_basis_def by blast

then show "generate_topology B S"

by (auto intro: generate_topology.intros dest: topological_basis_open)

next

fix S :: "'a set"

assume "generate_topology B S"

then show "open S"

by induct (auto dest: topological_basis_open[OF B])

qed

lemma basis_dense:

fixes B :: "'a set set"

and f :: "'a set => 'a"

assumes "topological_basis B"

and choosefrom_basis: "!!B'. B' ≠ {} ==> f B' ∈ B'"

shows "(∀X. open X --> X ≠ {} --> (∃B' ∈ B. f B' ∈ X))"

proof (intro allI impI)

fix X :: "'a set"

assume "open X" and "X ≠ {}"

from topological_basisE[OF `topological_basis B` `open X` choosefrom_basis[OF `X ≠ {}`]]

guess B' . note B' = this

then show "∃B'∈B. f B' ∈ X"

by (auto intro!: choosefrom_basis)

qed

end

lemma topological_basis_prod:

assumes A: "topological_basis A"

and B: "topological_basis B"

shows "topological_basis ((λ(a, b). a × b) ` (A × B))"

unfolding topological_basis_def

proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])

fix S :: "('a × 'b) set"

assume "open S"

then show "∃X⊆A × B. (\<Union>(a,b)∈X. a × b) = S"

proof (safe intro!: exI[of _ "{x∈A × B. fst x × snd x ⊆ S}"])

fix x y

assume "(x, y) ∈ S"

from open_prod_elim[OF `open S` this]

obtain a b where a: "open a""x ∈ a" and b: "open b" "y ∈ b" and "a × b ⊆ S"

by (metis mem_Sigma_iff)

moreover from topological_basisE[OF A a] guess A0 .

moreover from topological_basisE[OF B b] guess B0 .

ultimately show "(x, y) ∈ (\<Union>(a, b)∈{X ∈ A × B. fst X × snd X ⊆ S}. a × b)"

by (intro UN_I[of "(A0, B0)"]) auto

qed auto

qed (metis A B topological_basis_open open_Times)

subsection {* Countable Basis *}

locale countable_basis =

fixes B :: "'a::topological_space set set"

assumes is_basis: "topological_basis B"

and countable_basis: "countable B"

begin

lemma open_countable_basis_ex:

assumes "open X"

shows "∃B' ⊆ B. X = Union B'"

using assms countable_basis is_basis

unfolding topological_basis_def by blast

lemma open_countable_basisE:

assumes "open X"

obtains B' where "B' ⊆ B" "X = Union B'"

using assms open_countable_basis_ex

by (atomize_elim) simp

lemma countable_dense_exists:

"∃D::'a set. countable D ∧ (∀X. open X --> X ≠ {} --> (∃d ∈ D. d ∈ X))"

proof -

let ?f = "(λB'. SOME x. x ∈ B')"

have "countable (?f ` B)" using countable_basis by simp

with basis_dense[OF is_basis, of ?f] show ?thesis

by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI)

qed

lemma countable_dense_setE:

obtains D :: "'a set"

where "countable D" "!!X. open X ==> X ≠ {} ==> ∃d ∈ D. d ∈ X"

using countable_dense_exists by blast

end

lemma (in first_countable_topology) first_countable_basisE:

obtains A where "countable A" "!!a. a ∈ A ==> x ∈ a" "!!a. a ∈ A ==> open a"

"!!S. open S ==> x ∈ S ==> (∃a∈A. a ⊆ S)"

using first_countable_basis[of x]

apply atomize_elim

apply (elim exE)

apply (rule_tac x="range A" in exI)

apply auto

done

lemma (in first_countable_topology) first_countable_basis_Int_stableE:

obtains A where "countable A" "!!a. a ∈ A ==> x ∈ a" "!!a. a ∈ A ==> open a"

"!!S. open S ==> x ∈ S ==> (∃a∈A. a ⊆ S)"

"!!a b. a ∈ A ==> b ∈ A ==> a ∩ b ∈ A"

proof atomize_elim

from first_countable_basisE[of x] guess A' . note A' = this

def A ≡ "(λN. \<Inter>((λn. from_nat_into A' n) ` N)) ` (Collect finite::nat set set)"

then show "∃A. countable A ∧ (∀a. a ∈ A --> x ∈ a) ∧ (∀a. a ∈ A --> open a) ∧

(∀S. open S --> x ∈ S --> (∃a∈A. a ⊆ S)) ∧ (∀a b. a ∈ A --> b ∈ A --> a ∩ b ∈ A)"

proof (safe intro!: exI[where x=A])

show "countable A"

unfolding A_def by (intro countable_image countable_Collect_finite)

fix a

assume "a ∈ A"

then show "x ∈ a" "open a"

using A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into)

next

let ?int = "λN. \<Inter>(from_nat_into A' ` N)"

fix a b

assume "a ∈ A" "b ∈ A"

then obtain N M where "a = ?int N" "b = ?int M" "finite (N ∪ M)"

by (auto simp: A_def)

then show "a ∩ b ∈ A"

by (auto simp: A_def intro!: image_eqI[where x="N ∪ M"])

next

fix S

assume "open S" "x ∈ S"

then obtain a where a: "a∈A'" "a ⊆ S" using A' by blast

then show "∃a∈A. a ⊆ S" using a A'

by (intro bexI[where x=a]) (auto simp: A_def intro: image_eqI[where x="{to_nat_on A' a}"])

qed

qed

lemma (in topological_space) first_countableI:

assumes "countable A"

and 1: "!!a. a ∈ A ==> x ∈ a" "!!a. a ∈ A ==> open a"

and 2: "!!S. open S ==> x ∈ S ==> ∃a∈A. a ⊆ S"

shows "∃A::nat => 'a set. (∀i. x ∈ A i ∧ open (A i)) ∧ (∀S. open S ∧ x ∈ S --> (∃i. A i ⊆ S))"

proof (safe intro!: exI[of _ "from_nat_into A"])

fix i

have "A ≠ {}" using 2[of UNIV] by auto

show "x ∈ from_nat_into A i" "open (from_nat_into A i)"

using range_from_nat_into_subset[OF `A ≠ {}`] 1 by auto

next

fix S

assume "open S" "x∈S" from 2[OF this]

show "∃i. from_nat_into A i ⊆ S"

using subset_range_from_nat_into[OF `countable A`] by auto

qed

instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology

proof

fix x :: "'a × 'b"

from first_countable_basisE[of "fst x"] guess A :: "'a set set" . note A = this

from first_countable_basisE[of "snd x"] guess B :: "'b set set" . note B = this

show "∃A::nat => ('a × 'b) set.

(∀i. x ∈ A i ∧ open (A i)) ∧ (∀S. open S ∧ x ∈ S --> (∃i. A i ⊆ S))"

proof (rule first_countableI[of "(λ(a, b). a × b) ` (A × B)"], safe)

fix a b

assume x: "a ∈ A" "b ∈ B"

with A(2, 3)[of a] B(2, 3)[of b] show "x ∈ a × b" and "open (a × b)"

unfolding mem_Times_iff

by (auto intro: open_Times)

next

fix S

assume "open S" "x ∈ S"

from open_prod_elim[OF this] guess a' b' . note a'b' = this

moreover from a'b' A(4)[of a'] B(4)[of b']

obtain a b where "a ∈ A" "a ⊆ a'" "b ∈ B" "b ⊆ b'" by auto

ultimately show "∃a∈(λ(a, b). a × b) ` (A × B). a ⊆ S"

by (auto intro!: bexI[of _ "a × b"] bexI[of _ a] bexI[of _ b])

qed (simp add: A B)

qed

class second_countable_topology = topological_space +

assumes ex_countable_subbasis:

"∃B::'a::topological_space set set. countable B ∧ open = generate_topology B"

begin

lemma ex_countable_basis: "∃B::'a set set. countable B ∧ topological_basis B"

proof -

from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B"

by blast

let ?B = "Inter ` {b. finite b ∧ b ⊆ B }"

show ?thesis

proof (intro exI conjI)

show "countable ?B"

by (intro countable_image countable_Collect_finite_subset B)

{

fix S

assume "open S"

then have "∃B'⊆{b. finite b ∧ b ⊆ B}. (\<Union>b∈B'. \<Inter>b) = S"

unfolding B

proof induct

case UNIV

show ?case by (intro exI[of _ "{{}}"]) simp

next

case (Int a b)

then obtain x y where x: "a = UNION x Inter" "!!i. i ∈ x ==> finite i ∧ i ⊆ B"

and y: "b = UNION y Inter" "!!i. i ∈ y ==> finite i ∧ i ⊆ B"

by blast

show ?case

unfolding x y Int_UN_distrib2

by (intro exI[of _ "{i ∪ j| i j. i ∈ x ∧ j ∈ y}"]) (auto dest: x(2) y(2))

next

case (UN K)

then have "∀k∈K. ∃B'⊆{b. finite b ∧ b ⊆ B}. UNION B' Inter = k" by auto

then guess k unfolding bchoice_iff ..

then show "∃B'⊆{b. finite b ∧ b ⊆ B}. UNION B' Inter = \<Union>K"

by (intro exI[of _ "UNION K k"]) auto

next

case (Basis S)

then show ?case

by (intro exI[of _ "{{S}}"]) auto

qed

then have "(∃B'⊆Inter ` {b. finite b ∧ b ⊆ B}. \<Union>B' = S)"

unfolding subset_image_iff by blast }

then show "topological_basis ?B"

unfolding topological_space_class.topological_basis_def

by (safe intro!: topological_space_class.open_Inter)

(simp_all add: B generate_topology.Basis subset_eq)

qed

qed

end

sublocale second_countable_topology <

countable_basis "SOME B. countable B ∧ topological_basis B"

using someI_ex[OF ex_countable_basis]

by unfold_locales safe

instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology

proof

obtain A :: "'a set set" where "countable A" "topological_basis A"

using ex_countable_basis by auto

moreover

obtain B :: "'b set set" where "countable B" "topological_basis B"

using ex_countable_basis by auto

ultimately show "∃B::('a × 'b) set set. countable B ∧ open = generate_topology B"

by (auto intro!: exI[of _ "(λ(a, b). a × b) ` (A × B)"] topological_basis_prod

topological_basis_imp_subbasis)

qed

instance second_countable_topology ⊆ first_countable_topology

proof

fix x :: 'a

def B ≡ "SOME B::'a set set. countable B ∧ topological_basis B"

then have B: "countable B" "topological_basis B"

using countable_basis is_basis

by (auto simp: countable_basis is_basis)

then show "∃A::nat => 'a set.

(∀i. x ∈ A i ∧ open (A i)) ∧ (∀S. open S ∧ x ∈ S --> (∃i. A i ⊆ S))"

by (intro first_countableI[of "{b∈B. x ∈ b}"])

(fastforce simp: topological_space_class.topological_basis_def)+

qed

subsection {* Polish spaces *}

text {* Textbooks define Polish spaces as completely metrizable.

We assume the topology to be complete for a given metric. *}

class polish_space = complete_space + second_countable_topology

subsection {* General notion of a topology as a value *}

definition "istopology L <->

L {} ∧ (∀S T. L S --> L T --> L (S ∩ T)) ∧ (∀K. Ball K L --> L (\<Union> K))"

typedef 'a topology = "{L::('a set) => bool. istopology L}"

morphisms "openin" "topology"

unfolding istopology_def by blast

lemma istopology_open_in[intro]: "istopology(openin U)"

using openin[of U] by blast

lemma topology_inverse': "istopology U ==> openin (topology U) = U"

using topology_inverse[unfolded mem_Collect_eq] .

lemma topology_inverse_iff: "istopology U <-> openin (topology U) = U"

using topology_inverse[of U] istopology_open_in[of "topology U"] by auto

lemma topology_eq: "T1 = T2 <-> (∀S. openin T1 S <-> openin T2 S)"

proof

assume "T1 = T2"

then show "∀S. openin T1 S <-> openin T2 S" by simp

next

assume H: "∀S. openin T1 S <-> openin T2 S"

then have "openin T1 = openin T2" by (simp add: fun_eq_iff)

then have "topology (openin T1) = topology (openin T2)" by simp

then show "T1 = T2" unfolding openin_inverse .

qed

text{* Infer the "universe" from union of all sets in the topology. *}

definition "topspace T = \<Union>{S. openin T S}"

subsubsection {* Main properties of open sets *}

lemma openin_clauses:

fixes U :: "'a topology"

shows

"openin U {}"

"!!S T. openin U S ==> openin U T ==> openin U (S∩T)"

"!!K. (∀S ∈ K. openin U S) ==> openin U (\<Union>K)"

using openin[of U] unfolding istopology_def mem_Collect_eq by fast+

lemma openin_subset[intro]: "openin U S ==> S ⊆ topspace U"

unfolding topspace_def by blast

lemma openin_empty[simp]: "openin U {}"

by (simp add: openin_clauses)

lemma openin_Int[intro]: "openin U S ==> openin U T ==> openin U (S ∩ T)"

using openin_clauses by simp

lemma openin_Union[intro]: "(∀S ∈K. openin U S) ==> openin U (\<Union> K)"

using openin_clauses by simp

lemma openin_Un[intro]: "openin U S ==> openin U T ==> openin U (S ∪ T)"

using openin_Union[of "{S,T}" U] by auto

lemma openin_topspace[intro, simp]: "openin U (topspace U)"

by (simp add: openin_Union topspace_def)

lemma openin_subopen: "openin U S <-> (∀x ∈ S. ∃T. openin U T ∧ x ∈ T ∧ T ⊆ S)"

(is "?lhs <-> ?rhs")

proof

assume ?lhs

then show ?rhs by auto

next

assume H: ?rhs

let ?t = "\<Union>{T. openin U T ∧ T ⊆ S}"

have "openin U ?t" by (simp add: openin_Union)

also have "?t = S" using H by auto

finally show "openin U S" .

qed

subsubsection {* Closed sets *}

definition "closedin U S <-> S ⊆ topspace U ∧ openin U (topspace U - S)"

lemma closedin_subset: "closedin U S ==> S ⊆ topspace U"

by (metis closedin_def)

lemma closedin_empty[simp]: "closedin U {}"

by (simp add: closedin_def)

lemma closedin_topspace[intro, simp]: "closedin U (topspace U)"

by (simp add: closedin_def)

lemma closedin_Un[intro]: "closedin U S ==> closedin U T ==> closedin U (S ∪ T)"

by (auto simp add: Diff_Un closedin_def)

lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s∈S}"

by auto

lemma closedin_Inter[intro]:

assumes Ke: "K ≠ {}"

and Kc: "∀S ∈K. closedin U S"

shows "closedin U (\<Inter> K)"

using Ke Kc unfolding closedin_def Diff_Inter by auto

lemma closedin_Int[intro]: "closedin U S ==> closedin U T ==> closedin U (S ∩ T)"

using closedin_Inter[of "{S,T}" U] by auto

lemma Diff_Diff_Int: "A - (A - B) = A ∩ B"

by blast

lemma openin_closedin_eq: "openin U S <-> S ⊆ topspace U ∧ closedin U (topspace U - S)"

apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)

apply (metis openin_subset subset_eq)

done

lemma openin_closedin: "S ⊆ topspace U ==> (openin U S <-> closedin U (topspace U - S))"

by (simp add: openin_closedin_eq)

lemma openin_diff[intro]:

assumes oS: "openin U S"

and cT: "closedin U T"

shows "openin U (S - T)"

proof -

have "S - T = S ∩ (topspace U - T)" using openin_subset[of U S] oS cT

by (auto simp add: topspace_def openin_subset)

then show ?thesis using oS cT

by (auto simp add: closedin_def)

qed

lemma closedin_diff[intro]:

assumes oS: "closedin U S"

and cT: "openin U T"

shows "closedin U (S - T)"

proof -

have "S - T = S ∩ (topspace U - T)"

using closedin_subset[of U S] oS cT by (auto simp add: topspace_def)

then show ?thesis

using oS cT by (auto simp add: openin_closedin_eq)

qed

subsubsection {* Subspace topology *}

definition "subtopology U V = topology (λT. ∃S. T = S ∩ V ∧ openin U S)"

lemma istopology_subtopology: "istopology (λT. ∃S. T = S ∩ V ∧ openin U S)"

(is "istopology ?L")

proof -

have "?L {}" by blast

{

fix A B

assume A: "?L A" and B: "?L B"

from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa ∩ V" and Sb: "openin U Sb" "B = Sb ∩ V"

by blast

have "A ∩ B = (Sa ∩ Sb) ∩ V" "openin U (Sa ∩ Sb)"

using Sa Sb by blast+

then have "?L (A ∩ B)" by blast

}

moreover

{

fix K

assume K: "K ⊆ Collect ?L"

have th0: "Collect ?L = (λS. S ∩ V) ` Collect (openin U)"

apply (rule set_eqI)

apply (simp add: Ball_def image_iff)

apply metis

done

from K[unfolded th0 subset_image_iff]

obtain Sk where Sk: "Sk ⊆ Collect (openin U)" "K = (λS. S ∩ V) ` Sk"

by blast

have "\<Union>K = (\<Union>Sk) ∩ V"

using Sk by auto

moreover have "openin U (\<Union> Sk)"

using Sk by (auto simp add: subset_eq)

ultimately have "?L (\<Union>K)" by blast

}

ultimately show ?thesis

unfolding subset_eq mem_Collect_eq istopology_def by 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)

apply auto

done

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"

then have "∃T. openin U T ∧ S = T ∩ V"

using openin_subset[OF S] UV by auto

}

ultimately have "(∃T. openin U T ∧ S = T ∩ V) <-> openin U S"

by blast

}

then show ?thesis

unfolding topology_eq openin_subtopology by blast

qed

lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"

by (simp add: subtopology_superset)

lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"

by (simp add: subtopology_superset)

subsubsection {* The standard Euclidean topology *}

definition euclidean :: "'a::topological_space topology"

where "euclidean = topology open"

lemma open_openin: "open S <-> openin euclidean S"

unfolding euclidean_def

apply (rule cong[where x=S and y=S])

apply (rule topology_inverse[symmetric])

apply (auto simp add: istopology_def)

done

lemma topspace_euclidean: "topspace euclidean = UNIV"

apply (simp add: topspace_def)

apply (rule set_eqI)

apply (auto simp add: open_openin[symmetric])

done

lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"

by (simp add: topspace_euclidean topspace_subtopology)

lemma closed_closedin: "closed S <-> closedin euclidean S"

by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)

lemma open_subopen: "open S <-> (∀x∈S. ∃T. open T ∧ x ∈ T ∧ T ⊆ S)"

by (simp add: open_openin openin_subopen[symmetric])

text {* Basic "localization" results are handy for connectedness. *}

lemma openin_open: "openin (subtopology euclidean U) S <-> (∃T. open T ∧ (S = U ∩ T))"

by (auto simp add: openin_subtopology open_openin[symmetric])

lemma openin_open_Int[intro]: "open S ==> openin (subtopology euclidean U) (U ∩ S)"

by (auto simp add: openin_open)

lemma open_openin_trans[trans]:

"open S ==> open T ==> T ⊆ S ==> openin (subtopology euclidean S) T"

by (metis Int_absorb1 openin_open_Int)

lemma open_subset: "S ⊆ T ==> open S ==> openin (subtopology euclidean T) S"

by (auto simp add: openin_open)

lemma closedin_closed: "closedin (subtopology euclidean U) S <-> (∃T. closed T ∧ S = U ∩ T)"

by (simp add: closedin_subtopology closed_closedin Int_ac)

lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U ∩ S)"

by (metis closedin_closed)

lemma closed_closedin_trans:

"closed S ==> closed T ==> T ⊆ S ==> closedin (subtopology euclidean S) T"

apply (subgoal_tac "S ∩ T = T" )

apply auto

apply (frule closedin_closed_Int[of T S])

apply simp

done

lemma closed_subset: "S ⊆ T ==> closed S ==> closedin (subtopology euclidean T) S"

by (auto simp add: closedin_closed)

lemma openin_euclidean_subtopology_iff:

fixes S U :: "'a::metric_space set"

shows "openin (subtopology euclidean U) S <->

S ⊆ U ∧ (∀x∈S. ∃e>0. ∀x'∈U. dist x' x < e --> x'∈ S)"

(is "?lhs <-> ?rhs")

proof

assume ?lhs

then show ?rhs

unfolding openin_open open_dist by blast

next

def T ≡ "{x. ∃a∈S. ∃d>0. (∀y∈U. dist y a < d --> y ∈ S) ∧ dist x a < d}"

have 1: "∀x∈T. ∃e>0. ∀y. dist y x < e --> y ∈ T"

unfolding T_def

apply clarsimp

apply (rule_tac x="d - dist x a" in exI)

apply (clarsimp simp add: less_diff_eq)

apply (erule rev_bexI)

apply (rule_tac x=d in exI, clarify)

apply (erule le_less_trans [OF dist_triangle])

done

assume ?rhs then have 2: "S = U ∩ T"

unfolding T_def

apply auto

apply (drule (1) bspec, erule rev_bexI)

apply auto

done

from 1 2 show ?lhs

unfolding openin_open open_dist by 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)

apply arith

done

lemma open_contains_ball: "open S <-> (∀x∈S. ∃e>0. ball x e ⊆ S)"

unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

lemma openE[elim?]:

assumes "open S" "x∈S"

obtains e where "e>0" "ball x e ⊆ S"

using assms unfolding open_contains_ball by auto

lemma open_contains_ball_eq: "open S ==> ∀x. x∈S <-> (∃e>0. ball x e ⊆ S)"

by (metis open_contains_ball subset_eq centre_in_ball)

lemma ball_eq_empty[simp]: "ball x e = {} <-> e ≤ 0"

unfolding mem_ball set_eq_iff

apply (simp add: not_less)

apply (metis zero_le_dist order_trans dist_self)

done

lemma ball_empty[intro]: "e ≤ 0 ==> ball x e = {}" by simp

lemma euclidean_dist_l2:

fixes x y :: "'a :: euclidean_space"

shows "dist x y = setL2 (λi. dist (x • i) (y • i)) Basis"

unfolding dist_norm norm_eq_sqrt_inner setL2_def

by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)

definition "box a b = {x. ∀i∈Basis. a • i < x • i ∧ x • i < b • i}"

lemma rational_boxes:

fixes x :: "'a::euclidean_space"

assumes "e > 0"

shows "∃a b. (∀i∈Basis. a • i ∈ \<rat> ∧ b • i ∈ \<rat> ) ∧ x ∈ box a b ∧ box a b ⊆ ball x e"

proof -

def e' ≡ "e / (2 * sqrt (real (DIM ('a))))"

then have e: "e' > 0"

using assms by (auto intro!: divide_pos_pos simp: DIM_positive)

have "∀i. ∃y. y ∈ \<rat> ∧ y < x • i ∧ x • i - y < e'" (is "∀i. ?th i")

proof

fix i

from Rats_dense_in_real[of "x • i - e'" "x • i"] e

show "?th i" by auto

qed

from choice[OF this] guess a .. note a = this

have "∀i. ∃y. y ∈ \<rat> ∧ x • i < y ∧ y - x • i < e'" (is "∀i. ?th i")

proof

fix i

from Rats_dense_in_real[of "x • i" "x • i + e'"] e

show "?th i" by auto

qed

from choice[OF this] guess b .. note b = this

let ?a = "∑i∈Basis. a i *⇩_{R}i" and ?b = "∑i∈Basis. b i *⇩_{R}i"

show ?thesis

proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)

fix y :: 'a

assume *: "y ∈ box ?a ?b"

have "dist x y = sqrt (∑i∈Basis. (dist (x • i) (y • i))⇧^{2})"

unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

also have "… < sqrt (∑(i::'a)∈Basis. e^2 / real (DIM('a)))"

proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

fix i :: "'a"

assume i: "i ∈ Basis"

have "a i < y•i ∧ y•i < b i"

using * i by (auto simp: box_def)

moreover have "a i < x•i" "x•i - a i < e'"

using a by auto

moreover have "x•i < b i" "b i - x•i < e'"

using b by auto

ultimately have "¦x•i - y•i¦ < 2 * e'"

by auto

then have "dist (x • i) (y • i) < e/sqrt (real (DIM('a)))"

unfolding e'_def by (auto simp: dist_real_def)

then have "(dist (x • i) (y • i))⇧^{2}< (e/sqrt (real (DIM('a))))⇧^{2}"

by (rule power_strict_mono) auto

then show "(dist (x • i) (y • i))⇧^{2}< e⇧^{2}/ real DIM('a)"

by (simp add: power_divide)

qed auto

also have "… = e"

using `0 < e` by (simp add: real_eq_of_nat)

finally show "y ∈ ball x e"

by (auto simp: ball_def)

qed (insert a b, auto simp: box_def)

qed

lemma open_UNION_box:

fixes M :: "'a::euclidean_space set"

assumes "open M"

defines "a' ≡ λf :: 'a => real × real. (∑(i::'a)∈Basis. fst (f i) *⇩_{R}i)"

defines "b' ≡ λf :: 'a => real × real. (∑(i::'a)∈Basis. snd (f i) *⇩_{R}i)"

defines "I ≡ {f∈Basis ->⇩_{E}\<rat> × \<rat>. box (a' f) (b' f) ⊆ M}"

shows "M = (\<Union>f∈I. box (a' f) (b' f))"

proof -

{

fix x assume "x ∈ M"

obtain e where e: "e > 0" "ball x e ⊆ M"

using openE[OF `open M` `x ∈ M`] by auto

moreover obtain a b where ab:

"x ∈ box a b"

"∀i ∈ Basis. a • i ∈ \<rat>"

"∀i∈Basis. b • i ∈ \<rat>"

"box a b ⊆ ball x e"

using rational_boxes[OF e(1)] by metis

ultimately have "x ∈ (\<Union>f∈I. box (a' f) (b' f))"

by (intro UN_I[of "λi∈Basis. (a • i, b • i)"])

(auto simp: euclidean_representation I_def a'_def b'_def)

}

then show ?thesis by (auto simp: I_def)

qed

subsection{* Connectedness *}

lemma connected_local:

"connected S <->

¬ (∃e1 e2.

openin (subtopology euclidean S) e1 ∧

openin (subtopology euclidean S) e2 ∧

S ⊆ e1 ∪ e2 ∧

e1 ∩ e2 = {} ∧

e1 ≠ {} ∧

e2 ≠ {})"

unfolding connected_def openin_open

apply safe

apply blast+

done

lemma exists_diff:

fixes P :: "'a set => bool"

shows "(∃S. P(- S)) <-> (∃S. P S)" (is "?lhs <-> ?rhs")

proof -

{

assume "?lhs"

then have ?rhs by blast

}

moreover

{

fix S

assume H: "P S"

have "S = - (- S)" by auto

with H have "P (- (- S))" by metis

}

ultimately show ?thesis by metis

qed

lemma connected_clopen: "connected S <->

(∀T. openin (subtopology euclidean S) T ∧

closedin (subtopology euclidean S) T --> T = {} ∨ T = S)" (is "?lhs <-> ?rhs")

proof -

have "¬ connected S <->

(∃e1 e2. open e1 ∧ open (- e2) ∧ S ⊆ e1 ∪ (- e2) ∧ e1 ∩ (- e2) ∩ S = {} ∧ e1 ∩ S ≠ {} ∧ (- e2) ∩ S ≠ {})"

unfolding connected_def openin_open closedin_closed

apply (subst exists_diff)

apply blast

done

then have th0: "connected S <->

¬ (∃e2 e1. closed e2 ∧ open e1 ∧ S ⊆ e1 ∪ (- e2) ∧ e1 ∩ (- e2) ∩ S = {} ∧ e1 ∩ S ≠ {} ∧ (- e2) ∩ S ≠ {})"

(is " _ <-> ¬ (∃e2 e1. ?P e2 e1)")

apply (simp add: closed_def)

apply metis

done

have th1: "?rhs <-> ¬ (∃t' t. closed t'∧t = S∩t' ∧ t≠{} ∧ t≠S ∧ (∃t'. open t' ∧ t = S ∩ t'))"

(is "_ <-> ¬ (∃t' t. ?Q t' t)")

unfolding connected_def openin_open closedin_closed by auto

{

fix e2

{

fix e1

have "?P e2 e1 <-> (∃t. closed e2 ∧ t = S∩e2 ∧ open e1 ∧ t = S∩e1 ∧ t≠{} ∧ t ≠ S)"

by auto

}

then have "(∃e1. ?P e2 e1) <-> (∃t. ?Q e2 t)"

by metis

}

then have "∀e2. (∃e1. ?P e2 e1) <-> (∃t. ?Q e2 t)"

by blast

then show ?thesis

unfolding th0 th1 by simp

qed

subsection{* Limit points *}

definition (in topological_space) islimpt:: "'a => 'a set => bool" (infixr "islimpt" 60)

where "x islimpt S <-> (∀T. x∈T --> open T --> (∃y∈S. y∈T ∧ y≠x))"

lemma islimptI:

assumes "!!T. x ∈ T ==> open T ==> ∃y∈S. y ∈ T ∧ y ≠ x"

shows "x islimpt S"

using assms unfolding islimpt_def by auto

lemma islimptE:

assumes "x islimpt S" and "x ∈ T" and "open T"

obtains y where "y ∈ S" and "y ∈ T" and "y ≠ x"

using assms unfolding islimpt_def by auto

lemma islimpt_iff_eventually: "x islimpt S <-> ¬ eventually (λy. y ∉ S) (at x)"

unfolding islimpt_def eventually_at_topological by auto

lemma islimpt_subset: "x islimpt S ==> S ⊆ T ==> x islimpt T"

unfolding islimpt_def by fast

lemma islimpt_approachable:

fixes x :: "'a::metric_space"

shows "x islimpt S <-> (∀e>0. ∃x'∈S. x' ≠ x ∧ dist x' x < e)"

unfolding islimpt_iff_eventually eventually_at by fast

lemma islimpt_approachable_le:

fixes x :: "'a::metric_space"

shows "x islimpt S <-> (∀e>0. ∃x'∈ S. x' ≠ x ∧ dist x' x ≤ e)"

unfolding islimpt_approachable

using approachable_lt_le [where f="λy. dist y x" and P="λy. y ∉ S ∨ y = x",

THEN arg_cong [where f=Not]]

by (simp add: Bex_def conj_commute conj_left_commute)

lemma islimpt_UNIV_iff: "x islimpt UNIV <-> ¬ open {x}"

unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)

lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"

unfolding islimpt_def by blast

text {* A perfect space has no isolated points. *}

lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"

unfolding islimpt_UNIV_iff by (rule not_open_singleton)

lemma perfect_choose_dist:

fixes x :: "'a::{perfect_space, metric_space}"

shows "0 < r ==> ∃a. a ≠ x ∧ dist a x < r"

using islimpt_UNIV [of x]

by (simp add: islimpt_approachable)

lemma closed_limpt: "closed S <-> (∀x. x islimpt S --> x ∈ S)"

unfolding closed_def

apply (subst open_subopen)

apply (simp add: islimpt_def subset_eq)

apply (metis ComplE ComplI)

done

lemma islimpt_EMPTY[simp]: "¬ x islimpt {}"

unfolding islimpt_def by auto

lemma finite_set_avoid:

fixes a :: "'a::metric_space"

assumes fS: "finite S"

shows "∃d>0. ∀x∈S. x ≠ a --> d ≤ dist a x"

proof (induct rule: finite_induct[OF fS])

case 1

then show ?case by (auto intro: zero_less_one)

next

case (2 x F)

from 2 obtain d where d: "d >0" "∀x∈F. x≠a --> d ≤ dist a x"

by blast

show ?case

proof (cases "x = a")

case True

then show ?thesis using d by auto

next

case False

let ?d = "min d (dist a x)"

have dp: "?d > 0"

using False d(1) using dist_nz by auto

from d have d': "∀x∈F. x≠a --> ?d ≤ dist a x"

by auto

with dp False show ?thesis

by (auto intro!: exI[where x="?d"])

qed

qed

lemma islimpt_Un: "x islimpt (S ∪ T) <-> x islimpt S ∨ x islimpt T"

by (simp add: islimpt_iff_eventually eventually_conj_iff)

lemma discrete_imp_closed:

fixes S :: "'a::metric_space set"

assumes e: "0 < e"

and d: "∀x ∈ S. ∀y ∈ S. dist y x < e --> y = x"

shows "closed S"

proof -

{

fix x

assume C: "∀e>0. ∃x'∈S. x' ≠ x ∧ dist x' x < e"

from e have e2: "e/2 > 0" by arith

from C[rule_format, OF e2] obtain y where y: "y ∈ S" "y ≠ x" "dist y x < e/2"

by blast

let ?m = "min (e/2) (dist x y) "

from e2 y(2) have mp: "?m > 0"

by (simp add: dist_nz[symmetric])

from C[rule_format, OF mp] obtain z where z: "z ∈ S" "z ≠ x" "dist z x < ?m"

by blast

have th: "dist z y < e" using z y

by (intro dist_triangle_lt [where z=x], simp)

from d[rule_format, OF y(1) z(1) th] y z

have False by (auto simp add: dist_commute)}

then show ?thesis

by (metis islimpt_approachable closed_limpt [where 'a='a])

qed

subsection {* Interior of a Set *}

definition "interior S = \<Union>{T. open T ∧ T ⊆ S}"

lemma interiorI [intro?]:

assumes "open T" and "x ∈ T" and "T ⊆ S"

shows "x ∈ interior S"

using assms unfolding interior_def by fast

lemma interiorE [elim?]:

assumes "x ∈ interior S"

obtains T where "open T" and "x ∈ T" and "T ⊆ S"

using assms unfolding interior_def by fast

lemma open_interior [simp, intro]: "open (interior S)"

by (simp add: interior_def open_Union)

lemma interior_subset: "interior S ⊆ S"

by (auto simp add: interior_def)

lemma interior_maximal: "T ⊆ S ==> open T ==> T ⊆ interior S"

by (auto simp add: interior_def)

lemma interior_open: "open S ==> interior S = S"

by (intro equalityI interior_subset interior_maximal subset_refl)

lemma interior_eq: "interior S = S <-> open S"

by (metis open_interior interior_open)

lemma open_subset_interior: "open S ==> S ⊆ interior T <-> S ⊆ T"

by (metis interior_maximal interior_subset subset_trans)

lemma interior_empty [simp]: "interior {} = {}"

using open_empty by (rule interior_open)

lemma interior_UNIV [simp]: "interior UNIV = UNIV"

using open_UNIV by (rule interior_open)

lemma interior_interior [simp]: "interior (interior S) = interior S"

using open_interior by (rule interior_open)

lemma interior_mono: "S ⊆ T ==> interior S ⊆ interior T"

by (auto simp add: interior_def)

lemma interior_unique:

assumes "T ⊆ S" and "open T"

assumes "!!T'. T' ⊆ S ==> open T' ==> T' ⊆ T"

shows "interior S = T"

by (intro equalityI assms interior_subset open_interior interior_maximal)

lemma interior_inter [simp]: "interior (S ∩ T) = interior S ∩ interior T"

by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1

Int_lower2 interior_maximal interior_subset open_Int open_interior)

lemma mem_interior: "x ∈ interior S <-> (∃e>0. ball x e ⊆ S)"

using open_contains_ball_eq [where S="interior S"]

by (simp add: open_subset_interior)

lemma interior_limit_point [intro]:

fixes x :: "'a::perfect_space"

assumes x: "x ∈ interior S"

shows "x islimpt S"

using x islimpt_UNIV [of x]

unfolding interior_def islimpt_def

apply (clarsimp, rename_tac T T')

apply (drule_tac x="T ∩ T'" in spec)

apply (auto simp add: open_Int)

done

lemma interior_closed_Un_empty_interior:

assumes cS: "closed S"

and iT: "interior T = {}"

shows "interior (S ∪ T) = interior S"

proof

show "interior S ⊆ interior (S ∪ T)"

by (rule interior_mono) (rule Un_upper1)

show "interior (S ∪ T) ⊆ interior S"

proof

fix x

assume "x ∈ interior (S ∪ T)"

then obtain R where "open R" "x ∈ R" "R ⊆ S ∪ T" ..

show "x ∈ interior S"

proof (rule ccontr)

assume "x ∉ interior S"

with `x ∈ R` `open R` obtain y where "y ∈ R - S"

unfolding interior_def by fast

from `open R` `closed S` have "open (R - S)"

by (rule open_Diff)

from `R ⊆ S ∪ T` have "R - S ⊆ T"

by fast

from `y ∈ R - S` `open (R - S)` `R - S ⊆ T` `interior T = {}` show False

unfolding interior_def by fast

qed

qed

qed

lemma interior_Times: "interior (A × B) = interior A × interior B"

proof (rule interior_unique)

show "interior A × interior B ⊆ A × B"

by (intro Sigma_mono interior_subset)

show "open (interior A × interior B)"

by (intro open_Times open_interior)

fix T

assume "T ⊆ A × B" and "open T"

then show "T ⊆ interior A × interior B"

proof safe

fix x y

assume "(x, y) ∈ T"

then obtain C D where "open C" "open D" "C × D ⊆ T" "x ∈ C" "y ∈ D"

using `open T` unfolding open_prod_def by fast

then have "open C" "open D" "C ⊆ A" "D ⊆ B" "x ∈ C" "y ∈ D"

using `T ⊆ A × B` by auto

then show "x ∈ interior A" and "y ∈ interior B"

by (auto intro: interiorI)

qed

qed

subsection {* Closure of a Set *}

definition "closure S = S ∪ {x | x. x islimpt S}"

lemma interior_closure: "interior S = - (closure (- S))"

unfolding interior_def closure_def islimpt_def by auto

lemma closure_interior: "closure S = - interior (- S)"

unfolding interior_closure by simp

lemma closed_closure[simp, intro]: "closed (closure S)"

unfolding closure_interior by (simp add: closed_Compl)

lemma closure_subset: "S ⊆ closure S"

unfolding closure_def by simp

lemma closure_hull: "closure S = closed hull S"

unfolding hull_def closure_interior interior_def by auto

lemma closure_eq: "closure S = S <-> closed S"

unfolding closure_hull using closed_Inter by (rule hull_eq)

lemma closure_closed [simp]: "closed S ==> closure S = S"

unfolding closure_eq .

lemma closure_closure [simp]: "closure (closure S) = closure S"

unfolding closure_hull by (rule hull_hull)

lemma closure_mono: "S ⊆ T ==> closure S ⊆ closure T"

unfolding closure_hull by (rule hull_mono)

lemma closure_minimal: "S ⊆ T ==> closed T ==> closure S ⊆ T"

unfolding closure_hull by (rule hull_minimal)

lemma closure_unique:

assumes "S ⊆ T"

and "closed T"

and "!!T'. S ⊆ T' ==> closed T' ==> T ⊆ T'"

shows "closure S = T"

using assms unfolding closure_hull by (rule hull_unique)

lemma closure_empty [simp]: "closure {} = {}"

using closed_empty by (rule closure_closed)

lemma closure_UNIV [simp]: "closure UNIV = UNIV"

using closed_UNIV by (rule closure_closed)

lemma closure_union [simp]: "closure (S ∪ T) = closure S ∪ closure T"

unfolding closure_interior by simp

lemma closure_eq_empty: "closure S = {} <-> S = {}"

using closure_empty closure_subset[of S]

by blast

lemma closure_subset_eq: "closure S ⊆ S <-> closed S"

using closure_eq[of S] closure_subset[of S]

by simp

lemma open_inter_closure_eq_empty:

"open S ==> (S ∩ closure T) = {} <-> S ∩ T = {}"

using open_subset_interior[of S "- T"]

using interior_subset[of "- T"]

unfolding closure_interior

by auto

lemma open_inter_closure_subset:

"open S ==> (S ∩ (closure T)) ⊆ closure(S ∩ T)"

proof

fix x

assume as: "open S" "x ∈ S ∩ closure T"

{

assume *: "x islimpt T"

have "x islimpt (S ∩ T)"

proof (rule islimptI)

fix A

assume "x ∈ A" "open A"

with as have "x ∈ A ∩ S" "open (A ∩ S)"

by (simp_all add: open_Int)

with * obtain y where "y ∈ T" "y ∈ A ∩ S" "y ≠ x"

by (rule islimptE)

then have "y ∈ S ∩ T" "y ∈ A ∧ y ≠ x"

by simp_all

then show "∃y∈(S ∩ T). y ∈ A ∧ y ≠ x" ..

qed

}

then show "x ∈ closure (S ∩ T)" using as

unfolding closure_def

by blast

qed

lemma closure_complement: "closure (- S) = - interior S"

unfolding closure_interior by simp

lemma interior_complement: "interior (- S) = - closure S"

unfolding closure_interior by simp

lemma closure_Times: "closure (A × B) = closure A × closure B"

proof (rule closure_unique)

show "A × B ⊆ closure A × closure B"

by (intro Sigma_mono closure_subset)

show "closed (closure A × closure B)"

by (intro closed_Times closed_closure)

fix T

assume "A × B ⊆ T" and "closed T"

then show "closure A × closure B ⊆ T"

apply (simp add: closed_def open_prod_def, clarify)

apply (rule ccontr)

apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)

apply (simp add: closure_interior interior_def)

apply (drule_tac x=C in spec)

apply (drule_tac x=D in spec)

apply auto

done

qed

lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"

unfolding closure_def using islimpt_punctured by blast

subsection {* Frontier (aka boundary) *}

definition "frontier S = closure S - interior S"

lemma frontier_closed: "closed (frontier S)"

by (simp add: frontier_def closed_Diff)

lemma frontier_closures: "frontier S = (closure S) ∩ (closure(- S))"

by (auto simp add: frontier_def interior_closure)

lemma frontier_straddle:

fixes a :: "'a::metric_space"

shows "a ∈ frontier S <-> (∀e>0. (∃x∈S. dist a x < e) ∧ (∃x. x ∉ S ∧ dist a x < e))"

unfolding frontier_def closure_interior

by (auto simp add: mem_interior subset_eq ball_def)

lemma frontier_subset_closed: "closed S ==> frontier S ⊆ S"

by (metis frontier_def closure_closed Diff_subset)

lemma frontier_empty[simp]: "frontier {} = {}"

by (simp add: frontier_def)

lemma frontier_subset_eq: "frontier S ⊆ S <-> closed S"

proof-

{

assume "frontier S ⊆ S"

then have "closure S ⊆ S"

using interior_subset unfolding frontier_def by auto

then have "closed S"

using closure_subset_eq by auto

}

then show ?thesis using frontier_subset_closed[of S] ..

qed

lemma frontier_complement: "frontier(- S) = frontier S"

by (auto simp add: frontier_def closure_complement interior_complement)

lemma frontier_disjoint_eq: "frontier S ∩ S = {} <-> open S"

using frontier_complement frontier_subset_eq[of "- S"]

unfolding open_closed by auto

subsection {* Filters and the ``eventually true'' quantifier *}

definition indirection :: "'a::real_normed_vector => 'a => 'a filter"

(infixr "indirection" 70)

where "a indirection v = at a within {b. ∃c≥0. b - a = scaleR c v}"

text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}

lemma trivial_limit_within: "trivial_limit (at a within S) <-> ¬ a islimpt S"

proof

assume "trivial_limit (at a within S)"

then show "¬ a islimpt S"

unfolding trivial_limit_def

unfolding eventually_at_topological

unfolding islimpt_def

apply (clarsimp simp add: set_eq_iff)

apply (rename_tac T, rule_tac x=T in exI)

apply (clarsimp, drule_tac x=y in bspec, simp_all)

done

next

assume "¬ a islimpt S"

then show "trivial_limit (at a within S)"

unfolding trivial_limit_def

unfolding 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

lemma not_trivial_limit_within: "¬ trivial_limit (at x within S) = (x ∈ closure (S - {x}))"

using islimpt_in_closure

by (metis trivial_limit_within)

text {* Some property holds "sufficiently close" to the limit point. *}

lemma eventually_at2:

"eventually P (at a) <-> (∃d>0. ∀x. 0 < dist x a ∧ dist x a < d --> P x)"

unfolding eventually_at dist_nz by auto

lemma eventually_happens: "eventually P net ==> trivial_limit net ∨ (∃x. P x)"

unfolding trivial_limit_def

by (auto elim: eventually_rev_mp)

lemma trivial_limit_eventually: "trivial_limit net ==> eventually P net"

by simp

lemma trivial_limit_eq: "trivial_limit net <-> (∀P. eventually P net)"

by (simp add: filter_eq_iff)

text{* Combining theorems for "eventually" *}

lemma eventually_rev_mono:

"eventually P net ==> (∀x. P x --> Q x) ==> eventually Q net"

using eventually_mono [of P Q] by fast

lemma not_eventually: "(∀x. ¬ P x ) ==> ¬ trivial_limit net ==> ¬ eventually (λx. P x) net"

by (simp add: eventually_False)

subsection {* Limits *}

lemma Lim:

"(f ---> l) net <->

trivial_limit net ∨

(∀e>0. eventually (λx. dist (f x) l < e) net)"

unfolding tendsto_iff trivial_limit_eq by auto

text{* Show that they yield usual definitions in the various cases. *}

lemma Lim_within_le: "(f ---> l)(at a within S) <->

(∀e>0. ∃d>0. ∀x∈S. 0 < dist x a ∧ dist x a ≤ d --> dist (f x) l < e)"

by (auto simp add: tendsto_iff eventually_at_le dist_nz)

lemma Lim_within: "(f ---> l) (at a within S) <->

(∀e >0. ∃d>0. ∀x ∈ S. 0 < dist x a ∧ dist x a < d --> dist (f x) l < e)"

by (auto simp add: tendsto_iff eventually_at dist_nz)

lemma Lim_at: "(f ---> l) (at a) <->

(∀e >0. ∃d>0. ∀x. 0 < dist x a ∧ dist x a < d --> dist (f x) l < e)"

by (auto simp add: tendsto_iff eventually_at2)

lemma Lim_at_infinity:

"(f ---> l) at_infinity <-> (∀e>0. ∃b. ∀x. norm x ≥ b --> dist (f x) l < e)"

by (auto simp add: tendsto_iff eventually_at_infinity)

lemma Lim_eventually: "eventually (λx. f x = l) net ==> (f ---> l) net"

by (rule topological_tendstoI, auto elim: eventually_rev_mono)

text{* The expected monotonicity property. *}

lemma Lim_Un:

assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"

shows "(f ---> l) (at x within (S ∪ T))"

using assms unfolding at_within_union by (rule filterlim_sup)

lemma Lim_Un_univ:

"(f ---> l) (at x within S) ==> (f ---> l) (at x within T) ==>

S ∪ T = UNIV ==> (f ---> l) (at x)"

by (metis Lim_Un)

text{* Interrelations between restricted and unrestricted limits. *}

lemma Lim_at_within: (* FIXME: rename *)

"(f ---> l) (at x) ==> (f ---> l) (at x within S)"

by (metis order_refl filterlim_mono subset_UNIV at_le)

lemma eventually_within_interior:

assumes "x ∈ interior S"

shows "eventually P (at x within S) <-> eventually P (at x)"

(is "?lhs = ?rhs")

proof

from assms obtain T where T: "open T" "x ∈ T" "T ⊆ S" ..

{

assume "?lhs"

then obtain A where "open A" and "x ∈ A" and "∀y∈A. y ≠ x --> y ∈ S --> P y"

unfolding eventually_at_topological

by auto

with T have "open (A ∩ T)" and "x ∈ A ∩ T" and "∀y ∈ A ∩ T. y ≠ x --> P y"

by auto

then show "?rhs"

unfolding eventually_at_topological by auto

next

assume "?rhs"

then show "?lhs"

by (auto elim: eventually_elim1 simp: eventually_at_filter)

}

qed

lemma at_within_interior:

"x ∈ interior S ==> at x within S = at x"

unfolding filter_eq_iff by (intro allI eventually_within_interior)

lemma Lim_within_LIMSEQ:

fixes a :: "'a::first_countable_topology"

assumes "∀S. (∀n. S n ≠ a ∧ S n ∈ T) ∧ S ----> a --> (λn. X (S n)) ----> L"

shows "(X ---> L) (at a within T)"

using assms unfolding tendsto_def [where l=L]

by (simp add: sequentially_imp_eventually_within)

lemma Lim_right_bound:

fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} =>

'b::{linorder_topology, conditionally_complete_linorder}"

assumes mono: "!!a b. a ∈ I ==> b ∈ I ==> x < a ==> a ≤ b ==> f a ≤ f b"

and bnd: "!!a. a ∈ I ==> x < a ==> K ≤ f a"

shows "(f ---> Inf (f ` ({x<..} ∩ I))) (at x within ({x<..} ∩ I))"

proof (cases "{x<..} ∩ I = {}")

case True

then show ?thesis by simp

next

case False

show ?thesis

proof (rule order_tendstoI)

fix a

assume a: "a < Inf (f ` ({x<..} ∩ I))"

{

fix y

assume "y ∈ {x<..} ∩ I"

with False bnd have "Inf (f ` ({x<..} ∩ I)) ≤ f y"

by (auto intro: cInf_lower)

with a have "a < f y"

by (blast intro: less_le_trans)

}

then show "eventually (λx. a < f x) (at x within ({x<..} ∩ I))"

by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)

next

fix a

assume "Inf (f ` ({x<..} ∩ I)) < a"

from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y ∈ I" "f y < a"

by auto

then have "eventually (λx. x ∈ I --> f x < a) (at_right x)"

unfolding eventually_at_right by (metis less_imp_le le_less_trans mono)

then show "eventually (λx. f x < a) (at x within ({x<..} ∩ I))"

unfolding eventually_at_filter by eventually_elim simp

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 ≡ "λn. SOME y. y ∈ S ∧ y ∈ A n ∧ x ≠ y"

{

fix n

from `?lhs` have "∃y. y ∈ S ∧ y ∈ A n ∧ x ≠ y"

unfolding islimpt_def using A(1,2)[of n] by auto

then have "f n ∈ S ∧ f n ∈ A n ∧ x ≠ f n"

unfolding f_def by (rule someI_ex)

then have "f n ∈ S" "f n ∈ A n" "x ≠ f n" by auto

}

then have "∀n. f n ∈ S - {x}" by auto

moreover have "(λn. f n) ----> x"

proof (rule topological_tendstoI)

fix S

assume "open S" "x ∈ S"

from A(3)[OF this] `!!n. f n ∈ A n`

show "eventually (λx. f x ∈ S) sequentially"

by (auto elim!: eventually_elim1)

qed

ultimately show ?rhs by fast

next

assume ?rhs

then obtain f :: "nat => 'a" where f: "!!n. f n ∈ S - {x}" and lim: "f ----> x"

by auto

show ?lhs

unfolding islimpt_def

proof safe

fix T

assume "open T" "x ∈ T"

from lim[THEN topological_tendstoD, OF this] f

show "∃y∈S. y ∈ T ∧ y ≠ x"

unfolding eventually_sequentially by auto

qed

qed

lemma Lim_null:

fixes f :: "'a => 'b::real_normed_vector"

shows "(f ---> l) net <-> ((λx. f(x) - l) ---> 0) net"

by (simp add: Lim dist_norm)

lemma Lim_null_comparison:

fixes f :: "'a => 'b::real_normed_vector"

assumes "eventually (λx. norm (f x) ≤ g x) net" "(g ---> 0) net"

shows "(f ---> 0) net"

using assms(2)

proof (rule metric_tendsto_imp_tendsto)

show "eventually (λx. dist (f x) 0 ≤ dist (g x) 0) net"

using assms(1) by (rule eventually_elim1) (simp add: dist_norm)

qed

lemma Lim_transform_bound:

fixes f :: "'a => 'b::real_normed_vector"

and g :: "'a => 'c::real_normed_vector"

assumes "eventually (λn. norm (f n) ≤ norm (g n)) net"

and "(g ---> 0) net"

shows "(f ---> 0) net"

using assms(1) tendsto_norm_zero [OF assms(2)]

by (rule Lim_null_comparison)

text{* Deducing things about the limit from the elements. *}

lemma Lim_in_closed_set:

assumes "closed S"

and "eventually (λx. f(x) ∈ S) net"

and "¬ trivial_limit net" "(f ---> l) net"

shows "l ∈ S"

proof (rule ccontr)

assume "l ∉ S"

with `closed S` have "open (- S)" "l ∈ - S"

by (simp_all add: open_Compl)

with assms(4) have "eventually (λx. f x ∈ - S) net"

by (rule topological_tendstoD)

with assms(2) have "eventually (λx. False) net"

by (rule eventually_elim2) simp

with assms(3) show "False"

by (simp add: eventually_False)

qed

text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}

lemma Lim_dist_ubound:

assumes "¬(trivial_limit net)"

and "(f ---> l) net"

and "eventually (λx. dist a (f x) ≤ e) net"

shows "dist a l ≤ e"

proof -

have "dist a l ∈ {..e}"

proof (rule Lim_in_closed_set)

show "closed {..e}"

by simp

show "eventually (λx. dist a (f x) ∈ {..e}) net"

by (simp add: assms)

show "¬ trivial_limit net"

by fact

show "((λx. dist a (f x)) ---> dist a l) net"

by (intro tendsto_intros assms)

qed

then show ?thesis by simp

qed

lemma Lim_norm_ubound:

fixes f :: "'a => 'b::real_normed_vector"

assumes "¬(trivial_limit net)" "(f ---> l) net" "eventually (λx. norm(f x) ≤ e) net"

shows "norm(l) ≤ e"

proof -

have "norm l ∈ {..e}"

proof (rule Lim_in_closed_set)

show "closed {..e}"

by simp

show "eventually (λx. norm (f x) ∈ {..e}) net"

by (simp add: assms)

show "¬ trivial_limit net"

by fact

show "((λx. norm (f x)) ---> norm l) net"

by (intro tendsto_intros assms)

qed

then show ?thesis by simp

qed

lemma Lim_norm_lbound:

fixes f :: "'a => 'b::real_normed_vector"

assumes "¬ trivial_limit net"

and "(f ---> l) net"

and "eventually (λx. e ≤ norm (f x)) net"

shows "e ≤ norm l"

proof -

have "norm l ∈ {e..}"

proof (rule Lim_in_closed_set)

show "closed {e..}"

by simp

show "eventually (λx. norm (f x) ∈ {e..}) net"

by (simp add: assms)

show "¬ trivial_limit net"

by fact

show "((λx. norm (f x)) ---> norm l) net"

by (intro tendsto_intros assms)

qed

then show ?thesis by simp

qed

text{* Limit under bilinear function *}

lemma Lim_bilinear:

assumes "(f ---> l) net"

and "(g ---> m) net"

and "bounded_bilinear h"

shows "((λx. h (f x) (g x)) ---> (h l m)) net"

using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`

by (rule bounded_bilinear.tendsto)

text{* These are special for limits out of the same vector space. *}

lemma Lim_within_id: "(id ---> a) (at a within s)"

unfolding id_def by (rule tendsto_ident_at)

lemma Lim_at_id: "(id ---> a) (at a)"

unfolding id_def by (rule tendsto_ident_at)

lemma Lim_at_zero:

fixes a :: "'a::real_normed_vector"

and l :: "'b::topological_space"

shows "(f ---> l) (at a) <-> ((λx. f(a + x)) ---> l) (at 0)"

using LIM_offset_zero LIM_offset_zero_cancel ..

text{* It's also sometimes useful to extract the limit point from the filter. *}

abbreviation netlimit :: "'a::t2_space filter => 'a"

where "netlimit F ≡ Lim F (λx. x)"

lemma netlimit_within: "¬ trivial_limit (at a within S) ==> netlimit (at a within S) = a"

by (rule tendsto_Lim) (auto intro: tendsto_intros)

lemma netlimit_at:

fixes a :: "'a::{perfect_space,t2_space}"

shows "netlimit (at a) = a"

using netlimit_within [of a UNIV] by simp

lemma lim_within_interior:

"x ∈ interior S ==> (f ---> l) (at x within S) <-> (f ---> l) (at x)"

by (metis at_within_interior)

lemma netlimit_within_interior:

fixes x :: "'a::{t2_space,perfect_space}"

assumes "x ∈ interior S"

shows "netlimit (at x within S) = x"

using assms by (metis at_within_interior netlimit_at)

text{* Transformation of limit. *}

lemma Lim_transform:

fixes f g :: "'a::type => 'b::real_normed_vector"

assumes "((λx. f x - g x) ---> 0) net" "(f ---> l) net"

shows "(g ---> l) net"

using tendsto_diff [OF assms(2) assms(1)] by simp

lemma Lim_transform_eventually:

"eventually (λx. f x = g x) net ==> (f ---> l) net ==> (g ---> l) net"

apply (rule topological_tendstoI)

apply (drule (2) topological_tendstoD)

apply (erule (1) eventually_elim2, simp)

done

lemma Lim_transform_within:

assumes "0 < d"

and "∀x'∈S. 0 < dist x' x ∧ dist x' x < d --> f x' = g x'"

and "(f ---> l) (at x within S)"

shows "(g ---> l) (at x within S)"

proof (rule Lim_transform_eventually)

show "eventually (λx. f x = g x) (at x within S)"

using assms(1,2) by (auto simp: dist_nz eventually_at)

show "(f ---> l) (at x within S)" by fact

qed

lemma Lim_transform_at:

assumes "0 < d"

and "∀x'. 0 < dist x' x ∧ dist x' x < d --> f x' = g x'"

and "(f ---> l) (at x)"

shows "(g ---> l) (at x)"

using _ assms(3)

proof (rule Lim_transform_eventually)

show "eventually (λx. f x = g x) (at x)"

unfolding eventually_at2

using assms(1,2) by auto

qed

text{* Common case assuming being away from some crucial point like 0. *}

lemma Lim_transform_away_within:

fixes a b :: "'a::t1_space"

assumes "a ≠ b"

and "∀x∈S. x ≠ a ∧ x ≠ b --> f x = g x"

and "(f ---> l) (at a within S)"

shows "(g ---> l) (at a within S)"

proof (rule Lim_transform_eventually)

show "(f ---> l) (at a within S)" by fact

show "eventually (λx. f x = g x) (at a within S)"

unfolding eventually_at_topological

by (rule exI [where x="- {b}"], simp add: open_Compl assms)

qed

lemma Lim_transform_away_at:

fixes a b :: "'a::t1_space"

assumes ab: "a≠b"

and fg: "∀x. x ≠ a ∧ x ≠ b --> f x = g x"

and fl: "(f ---> l) (at a)"

shows "(g ---> l) (at a)"

using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp

text{* Alternatively, within an open set. *}

lemma Lim_transform_within_open:

assumes "open S" and "a ∈ S"

and "∀x∈S. x ≠ a --> f x = g x"

and "(f ---> l) (at a)"

shows "(g ---> l) (at a)"

proof (rule Lim_transform_eventually)

show "eventually (λx. f x = g x) (at a)"

unfolding eventually_at_topological

using assms(1,2,3) by auto

show "(f ---> l) (at a)" by fact

qed

text{* A congruence rule allowing us to transform limits assuming not at point. *}

(* FIXME: Only one congruence rule for tendsto can be used at a time! *)

lemma Lim_cong_within(*[cong add]*):

assumes "a = b"

and "x = y"

and "S = T"

and "!!x. x ≠ b ==> x ∈ T ==> f x = g x"

shows "(f ---> x) (at a within S) <-> (g ---> y) (at b within T)"

unfolding tendsto_def eventually_at_topological

using assms by simp

lemma Lim_cong_at(*[cong add]*):

assumes "a = b" "x = y"

and "!!x. x ≠ a ==> f x = g x"

shows "((λx. f x) ---> x) (at a) <-> ((g ---> y) (at a))"

unfolding tendsto_def eventually_at_topological

using assms by simp

text{* Useful lemmas on closure and set of possible sequential limits.*}

lemma closure_sequential:

fixes l :: "'a::first_countable_topology"

shows "l ∈ closure S <-> (∃x. (∀n. x n ∈ S) ∧ (x ---> l) sequentially)"

(is "?lhs = ?rhs")

proof

assume "?lhs"

moreover

{

assume "l ∈ S"

then have "?rhs" using tendsto_const[of l sequentially] by auto

}

moreover

{

assume "l islimpt S"

then have "?rhs" unfolding islimpt_sequential by auto

}

ultimately show "?rhs"

unfolding closure_def by auto

next

assume "?rhs"

then show "?lhs" unfolding closure_def islimpt_sequential by auto

qed

lemma closed_sequential_limits:

fixes S :: "'a::first_countable_topology set"

shows "closed S <-> (∀x l. (∀n. x n ∈ S) ∧ (x ---> l) sequentially --> l ∈ S)"

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)

apply (metis dist_self)

done

lemma closed_approachable:

fixes S :: "'a::metric_space set"

shows "closed S ==> (∀e>0. ∃y∈S. dist y x < e) <-> x ∈ S"

by (metis closure_closed closure_approachable)

lemma closure_contains_Inf:

fixes S :: "real set"

assumes "S ≠ {}" "∀x∈S. B ≤ x"

shows "Inf S ∈ closure S"

proof -

have *: "∀x∈S. Inf S ≤ x"

using cInf_lower_EX[of _ S] assms by metis

{

fix e :: real

assume "e > 0"

then have "Inf S < Inf S + e" by simp

with assms obtain x where "x ∈ S" "x < Inf S + e"

by (subst (asm) cInf_less_iff[of _ B]) auto

with * have "∃x∈S. dist x (Inf S) < e"

by (intro bexI[of _ x]) (auto simp add: dist_real_def)

}

then show ?thesis unfolding closure_approachable by auto

qed

lemma closed_contains_Inf:

fixes S :: "real set"

assumes "S ≠ {}" "∀x∈S. B ≤ x"

and "closed S"

shows "Inf S ∈ S"

by (metis closure_contains_Inf closure_closed assms)

lemma not_trivial_limit_within_ball:

"¬ trivial_limit (at x within S) <-> (∀e>0. S ∩ ball x e - {x} ≠ {})"

(is "?lhs = ?rhs")

proof -

{

assume "?lhs"

{

fix e :: real

assume "e > 0"

then obtain y where "y ∈ S - {x}" and "dist y x < e"

using `?lhs` not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

by auto

then have "y ∈ S ∩ ball x e - {x}"

unfolding ball_def by (simp add: dist_commute)

then have "S ∩ ball x e - {x} ≠ {}" by blast

}

then have "?rhs" by auto

}

moreover

{

assume "?rhs"

{

fix e :: real

assume "e > 0"

then obtain y where "y ∈ S ∩ ball x e - {x}"

using `?rhs` by blast

then have "y ∈ S - {x}" and "dist y x < e"

unfolding ball_def by (simp_all add: dist_commute)

then have "∃y ∈ S - {x}. dist y x < e"

by auto

}

then have "?lhs"

using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

by auto

}

ultimately show ?thesis by auto

qed

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: "0 ≤ infdist x A"

by (auto simp add: infdist_def intro: cInf_greatest)

lemma infdist_le:

assumes "a ∈ A"

and "d = dist x a"

shows "infdist x A ≤ d"

using assms by (auto intro!: cInf_lower[where z=0] simp add: infdist_def)

lemma infdist_zero[simp]:

assumes "a ∈ A"

shows "infdist a A = 0"

proof -

from infdist_le[OF assms, of "dist a a"] have "infdist a A ≤ 0"

by auto

with infdist_nonneg[of a A] assms show "infdist a A = 0"

by auto

qed

lemma infdist_triangle: "infdist x A ≤ infdist y A + dist x y"

proof (cases "A = {}")

case True

then show ?thesis by (simp add: infdist_def)

next

case False

then obtain a where "a ∈ A" by auto

have "infdist x A ≤ Inf {dist x y + dist y a |a. a ∈ A}"

proof (rule cInf_greatest)

from `A ≠ {}` show "{dist x y + dist y a |a. a ∈ A} ≠ {}"

by simp

fix d

assume "d ∈ {dist x y + dist y a |a. a ∈ A}"

then obtain a where d: "d = dist x y + dist y a" "a ∈ A"

by auto

show "infdist x A ≤ d"

unfolding infdist_notempty[OF `A ≠ {}`]

proof (rule cInf_lower2)

show "dist x a ∈ {dist x a |a. a ∈ A}"

using `a ∈ A` by auto

show "dist x a ≤ d"

unfolding d by (rule dist_triangle)

fix d

assume "d ∈ {dist x a |a. a ∈ A}"

then obtain a where "a ∈ A" "d = dist x a"

by auto

then show "infdist x A ≤ d"

by (rule infdist_le)

qed

qed

also have "… = dist x y + infdist y A"

proof (rule cInf_eq, safe)

fix a

assume "a ∈ A"

then show "dist x y + infdist y A ≤ dist x y + dist y a"

by (auto intro: infdist_le)

next

fix i

assume inf: "!!d. d ∈ {dist x y + dist y a |a. a ∈ A} ==> i ≤ d"

then have "i - dist x y ≤ infdist y A"

unfolding infdist_notempty[OF `A ≠ {}`] using `a ∈ A`

by (intro cInf_greatest) (auto simp: field_simps)

then show "i ≤ dist x y + infdist y A"

by simp

qed

finally show ?thesis by simp

qed

lemma in_closure_iff_infdist_zero:

assumes "A ≠ {}"

shows "x ∈ closure A <-> infdist x A = 0"

proof

assume "x ∈ closure A"

show "infdist x A = 0"

proof (rule ccontr)

assume "infdist x A ≠ 0"

with infdist_nonneg[of x A] have "infdist x A > 0"

by auto

then have "ball x (infdist x A) ∩ closure A = {}"

apply auto

apply (metis `0 < infdist x A` `x ∈ closure A` closure_approachable dist_commute

eucl_less_not_refl euclidean_trans(2) infdist_le)

done

then have "x ∉ closure A"

by (metis `0 < infdist x A` centre_in_ball disjoint_iff_not_equal)

then show False using `x ∈ closure A` by simp

qed

next

assume x: "infdist x A = 0"

then obtain a where "a ∈ A"

by atomize_elim (metis all_not_in_conv assms)

show "x ∈ closure A"

unfolding closure_approachable

apply safe

proof (rule ccontr)

fix e :: real

assume "e > 0"

assume "¬ (∃y∈A. dist y x < e)"

then have "infdist x A ≥ e" using `a ∈ A`

unfolding infdist_def

by (force simp: dist_commute intro: cInf_greatest)

with x `e > 0` show False by auto

qed

qed

lemma in_closed_iff_infdist_zero:

assumes "closed A" "A ≠ {}"

shows "x ∈ A <-> infdist x A = 0"

proof -

have "x ∈ closure A <-> infdist x A = 0"

by (rule in_closure_iff_infdist_zero) fact

with assms show ?thesis by simp

qed

lemma tendsto_infdist [tendsto_intros]:

assumes f: "(f ---> l) F"

shows "((λx. infdist (f x) A) ---> infdist l A) F"

proof (rule tendstoI)

fix e ::real

assume "e > 0"

from tendstoD[OF f this]

show "eventually (λx. dist (infdist (f x) A) (infdist l A) < e) F"

proof (eventually_elim)

fix x

from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]

have "dist (infdist (f x) A) (infdist l A) ≤ dist (f x) l"

by (simp add: dist_commute dist_real_def)

also assume "dist (f x) l < e"

finally show "dist (infdist (f x) A) (infdist l A) < e" .

qed

qed

text{* Some other lemmas about sequences. *}

lemma sequentially_offset: (* TODO: move to Topological_Spaces.thy *)

assumes "eventually (λi. P i) sequentially"

shows "eventually (λi. P (i + k)) sequentially"

using assms by (rule eventually_sequentially_seg [THEN iffD2])

lemma seq_offset_neg: (* TODO: move to Topological_Spaces.thy *)

"(f ---> l) sequentially ==> ((λi. f(i - k)) ---> l) sequentially"

apply (erule filterlim_compose)

apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially)

apply arith

done

lemma seq_harmonic: "((λn. inverse (real n)) ---> 0) sequentially"

using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc) (* TODO: move to Limits.thy *)

subsection {* More properties of closed balls *}

lemma closed_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"

then have "∃d>0. cball x d ⊆ S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)

}

moreover

{

fix x and e::real

assume "x∈S" "e>0" "cball x e ⊆ S"

then have "∃d>0. ball x d ⊆ S"

unfolding subset_eq

apply(rule_tac x="e/2" in exI)

apply auto

done

}

ultimately show ?thesis

unfolding open_contains_ball by auto

qed

lemma open_contains_cball_eq: "open S ==> (∀x. x ∈ S <-> (∃e>0. cball x e ⊆ S))"

by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

lemma mem_interior_cball: "x ∈ interior S <-> (∃e>0. cball x e ⊆ S)"

apply (simp add: interior_def, safe)

apply (force simp add: open_contains_cball)

apply (rule_tac x="ball x e" in exI)

apply (simp add: subset_trans [OF ball_subset_cball])

done

lemma islimpt_ball:

fixes x y :: "'a::{real_normed_vector,perfect_space}"

shows "y islimpt ball x e <-> 0 < e ∧ y ∈ cball x e"

(is "?lhs = ?rhs")

proof

assume "?lhs"

{

assume "e ≤ 0"

then have *:"ball x e = {}"

using ball_eq_empty[of x e] by auto

have False using `?lhs`

unfolding * using islimpt_EMPTY[of y] by auto

}

then have "e > 0" by (metis not_less)

moreover

have "y ∈ cball x e"

using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]

ball_subset_cball[of x e] `?lhs`

unfolding closed_limpt by auto

ultimately show "?rhs" by auto

next

assume "?rhs"

then have "e > 0" by auto

{

fix d :: real

assume "d > 0"

have "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"

proof (cases "d ≤ dist x y")

case True

then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"

proof (cases "x = y")

case True

then have False

using `d ≤ dist x y` `d>0` by auto

then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"

by auto

next

case False

have "dist x (y - (d / (2 * dist y x)) *⇩_{R}(y - x)) =

norm (x - y + (d / (2 * norm (y - x))) *⇩_{R}(y - x))"

unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]

by auto

also have "… = ¦- 1 + d / (2 * norm (x - y))¦ * norm (x - y)"

using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]

unfolding scaleR_minus_left scaleR_one

by (auto simp add: norm_minus_commute)

also have "… = ¦- norm (x - y) + d / 2¦"

unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]

unfolding distrib_right using `x≠y`[unfolded dist_nz, unfolded dist_norm]

by auto

also have "… ≤ e - d/2" using `d ≤ dist x y` and `d>0` and `?rhs`

by (auto simp add: dist_norm)

finally have "y - (d / (2 * dist y x)) *⇩_{R}(y - x) ∈ ball x e" using `d>0`

by auto

moreover

have "(d / (2*dist y x)) *⇩_{R}(y - x) ≠ 0"

using `x≠y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff

by (auto simp add: dist_commute)

moreover

have "dist (y - (d / (2 * dist y x)) *⇩_{R}(y - x)) y < d"

unfolding dist_norm

apply simp

unfolding norm_minus_cancel

using `d > 0` `x≠y`[unfolded dist_nz] dist_commute[of x y]

unfolding dist_norm

apply auto

done

ultimately show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"

apply (rule_tac x = "y - (d / (2*dist y x)) *⇩_{R}(y - x)" in bexI)

apply auto

done

qed

next

case False

then have "d > dist x y" by auto

show "∃x' ∈ ball x e. x' ≠ y ∧ dist x' y < d"

proof (cases "x = y")

case True

obtain z where **: "z ≠ y" "dist z y < min e d"

using perfect_choose_dist[of "min e d" y]

using `d > 0` `e>0` by auto

show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"

unfolding `x = y`

using `z ≠ y` **

apply (rule_tac x=z in bexI)

apply (auto simp add: dist_commute)

done

next

case False

then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"

using `d>0` `d > dist x y` `?rhs`

apply (rule_tac x=x in bexI)

apply auto

done

qed

qed

}

then show "?lhs"

unfolding mem_cball islimpt_approachable mem_ball by auto

qed

lemma closure_ball_lemma:

fixes x y :: "'a::real_normed_vector"

assumes "x ≠ y"

shows "y islimpt ball x (dist x y)"

proof (rule islimptI)

fix T

assume "y ∈ T" "open T"

then obtain r where "0 < r" "∀z. dist z y < r --> z ∈ T"

unfolding open_dist by fast

(* choose point between x and y, within distance r of y. *)

def k ≡ "min 1 (r / (2 * dist x y))"

def z ≡ "y + scaleR k (x - y)"

have z_def2: "z = x + scaleR (1 - k) (y - x)"

unfolding z_def by (simp add: algebra_simps)

have "dist z y < r"

unfolding z_def k_def using `0 < r`

by (simp add: dist_norm min_def)

then have "z ∈ T"

using `∀z. dist z y < r --> z ∈ T` by simp

have "dist x z < dist x y"

unfolding z_def2 dist_norm

apply (simp add: norm_minus_commute)

apply (simp only: dist_norm [symmetric])

apply (subgoal_tac "¦1 - k¦ * dist x y < 1 * dist x y", simp)

apply (rule mult_strict_right_mono)

apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x ≠ y`)

apply (simp add: zero_less_dist_iff `x ≠ y`)

done

then have "z ∈ ball x (dist x y)"

by simp

have "z ≠ y"

unfolding z_def k_def using `x ≠ y` `0 < r`

by (simp add: min_def)

show "∃z∈ball x (dist x y). z ∈ T ∧ z ≠ y"

using `z ∈ ball x (dist x y)` `z ∈ T` `z ≠ y`

by fast

qed

lemma closure_ball:

fixes x :: "'a::real_normed_vector"

shows "0 < e ==> closure (ball x e) = cball x e"

apply (rule equalityI)

apply (rule closure_minimal)

apply (rule ball_subset_cball)

apply (rule closed_cball)

apply (rule subsetI, rename_tac y)

apply (simp add: le_less [where 'a=real])

apply (erule disjE)

apply (rule subsetD [OF closure_subset], simp)

apply (simp add: closure_def)

apply clarify

apply (rule closure_ball_lemma)

apply (simp add: zero_less_dist_iff)

done

(* In a trivial vector space, this fails for e = 0. *)

lemma interior_cball:

fixes x :: "'a::{real_normed_vector, perfect_space}"

shows "interior (cball x e) = ball x e"

proof (cases "e ≥ 0")

case False note cs = this

from cs have "ball x e = {}"

using ball_empty[of e x] by auto

moreover

{

fix y

assume "y ∈ cball x e"

then have False

unfolding mem_cball using dist_nz[of x y] cs by auto

}

then have "cball x e = {}" by auto

then have "interior (cball x e) = {}"

using interior_empty by auto

ultimately show ?thesis by blast

next

case True note cs = this

have "ball x e ⊆ cball x e"

using ball_subset_cball by auto

moreover

{

fix S y

assume as: "S ⊆ cball x e" "open S" "y∈S"

then obtain d where "d>0" and d: "∀x'. dist x' y < d --> x' ∈ S"

unfolding open_dist by blast

then obtain xa where xa_y: "xa ≠ y" and xa: "dist xa y < d"

using perfect_choose_dist [of d] by auto

have "xa ∈ S"

using d[THEN spec[where x = xa]]

using xa by (auto simp add: dist_commute)

then have xa_cball: "xa ∈ cball x e"

using as(1) by auto

then have "y ∈ ball x e"

proof (cases "x = y")

case True

then have "e > 0"

using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball]

by (auto simp add: dist_commute)

then show "y ∈ ball x e"

using `x = y ` by simp

next

case False

have "dist (y + (d / 2 / dist y x) *⇩_{R}(y - x)) y < d"

unfolding dist_norm

using `d>0` norm_ge_zero[of "y - x"] `x ≠ y` by auto

then have *: "y + (d / 2 / dist y x) *⇩_{R}(y - x) ∈ cball x e"

using d as(1)[unfolded subset_eq] by blast

have "y - x ≠ 0" using `x ≠ y` by auto

then have **:"d / (2 * norm (y - x)) > 0"

unfolding zero_less_norm_iff[symmetric]

using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto

have "dist (y + (d / 2 / dist y x) *⇩_{R}(y - x)) x =

norm (y + (d / (2 * norm (y - x))) *⇩_{R}y - (d / (2 * norm (y - x))) *⇩_{R}x - x)"

by (auto simp add: dist_norm algebra_simps)

also have "… = norm ((1 + d / (2 * norm (y - x))) *⇩_{R}(y - x))"

by (auto simp add: algebra_simps)

also have "… = ¦1 + d / (2 * norm (y - x))¦ * norm (y - x)"

using ** by auto

also have "… = (dist y x) + d/2"

using ** by (auto simp add: distrib_right dist_norm)

finally have "e ≥ dist x y +d/2"

using *[unfolded mem_cball] by (auto simp add: dist_commute)

then show "y ∈ ball x e"

unfolding mem_ball using `d>0` by auto

qed

}

then have "∀S ⊆ cball x e. open S --> S ⊆ ball x e"

by auto

ultimately show ?thesis

using interior_unique[of "ball x e" "cball x e"]

using open_ball[of x e]

by auto

qed

lemma frontier_ball:

fixes a :: "'a::real_normed_vector"

shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"

apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)

apply (simp add: set_eq_iff)

apply arith

done

lemma frontier_cball:

fixes a :: "'a::{real_normed_vector, perfect_space}"

shows "frontier (cball a e) = {x. dist a x = e}"

apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)

apply (simp add: set_eq_iff)

apply arith

done

lemma cball_eq_empty: "cball x e = {} <-> e < 0"

apply (simp add: set_eq_iff not_le)

apply (metis zero_le_dist dist_self order_less_le_trans)

done

lemma cball_empty: "e < 0 ==> cball x e = {}"

by (simp add: cball_eq_empty)

lemma cball_eq_sing:

fixes x :: "'a::{metric_space,perfect_space}"

shows "cball x e = {x} <-> e = 0"

proof (rule linorder_cases)

assume e: "0 < e"

obtain a where "a ≠ x" "dist a x < e"

using perfect_choose_dist [OF e] by auto

then have "a ≠ x" "dist x a ≤ e"

by (auto simp add: dist_commute)

with e show ?thesis by (auto simp add: set_eq_iff)

qed auto

lemma cball_sing:

fixes x :: "'a::metric_space"

shows "e = 0 ==> cball x e = {x}"

by (auto simp add: set_eq_iff)

subsection {* Boundedness *}

(* FIXME: This has to be unified with BSEQ!! *)

definition (in metric_space) bounded :: "'a set => bool"

where "bounded S <-> (∃x e. ∀y∈S. dist x y ≤ e)"

lemma bounded_subset_cball: "bounded S <-> (∃e x. S ⊆ cball x e)"

unfolding bounded_def subset_eq by auto

lemma bounded_any_center: "bounded S <-> (∃e. ∀y∈S. dist a y ≤ e)"

unfolding bounded_def

apply safe

apply (rule_tac x="dist a x + e" in exI)

apply clarify

apply (drule (1) bspec)

apply (erule order_trans [OF dist_triangle add_left_mono])

apply auto

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

apply auto

done

lemma bounded_empty [simp]: "bounded {}"

by (simp add: bounded_def)

lemma bounded_subset: "bounded T ==> S ⊆ T ==> bounded S"

by (metis bounded_def subset_eq)

lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"

by (metis bounded_subset interior_subset)

lemma bounded_closure[intro]:

assumes "bounded S"

shows "bounded (closure S)"

proof -

from assms obtain x and a where a: "∀y∈S. dist x y ≤ a"

unfolding bounded_def by auto

{

fix y

assume "y ∈ closure S"

then obtain f where f: "∀n. f n ∈ S" "(f ---> y) sequentially"

unfolding closure_sequential by auto

have "∀n. f n ∈ S --> dist x (f n) ≤ a" using a by simp

then have "eventually (λn. dist x (f n) ≤ a) sequentially"

by (rule eventually_mono, simp add: f(1))

have "dist x y ≤ a"

apply (rule Lim_dist_ubound [of sequentially f])

apply (rule trivial_limit_sequentially)

apply (rule f(2))

apply fact

done

}

then show ?thesis

unfolding bounded_def by auto

qed

lemma bounded_cball[simp,intro]: "bounded (cball x e)"

apply (simp add: bounded_def)

apply (rule_tac x=x in exI)

apply (rule_tac x=e in exI)

apply auto

done

lemma bounded_ball[simp,intro]: "bounded (ball x e)"

by (metis ball_subset_cball bounded_cball bounded_subset)

lemma bounded_Un[simp]: "bounded (S ∪ T) <-> bounded S ∧ bounded T"

apply (auto simp add: bounded_def)

apply (rename_tac x y r s)

apply (rule_tac x=x in exI)

apply (rule_tac x="max r (dist x y + s)" in exI)

apply (rule ballI)

apply safe

apply (drule (1) bspec)

apply simp

apply (drule (1) bspec)

apply (rule min_max.le_supI2)

apply (erule order_trans [OF dist_triangle add_left_mono])

done

lemma bounded_Union[intro]: "finite F ==> ∀S∈F. bounded S ==> bounded (\<Union>F)"

by (induct rule: finite_induct[of F]) auto

lemma bounded_UN [intro]: "finite A ==> ∀x∈A. bounded (B x) ==> bounded (\<Union>x∈A. B x)"

by (induct set: finite) auto

lemma bounded_insert [simp]: "bounded (insert x S) <-> bounded S"

proof -

have "∀y∈{x}. dist x y ≤ 0"

by simp

then have "bounded {x}"

unfolding bounded_def by fast

then show ?thesis

by (metis insert_is_Un bounded_Un)

qed

lemma finite_imp_bounded [intro]: "finite S ==> bounded S"

by (induct set: finite) simp_all

lemma bounded_pos: "bounded S <-> (∃b>0. ∀x∈ S. norm x ≤ b)"

apply (simp add: bounded_iff)

apply (subgoal_tac "!!x (y::real). 0 < 1 + abs y ∧ (x ≤ y --> x ≤ 1 + abs y)")

apply metis

apply arith

done

lemma Bseq_eq_bounded:

fixes f :: "nat => 'a::real_normed_vector"

shows "Bseq f <-> bounded (range f)"

unfolding Bseq_def bounded_pos by auto

lemma bounded_Int[intro]: "bounded S ∨ bounded T ==> bounded (S ∩ T)"

by (metis Int_lower1 Int_lower2 bounded_subset)

lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"

by (metis Diff_subset bounded_subset)

lemma not_bounded_UNIV[simp, intro]:

"¬ bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"

proof (auto simp add: bounded_pos not_le)

obtain x :: 'a where "x ≠ 0"

using perfect_choose_dist [OF zero_less_one] by fast

fix b :: real

assume b: "b >0"

have b1: "b +1 ≥ 0"

using b by simp

with `x ≠ 0` have "b < norm (scaleR (b + 1) (sgn x))"

by (simp add: norm_sgn)

then show "∃x::'a. b < norm x" ..

qed

lemma bounded_linear_image:

assumes "bounded S"

and "bounded_linear f"

shows "bounded (f ` S)"

proof -

from assms(1) obtain b where b: "b > 0" "∀x∈S. norm x ≤ b"

unfolding bounded_pos by auto

from assms(2) obtain B where B: "B > 0" "∀x. norm (f x) ≤ B * norm x"

using bounded_linear.pos_bounded by (auto simp add: mult_ac)

{

fix x

assume "x ∈ S"

then have "norm x ≤ b"

using b by auto

then have "norm (f x) ≤ B * b"

using B(2)

apply (erule_tac x=x in allE)

apply (metis B(1) B(2) order_trans mult_le_cancel_left_pos)

done

}

then show ?thesis

unfolding bounded_pos

apply (rule_tac x="b*B" in exI)

using b B mult_pos_pos [of b B]

apply (auto simp add: mult_commute)

done

qed

lemma bounded_scaling:

fixes S :: "'a::real_normed_vector set"

shows "bounded S ==> bounded ((λx. c *⇩_{R}x) ` S)"

apply (rule bounded_linear_image)

apply assumption

apply (rule bounded_linear_scaleR_right)

done

lemma bounded_translation:

fixes S :: "'a::real_normed_vector set"

assumes "bounded S"

shows "bounded ((λx. a + x) ` S)"

proof -

from assms obtain b where b: "b > 0" "∀x∈S. norm x ≤ b"

unfolding bounded_pos by auto

{

fix x

assume "x ∈ S"

then have "norm (a + x) ≤ b + norm a"

using norm_triangle_ineq[of a x] b by auto

}

then show ?thesis

unfolding bounded_pos

using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]

by (auto intro!: exI[of _ "b + norm a"])

qed

text{* Some theorems on sups and infs using the notion "bounded". *}

lemma bounded_real:

fixes S :: "real set"

shows "bounded S <-> (∃a. ∀x∈S. abs x ≤ a)"

by (simp add: bounded_iff)

lemma bounded_has_Sup:

fixes S :: "real set"

assumes "bounded S"

and "S ≠ {}"

shows "∀x∈S. x ≤ Sup S"

and "∀b. (∀x∈S. x ≤ b) --> Sup S ≤ b"

proof

fix x

assume "x∈S"

then show "x ≤ Sup S"

by (metis cSup_upper abs_le_D1 assms(1) bounded_real)

next

show "∀b. (∀x∈S. x ≤ b) --> Sup S ≤ b"

using assms by (metis cSup_least)

qed

lemma Sup_insert:

fixes S :: "real set"

shows "bounded S ==> Sup (insert x S) = (if S = {} then x else max x (Sup S))"

apply (subst cSup_insert_If)

apply (rule bounded_has_Sup(1)[of S, rule_format])

apply (auto simp: sup_max)

done

lemma Sup_insert_finite:

fixes S :: "real set"

shows "finite S ==> Sup (insert x S) = (if S = {} then x else max x (Sup S))"

apply (rule Sup_insert)

apply (rule finite_imp_bounded)

apply simp

done

lemma bounded_has_Inf:

fixes S :: "real set"

assumes "bounded S"

and "S ≠ {}"

shows "∀x∈S. x ≥ Inf S"

and "∀b. (∀x∈S. x ≥ b) --> Inf S ≥ b"

proof

fix x

assume "x ∈ S"

from assms(1) obtain a where a: "∀x∈S. ¦x¦ ≤ a"

unfolding bounded_real by auto

then show "x ≥ Inf S" using `x ∈ S`

by (metis cInf_lower_EX abs_le_D2 minus_le_iff)

next

show "∀b. (∀x∈S. x ≥ b) --> Inf S ≥ b"

using assms by (metis cInf_greatest)

qed

lemma Inf_insert:

fixes S :: "real set"

shows "bounded S ==> Inf (insert x S) = (if S = {} then x else min x (Inf S))"

apply (subst cInf_insert_if)

apply (rule bounded_has_Inf(1)[of S, rule_format])

apply (auto simp: inf_min)

done

lemma Inf_insert_finite:

fixes S :: "real set"

shows "finite S ==> Inf (insert x S) = (if S = {} then x else min x (Inf S))"

apply (rule Inf_insert)

apply (rule finite_imp_bounded)

apply simp

done

subsection {* Compactness *}

subsubsection {* Bolzano-Weierstrass property *}

lemma heine_borel_imp_bolzano_weierstrass:

assumes "compact s"

and "infinite t"

and "t ⊆ s"

shows "∃x ∈ s. x islimpt t"

proof (rule ccontr)

assume "¬ (∃x ∈ s. x islimpt t)"

then obtain f where f: "∀x∈s. x ∈ f x ∧ open (f x) ∧ (∀y∈t. y ∈ f x --> y = x)"

unfolding islimpt_def

using bchoice[of s "λ x T. x ∈ T ∧ open T ∧ (∀y∈t. y ∈ T --> y = x)"]

by auto

obtain g where g: "g ⊆ {t. ∃x. x ∈ s ∧ t = f x}" "finite g" "s ⊆ \<Union>g"

using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. ∃x. x∈s ∧ t = f x}"]]

using f by auto

from g(1,3) have g':"∀x∈g. ∃xa ∈ s. x = f xa"

by auto

{

fix x y

assume "x ∈ t" "y ∈ t" "f x = f y"

then have "x ∈ f x" "y ∈ f x --> y = x"

using f[THEN bspec[where x=x]] and `t ⊆ s` by auto

then have "x = y"

using `f x = f y` and f[THEN bspec[where x=y]] and `y ∈ t` and `t ⊆ s`

by auto

}

then have "inj_on f t"

unfolding inj_on_def by simp

then have "infinite (f ` t)"

using assms(2) using finite_imageD by auto

moreover

{

fix x

assume "x ∈ t" "f x ∉ g"

from g(3) assms(3) `x ∈ t` obtain h where "h ∈ g" and "x ∈ h"

by auto

then obtain y where "y ∈ s" "h = f y"

using g'[THEN bspec[where x=h]] by auto

then have "y = x"

using f[THEN bspec[where x=y]] and `x∈t` and `x∈h`[unfolded `h = f y`]

by auto

then have False

using `f x ∉ g` `h ∈ g` unfolding `h = f y`

by auto

}

then have "f ` t ⊆ g" by auto

ultimately show False

using g(2) using finite_subset by auto

qed

lemma acc_point_range_imp_convergent_subsequence:

fixes l :: "'a :: first_countable_topology"

assumes l: "∀U. l∈U --> open U --> infinite (U ∩ range f)"

shows "∃r. subseq r ∧ (f o r) ----> l"

proof -

from countable_basis_at_decseq[of l] guess A . note A = this

def s ≡ "λn i. SOME j. i < j ∧ f j ∈ A (Suc n)"

{

fix n i

have "infinite (A (Suc n) ∩ range f - f`{.. i})"

using l A by auto

then have "∃x. x ∈ A (Suc n) ∩ range f - f`{.. i}"

unfolding ex_in_conv by (intro notI) simp

then have "∃j. f j ∈ A (Suc n) ∧ j ∉ {.. i}"

by auto

then have "∃a. i < a ∧ f a ∈ A (Suc n)"

by (auto simp: not_le)

then have "i < s n i" "f (s n i) ∈ A (Suc n)"

unfolding s_def by (auto intro: someI2_ex)

}

note s = this

def r ≡ "nat_rec (s 0 0) s"

have "subseq r"

by (auto simp: r_def s subseq_Suc_iff)

moreover

have "(λn. f (r n)) ----> l"

proof (rule topological_tendstoI)

fix S

assume "open S" "l ∈ S"

with A(3) have "eventually (λi. A i ⊆ S) sequentially"

by auto

moreover

{

fix i

assume "Suc 0 ≤ i"

then have "f (r i) ∈ A i"

by (cases i) (simp_all add: r_def s)

}

then have "eventually (λi. f (r i) ∈ A i) sequentially"

by (auto simp: eventually_sequentially)

ultimately show "eventually (λi. f (r i) ∈ S) sequentially"

by eventually_elim auto

qed

ultimately show "∃r. subseq r ∧ (f o r) ----> l"

by (auto simp: convergent_def comp_def)

qed

lemma sequence_infinite_lemma:

fixes f :: "nat => 'a::t1_space"

assumes "∀n. f n ≠ l"

and "(f ---> l) sequentially"

shows "infinite (range f)"

proof

assume "finite (range f)"

then have "closed (range f)"

by (rule finite_imp_closed)

then have "open (- range f)"

by (rule open_Compl)

from assms(1) have "l ∈ - range f"

by auto

from assms(2) have "eventually (λn. f n ∈ - range f) sequentially"

using `open (- range f)` `l ∈ - range f`

by (rule topological_tendstoD)

then show False

unfolding eventually_sequentially

by auto

qed

lemma closure_insert:

fixes x :: "'a::t1_space"

shows "closure (insert x s) = insert x (closure s)"

apply (rule closure_unique)

apply (rule insert_mono [OF closure_subset])

apply (rule closed_insert [OF closed_closure])

apply (simp add: closure_minimal)

done

lemma islimpt_insert:

fixes x :: "'a::t1_space"

shows "x islimpt (insert a s) <-> x islimpt s"

proof

assume *: "x islimpt (insert a s)"

show "x islimpt s"

proof (rule islimptI)

fix t

assume t: "x ∈ t" "open t"

show "∃y∈s. y ∈ t ∧ y ≠ x"

proof (cases "x = a")

case True

obtain y where "y ∈ insert a s" "y ∈ t" "y ≠ x"

using * t by (rule islimptE)

with `x = a` show ?thesis by auto

next

case False

with t have t': "x ∈ t - {a}" "open (t - {a})"

by (simp_all add: open_Diff)

obtain y where "y ∈ insert a s" "y ∈ t - {a}" "y ≠ x"

using * t' by (rule islimptE)

then show ?thesis by auto

qed

qed

next

assume "x islimpt s"

then show "x islimpt (insert a s)"

by (rule islimpt_subset) auto

qed

lemma islimpt_finite:

fixes x :: "'a::t1_space"

shows "finite s ==> ¬ x islimpt s"

by (induct set: finite) (simp_all add: islimpt_insert)

lemma islimpt_union_finite:

fixes x :: "'a::t1_space"

shows "finite s ==> x islimpt (s ∪ t) <-> x islimpt t"

by (simp add: islimpt_Un islimpt_finite)

lemma islimpt_eq_acc_point:

fixes l :: "'a :: t1_space"

shows "l islimpt S <-> (∀U. l∈U --> open U --> infinite (U ∩ S))"

proof (safe intro!: islimptI)

fix U

assume "l islimpt S" "l ∈ U" "open U" "finite (U ∩ S)"

then have "l islimpt S" "l ∈ (U - (U ∩ S - {l}))" "open (U - (U ∩ S - {l}))"

by (auto intro: finite_imp_closed)

then show False

by (rule islimptE) auto

next

fix T

assume *: "∀U. l∈U --> open U --> infinite (U ∩ S)" "l ∈ T" "open T"

then have "infinite (T ∩ S - {l})"

by auto

then have "∃x. x ∈ (T ∩ S - {l})"

unfolding ex_in_conv by (intro notI) simp

then show "∃y∈S. y ∈ T ∧ y ≠ l"

by auto

qed

lemma islimpt_range_imp_convergent_subsequence:

fixes l :: "'a :: {t1_space, first_countable_topology}"

assumes l: "l islimpt (range f)"

shows "∃r. subseq r ∧ (f o r) ----> l"

using l unfolding islimpt_eq_acc_point

by (rule acc_point_range_imp_convergent_subsequence)

lemma sequence_unique_limpt:

fixes f :: "nat => 'a::t2_space"

assumes "(f ---> l) sequentially"

and "l' islimpt (range f)"

shows "l' = l"

proof (rule ccontr)

assume "l' ≠ l"

obtain s t where "open s" "open t" "l' ∈ s" "l ∈ t" "s ∩ t = {}"

using hausdorff [OF `l' ≠ l`] by auto

have "eventually (λn. f n ∈ t) sequentially"

using assms(1) `open t` `l ∈ t` by (rule topological_tendstoD)

then obtain N where "∀n≥N. f n ∈ t"

unfolding eventually_sequentially by auto

have "UNIV = {..<N} ∪ {N..}"

by auto

then have "l' islimpt (f ` ({..<N} ∪ {N..}))"

using assms(2) by simp

then have "l' islimpt (f ` {..<N} ∪ f ` {N..})"

by (simp add: image_Un)

then have "l' islimpt (f ` {N..})"

by (simp add: islimpt_union_finite)

then obtain y where "y ∈ f ` {N..}" "y ∈ s" "y ≠ l'"

using `l' ∈ s` `open s` by (rule islimptE)

then obtain n where "N ≤ n" "f n ∈ s" "f n ≠ l'"

by auto

with `∀n≥N. f n ∈ t` have "f n ∈ s ∩ t"

by simp

with `s ∩ t = {}` show False

by simp

qed

lemma bolzano_weierstrass_imp_closed:

fixes s :: "'a::{first_countable_topology,t2_space} set"

assumes "∀t. infinite t ∧ t ⊆ s --> (∃x ∈ s. x islimpt t)"

shows "closed s"

proof -

{

fix x l

assume as: "∀n::nat. x n ∈ s" "(x ---> l) sequentially"

then have "l ∈ s"

proof (cases "∀n. x n ≠ l")

case False

then show "l∈s" using as(1) by auto

next

case True note cas = this

with as(2) have "infinite (range x)"

using sequence_infinite_lemma[of x l] by auto

then obtain l' where "l'∈s" "l' islimpt (range x)"

using assms[THEN spec[where x="range x"]] as(1) by auto

then show "l∈s" using sequence_unique_limpt[of x l l']

using as cas by auto

qed

}

then show ?thesis

unfolding closed_sequential_limits by fast

qed

lemma compact_imp_bounded:

assumes "compact U"

shows "bounded U"

proof -

have "compact U" "∀x∈U. open (ball x 1)" "U ⊆ (\<Union>x∈U. ball x 1)"

using assms by auto

then obtain D where D: "D ⊆ U" "finite D" "U ⊆ (\<Union>x∈D. ball x 1)"

by (rule compactE_image)

from `finite D` have "bounded (\<Union>x∈D. ball x 1)"

by (simp add: bounded_UN)

then show "bounded U" using `U ⊆ (\<Union>x∈D. ball x 1)`

by (rule bounded_subset)

qed

text{* In particular, some common special cases. *}

lemma compact_union [intro]:

assumes "compact s"

and "compact t"

shows " compact (s ∪ t)"

proof (rule compactI)

fix f

assume *: "Ball f open" "s ∪ t ⊆ \<Union>f"

from * `compact s` obtain s' where "s' ⊆ f ∧ finite s' ∧ s ⊆ \<Union>s'"

unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

moreover

from * `compact t` obtain t' where "t' ⊆ f ∧ finite t' ∧ t ⊆ \<Union>t'"

unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

ultimately show "∃f'⊆f. finite f' ∧ s ∪ t ⊆ \<Union>f'"

by (auto intro!: exI[of _ "s' ∪ t'"])

qed

lemma compact_Union [intro]: "finite S ==> (!!T. T ∈ S ==> compact T) ==> compact (\<Union>S)"

by (induct set: finite) auto

lemma compact_UN [intro]:

"finite A ==> (!!x. x ∈ A ==> compact (B x)) ==> compact (\<Union>x∈A. B x)"

unfolding SUP_def by (rule compact_Union) auto

lemma closed_inter_compact [intro]:

assumes "closed s"

and "compact t"

shows "compact (s ∩ t)"

using compact_inter_closed [of t s] assms

by (simp add: Int_commute)

lemma compact_inter [intro]:

fixes s t :: "'a :: t2_space set"

assumes "compact s"

and "compact t"

shows "compact (s ∩ t)"

using assms by (intro compact_inter_closed compact_imp_closed)

lemma compact_sing [simp]: "compact {a}"

unfolding compact_eq_heine_borel by auto

lemma compact_insert [simp]:

assumes "compact s"

shows "compact (insert x s)"

proof -

have "compact ({x} ∪ s)"

using compact_sing assms by (rule compact_union)

then show ?thesis by simp

qed

lemma finite_imp_compact: "finite s ==> compact s"

by (induct set: finite) simp_all

lemma open_delete:

fixes s :: "'a::t1_space set"

shows "open s ==> open (s - {x})"

by (simp add: open_Diff)

text{* Finite intersection property *}

lemma inj_setminus: "inj_on uminus (A::'a set set)"

by (auto simp: inj_on_def)

lemma compact_fip:

"compact U <->

(∀A. (∀a∈A. closed a) --> (∀B ⊆ A. finite B --> U ∩ \<Inter>B ≠ {}) --> U ∩ \<Inter>A ≠ {})"

(is "_ <-> ?R")

proof (safe intro!: compact_eq_heine_borel[THEN iffD2])

fix A

assume "compact U"

and A: "∀a∈A. closed a" "U ∩ \<Inter>A = {}"

and fi: "∀B ⊆ A. finite B --> U ∩ \<Inter>B ≠ {}"

from A have "(∀a∈uminus`A. open a) ∧ U ⊆ \<Union>(uminus`A)"

by auto

with `compact U` obtain B where "B ⊆ A" "finite (uminus`B)" "U ⊆ \<Union>(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

assume "∀a∈A. open a" "U ⊆ \<Union>A"

then have "U ∩ \<Inter>(uminus`A) = {}" "∀a∈uminus`A. closed a"

by auto

with `?R` obtain B where "B ⊆ A" "finite (uminus`B)" "U ∩ \<Inter>(uminus`B) = {}"

by (metis subset_image_iff)

then show "∃T⊆A. finite T ∧ U ⊆ \<Union>T"

by (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)

qed

lemma compact_imp_fip:

"compact s ==> ∀t ∈ f. closed t ==> ∀f'. finite f' ∧ f' ⊆ f --> (s ∩ (\<Inter> f') ≠ {}) ==>

s ∩ (\<Inter> f) ≠ {}"

unfolding compact_fip by auto

text{*Compactness expressed with filters*}

definition "filter_from_subbase B = Abs_filter (λP. ∃X ⊆ B. finite X ∧ Inf X ≤ P)"

lemma eventually_filter_from_subbase:

"eventually P (filter_from_subbase B) <-> (∃X ⊆ B. finite X ∧ Inf X ≤ P)"

(is "_ <-> ?R P")

unfolding filter_from_subbase_def

proof (rule eventually_Abs_filter is_filter.intro)+

show "?R (λx. True)"

by (rule exI[of _ "{}"]) (simp add: le_fun_def)

next

fix P Q assume "?R P" then guess X ..

moreover assume "?R Q" then guess Y ..

ultimately show "?R (λx. P x ∧ Q x)"

by (intro exI[of _ "X ∪ Y"]) 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 (λx. x ∈ U) F --> (∃x∈U. inf (nhds x) F ≠ bot))"

proof (intro allI iffI impI compact_fip[THEN iffD2] notI)

fix F

assume "compact U"

assume F: "F ≠ bot" "eventually (λx. x ∈ U) F"

then have "U ≠ {}"

by (auto simp: eventually_False)

def Z ≡ "closure ` {A. eventually (λx. x ∈ A) F}"

then have "∀z∈Z. closed z"

by auto

moreover

have ev_Z: "!!z. z ∈ Z ==> eventually (λx. x ∈ z) F"

unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])

have "(∀B ⊆ Z. finite B --> U ∩ \<Inter>B ≠ {})"

proof (intro allI impI)

fix B assume "finite B" "B ⊆ Z"

with `finite B` ev_Z have "eventually (λx. ∀b∈B. x ∈ b) F"

by (auto intro!: eventually_Ball_finite)

with F(2) have "eventually (λx. x ∈ U ∩ (\<Inter>B)) F"

by eventually_elim auto

with F show "U ∩ \<Inter>B ≠ {}"

by (intro notI) (simp add: eventually_False)

qed

ultimately have "U ∩ \<Inter>Z ≠ {}"

using `compact U` unfolding compact_fip by blast

then obtain x where "x ∈ U" and x: "!!z. z ∈ Z ==> x ∈ z"

by auto

have "!!P. eventually P (inf (nhds x) F) ==> P ≠ bot"

unfolding eventually_inf eventually_nhds

proof safe

fix P Q R S

assume "eventually R F" "open S" "x ∈ S"

with open_inter_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]

have "S ∩ {x. R x} ≠ {}" by (auto simp: Z_def)

moreover assume "Ball S Q" "∀x. Q x ∧ R x --> bot x"

ultimately show False by (auto simp: set_eq_iff)

qed

with `x ∈ U` show "∃x∈U. inf (nhds x) F ≠ bot"

by (metis eventually_bot)

next

fix A

assume A: "∀a∈A. closed a" "∀B⊆A. finite B --> U ∩ \<Inter>B ≠ {}" "U ∩ \<Inter>A = {}"

def P' ≡ "(λa (x::'a). x ∈ a)"

then have inj_P': "!!A. inj_on P' A"

by (auto intro!: inj_onI simp: fun_eq_iff)

def F ≡ "filter_from_subbase (P' ` insert U A)"

have "F ≠ bot"

unfolding F_def

proof (safe intro!: filter_from_subbase_not_bot)

fix X

assume "X ⊆ P' ` insert U A" "finite X" "Inf X = bot"

then obtain B where "B ⊆ insert U A" "finite B" and B: "Inf (P' ` B) = bot"

unfolding subset_image_iff by (auto intro: inj_P' finite_imageD)

with A(2)[THEN spec, of "B - {U}"] have "U ∩ \<Inter>(B - {U}) ≠ {}"

by auto

with B show False

by (auto simp: P'_def fun_eq_iff)

qed

moreover have "eventually (λx. x ∈ U) F"

unfolding F_def by (rule eventually_filter_from_subbaseI) (auto simp: P'_def)

moreover

assume "∀F. F ≠ bot --> eventually (λx. x ∈ U) F --> (∃x∈U. inf (nhds x) F ≠ bot)"

ultimately obtain x where "x ∈ U" and x: "inf (nhds x) F ≠ bot"

by auto

{

fix V

assume "V ∈ A"

then have V: "eventually (λx. x ∈ V) F"

by (auto simp add: F_def image_iff P'_def intro!: eventually_filter_from_subbaseI)

have "x ∈ closure V"

unfolding closure_iff_nhds_not_empty

proof (intro impI allI)

fix S A

assume "open S" "x ∈ S" "S ⊆ A"

then have "eventually (λx. x ∈ A) (nhds x)"

by (auto simp: eventually_nhds)

with V have "eventually (λx. x ∈ V ∩ A) (inf (nhds x) F)"

by (auto simp: eventually_inf)

with x show "V ∩ A ≠ {}"

by (auto simp del: Int_iff simp add: trivial_limit_def)

qed

then have "x ∈ V"

using `V ∈ A` A(1) by simp

}

with `x∈U` have "x ∈ U ∩ \<Inter>A" by auto

with `U ∩ \<Inter>A = {}` show False by auto

qed

definition "countably_compact U <->

(∀A. countable A --> (∀a∈A. open a) --> U ⊆ \<Union>A --> (∃T⊆A. finite T ∧ U ⊆ \<Union>T))"

lemma countably_compactE:

assumes "countably_compact s" and "∀t∈C. open t" and "s ⊆ \<Union>C" "countable C"

obtains C' where "C' ⊆ C" and "finite C'" and "s ⊆ \<Union>C'"

using assms unfolding countably_compact_def by metis

lemma countably_compactI:

assumes "!!C. ∀t∈C. open t ==> s ⊆ \<Union>C ==> countable C ==> (∃C'⊆C. finite C' ∧ s ⊆ \<Union>C')"

shows "countably_compact s"

using assms unfolding countably_compact_def by metis

lemma compact_imp_countably_compact: "compact U ==> countably_compact U"

by (auto simp: compact_eq_heine_borel countably_compact_def)

lemma countably_compact_imp_compact:

assumes "countably_compact U"

and ccover: "countable B" "∀b∈B. open b"

and basis: "!!T x. open T ==> x ∈ T ==> x ∈ U ==> ∃b∈B. x ∈ b ∧ b ∩ U ⊆ T"

shows "compact U"

using `countably_compact U`

unfolding compact_eq_heine_borel countably_compact_def

proof safe

fix A

assume A: "∀a∈A. open a" "U ⊆ \<Union>A"

assume *: "∀A. countable A --> (∀a∈A. open a) --> U ⊆ \<Union>A --> (∃T⊆A. finite T ∧ U ⊆ \<Union>T)"

moreover def C ≡ "{b∈B. ∃a∈A. b ∩ U ⊆ a}"

ultimately have "countable C" "∀a∈C. open a"

unfolding C_def using ccover by auto

moreover

have "\<Union>A ∩ U ⊆ \<Union>C"

proof safe

fix x a

assume "x ∈ U" "x ∈ a" "a ∈ A"

with basis[of a x] A obtain b where "b ∈ B" "x ∈ b" "b ∩ U ⊆ a"

by blast

with `a ∈ A` show "x ∈ \<Union>C"

unfolding C_def by auto

qed

then have "U ⊆ \<Union>C" using `U ⊆ \<Union>A` by auto

ultimately obtain T where T: "T⊆C" "finite T" "U ⊆ \<Union>T"

using * by metis

then have "∀t∈T. ∃a∈A. t ∩ U ⊆ a"

by (auto simp: C_def)

then guess f unfolding bchoice_iff Bex_def ..

with T show "∃T⊆A. finite T ∧ U ⊆ \<Union>T"

unfolding C_def by (intro exI[of _ "f`T"]) fastforce

qed

lemma countably_compact_imp_compact_second_countable:

"countably_compact U ==> compact (U :: 'a :: second_countable_topology set)"

proof (rule countably_compact_imp_compact)

fix T and x :: 'a

assume "open T" "x ∈ T"

from topological_basisE[OF is_basis this] 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 ⊆ \<Union>A" "countable A"

have subseq: "!!X. range X ⊆ U ==> ∃r x. x ∈ U ∧ subseq r ∧ (X o r) ----> x"

using `seq_compact U` by (fastforce simp: seq_compact_def subset_eq)

show "∃T⊆A. finite T ∧ U ⊆ \<Union>T"

proof cases

assume "finite A"

with A show ?thesis by auto

next

assume "infinite A"

then have "A ≠ {}" by auto

show ?thesis

proof (rule ccontr)

assume "¬ (∃T⊆A. finite T ∧ U ⊆ \<Union>T)"

then have "∀T. ∃x. T ⊆ A ∧ finite T --> (x ∈ U - \<Union>T)"

by auto

then obtain X' where T: "!!T. T ⊆ A ==> finite T ==> X' T ∈ U - \<Union>T"

by metis

def X ≡ "λn. X' (from_nat_into A ` {.. n})"

have X: "!!n. X n ∈ U - (\<Union>i≤n. from_nat_into A i)"

using `A ≠ {}` unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)

then have "range X ⊆ U"

by auto

with subseq[of X] obtain r x where "x ∈ U" and r: "subseq r" "(X o r) ----> x"

by auto

from `x∈U` `U ⊆ \<Union>A` from_nat_into_surj[OF `countable A`]

obtain n where "x ∈ from_nat_into A n" by auto

with r(2) A(1) from_nat_into[OF `A ≠ {}`, of n]

have "eventually (λi. X (r i) ∈ from_nat_into A n) sequentially"

unfolding tendsto_def by (auto simp: comp_def)

then obtain N where "!!i. N ≤ i ==> X (r i) ∈ from_nat_into A n"

by (auto simp: eventually_sequentially)

moreover from X have "!!i. n ≤ r i ==> X (r i) ∉ from_nat_into A n"

by auto

moreover from `subseq r`[THEN seq_suble, of "max n N"] have "∃i. n ≤ r i ∧ N ≤ i"

by (auto intro!: exI[of _ "max n N"])

ultimately show False

by auto

qed

qed

qed

lemma compact_imp_seq_compact:

fixes U :: "'a :: first_countable_topology set"

assumes "compact U"

shows "seq_compact U"

unfolding seq_compact_def

proof safe

fix X :: "nat => 'a"

assume "∀n. X n ∈ U"

then have "eventually (λx. x ∈ U) (filtermap X sequentially)"

by (auto simp: eventually_filtermap)

moreover

have "filtermap X sequentially ≠ bot"

by (simp add: trivial_limit_def eventually_filtermap)

ultimately

obtain x where "x ∈ U" and x: "inf (nhds x) (filtermap X sequentially) ≠ bot" (is "?F ≠ _")

using `compact U` by (auto simp: compact_filter)

from countable_basis_at_decseq[of x] guess A . note A = this

def s ≡ "λn i. SOME j. i < j ∧ X j ∈ A (Suc n)"

{

fix n i

have "∃a. i < a ∧ X a ∈ A (Suc n)"

proof (rule ccontr)

assume "¬ (∃a>i. X a ∈ A (Suc n))"

then have "!!a. Suc i ≤ a ==> X a ∉ A (Suc n)"

by auto

then have "eventually (λx. x ∉ A (Suc n)) (filtermap X sequentially)"

by (auto simp: eventually_filtermap eventually_sequentially)

moreover have "eventually (λx. x ∈ A (Suc n)) (nhds x)"

using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)

ultimately have "eventually (λx. False) ?F"

by (auto simp add: eventually_inf)

with x show False

by (simp add: eventually_False)

qed

then have "i < s n i" "X (s n i) ∈ A (Suc n)"

unfolding s_def by (auto intro: someI2_ex)

}

note s = this

def r ≡ "nat_rec (s 0 0) s"

have "subseq r"

by (auto simp: r_def s subseq_Suc_iff)

moreover

have "(λn. X (r n)) ----> x"

proof (rule topological_tendstoI)

fix S

assume "open S" "x ∈ S"

with A(3) have "eventually (λi. A i ⊆ S) sequentially"

by auto

moreover

{

fix i

assume "Suc 0 ≤ i"

then have "X (r i) ∈ A i"

by (cases i) (simp_all add: r_def s)

}

then have "eventually (λi. X (r i) ∈ A i) sequentially"

by (auto simp: eventually_sequentially)

ultimately show "eventually (λi. X (r i) ∈ S) sequentially"

by eventually_elim auto

qed

ultimately show "∃x ∈ U. ∃r. subseq r ∧ (X o r) ----> x"

using `x ∈ U` by (auto simp: convergent_def comp_def)

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 o r) ---> l) sequentially"

using assms unfolding seq_compact_def by fast

lemma countably_compact_imp_acc_point:

assumes "countably_compact s"

and "countable t"

and "infinite t"

and "t ⊆ s"

shows "∃x∈s. ∀U. x∈U ∧ open U --> infinite (U ∩ t)"

proof (rule ccontr)

def C ≡ "(λF. interior (F ∪ (- t))) ` {F. finite F ∧ F ⊆ t }"

note `countably_compact s`

moreover have "∀t∈C. open t"

by (auto simp: C_def)

moreover

assume "¬ (∃x∈s. ∀U. x∈U ∧ open U --> infinite (U ∩ t))"

then have s: "!!x. x ∈ s ==> ∃U. x∈U ∧ open U ∧ finite (U ∩ t)" by metis

have "s ⊆ \<Union>C"

using `t ⊆ s`

unfolding C_def Union_image_eq

apply (safe dest!: s)

apply (rule_tac a="U ∩ t" in UN_I)

apply (auto intro!: interiorI simp add: finite_subset)

done

moreover

from `countable t` have "countable C"

unfolding C_def by (auto intro: countable_Collect_finite_subset)

ultimately guess D by (rule countably_compactE)

then obtain E where E: "E ⊆ {F. finite F ∧ F ⊆ t }" "finite E"

and s: "s ⊆ (\<Union>F∈E. interior (F ∪ (- t)))"

by (metis (lifting) Union_image_eq finite_subset_image C_def)

from s `t ⊆ s` have "t ⊆ \<Union>E"

using interior_subset by blast

moreover have "finite (\<Union>E)"

using E by auto

ultimately show False using `infinite t`

by (auto simp: finite_subset)

qed

lemma countable_acc_point_imp_seq_compact:

fixes s :: "'a::first_countable_topology set"

assumes "∀t. infinite t ∧ countable t ∧ t ⊆ s -->

(∃x∈s. ∀U. x∈U ∧ open U --> infinite (U ∩ t))"

shows "seq_compact s"

proof -

{

fix f :: "nat => 'a"

assume f: "∀n. f n ∈ s"

have "∃l∈s. ∃r. subseq r ∧ ((f o r) ---> l) sequentially"

proof (cases "finite (range f)")

case True

obtain l where "infinite {n. f n = f l}"

using pigeonhole_infinite[OF _ True] by auto

then obtain r where "subseq r" and fr: "∀n. f (r n) = f l"

using infinite_enumerate by blast

then have "subseq r ∧ (f o r) ----> f l"

by (simp add: fr tendsto_const o_def)

with f show "∃l∈s. ∃r. subseq r ∧ (f o r) ----> l"

by auto

next

case False

with f assms have "∃x∈s. ∀U. x∈U ∧ open U --> infinite (U ∩ range f)"

by auto

then obtain l where "l ∈ s" "∀U. l∈U ∧ open U --> infinite (U ∩ range f)" ..

from this(2) have "∃r. subseq r ∧ ((f o r) ---> l) sequentially"

using acc_point_range_imp_convergent_subsequence[of l f] by auto

with `l ∈ s` show "∃l∈s. ∃r. subseq r ∧ ((f o r) ---> l) sequentially" ..

qed

}

then show ?thesis

unfolding seq_compact_def by 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 metis

fun helper_1 :: "('a::metric_space set) => real => nat => 'a"

where

"helper_1 s e n = (SOME y::'a. y ∈ s ∧ (∀m<n. ¬ (dist (helper_1 s e m) y < e)))"

declare helper_1.simps[simp del]

lemma seq_compact_imp_totally_bounded:

assumes "seq_compact s"

shows "∀e>0. ∃k. finite k ∧ k ⊆ s ∧ s ⊆ (\<Union>((λx. ball x e) ` k))"

proof (rule, rule, rule ccontr)

fix e::real

assume "e > 0"

assume assm: "¬ (∃k. finite k ∧ k ⊆ s ∧ s ⊆ \<Union>((λx. ball x e) ` k))"

def x ≡ "helper_1 s e"

{

fix n

have "x n ∈ s ∧ (∀m<n. ¬ dist (x m) (x n) < e)"

proof (induct n rule: nat_less_induct)

fix n

def Q ≡ "(λy. y ∈ s ∧ (∀m<n. ¬ dist (x m) y < e))"

assume as: "∀m<n. x m ∈ s ∧ (∀ma<m. ¬ dist (x ma) (x m) < e)"

have "¬ s ⊆ (\<Union>x∈x ` {0..<n}. ball x e)"

using assm

apply simp

apply (erule_tac x="x ` {0 ..< n}" in allE)

using as

apply auto

done

then obtain z where z:"z∈s" "z ∉ (\<Union>x∈x ` {0..<n}. ball x e)"

unfolding subset_eq by auto

have "Q (x n)"

unfolding x_def and helper_1.simps[of s e n]

apply (rule someI2[where a=z])

unfolding x_def[symmetric] and Q_def

using z

apply auto

done

then show "x n ∈ s ∧ (∀m<n. ¬ dist (x m) (x n) < e)"

unfolding Q_def by auto

qed

}

then have "∀n::nat. x n ∈ s" and x:"∀n. ∀m < n. ¬ (dist (x m) (x n) < e)"

by blast+

then obtain l r where "l∈s" and r:"subseq r" and "((x o r) ---> l) sequentially"

using assms(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto

from this(3) have "Cauchy (x o r)"

using LIMSEQ_imp_Cauchy by auto

then obtain N::nat where N:"∀m n. N ≤ m ∧ N ≤ n --> dist ((x o r) m) ((x o r) n) < e"

unfolding cauchy_def using `e>0` by auto

show False

using N[THEN spec[where x=N], THEN spec[where x="N+1"]]

using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]

using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]]

by 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 ≡ "(λ(x, r). ball x r) ` ((\<Union>e ∈ \<rat> ∩ {0 <..}. f e) × \<rat>)"

have "countably_compact s"

using `seq_compact s` by (rule seq_compact_imp_countably_compact)

then show "compact s"

proof (rule countably_compact_imp_compact)

show "countable K"

unfolding K_def using f

by (auto intro: countable_finite countable_subset countable_rat

intro!: countable_image countable_SIGMA countable_UN)

show "∀b∈K. open b" by (auto simp: K_def)

next

fix T x

assume T: "open T" "x ∈ T" and x: "x ∈ s"

from openE[OF T] obtain e where "0 < e" "ball x e ⊆ T"

by auto

then have "0 < e / 2" "ball x (e / 2) ⊆ T"

by auto

from Rats_dense_in_real[OF `0 < e / 2`] obtain r where "r ∈ \<rat>" "0 < r" "r < e / 2"

by auto

from f[rule_format, of r] `0 < r` `x ∈ s` obtain k where "k ∈ f r" "x ∈ ball k r"

unfolding Union_image_eq by auto

from `r ∈ \<rat>` `0 < r` `k ∈ f r` have "ball k r ∈ K"

by (auto simp: K_def)

then show "∃b∈K. x ∈ b ∧ b ∩ s ⊆ T"

proof (rule bexI[rotated], safe)

fix y

assume "y ∈ ball k r"

with `r < e / 2` `x ∈ ball k r` have "dist x y < e"

by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute)

with `ball x e ⊆ T` show "y ∈ T"

by auto

next

show "x ∈ ball k r" by fact

qed

qed

qed

lemma compact_eq_seq_compact_metric:

"compact (s :: 'a::metric_space set) <-> seq_compact s"

using compact_imp_seq_compact seq_compact_imp_heine_borel by blast

lemma compact_def:

"compact (S :: 'a::metric_space set) <->

(∀f. (∀n. f n ∈ S) --> (∃l∈S. ∃r. subseq r ∧ (f o r) ----> l))"

unfolding compact_eq_seq_compact_metric seq_compact_def by auto

subsubsection {* Complete the chain of compactness variants *}

lemma compact_eq_bolzano_weierstrass:

fixes s :: "'a::metric_space set"

shows "compact s <-> (∀t. infinite t ∧ t ⊆ s --> (∃x ∈ s. x islimpt t))"

(is "?lhs = ?rhs")

proof

assume ?lhs

then show ?rhs

using heine_borel_imp_bolzano_weierstrass[of s] by auto

next

assume ?rhs

then show ?lhs

unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)

qed

lemma bolzano_weierstrass_imp_bounded:

"∀t. infinite t ∧ t ⊆ s --> (∃x ∈ s. x islimpt t) ==> bounded s"

using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .

text {*

A metric space (or topological vector space) is said to have the

Heine-Borel property if every closed and bounded subset is compact.

*}

class heine_borel = metric_space +

assumes bounded_imp_convergent_subsequence:

"bounded (range f) ==> ∃l r. subseq r ∧ ((f o r) ---> l) sequentially"

lemma bounded_closed_imp_seq_compact:

fixes s::"'a::heine_borel set"

assumes "bounded s"

and "closed s"

shows "seq_compact s"

proof (unfold seq_compact_def, clarify)

fix f :: "nat => 'a"

assume f: "∀n. f n ∈ s"

with `bounded s` have "bounded (range f)"

by (auto intro: bounded_subset)

obtain l r where r: "subseq r" and l: "((f o r) ---> l) sequentially"

using bounded_imp_convergent_subsequence [OF `bounded (range f)`] by auto

from f have fr: "∀n. (f o r) n ∈ s"

by simp

have "l ∈ s" using `closed s` fr l

unfolding closed_sequential_limits by blast

show "∃l∈s. ∃r. subseq r ∧ ((f o r) ---> l) sequentially"

using `l ∈ s` r l by blast

qed

lemma compact_eq_bounded_closed:

fixes s :: "'a::heine_borel set"

shows "compact s <-> bounded s ∧ closed s"

(is "?lhs = ?rhs")

proof

assume ?lhs

then show ?rhs

using compact_imp_closed compact_imp_bounded

by blast

next

assume ?rhs

then show ?lhs

using bounded_closed_imp_seq_compact[of s]

unfolding compact_eq_seq_compact_metric

by auto

qed

(* TODO: is this lemma necessary? *)

lemma bounded_increasing_convergent:

fixes s :: "nat => real"

shows "bounded {s n| n. True} ==> ∀n. s n ≤ s (Suc n) ==> ∃l. s ----> l"

using Bseq_mono_convergent[of s] incseq_Suc_iff[of s]

by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)

instance real :: heine_borel

proof

fix f :: "nat => real"

assume f: "bounded (range f)"

obtain r where r: "subseq r" "monoseq (f o r)"

unfolding comp_def by (metis seq_monosub)

then have "Bseq (f o r)"

unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset)

with r show "∃l r. subseq r ∧ (f o r) ----> l"

using Bseq_monoseq_convergent[of "f o r"] by (auto simp: convergent_def)

qed

lemma compact_lemma:

fixes f :: "nat => 'a::euclidean_space"

assumes "bounded (range f)"

shows "∀d⊆Basis. ∃l::'a. ∃ r.

subseq r ∧ (∀e>0. eventually (λn. ∀i∈d. dist (f (r n) • i) (l • i) < e) sequentially)"

proof safe

fix d :: "'a set"

assume d: "d ⊆ Basis"

with finite_Basis have "finite d"

by (blast intro: finite_subset)

from this d show "∃l::'a. ∃r. subseq r ∧

(∀e>0. eventually (λn. ∀i∈d. dist (f (r n) • i) (l • i) < e) sequentially)"

proof (induct d)

case empty

then show ?case

unfolding subseq_def by auto

next

case (insert k d)

have k[intro]: "k ∈ Basis"

using insert by auto

have s': "bounded ((λx. x • k) ` range f)"

using `bounded (range f)`

by (auto intro!: bounded_linear_image bounded_linear_inner_left)

obtain l1::"'a" and r1 where r1: "subseq r1"

and lr1: "∀e > 0. eventually (λn. ∀i∈d. dist (f (r1 n) • i) (l1 • i) < e) sequentially"

using insert(3) using insert(4) by auto

have f': "∀n. f (r1 n) • k ∈ (λx. x • k) ` range f"

by simp

have "bounded (range (λi. f (r1 i) • k))"

by (metis (lifting) bounded_subset f' image_subsetI s')

then obtain l2 r2 where r2:"subseq r2" and lr2:"((λi. f (r1 (r2 i)) • k) ---> l2) sequentially"

using bounded_imp_convergent_subsequence[of "λi. f (r1 i) • k"]

by (auto simp: o_def)

def r ≡ "r1 o r2"

have r:"subseq r"

using r1 and r2 unfolding r_def o_def subseq_def by auto

moreover

def l ≡ "(∑i∈Basis. (if i = k then l2 else l1•i) *⇩_{R}i)::'a"

{

fix e::real

assume "e > 0"

from lr1 `e > 0` have N1: "eventually (λn. ∀i∈d. dist (f (r1 n) • i) (l1 • i) < e) sequentially"

by blast

from lr2 `e > 0` have N2:"eventually (λn. dist (f (r1 (r2 n)) • k) l2 < e) sequentially"

by (rule tendstoD)

from r2 N1 have N1': "eventually (λn. ∀i∈d. dist (f (r1 (r2 n)) • i) (l1 • i) < e) sequentially"

by (rule eventually_subseq)

have "eventually (λn. ∀i∈(insert k d). dist (f (r n) • i) (l • i) < e) sequentially"

using N1' N2

by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)

}

ultimately show ?case by auto

qed

qed

instance euclidean_space ⊆ heine_borel

proof

fix f :: "nat => 'a"

assume f: "bounded (range f)"

then obtain l::'a and r where r: "subseq r"

and l: "∀e>0. eventually (λn. ∀i∈Basis. dist (f (r n) • i) (l • i) < e) sequentially"

using compact_lemma [OF f] by blast

{

fix e::real

assume "e > 0"

then have "e / real_of_nat DIM('a) > 0"

by (auto intro!: divide_pos_pos DIM_positive)

with l have "eventually (λn. ∀i∈Basis. dist (f (r n) • i) (l • i) < e / (real_of_nat DIM('a))) sequentially"

by simp

moreover

{

fix n

assume n: "∀i∈Basis. dist (f (r n) • i) (l • i) < e / (real_of_nat DIM('a))"

have "dist (f (r n)) l ≤ (∑i∈Basis. dist (f (r n) • i) (l • i))"

apply (subst euclidean_dist_l2)

using zero_le_dist

apply (rule setL2_le_setsum)

done

also have "… < (∑i∈(Basis::'a set). e / (real_of_nat DIM('a)))"

apply (rule setsum_strict_mono)

using n

apply auto

done

finally have "dist (f (r n)) l < e"

by auto

}

ultimately have "eventually (λn. dist (f (r n)) l < e) sequentially"

by (rule eventually_elim1)

}

then have *: "((f o r) ---> l) sequentially"

unfolding o_def tendsto_iff by simp

with r show "∃l r. subseq r ∧ ((f o r) ---> l) sequentially"

by 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 o f))"

unfolding image_comp by (rule bounded_fst)

obtain l1 r1 where r1: "subseq r1" and l1: "(λn. fst (f (r1 n))) ----> l1"

using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast

from f have s2: "bounded (range (snd o f o r1))"

by (auto simp add: image_comp intro: bounded_snd bounded_subset)

obtain l2 r2 where r2: "subseq r2" and l2: "((λn. snd (f (r1 (r2 n)))) ---> l2) sequentially"

using bounded_imp_convergent_subsequence [OF s2]

unfolding o_def by fast

have l1': "((λn. fst (f (r1 (r2 n)))) ---> l1) sequentially"

using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .

have l: "((f o (r1 o r2)) ---> (l1, l2)) sequentially"

using tendsto_Pair [OF l1' l2] unfolding o_def by simp

have r: "subseq (r1 o r2)"

using r1 r2 unfolding subseq_def by simp

show "∃l r. subseq r ∧ ((f o r) ---> l) sequentially"

using l r by 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 o r) ----> l"

using assms unfolding compact_def by blast

note lr' = seq_suble [OF lr(2)]

{

fix e :: real

assume "e > 0"

from as(2) obtain N where N:"∀m n. N ≤ m ∧ N ≤ n --> dist (f m) (f n) < e/2"

unfolding cauchy_def

using `e > 0`

apply (erule_tac x="e/2" in allE)

apply auto

done

from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]]

obtain M where M:"∀n≥M. dist ((f o r) n) l < e/2"

using `e > 0` by auto

{

fix n :: nat

assume n: "n ≥ max N M"

have "dist ((f o r) n) l < e/2"

using n M by auto

moreover have "r n ≥ N"

using lr'[of n] n by auto

then have "dist (f n) ((f o r) n) < e / 2"

using N and n by auto

ultimately have "dist (f n) l < e"

using dist_triangle_half_r[of "f (r n)" "f n" e l]

by (auto simp add: dist_commute)

}

then have "∃N. ∀n≥N. dist (f n) l < e" by blast

}

then have "∃l∈s. (f ---> l) sequentially" using `l∈s`

unfolding LIMSEQ_def by auto

}

then show ?thesis unfolding complete_def by 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 ⊆ (\<Union>((λx. ball x e) ` k)))"

(is "_ <-> ?rhs")

proof

assume assms: "?rhs"

then obtain k where k: "!!e. 0 < e ==> finite (k e)" "!!e. 0 < e ==> s ⊆ (\<Union>x∈k e. ball x e)"

by (auto simp: choice_iff')

show "compact s"

proof cases

assume "s = {}"

then show "compact s" by (simp add: compact_def)

next

assume "s ≠ {}"

show ?thesis

unfolding compact_def

proof safe

fix f :: "nat => 'a"

assume f: "∀n. f n ∈ s"

def e ≡ "λn. 1 / (2 * Suc n)"

then have [simp]: "!!n. 0 < e n" by auto

def B ≡ "λn U. SOME b. infinite {n. f n ∈ b} ∧ (∃x. b ⊆ ball x (e n) ∩ U)"

{

fix n U

assume "infinite {n. f n ∈ U}"

then have "∃b∈k (e n). infinite {i∈{n. f n ∈ U}. f i ∈ ball b (e n)}"

using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)

then guess a ..

then have "∃b. infinite {i. f i ∈ b} ∧ (∃x. b ⊆ ball x (e n) ∩ U)"

by (intro exI[of _ "ball a (e n) ∩ U"] exI[of _ a]) (auto simp: ac_simps)

from someI_ex[OF this]

have "infinite {i. f i ∈ B n U}" "∃x. B n U ⊆ ball x (e n) ∩ U"

unfolding B_def by auto

}

note B = this

def F ≡ "nat_rec (B 0 UNIV) B"

{

fix n

have "infinite {i. f i ∈ F n}"

by (induct n) (auto simp: F_def B)

}

then have F: "!!n. ∃x. F (Suc n) ⊆ ball x (e n) ∩ F n"

using B by (simp add: F_def)

then have F_dec: "!!m n. m ≤ n ==> F n ⊆ F m"

using decseq_SucI[of F] by (auto simp: decseq_def)

obtain sel where sel: "!!k i. i < sel k i" "!!k i. f (sel k i) ∈ F k"

proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)

fix k i

have "infinite ({n. f n ∈ F k} - {.. i})"

using `infinite {n. f n ∈ F k}` by auto

from infinite_imp_nonempty[OF this]

show "∃x>i. f x ∈ F k"

by (simp add: set_eq_iff not_le conj_commute)

qed

def t ≡ "nat_rec (sel 0 0) (λn i. sel (Suc n) i)"

have "subseq t"

unfolding subseq_Suc_iff by (simp add: t_def sel)

moreover have "∀i. (f o t) i ∈ s"

using f by auto

moreover

{

fix n

have "(f o t) n ∈ F n"

by (cases n) (simp_all add: t_def sel)

}

note t = this

have "Cauchy (f o t)"

proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)

fix r :: real and N n m

assume "1 / Suc N < r" "Suc N ≤ n" "Suc N ≤ m"

then have "(f o t) n ∈ F (Suc N)" "(f o t) m ∈ F (Suc N)" "2 * e N < r"

using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc)

with F[of N] obtain x where "dist x ((f o t) n) < e N" "dist x ((f o t) m) < e N"

by (auto simp: subset_eq)

with dist_triangle[of "(f o t) m" "(f o t) n" x] `2 * e N < r`

show "dist ((f o t) m) ((f o t) n) < r"

by (simp add: dist_commute)

qed

ultimately show "∃l∈s. ∃r. subseq r ∧ (f o r) ----> l"

using assms unfolding complete_def by blast

qed

qed

qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)

lemma cauchy: "Cauchy s <-> (∀e>0.∃ N::nat. ∀n≥N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")

proof -

{

assume ?rhs

{

fix e::real

assume "e>0"

with `?rhs` obtain N where N:"∀n≥N. dist (s n) (s N) < e/2"

by (erule_tac x="e/2" in allE) auto

{

fix n m

assume nm:"N ≤ m ∧ N ≤ n"

then have "dist (s m) (s n) < e" using N

using dist_triangle_half_l[of "s m" "s N" "e" "s n"]

by blast

}

then have "∃N. ∀m n. N ≤ m ∧ N ≤ n --> dist (s m) (s n) < e"

by blast

}

then have ?lhs

unfolding cauchy_def

by blast

}

then show ?thesis

unfolding cauchy_def

using dist_triangle_half_l

by blast

qed

lemma cauchy_imp_bounded:

assumes "Cauchy s"

shows "bounded (range s)"

proof -

from assms obtain N :: nat where "∀m n. N ≤ m ∧ N ≤ n --> dist (s m) (s n) < 1"

unfolding cauchy_def

apply (erule_tac x= 1 in allE)

apply auto

done

then have N:"∀n. N ≤ n --> dist (s N) (s n) < 1" by auto

moreover

have "bounded (s ` {0..N})"

using finite_imp_bounded[of "s ` {1..N}"] by auto

then obtain a where a:"∀x∈s ` {0..N}. dist (s N) x ≤ a"

unfolding bounded_any_center [where a="s N"] by auto

ultimately show "?thesis"

unfolding bounded_any_center [where a="s N"]

apply (rule_tac x="max a 1" in exI)

apply auto

apply (erule_tac x=y in allE)

apply (erule_tac x=y in ballE)

apply auto

done

qed

instance heine_borel < complete_space

proof

fix f :: "nat => 'a" assume "Cauchy f"

then have "bounded (range f)"

by (rule cauchy_imp_bounded)

then have "compact (closure (range f))"

unfolding compact_eq_bounded_closed by auto

then have "complete (closure (range f))"

by (rule compact_imp_complete)

moreover have "∀n. f n ∈ closure (range f)"

using closure_subset [of "range f"] by auto

ultimately have "∃l∈closure (range f). (f ---> l) sequentially"

using `Cauchy f` unfolding complete_def by auto

then show "convergent f"

unfolding convergent_def by auto

qed

instance euclidean_space ⊆ banach ..

lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"

proof (simp add: complete_def, rule, rule)

fix f :: "nat => 'a"

assume "Cauchy f"

then have "convergent f" by (rule Cauchy_convergent)

then show "∃l. f ----> l" unfolding convergent_def .

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

then have "x ∈ s"

using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto

}

then show "closed s" unfolding closed_limpt by auto

qed

lemma complete_eq_closed:

fixes s :: "'a::complete_space set"

shows "complete s <-> closed s" (is "?lhs = ?rhs")

proof

assume ?lhs

then show ?rhs by (rule complete_imp_closed)

next

assume ?rhs

{

fix f

assume as:"∀n::nat. f n ∈ s" "Cauchy f"

then obtain l where "(f ---> l) sequentially"

using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto

then have "∃l∈s. (f ---> l) sequentially"

using `?rhs`[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]]

using as(1) by auto

}

then show ?lhs unfolding complete_def by 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)"

and "∀n. (s n ≠ {})"

and "(∀m n. m ≤ n --> s n ⊆ s m)"

and "bounded(s 0)"

shows "∃a::'a::heine_borel. ∀n::nat. a ∈ s(n)"

proof -

from assms(2) obtain x where x:"∀n::nat. x n ∈ s n"

using choice[of "λn x. x∈ s n"] by auto

from assms(4,1) have *:"seq_compact (s 0)"

using bounded_closed_imp_seq_compact[of "s 0"] by auto

then obtain l r where lr:"l∈s 0" "subseq r" "((x o r) ---> l) sequentially"

unfolding seq_compact_def

apply (erule_tac x=x in allE)

using x using assms(3)

apply blast

done

{

fix n :: nat

{

fix e :: real

assume "e>0"

with lr(3) obtain N where N:"∀m≥N. dist ((x o r) m) l < e"

unfolding LIMSEQ_def by auto

then have "dist ((x o r) (max N n)) l < e" by auto

moreover

have "r (max N n) ≥ n" using lr(2) using seq_suble[of r "max N n"]

by auto

then have "(x o r) (max N n) ∈ s n"

using x

apply (erule_tac x=n in allE)

using x

apply (erule_tac x="r (max N n)" in allE)

using assms(3)

apply (erule_tac x=n in allE)

apply (erule_tac x="r (max N n)" in allE)

apply auto

done

ultimately have "∃y∈s n. dist y l < e"

by auto

}

then have "l ∈ s n"

using closed_approachable[of "s n" l] assms(1) by blast

}

then show ?thesis by 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

then have "∃t. ∀n. t n ∈ s n"

using choice[of "λ n x. x ∈ s n"] by auto

then obtain t where t: "∀n. t n ∈ s n" by auto

{

fix e :: real

assume "e > 0"

then obtain N where N:"∀x∈s N. ∀y∈s N. dist x y < e"

using assms(4) by auto

{

fix m n :: nat

assume "N ≤ m ∧ N ≤ n"

then have "t m ∈ s N" "t n ∈ s N"

using assms(3) t unfolding subset_eq t by blast+

then have "dist (t m) (t n) < e"

using N by auto

}

then have "∃N. ∀m n. N ≤ m ∧ N ≤ n --> dist (t m) (t n) < e"

by auto

}

then have "Cauchy t"

unfolding cauchy_def by auto

then obtain l where l:"(t ---> l) sequentially"

using complete_univ unfolding complete_def by auto

{

fix n :: nat

{

fix e :: real

assume "e > 0"

then obtain N :: nat where N: "∀n≥N. dist (t n) l < e"

using l[unfolded LIMSEQ_def] by auto

have "t (max n N) ∈ s n"

using assms(3)

unfolding subset_eq

apply (erule_tac x=n in allE)

apply (erule_tac x="max n N" in allE)

using t

apply auto

done

then have "∃y∈s n. dist y l < e"

apply (rule_tac x="t (max n N)" in bexI)

using N

apply auto

done

}

then have "l ∈ s n"

using closed_approachable[of "s n" l] assms(1) by auto

}

then show ?thesis by auto

qed

text {* Strengthen it to the intersection actually being a singleton. *}

lemma decreasing_closed_nest_sing:

fixes s :: "nat => 'a::complete_space set"

assumes

"∀n. closed(s n)"

"∀n. s n ≠ {}"

"∀m n. m ≤ n --> s n ⊆ s m"

"∀e>0. ∃n. ∀x ∈ (s n). ∀ y∈(s n). dist x y < e"

shows "∃a. \<Inter>(range s) = {a}"

proof -

obtain a where a: "∀n. a ∈ s n"

using decreasing_closed_nest[of s] using assms by auto

{

fix b

assume b: "b ∈ \<Inter>(range s)"

{

fix e :: real

assume "e > 0"

then have "dist a b < e"

using assms(4) and b and a by blast

}

then have "dist a b = 0"

by (metis dist_eq_0_iff dist_nz less_le)

}

with a have "\<Inter>(range s) = {a}"

unfolding image_def by auto

then show ?thesis ..

qed

text{* Cauchy-type criteria for uniform convergence. *}

lemma uniformly_convergent_eq_cauchy:

fixes s::"nat => 'b => 'a::complete_space"

shows

"(∃l. ∀e>0. ∃N. ∀n x. N ≤ n ∧ P x --> dist(s n x)(l x) < e) <->

(∀e>0. ∃N. ∀m n x. N ≤ m ∧ N ≤ n ∧ P x --> dist (s m x) (s n x) < e)"

(is "?lhs = ?rhs")

proof

assume ?lhs

then obtain l where l:"∀e>0. ∃N. ∀n x. N ≤ n ∧ P x --> dist (s n x) (l x) < e"

by auto

{

fix e :: real

assume "e > 0"

then obtain N :: nat where N: "∀n x. N ≤ n ∧ P x --> dist (s n x) (l x) < e / 2"

using l[THEN spec[where x="e/2"]] by auto

{

fix n m :: nat and x :: "'b"

assume "N ≤ m ∧ N ≤ n ∧ P x"

then have "dist (s m x) (s n x) < e"

using N[THEN spec[where x=m], THEN spec[where x=x]]

using N[THEN spec[where x=n], THEN spec[where x=x]]

using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto

}

then have "∃N. ∀m n x. N ≤ m ∧ N ≤ n ∧ P x --> dist (s m x) (s n x) < e" by auto

}

then show ?rhs by auto

next

assume ?rhs

then have "∀x. P x --> Cauchy (λn. s n x)"

unfolding cauchy_def

apply auto

apply (erule_tac x=e in allE)

apply auto

done

then obtain l where l: "∀x. P x --> ((λn. s n x) ---> l x) sequentially"

unfolding convergent_eq_cauchy[symmetric]

using choice[of "λx l. P x --> ((λn. s n x) ---> l) sequentially"]

by auto

{

fix e :: real

assume "e > 0"

then obtain N where N:"∀m n x. N ≤ m ∧ N ≤ n ∧ P x --> dist (s m x) (s n x) < e/2"

using `?rhs`[THEN spec[where x="e/2"]] by auto

{

fix x

assume "P x"

then obtain M where M:"∀n≥M. dist (s n x) (l x) < e/2"

using l[THEN spec[where x=x], unfolded LIMSEQ_def] and `e > 0`

by (auto elim!: allE[where x="e/2"])

fix n :: nat

assume "n ≥ N"

then have "dist(s n x)(l x) < e"

using `P x`and N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]

using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"]

by (auto simp add: dist_commute)

}

then have "∃N. ∀n x. N ≤ n ∧ P x --> dist(s n x)(l x) < e"

by auto

}

then show ?lhs by auto

qed

lemma uniformly_cauchy_imp_uniformly_convergent:

fixes s :: "nat => 'a => 'b::complete_space"

assumes "∀e>0.∃N. ∀m (n::nat) x. N ≤ m ∧ N ≤ n ∧ P x --> dist(s m x)(s n x) < e"

and "∀x. P x --> (∀e>0. ∃N. ∀n. N ≤ n --> dist(s n x)(l x) < e)"

shows "∀e>0. ∃N. ∀n x. N ≤ n ∧ P x --> dist(s n x)(l x) < e"

proof -

obtain l' where l:"∀e>0. ∃N. ∀n x. N ≤ n ∧ P x --> dist (s n x) (l' x) < e"

using assms(1) unfolding uniformly_convergent_eq_cauchy[symmetric] by auto

moreover

{

fix x

assume "P x"

then have "l x = l' x"

using tendsto_unique[OF trivial_limit_sequentially, of "λn. s n x" "l x" "l' x"]

using l and assms(2) unfolding LIMSEQ_def by blast

}

ultimately show ?thesis by auto

qed

subsection {* Continuity *}

text{* Derive the epsilon-delta forms, which we often use as "definitions" *}

lemma continuous_within_eps_delta:

"continuous (at x within s) f <-> (∀e>0. ∃d>0. ∀x'∈ s. dist x' x < d --> dist (f x') (f x) < e)"

unfolding continuous_within and Lim_within

apply auto

unfolding dist_nz[symmetric]

apply (auto del: allE elim!:allE)

apply(rule_tac x=d in exI)

apply auto

done

lemma continuous_at_eps_delta:

"continuous (at x) f <-> (∀e > 0. ∃d > 0. ∀x'. dist x' x < d --> dist (f x') (f x) < e)"

using continuous_within_eps_delta [of x UNIV f] by simp

text{* Versions in terms of open balls. *}

lemma continuous_within_ball:

"continuous (at x within s) f <->

(∀e > 0. ∃d > 0. f ` (ball x d ∩ s) ⊆ ball (f x) e)"

(is "?lhs = ?rhs")

proof

assume ?lhs

{

fix e :: real

assume "e > 0"

then obtain d where d: "d>0" "∀xa∈s. 0 < dist xa x ∧ dist xa x < d --> dist (f xa) (f x) < e"

using `?lhs`[unfolded continuous_within Lim_within] by auto

{

fix y

assume "y ∈ f ` (ball x d ∩ s)"

then have "y ∈ ball (f x) e"

using d(2)

unfolding dist_nz[symmetric]

apply (auto simp add: dist_commute)

apply (erule_tac x=xa in ballE)

apply auto

using `e > 0`

apply auto

done

}

then have "∃d>0. f ` (ball x d ∩ s) ⊆ ball (f x) e"

using `d > 0`

unfolding subset_eq ball_def by (auto simp add: dist_commute)

}

then show ?rhs by auto

next

assume ?rhs

then show ?lhs

unfolding continuous_within Lim_within ball_def subset_eq

apply (auto simp add: dist_commute)

apply (erule_tac x=e in allE)

apply auto

done

qed

lemma continuous_at_ball:

"continuous (at x) f <-> (∀e>0. ∃d>0. f ` (ball x d) ⊆ ball (f x) e)" (is "?lhs = ?rhs")

proof

assume ?lhs

then show ?rhs

unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

apply auto

apply (erule_tac x=e in allE)

apply auto

apply (rule_tac x=d in exI)

apply auto

apply (erule_tac x=xa in allE)

apply (auto simp add: dist_commute dist_nz)

unfolding dist_nz[symmetric]

apply auto

done

next

assume ?rhs

then show ?lhs

unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

apply auto

apply (erule_tac x=e in allE)

apply auto

apply (rule_tac x=d in exI)

apply auto

apply (erule_tac x="f xa" in allE)

apply (auto simp add: dist_commute dist_nz)

done

qed

text{* Define setwise continuity in terms of limits within the set. *}

lemma continuous_on_iff:

"continuous_on s f <->

(∀x∈s. ∀e>0. ∃d>0. ∀x'∈s. dist x' x < d --> dist (f x') (f x) < e)"

unfolding continuous_on_def Lim_within

apply (intro ball_cong [OF refl] all_cong ex_cong)

apply (rename_tac y, case_tac "y = x")

apply simp

apply (simp add: dist_nz)

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"

by simp

lemmas continuous_on = continuous_on_def -- "legacy theorem name"

lemma continuous_within_subset:

"continuous (at x within s) f ==> t ⊆ s ==> continuous (at x within t) f"

unfolding continuous_within by(metis tendsto_within_subset)

lemma continuous_on_interior:

"continuous_on s f ==> x ∈ interior s ==> continuous (at x) f"

apply (erule interiorE)

apply (drule (1) continuous_on_subset)

apply (simp add: continuous_on_eq_continuous_at)

done

lemma continuous_on_eq:

"(∀x ∈ s. f x = g x) ==> continuous_on s f ==> continuous_on s g"

unfolding continuous_on_def tendsto_def eventually_at_topological

by simp

text {* Characterization of various kinds of continuity in terms of sequences. *}

lemma continuous_within_sequentially:

fixes f :: "'a::metric_space => 'b::topological_space"

shows "continuous (at a within s) f <->

(∀x. (∀n::nat. x n ∈ s) ∧ (x ---> a) sequentially

--> ((f o x) ---> f a) sequentially)"

(is "?lhs = ?rhs")

proof

assume ?lhs

{

fix x :: "nat => 'a"

assume x: "∀n. x n ∈ s" "∀e>0. eventually (λn. dist (x n) a < e) sequentially"

fix T :: "'b set"

assume "open T" and "f a ∈ T"

with `?lhs` obtain d where "d>0" and d:"∀x∈s. 0 < dist x a ∧ dist x a < d --> f x ∈ T"

unfolding continuous_within tendsto_def eventually_at by (auto simp: dist_nz)

have "eventually (λn. dist (x n) a < d) sequentially"

using x(2) `d>0` by simp

then have "eventually (λn. (f o x) n ∈ T) sequentially"

proof eventually_elim

case (elim n)

then show ?case

using d x(1) `f a ∈ T` unfolding dist_nz[symmetric] by auto

qed

}

then show ?rhs

unfolding tendsto_iff tendsto_def by simp

next

assume ?rhs

then show ?lhs

unfolding continuous_within tendsto_def [where l="f a"]

by (simp add: sequentially_imp_eventually_within)

qed

lemma continuous_at_sequentially:

fixes f :: "'a::metric_space => 'b::topological_space"

shows "continuous (at a) f <->

(∀x. (x ---> a) sequentially --> ((f o x) ---> f a) sequentially)"

using continuous_within_sequentially[of a UNIV f] by simp

lemma continuous_on_sequentially:

fixes f :: "'a::metric_space => 'b::topological_space"

shows "continuous_on s f <->

(∀x. ∀a ∈ s. (∀n. x(n) ∈ s) ∧ (x ---> a) sequentially

--> ((f o x) ---> f a) sequentially)"

(is "?lhs = ?rhs")

proof

assume ?rhs

then show ?lhs

using continuous_within_sequentially[of _ s f]

unfolding continuous_on_eq_continuous_within

by auto

next

assume ?lhs

then show ?rhs

unfolding continuous_on_eq_continuous_within

using continuous_within_sequentially[of _ s f]

by auto

qed

lemma uniformly_continuous_on_sequentially:

"uniformly_continuous_on s f <-> (∀x y. (∀n. x n ∈ s) ∧ (∀n. y n ∈ s) ∧

((λn. dist (x n) (y n)) ---> 0) sequentially

--> ((λn. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")

proof

assume ?lhs

{

fix x y

assume x: "∀n. x n ∈ s"

and y: "∀n. y n ∈ s"

and xy: "((λn. dist (x n) (y n)) ---> 0) sequentially"

{

fix e :: real

assume "e > 0"

then obtain d where "d > 0" and d: "∀x∈s. ∀x'∈s. dist x' x < d --> dist (f x') (f x) < e"

using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto

obtain N where N: "∀n≥N. dist (x n) (y n) < d"

using xy[unfolded LIMSEQ_def dist_norm] and `d>0` by auto

{

fix n

assume "n≥N"

then have "dist (f (x n)) (f (y n)) < e"

using N[THEN spec[where x=n]]

using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]]

using x and y

unfolding dist_commute

by simp

}

then have "∃N. ∀n≥N. dist (f (x n)) (f (y n)) < e"

by auto

}

then have "((λn. dist (f(x n)) (f(y n))) ---> 0) sequentially"

unfolding LIMSEQ_def and dist_real_def by auto

}

then show ?rhs by auto

next

assume ?rhs

{

assume "¬ ?lhs"

then obtain e where "e > 0" "∀d>0. ∃x∈s. ∃x'∈s. dist x' x < d ∧ ¬ dist (f x') (f x) < e"

unfolding uniformly_continuous_on_def by auto

then obtain fa where fa:

"∀x. 0 < x --> fst (fa x) ∈ s ∧ snd (fa x) ∈ s ∧ dist (fst (fa x)) (snd (fa x)) < x ∧ ¬ dist (f (fst (fa x))) (f (snd (fa x))) < e"

using choice[of "λd x. d>0 --> fst x ∈ s ∧ snd x ∈ s ∧ dist (snd x) (fst x) < d ∧ ¬ dist (f (snd x)) (f (fst x)) < e"]

unfolding Bex_def

by (auto simp add: dist_commute)

def x ≡ "λn::nat. fst (fa (inverse (real n + 1)))"

def y ≡ "λn::nat. snd (fa (inverse (real n + 1)))"

have xyn: "∀n. x n ∈ s ∧ y n ∈ s"

and xy0: "∀n. dist (x n) (y n) < inverse (real n + 1)"

and fxy:"∀n. ¬ dist (f (x n)) (f (y n)) < e"

unfolding x_def and y_def using fa

by auto

{

fix e :: real

assume "e > 0"

then obtain N :: nat where "N ≠ 0" and N: "0 < inverse (real N) ∧ inverse (real N) < e"

unfolding real_arch_inv[of e] by auto

{

fix n :: nat

assume "n ≥ N"

then have "inverse (real n + 1) < inverse (real N)"

using real_of_nat_ge_zero and `N≠0` by auto

also have "… < e" using N by auto

finally have "inverse (real n + 1) < e" by auto

then have "dist (x n) (y n) < e"

using xy0[THEN spec[where x=n]] by auto

}

then have "∃N. ∀n≥N. dist (x n) (y n) < e" by auto

}

then have "∀e>0. ∃N. ∀n≥N. dist (f (x n)) (f (y n)) < e"

using `?rhs`[THEN spec[where x=x], THEN spec[where x=y]] and xyn

unfolding LIMSEQ_def dist_real_def by auto

then have False using fxy and `e>0` by auto

}

then show ?lhs

unfolding uniformly_continuous_on_def by blast

qed

text{* The usual transformation theorems. *}

lemma continuous_transform_within:

fixes f g :: "'a::metric_space => 'b::topological_space"

assumes "0 < d"

and "x ∈ s"

and "∀x' ∈ s. dist x' x < d --> f x' = g x'"

and "continuous (at x within s) f"

shows "continuous (at x within s) g"

unfolding continuous_within

proof (rule Lim_transform_within)

show "0 < d" by fact

show "∀x'∈s. 0 < dist x' x ∧ dist x' x < d --> f x' = g x'"

using assms(3) by auto

have "f x = g x"

using assms(1,2,3) by auto

then show "(f ---> g x) (at x within s)"

using assms(4) unfolding continuous_within by simp

qed

lemma continuous_transform_at:

fixes f g :: "'a::metric_space => 'b::topological_space"

assumes "0 < d"

and "∀x'. dist x' x < d --> f x' = g x'"

and "continuous (at x) f"

shows "continuous (at x) g"

using continuous_transform_within [of d x UNIV f g] assms by simp

subsubsection {* Structural rules for pointwise continuity *}

lemmas continuous_within_id = continuous_ident

lemmas continuous_at_id = isCont_ident

lemma continuous_infdist[continuous_intros]:

assumes "continuous F f"

shows "continuous F (λx. infdist (f x) A)"

using assms unfolding continuous_def by (rule tendsto_infdist)

lemma continuous_infnorm[continuous_intros]:

"continuous F f ==> continuous F (λx. infnorm (f x))"

unfolding continuous_def by (rule tendsto_infnorm)

lemma continuous_inner[continuous_intros]:

assumes "continuous F f"

and "continuous F g"

shows "continuous F (λx. inner (f x) (g x))"

using assms unfolding continuous_def by (rule tendsto_inner)

lemmas continuous_at_inverse = isCont_inverse

subsubsection {* Structural rules for setwise continuity *}

lemma continuous_on_infnorm[continuous_on_intros]:

"continuous_on s f ==> continuous_on s (λx. infnorm (f x))"

unfolding continuous_on by (fast intro: tendsto_infnorm)

lemma continuous_on_inner[continuous_on_intros]:

fixes g :: "'a::topological_space => 'b::real_inner"

assumes "continuous_on s f"

and "continuous_on s g"

shows "continuous_on s (λx. inner (f x) (g x))"

using bounded_bilinear_inner assms

by (rule bounded_bilinear.continuous_on)

subsubsection {* Structural rules for uniform continuity *}

lemma uniformly_continuous_on_id[continuous_on_intros]:

"uniformly_continuous_on s (λx. x)"

unfolding uniformly_continuous_on_def by auto

lemma uniformly_continuous_on_const[continuous_on_intros]:

"uniformly_continuous_on s (λx. c)"

unfolding uniformly_continuous_on_def by simp

lemma uniformly_continuous_on_dist[continuous_on_intros]:

fixes f g :: "'a::metric_space => 'b::metric_space"

assumes "uniformly_continuous_on s f"

and "uniformly_continuous_on s g"

shows "uniformly_continuous_on s (λx. dist (f x) (g x))"

proof -

{

fix a b c d :: 'b

have "¦dist a b - dist c d¦ ≤ dist a c + dist b d"

using dist_triangle2 [of a b c] dist_triangle2 [of b c d]

using dist_triangle3 [of c d a] dist_triangle [of a d b]

by arith

} note le = this

{

fix x y

assume f: "(λn. dist (f (x n)) (f (y n))) ----> 0"

assume g: "(λn. dist (g (x n)) (g (y n))) ----> 0"

have "(λn. ¦dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))¦) ----> 0"

by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],

simp add: le)

}

then show ?thesis

using assms unfolding uniformly_continuous_on_sequentially

unfolding dist_real_def by simp

qed

lemma uniformly_continuous_on_norm[continuous_on_intros]:

assumes "uniformly_continuous_on s f"

shows "uniformly_continuous_on s (λx. norm (f x))"

unfolding norm_conv_dist using assms

by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

lemma (in bounded_linear) uniformly_continuous_on[continuous_on_intros]:

assumes "uniformly_continuous_on s g"

shows "uniformly_continuous_on s (λx. f (g x))"

using assms unfolding uniformly_continuous_on_sequentially

unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]

by (auto intro: tendsto_zero)

lemma uniformly_continuous_on_cmul[continuous_on_intros]:

fixes f :: "'a::metric_space => 'b::real_normed_vector"

assumes "uniformly_continuous_on s f"

shows "uniformly_continuous_on s (λx. c *⇩_{R}f(x))"

using bounded_linear_scaleR_right assms

by (rule bounded_linear.uniformly_continuous_on)

lemma dist_minus:

fixes x y :: "'a::real_normed_vector"

shows "dist (- x) (- y) = dist x y"

unfolding dist_norm minus_diff_minus norm_minus_cancel ..

lemma uniformly_continuous_on_minus[continuous_on_intros]:

fixes f :: "'a::metric_space => 'b::real_normed_vector"

shows "uniformly_continuous_on s f ==> uniformly_continuous_on s (λx. - f x)"

unfolding uniformly_continuous_on_def dist_minus .

lemma uniformly_continuous_on_add[continuous_on_intros]:

fixes f g :: "'a::metric_space => 'b::real_normed_vector"

assumes "uniformly_continuous_on s f"

and "uniformly_continuous_on s g"

shows "uniformly_continuous_on s (λx. f x + g x)"

using assms

unfolding uniformly_continuous_on_sequentially

unfolding dist_norm tendsto_norm_zero_iff add_diff_add

by (auto intro: tendsto_add_zero)

lemma uniformly_continuous_on_diff[continuous_on_intros]:

fixes f :: "'a::metric_space => 'b::real_normed_vector"

assumes "uniformly_continuous_on s f"

and "uniformly_continuous_on s g"

shows "uniformly_continuous_on s (λx. f x - g x)"

unfolding ab_diff_minus using assms

by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)

text{* Continuity of all kinds is preserved under composition. *}

lemmas continuous_at_compose = isCont_o

lemma uniformly_continuous_on_compose[continuous_on_intros]:

assumes "uniformly_continuous_on s f" "uniformly_continuous_on (f ` s) g"

shows "uniformly_continuous_on s (g o f)"

proof -

{

fix e :: real

assume "e > 0"

then obtain d where "d > 0"

and d: "∀x∈f ` s. ∀x'∈f ` s. dist x' x < d --> dist (g x') (g x) < e"

using assms(2) unfolding uniformly_continuous_on_def by auto

obtain d' where "d'>0" "∀x∈s. ∀x'∈s. dist x' x < d' --> dist (f x') (f x) < d"

using `d > 0` using assms(1) unfolding uniformly_continuous_on_def by auto

then have "∃d>0. ∀x∈s. ∀x'∈s. dist x' x < d --> dist ((g o f) x') ((g o f) x) < e"

using `d>0` using d by auto

}

then show ?thesis

using assms unfolding uniformly_continuous_on_def by auto

qed

text{* Continuity in terms of open preimages. *}

lemma continuous_at_open:

"continuous (at x) f <-> (∀t. open t ∧ f x ∈ t --> (∃s. open s ∧ x ∈ s ∧ (∀x' ∈ s. (f x') ∈ t)))"

unfolding continuous_within_topological [of x UNIV f]

unfolding imp_conjL

by (intro all_cong imp_cong ex_cong conj_cong refl) auto

lemma continuous_imp_tendsto:

assumes "continuous (at x0) f"

and "x ----> x0"

shows "(f o x) ----> (f x0)"

proof (rule topological_tendstoI)

fix S

assume "open S" "f x0 ∈ S"

then obtain T where T_def: "open T" "x0 ∈ T" "∀x∈T. f x ∈ S"

using assms continuous_at_open by metis

then have "eventually (λn. x n ∈ T) sequentially"

using assms T_def by (auto simp: tendsto_def)

then show "eventually (λn. (f o x) n ∈ S) sequentially"

using T_def by (auto elim!: eventually_elim1)

qed

lemma continuous_on_open:

"continuous_on s f <->

(∀t. openin (subtopology euclidean (f ` s)) t -->

openin (subtopology euclidean s) {x ∈ s. f x ∈ t})"

unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute

by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

text {* Similarly in terms of closed sets. *}

lemma continuous_on_closed:

"continuous_on s f <->

(∀t. closedin (subtopology euclidean (f ` s)) t -->

closedin (subtopology euclidean s) {x ∈ s. f x ∈ t})"

unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute

by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

text {* Half-global and completely global cases. *}

lemma continuous_open_in_preimage:

assumes "continuous_on s f" "open t"

shows "openin (subtopology euclidean s) {x ∈ s. f x ∈ t}"

proof -

have *: "∀x. x ∈ s ∧ f x ∈ t <-> x ∈ s ∧ f x ∈ (t ∩ f ` s)"

by auto

have "openin (subtopology euclidean (f ` s)) (t ∩ f ` s)"

using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto

then show ?thesis

using assms(1)[unfolded continuous_on_open, THEN spec[where x="t ∩ f ` s"]]

using * by auto

qed

lemma continuous_closed_in_preimage:

assumes "continuous_on s f" and "closed t"

shows "closedin (subtopology euclidean s) {x ∈ s. f x ∈ t}"

proof -

have *: "∀x. x ∈ s ∧ f x ∈ t <-> x ∈ s ∧ f x ∈ (t ∩ f ` s)"

by auto

have "closedin (subtopology euclidean (f ` s)) (t ∩ f ` s)"

using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute

by auto

then show ?thesis

using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t ∩ f ` s"]]

using * by auto

qed

lemma continuous_open_preimage:

assumes "continuous_on s f"

and "open s"

and "open t"

shows "open {x ∈ s. f x ∈ t}"

proof-

obtain T where T: "open T" "{x ∈ s. f x ∈ t} = s ∩ T"

using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto

then show ?thesis

using open_Int[of s T, OF assms(2)] by auto

qed

lemma continuous_closed_preimage:

assumes "continuous_on s f"

and "closed s"

and "closed t"

shows "closed {x ∈ s. f x ∈ t}"

proof-

obtain T where "closed T" "{x ∈ s. f x ∈ t} = s ∩ T"

using continuous_closed_in_preimage[OF assms(1,3)]

unfolding closedin_closed by auto

then show ?thesis using closed_Int[of s T, OF assms(2)] by auto

qed

lemma continuous_open_preimage_univ:

"∀x. continuous (at x) f ==> open s ==> open {x. f x ∈ s}"

using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

lemma continuous_closed_preimage_univ:

"(∀x. continuous (at x) f) ==> closed s ==> closed {x. f x ∈ s}"

using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

lemma continuous_open_vimage: "∀x. continuous (at x) f ==> open s ==> open (f -` s)"

unfolding vimage_def by (rule continuous_open_preimage_univ)

lemma continuous_closed_vimage: "∀x. continuous (at x) f ==> closed s ==> closed (f -` s)"

unfolding vimage_def by (rule continuous_closed_preimage_univ)

lemma interior_image_subset:

assumes "∀x. continuous (at x) f"

and "inj f"

shows "interior (f ` s) ⊆ f ` (interior s)"

proof

fix x assume "x ∈ interior (f ` s)"

then obtain T where as: "open T" "x ∈ T" "T ⊆ f ` s" ..

then have "x ∈ f ` s" by auto

then obtain y where y: "y ∈ s" "x = f y" by auto

have "open (vimage f T)"

using assms(1) `open T` by (rule continuous_open_vimage)

moreover have "y ∈ vimage f T"

using `x = f y` `x ∈ T` by simp

moreover have "vimage f T ⊆ s"

using `T ⊆ image f s` `inj f` unfolding inj_on_def subset_eq by auto

ultimately have "y ∈ interior s" ..

with `x = f y` show "x ∈ f ` interior s" ..

qed

text {* Equality of continuous functions on closure and related results. *}

lemma continuous_closed_in_preimage_constant:

fixes f :: "_ => 'b::t1_space"

shows "continuous_on s f ==> closedin (subtopology euclidean s) {x ∈ s. f x = a}"

using continuous_closed_in_preimage[of s f "{a}"] by auto

lemma continuous_closed_preimage_constant:

fixes f :: "_ => 'b::t1_space"

shows "continuous_on s f ==> closed s ==> closed {x ∈ s. f x = a}"

using continuous_closed_preimage[of s f "{a}"] by auto

lemma continuous_constant_on_closure:

fixes f :: "_ => 'b::t1_space"

assumes "continuous_on (closure s) f"

and "∀x ∈ s. f x = a"

shows "∀x ∈ (closure s). f x = a"

using continuous_closed_preimage_constant[of "closure s" f a]

assms closure_minimal[of s "{x ∈ closure s. f x = a}"] closure_subset

unfolding subset_eq

by auto

lemma image_closure_subset:

assumes "continuous_on (closure s) f"

and "closed t"

and "(f ` s) ⊆ t"

shows "f ` (closure s) ⊆ t"

proof -

have "s ⊆ {x ∈ closure s. f x ∈ t}"

using assms(3) closure_subset by auto

moreover have "closed {x ∈ closure s. f x ∈ t}"

using continuous_closed_preimage[OF assms(1)] and assms(2) by auto

ultimately have "closure s = {x ∈ closure s . f x ∈ t}"

using closure_minimal[of s "{x ∈ closure s. f x ∈ t}"] by auto

then show ?thesis by auto

qed

lemma continuous_on_closure_norm_le:

fixes f :: "'a::metric_space => 'b::real_normed_vector"

assumes "continuous_on (closure s) f"

and "∀y ∈ s. norm(f y) ≤ b"

and "x ∈ (closure s)"

shows "norm (f x) ≤ b"

proof -

have *: "f ` s ⊆ cball 0 b"

using assms(2)[unfolded mem_cball_0[symmetric]] by auto

show ?thesis

using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)

unfolding subset_eq

apply (erule_tac x="f x" in ballE)

apply (auto simp add: dist_norm)

done

qed

text {* Making a continuous function avoid some value in a neighbourhood. *}

lemma continuous_within_avoid:

fixes f :: "'a::metric_space => 'b::t1_space"

assumes "continuous (at x within s) f"

and "f x ≠ a"

shows "∃e>0. ∀y ∈ s. dist x y < e --> f y ≠ a"

proof -

obtain U where "open U" and "f x ∈ U" and "a ∉ U"

using t1_space [OF `f x ≠ a`] by fast

have "(f ---> f x) (at x within s)"

using assms(1) by (simp add: continuous_within)

then have "eventually (λy. f y ∈ U) (at x within s)"

using `open U` and `f x ∈ U`

unfolding tendsto_def by fast

then have "eventually (λy. f y ≠ a) (at x within s)"

using `a ∉ U` by (fast elim: eventually_mono [rotated])

then show ?thesis

using `f x ≠ a` by (auto simp: dist_commute zero_less_dist_iff eventually_at)

qed

lemma continuous_at_avoid:

fixes f :: "'a::metric_space => 'b::t1_space"

assumes "continuous (at x) f"

and "f x ≠ a"

shows "∃e>0. ∀y. dist x y < e --> f y ≠ a"

using assms continuous_within_avoid[of x UNIV f a] by simp

lemma continuous_on_avoid:

fixes f :: "'a::metric_space => 'b::t1_space"

assumes "continuous_on s f"

and "x ∈ s"

and "f x ≠ a"

shows "∃e>0. ∀y ∈ s. dist x y < e --> f y ≠ a"

using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x],

OF assms(2)] continuous_within_avoid[of x s f a]

using assms(3)

by auto

lemma continuous_on_open_avoid:

fixes f :: "'a::metric_space => 'b::t1_space"

assumes "continuous_on s f"

and "open s"

and "x ∈ s"

and "f x ≠ a"

shows "∃e>0. ∀y. dist x y < e --> f y ≠ a"

using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]

using continuous_at_avoid[of x f a] assms(4)

by auto

text {* Proving a function is constant by proving open-ness of level set. *}

lemma continuous_levelset_open_in_cases:

fixes f :: "_ => 'b::t1_space"

shows "connected s ==> continuous_on s f ==>

openin (subtopology euclidean s) {x ∈ s. f x = a}

==> (∀x ∈ s. f x ≠ a) ∨ (∀x ∈ s. f x = a)"

unfolding connected_clopen

using continuous_closed_in_preimage_constant by auto

lemma continuous_levelset_open_in:

fixes f :: "_ => 'b::t1_space"

shows "connected s ==> continuous_on s f ==>

openin (subtopology euclidean s) {x ∈ s. f x = a} ==>

(∃x ∈ s. f x = a) ==> (∀x ∈ s. f x = a)"

using continuous_levelset_open_in_cases[of s f ]

by meson

lemma continuous_levelset_open:

fixes f :: "_ => 'b::t1_space"

assumes "connected s"

and "continuous_on s f"

and "open {x ∈ s. f x = a}"

and "∃x ∈ s. f x = a"

shows "∀x ∈ s. f x = a"

using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open]

using assms (3,4)

by fast

text {* Some arithmetical combinations (more to prove). *}

lemma open_scaling[intro]:

fixes s :: "'a::real_normed_vector set"

assumes "c ≠ 0"

and "open s"

shows "open((λx. c *⇩_{R}x) ` s)"

proof -

{

fix x

assume "x ∈ s"

then obtain e where "e>0"

and e:"∀x'. dist x' x < e --> x' ∈ s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]

by auto

have "e * abs c > 0"

using assms(1)[unfolded zero_less_abs_iff[symmetric]]

using mult_pos_pos[OF `e>0`]

by auto

moreover

{

fix y

assume "dist y (c *⇩_{R}x) < e * ¦c¦"

then have "norm ((1 / c) *⇩_{R}y - x) < e"

unfolding dist_norm

using norm_scaleR[of c "(1 / c) *⇩_{R}y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)

then have "y ∈ op *⇩_{R}c ` s"

using rev_image_eqI[of "(1 / c) *⇩_{R}y" s y "op *⇩_{R}c"]

using e[THEN spec[where x="(1 / c) *⇩_{R}y"]]

using assms(1)

unfolding dist_norm scaleR_scaleR

by auto

}

ultimately have "∃e>0. ∀x'. dist x' (c *⇩_{R}x) < e --> x' ∈ op *⇩_{R}c ` s"

apply (rule_tac x="e * abs c" in exI)

apply auto

done

}

then show ?thesis unfolding open_dist by auto

qed

lemma minus_image_eq_vimage:

fixes A :: "'a::ab_group_add set"

shows "(λx. - x) ` A = (λx. - x) -` A"

by (auto intro!: image_eqI [where f="λx. - x"])

lemma open_negations:

fixes s :: "'a::real_normed_vector set"

shows "open s ==> open ((λ x. -x) ` s)"

unfolding scaleR_minus1_left [symmetric]

by (rule open_scaling, auto)

lemma open_translation:

fixes s :: "'a::real_normed_vector set"

assumes "open s"

shows "open((λx. a + x) ` s)"

proof -

{

fix x

have "continuous (at x) (λx. x - a)"

by (intro continuous_diff continuous_at_id continuous_const)

}

moreover have "{x. x - a ∈ s} = op + a ` s"

by force

ultimately show ?thesis using continuous_open_preimage_univ[of "λx. x - a" s]

using assms by auto

qed

lemma open_affinity:

fixes s :: "'a::real_normed_vector set"

assumes "open s" "c ≠ 0"

shows "open ((λx. a + c *⇩_{R}x) ` s)"

proof -

have *: "(λx. a + c *⇩_{R}x) = (λx. a + x) o (λx. c *⇩_{R}x)"

unfolding o_def ..

have "op + a ` op *⇩_{R}c ` s = (op + a o op *⇩_{R}c) ` s"

by auto

then show ?thesis

using assms open_translation[of "op *⇩_{R}c ` s" a]

unfolding *

by auto

qed

lemma interior_translation:

fixes s :: "'a::real_normed_vector set"

shows "interior ((λx. a + x) ` s) = (λx. a + x) ` (interior s)"

proof (rule set_eqI, rule)

fix x

assume "x ∈ interior (op + a ` s)"

then obtain e where "e > 0" and e: "ball x e ⊆ op + a ` s"

unfolding mem_interior by auto

then have "ball (x - a) e ⊆ s"

unfolding subset_eq Ball_def mem_ball dist_norm

apply auto

apply (erule_tac x="a + xa" in allE)

unfolding ab_group_add_class.diff_diff_eq[symmetric]

apply auto

done

then show "x ∈ op + a ` interior s"

unfolding image_iff

apply (rule_tac x="x - a" in bexI)

unfolding mem_interior

using `e > 0`

apply auto

done

next

fix x

assume "x ∈ op + a ` interior s"

then obtain y e where "e > 0" and e: "ball y e ⊆ s" and y: "x = a + y"

unfolding image_iff Bex_def mem_interior by auto

{

fix z

have *: "a + y - z = y + a - z" by auto

assume "z ∈ ball x e"

then have "z - a ∈ s"

using e[unfolded subset_eq, THEN bspec[where x="z - a"]]

unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *

by auto

then have "z ∈ op + a ` s"

unfolding image_iff by (auto intro!: bexI[where x="z - a"])

}

then have "ball x e ⊆ op + a ` s"

unfolding subset_eq by auto

then show "x ∈ interior (op + a ` s)"

unfolding mem_interior using `e > 0` by auto

qed

text {* Topological properties of linear functions. *}

lemma linear_lim_0:

assumes "bounded_linear f"

shows "(f ---> 0) (at (0))"

proof -

interpret f: bounded_linear f by fact

have "(f ---> f 0) (at 0)"

using tendsto_ident_at by (rule f.tendsto)

then show ?thesis unfolding f.zero .

qed

lemma linear_continuous_at:

assumes "bounded_linear f"

shows "continuous (at a) f"

unfolding continuous_at using assms

apply (rule bounded_linear.tendsto)

apply (rule tendsto_ident_at)

done

lemma linear_continuous_within:

"bounded_linear f ==> continuous (at x within s) f"

using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

lemma linear_continuous_on:

"bounded_linear f ==> continuous_on s f"

using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

text {* Also bilinear functions, in composition form. *}

lemma bilinear_continuous_at_compose:

"continuous (at x) f ==> continuous (at x) g ==> bounded_bilinear h ==>

continuous (at x) (λx. h (f x) (g x))"

unfolding continuous_at

using Lim_bilinear[of f "f x" "(at x)" g "g x" h]

by auto

lemma bilinear_continuous_within_compose:

"continuous (at x within s) f ==> continuous (at x within s) g ==> bounded_bilinear h ==>

continuous (at x within s) (λx. h (f x) (g x))"

unfolding continuous_within

using Lim_bilinear[of f "f x"]

by auto

lemma bilinear_continuous_on_compose:

"continuous_on s f ==> continuous_on s g ==> bounded_bilinear h ==>

continuous_on s (λx. h (f x) (g x))"

unfolding continuous_on_def

by (fast elim: bounded_bilinear.tendsto)

text {* Preservation of compactness and connectedness under continuous function. *}

lemma compact_eq_openin_cover:

"compact S <->

(∀C. (∀c∈C. openin (subtopology euclidean S) c) ∧ S ⊆ \<Union>C -->

(∃D⊆C. finite D ∧ S ⊆ \<Union>D))"

proof safe

fix C

assume "compact S" and "∀c∈C. openin (subtopology euclidean S) c" and "S ⊆ \<Union>C"

then have "∀c∈{T. open T ∧ S ∩ T ∈ C}. open c" and "S ⊆ \<Union>{T. open T ∧ S ∩ T ∈ C}"

unfolding openin_open by force+

with `compact S` obtain D where "D ⊆ {T. open T ∧ S ∩ T ∈ C}" and "finite D" and "S ⊆ \<Union>D"

by (rule compactE)

then have "image (λT. S ∩ T) D ⊆ C ∧ finite (image (λT. S ∩ T) D) ∧ S ⊆ \<Union>(image (λT. S ∩ T) D)"

by auto

then show "∃D⊆C. finite D ∧ S ⊆ \<Union>D" ..

next

assume 1: "∀C. (∀c∈C. openin (subtopology euclidean S) c) ∧ S ⊆ \<Union>C -->

(∃D⊆C. finite D ∧ S ⊆ \<Union>D)"

show "compact S"

proof (rule compactI)

fix C

let ?C = "image (λT. S ∩ T) C"

assume "∀t∈C. open t" and "S ⊆ \<Union>C"

then have "(∀c∈?C. openin (subtopology euclidean S) c) ∧ S ⊆ \<Union>?C"

unfolding openin_open by auto

with 1 obtain D where "D ⊆ ?C" and "finite D" and "S ⊆ \<Union>D"

by metis

let ?D = "inv_into C (λT. S ∩ T) ` D"

have "?D ⊆ C ∧ finite ?D ∧ S ⊆ \<Union>?D"

proof (intro conjI)

from `D ⊆ ?C` show "?D ⊆ C"

by (fast intro: inv_into_into)

from `finite D` show "finite ?D"

by (rule finite_imageI)

from `S ⊆ \<Union>D` show "S ⊆ \<Union>?D"

apply (rule subset_trans)

apply clarsimp

apply (frule subsetD [OF `D ⊆ ?C`, THEN f_inv_into_f])

apply (erule rev_bexI, fast)

done

qed

then show "∃D⊆C. finite D ∧ S ⊆ \<Union>D" ..

qed

qed

lemma connected_continuous_image:

assumes "continuous_on s f"

and "connected s"

shows "connected(f ` s)"

proof -

{

fix T

assume as:

"T ≠ {}"

"T ≠ f ` s"

"openin (subtopology euclidean (f ` s)) T"

"closedin (subtopology euclidean (f ` s)) T"

have "{x ∈ s. f x ∈ T} = {} ∨ {x ∈ s. f x ∈ T} = s"

using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]

using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]

using assms(2)[unfolded connected_clopen, THEN spec[where x="{x ∈ s. f x ∈ T}"]] as(3,4) by auto

then have False using as(1,2)

using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto

}

then show ?thesis

unfolding connected_clopen by auto

qed

text {* Continuity implies uniform continuity on a compact domain. *}

lemma compact_uniformly_continuous:

assumes f: "continuous_on s f"

and s: "compact s"

shows "uniformly_continuous_on s f"

unfolding uniformly_continuous_on_def

proof (cases, safe)

fix e :: real

assume "0 < e" "s ≠ {}"

def [simp]: R ≡ "{(y, d). y ∈ s ∧ 0 < d ∧ ball y d ∩ s ⊆ {x ∈ s. f x ∈ ball (f y) (e/2) } }"

let ?b = "(λ(y, d). ball y (d/2))"

have "(∀r∈R. open (?b r))" "s ⊆ (\<Union>r∈R. ?b r)"

proof safe

fix y

assume "y ∈ s"

from continuous_open_in_preimage[OF f open_ball]

obtain T where "open T" and T: "{x ∈ s. f x ∈ ball (f y) (e/2)} = T ∩ s"

unfolding openin_subtopology open_openin by metis

then obtain d where "ball y d ⊆ T" "0 < d"

using `0 < e` `y ∈ s` by (auto elim!: openE)

with T `y ∈ s` show "y ∈ (\<Union>r∈R. ?b r)"

by (intro UN_I[of "(y, d)"]) auto

qed auto

with s obtain D where D: "finite D" "D ⊆ R" "s ⊆ (\<Union>(y, d)∈D. ball y (d/2))"

by (rule compactE_image)

with `s ≠ {}` have [simp]: "!!x. x < Min (snd ` D) <-> (∀(y, d)∈D. x < d)"

by (subst Min_gr_iff) auto

show "∃d>0. ∀x∈s. ∀x'∈s. dist x' x < d --> dist (f x') (f x) < e"

proof (rule, safe)

fix x x'

assume in_s: "x' ∈ s" "x ∈ s"

with D obtain y d where x: "x ∈ ball y (d/2)" "(y, d) ∈ D"

by blast

moreover assume "dist x x' < Min (snd`D) / 2"

ultimately have "dist y x' < d"

by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute)

with D x in_s show "dist (f x) (f x') < e"

by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq)

qed (insert D, auto)

qed auto

text {* A uniformly convergent limit of continuous functions is continuous. *}

lemma continuous_uniform_limit:

fixes f :: "'a => 'b::metric_space => 'c::metric_space"

assumes "¬ trivial_limit F"

and "eventually (λn. continuous_on s (f n)) F"

and "∀e>0. eventually (λn. ∀x∈s. dist (f n x) (g x) < e) F"

shows "continuous_on s g"

proof -

{

fix x and e :: real

assume "x∈s" "e>0"

have "eventually (λn. ∀x∈s. dist (f n x) (g x) < e / 3) F"

using `e>0` assms(3)[THEN spec[where x="e/3"]] by auto

from eventually_happens [OF eventually_conj [OF this assms(2)]]

obtain n where n:"∀x∈s. dist (f n x) (g x) < e / 3" "continuous_on s (f n)"

using assms(1) by blast

have "e / 3 > 0" using `e>0` by auto

then obtain d where "d>0" and d:"∀x'∈s. dist x' x < d --> dist (f n x') (f n x) < e / 3"

using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x∈s`, THEN spec[where x="e/3"]] by blast

{

fix y

assume "y ∈ s" and "dist y x < d"

then have "dist (f n y) (f n x) < e / 3"

by (rule d [rule_format])

then have "dist (f n y) (g x) < 2 * e / 3"

using dist_triangle [of "f n y" "g x" "f n x"]

using n(1)[THEN bspec[where x=x], OF `x∈s`]

by auto

then have "dist (g y) (g x) < e"

using n(1)[THEN bspec[where x=y], OF `y∈s`]

using dist_triangle3 [of "g y" "g x" "f n y"]

by auto

}

then have "∃d>0. ∀x'∈s. dist x' x < d --> dist (g x') (g x) < e"

using `d>0` by auto

}

then show ?thesis

unfolding continuous_on_iff by auto

qed

subsection {* Topological stuff lifted from and dropped to R *}

lemma open_real:

fixes s :: "real set"

shows "open s <-> (∀x ∈ s. ∃e>0. ∀x'. abs(x' - x) < e --> x' ∈ s)"

unfolding open_dist dist_norm by simp

lemma islimpt_approachable_real:

fixes s :: "real set"

shows "x islimpt s <-> (∀e>0. ∃x'∈ s. x' ≠ x ∧ abs(x' - x) < e)"

unfolding islimpt_approachable dist_norm by simp

lemma closed_real:

fixes s :: "real set"

shows "closed s <-> (∀x. (∀e>0. ∃x' ∈ s. x' ≠ x ∧ abs(x' - x) < e) --> x ∈ s)"

unfolding closed_limpt islimpt_approachable dist_norm by simp

lemma continuous_at_real_range:

fixes f :: "'a::real_normed_vector => real"

shows "continuous (at x) f <-> (∀e>0. ∃d>0. ∀x'. norm(x' - x) < d --> abs(f x' - f x) < e)"

unfolding continuous_at

unfolding Lim_at

unfolding dist_nz[symmetric]

unfolding dist_norm

apply auto

apply (erule_tac x=e in allE)

apply auto

apply (rule_tac x=d in exI)

apply auto

apply (erule_tac x=x' in allE)

apply auto

apply (erule_tac x=e in allE)

apply auto

done

lemma continuous_on_real_range:

fixes f :: "'a::real_normed_vector => real"

shows "continuous_on s f <->

(∀x ∈ s. ∀e>0. ∃d>0. (∀x' ∈ s. norm(x' - x) < d --> abs(f x' - f x) < e))"

unfolding continuous_on_iff dist_norm by simp

text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

lemma distance_attains_sup:

assumes "compact s" "s ≠ {}"

shows "∃x∈s. ∀y∈s. dist a y ≤ dist a x"

proof (rule continuous_attains_sup [OF assms])

{

fix x

assume "x∈s"

have "(dist a ---> dist a x) (at x within s)"

by (intro tendsto_dist tendsto_const tendsto_ident_at)

}

then show "continuous_on s (dist a)"

unfolding continuous_on ..

qed

text {* For \emph{minimal} distance, we only need closure, not compactness. *}

lemma distance_attains_inf:

fixes a :: "'a::heine_borel"

assumes "closed s"

and "s ≠ {}"

shows "∃x∈s. ∀y∈s. dist a x ≤ dist a y"

proof -

from assms(2) obtain b where "b ∈ s" by auto

let ?B = "s ∩ cball a (dist b a)"

have "?B ≠ {}" using `b ∈ s`

by (auto simp add: dist_commute)

moreover have "continuous_on ?B (dist a)"

by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_at_id continuous_const)

moreover have "compact ?B"

by (intro closed_inter_compact `closed s` compact_cball)

ultimately obtain x where "x ∈ ?B" "∀y∈?B. dist a x ≤ dist a y"

by (metis continuous_attains_inf)

then show ?thesis by fastforce

qed

subsection {* Pasted sets *}

lemma bounded_Times:

assumes "bounded s" "bounded t"

shows "bounded (s × t)"

proof -

obtain x y a b where "∀z∈s. dist x z ≤ a" "∀z∈t. dist y z ≤ b"

using assms [unfolded bounded_def] by auto

then have "∀z∈s × t. dist (x, y) z ≤ sqrt (a⇧^{2}+ b⇧^{2})"

by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

qed

lemma mem_Times_iff: "x ∈ A × B <-> fst x ∈ A ∧ snd x ∈ B"

by (induct x) simp

lemma seq_compact_Times: "seq_compact s ==> seq_compact t ==> seq_compact (s × t)"

unfolding seq_compact_def

apply clarify

apply (drule_tac x="fst o f" in spec)

apply (drule mp, simp add: mem_Times_iff)

apply (clarify, rename_tac l1 r1)

apply (drule_tac x="snd o f o r1" in spec)

apply (drule mp, simp add: mem_Times_iff)

apply (clarify, rename_tac l2 r2)

apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

apply (rule_tac x="r1 o r2" in exI)

apply (rule conjI, simp add: subseq_def)

apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)

apply (drule (1) tendsto_Pair) back

apply (simp add: o_def)

done

lemma compact_Times:

assumes "compact s" "compact t"

shows "compact (s × t)"

proof (rule compactI)

fix C

assume C: "∀t∈C. open t" "s × t ⊆ \<Union>C"

have "∀x∈s. ∃a. open a ∧ x ∈ a ∧ (∃d⊆C. finite d ∧ a × t ⊆ \<Union>d)"

proof

fix x

assume "x ∈ s"

have "∀y∈t. ∃a b c. c ∈ C ∧ open a ∧ open b ∧ x ∈ a ∧ y ∈ b ∧ a × b ⊆ c" (is "∀y∈t. ?P y")

proof

fix y

assume "y ∈ t"

with `x ∈ s` C obtain c where "c ∈ C" "(x, y) ∈ c" "open c" by auto

then show "?P y" by (auto elim!: open_prod_elim)

qed

then obtain a b c where b: "!!y. y ∈ t ==> open (b y)"

and c: "!!y. y ∈ t ==> c y ∈ C ∧ open (a y) ∧ open (b y) ∧ x ∈ a y ∧ y ∈ b y ∧ a y × b y ⊆ c y"

by metis

then have "∀y∈t. open (b y)" "t ⊆ (\<Union>y∈t. b y)" by auto

from compactE_image[OF `compact t` this] obtain D where D: "D ⊆ t" "finite D" "t ⊆ (\<Union>y∈D. b y)"

by auto

moreover from D c have "(\<Inter>y∈D. a y) × t ⊆ (\<Union>y∈D. c y)"

by (fastforce simp: subset_eq)

ultimately show "∃a. open a ∧ x ∈ a ∧ (∃d⊆C. finite d ∧ a × t ⊆ \<Union>d)"

using c by (intro exI[of _ "c`D"] exI[of _ "\<Inter>(a`D)"] conjI) (auto intro!: open_INT)

qed

then obtain a d where a: "∀x∈s. open (a x)" "s ⊆ (\<Union>x∈s. a x)"

and d: "!!x. x ∈ s ==> d x ⊆ C ∧ finite (d x) ∧ a x × t ⊆ \<Union>d x"

unfolding subset_eq UN_iff by metis

moreover

from compactE_image[OF `compact s` a]

obtain e where e: "e ⊆ s" "finite e" and s: "s ⊆ (\<Union>x∈e. a x)"

by auto

moreover

{

from s have "s × t ⊆ (\<Union>x∈e. a x × t)"

by auto

also have "… ⊆ (\<Union>x∈e. \<Union>d x)"

using d `e ⊆ s` by (intro UN_mono) auto

finally have "s × t ⊆ (\<Union>x∈e. \<Union>d x)" .

}

ultimately show "∃C'⊆C. finite C' ∧ s × t ⊆ \<Union>C'"

by (intro exI[of _ "(\<Union>x∈e. d x)"]) (auto simp add: subset_eq)

qed

text{* Hence some useful properties follow quite easily. *}

lemma compact_scaling:

fixes s :: "'a::real_normed_vector set"

assumes "compact s"

shows "compact ((λx. c *⇩_{R}x) ` s)"

proof -

let ?f = "λx. scaleR c x"

have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)

show ?thesis

using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

using linear_continuous_at[OF *] assms

by auto

qed

lemma compact_negations:

fixes s :: "'a::real_normed_vector set"

assumes "compact s"

shows "compact ((λx. - x) ` s)"

using compact_scaling [OF assms, of "- 1"] by auto

lemma compact_sums:

fixes s t :: "'a::real_normed_vector set"

assumes "compact s"

and "compact t"

shows "compact {x + y | x y. x ∈ s ∧ y ∈ t}"

proof -

have *: "{x + y | x y. x ∈ s ∧ y ∈ t} = (λz. fst z + snd z) ` (s × t)"

apply auto

unfolding image_iff

apply (rule_tac x="(xa, y)" in bexI)

apply auto

done

have "continuous_on (s × t) (λz. fst z + snd z)"

unfolding continuous_on by (rule ballI) (intro tendsto_intros)

then show ?thesis

unfolding * using compact_continuous_image compact_Times [OF assms] by auto

qed

lemma compact_differences:

fixes s t :: "'a::real_normed_vector set"

assumes "compact s"

and "compact t"

shows "compact {x - y | x y. x ∈ s ∧ y ∈ t}"

proof-

have "{x - y | x y. x∈s ∧ y ∈ t} = {x + y | x y. x ∈ s ∧ y ∈ (uminus ` t)}"

apply auto

apply (rule_tac x= xa in exI)

apply auto

apply (rule_tac x=xa in exI)

apply auto

done

then show ?thesis

using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

qed

lemma compact_translation:

fixes s :: "'a::real_normed_vector set"

assumes "compact s"

shows "compact ((λx. a + x) ` s)"

proof -

have "{x + y |x y. x ∈ s ∧ y ∈ {a}} = (λx. a + x) ` s"

by auto

then show ?thesis

using compact_sums[OF assms compact_sing[of a]] by auto

qed

lemma compact_affinity:

fixes s :: "'a::real_normed_vector set"

assumes "compact s"

shows "compact ((λx. a + c *⇩_{R}x) ` s)"

proof -

have "op + a ` op *⇩_{R}c ` s = (λx. a + c *⇩_{R}x) ` s"

by auto

then show ?thesis

using compact_translation[OF compact_scaling[OF assms], of a c] by auto

qed

text {* Hence we get the following. *}

lemma compact_sup_maxdistance:

fixes s :: "'a::metric_space set"

assumes "compact s"

and "s ≠ {}"

shows "∃x∈s. ∃y∈s. ∀u∈s. ∀v∈s. dist u v ≤ dist x y"

proof -

have "compact (s × s)"

using `compact s` by (intro compact_Times)

moreover have "s × s ≠ {}"

using `s ≠ {}` by auto

moreover have "continuous_on (s × s) (λx. dist (fst x) (snd x))"

by (intro continuous_at_imp_continuous_on ballI continuous_intros)

ultimately show ?thesis

using continuous_attains_sup[of "s × s" "λx. dist (fst x) (snd x)"] by auto

qed

text {* We can state this in terms of diameter of a set. *}

definition "diameter s = (if s = {} then 0::real else Sup {dist x y | x y. x ∈ s ∧ y ∈ s})"

lemma diameter_bounded_bound:

fixes s :: "'a :: metric_space set"

assumes s: "bounded s" "x ∈ s" "y ∈ s"

shows "dist x y ≤ diameter s"

proof -

let ?D = "{dist x y |x y. x ∈ s ∧ y ∈ s}"

from s obtain z d where z: "!!x. x ∈ s ==> dist z x ≤ d"

unfolding bounded_def by auto

have "dist x y ≤ Sup ?D"

proof (rule cSup_upper, safe)

fix a b

assume "a ∈ s" "b ∈ s"

with z[of a] z[of b] dist_triangle[of a b z]

show "dist a b ≤ 2 * d"

by (simp add: dist_commute)

qed (insert s, auto)

with `x ∈ s` show ?thesis

by (auto simp add: diameter_def)

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 cSup_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"

and "∀d>0. d < diameter s --> (∃x∈s. ∃y∈s. dist x y > d)"

using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms

by auto

lemma diameter_compact_attained:

assumes "compact s"

and "s ≠ {}"

shows "∃x∈s. ∃y∈s. dist x y = diameter s"

proof -

have b: "bounded s" using assms(1)

by (rule compact_imp_bounded)

then obtain x y where xys: "x∈s" "y∈s"

and xy: "∀u∈s. ∀v∈s. dist u v ≤ dist x y"

using compact_sup_maxdistance[OF assms] by auto

then have "diameter s ≤ dist x y"

unfolding diameter_def

apply clarsimp

apply (rule cSup_least)

apply fast+

done

then show ?thesis

by (metis b diameter_bounded_bound order_antisym xys)

qed

text {* Related results with closure as the conclusion. *}

lemma closed_scaling:

fixes s :: "'a::real_normed_vector set"

assumes "closed s"

shows "closed ((λx. c *⇩_{R}x) ` s)"

proof (cases "c = 0")

case True then show ?thesis

by (auto simp add: image_constant_conv)

next

case False

from assms have "closed ((λx. inverse c *⇩_{R}x) -` s)"

by (simp add: continuous_closed_vimage)

also have "(λx. inverse c *⇩_{R}x) -` s = (λx. c *⇩_{R}x) ` s"

using `c ≠ 0` by (auto elim: image_eqI [rotated])

finally show ?thesis .

qed

lemma closed_negations:

fixes s :: "'a::real_normed_vector set"

assumes "closed s"

shows "closed ((λx. -x) ` s)"

using closed_scaling[OF assms, of "- 1"] by simp

lemma compact_closed_sums:

fixes s :: "'a::real_normed_vector set"

assumes "compact s" and "closed t"

shows "closed {x + y | x y. x ∈ s ∧ y ∈ t}"

proof -

let ?S = "{x + y |x y. x ∈ s ∧ y ∈ t}"

{

fix x l

assume as: "∀n. x n ∈ ?S" "(x ---> l) sequentially"

from as(1) obtain f where f: "∀n. x n = fst (f n) + snd (f n)" "∀n. fst (f n) ∈ s" "∀n. snd (f n) ∈ t"

using choice[of "λn y. x n = (fst y) + (snd y) ∧ fst y ∈ s ∧ snd y ∈ t"] by auto

obtain l' r where "l'∈s" and r: "subseq r" and lr: "(((λn. fst (f n)) o r) ---> l') sequentially"

using assms(1)[unfolded compact_def, THEN spec[where x="λ n. fst (f n)"]] using f(2) by auto

have "((λn. snd (f (r n))) ---> l - l') sequentially"

using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)

unfolding o_def

by auto

then have "l - l' ∈ t"

using assms(2)[unfolded closed_sequential_limits,

THEN spec[where x="λ n. snd (f (r n))"],

THEN spec[where x="l - l'"]]

using f(3)

by auto

then have "l ∈ ?S"

using `l' ∈ s`

apply auto

apply (rule_tac x=l' in exI)

apply (rule_tac x="l - l'" in exI)

apply auto

done

}

then show ?thesis

unfolding closed_sequential_limits by fast

qed

lemma closed_compact_sums:

fixes s t :: "'a::real_normed_vector set"

assumes "closed s"

and "compact t"

shows "closed {x + y | x y. x ∈ s ∧ y ∈ t}"

proof -

have "{x + y |x y. x ∈ t ∧ y ∈ s} = {x + y |x y. x ∈ s ∧ y ∈ t}"

apply auto

apply (rule_tac x=y in exI)

apply auto

apply (rule_tac x=y in exI)

apply auto

done

then show ?thesis

using compact_closed_sums[OF assms(2,1)] by simp

qed

lemma compact_closed_differences:

fixes s t :: "'a::real_normed_vector set"

assumes "compact s"

and "closed t"

shows "closed {x - y | x y. x ∈ s ∧ y ∈ t}"

proof -

have "{x + y |x y. x ∈ s ∧ y ∈ uminus ` t} = {x - y |x y. x ∈ s ∧ y ∈ t}"

apply auto

apply (rule_tac x=xa in exI)

apply auto

apply (rule_tac x=xa in exI)

apply auto

done

then show ?thesis

using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto

qed

lemma closed_compact_differences:

fixes s t :: "'a::real_normed_vector set"

assumes "closed s"

and "compact t"

shows "closed {x - y | x y. x ∈ s ∧ y ∈ t}"

proof -

have "{x + y |x y. x ∈ s ∧ y ∈ uminus ` t} = {x - y |x y. x ∈ s ∧ y ∈ t}"

apply auto

apply (rule_tac x=xa in exI)

apply auto

apply (rule_tac x=xa in exI)

apply auto

done

then show ?thesis

using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp

qed

lemma closed_translation:

fixes a :: "'a::real_normed_vector"

assumes "closed s"

shows "closed ((λx. a + x) ` s)"

proof -

have "{a + y |y. y ∈ s} = (op + a ` s)" by auto

then show ?thesis

using compact_closed_sums[OF compact_sing[of a] assms] by auto

qed

lemma translation_Compl:

fixes a :: "'a::ab_group_add"

shows "(λx. a + x) ` (- t) = - ((λx. a + x) ` t)"

apply (auto simp add: image_iff)

apply (rule_tac x="x - a" in bexI)

apply auto

done

lemma translation_UNIV:

fixes a :: "'a::ab_group_add"

shows "range (λx. a + x) = UNIV"

apply (auto simp add: image_iff)

apply (rule_tac x="x - a" in exI)

apply auto

done

lemma translation_diff:

fixes a :: "'a::ab_group_add"

shows "(λx. a + x) ` (s - t) = ((λx. a + x) ` s) - ((λx. a + x) ` t)"

by auto

lemma closure_translation:

fixes a :: "'a::real_normed_vector"

shows "closure ((λx. a + x) ` s) = (λx. a + x) ` (closure s)"

proof -

have *: "op + a ` (- s) = - op + a ` s"

apply auto

unfolding image_iff

apply (rule_tac x="x - a" in bexI)

apply auto

done

show ?thesis

unfolding closure_interior translation_Compl

using interior_translation[of a "- s"]

unfolding *

by auto

qed

lemma frontier_translation:

fixes a :: "'a::real_normed_vector"

shows "frontier((λx. a + x) ` s) = (λx. a + x) ` (frontier s)"

unfolding frontier_def translation_diff interior_translation closure_translation

by auto

subsection {* Separation between points and sets *}

lemma separate_point_closed:

fixes s :: "'a::heine_borel set"

assumes "closed s"

and "a ∉ s"

shows "∃d>0. ∀x∈s. d ≤ dist a x"

proof (cases "s = {}")

case True

then show ?thesis by(auto intro!: exI[where x=1])

next

case False

from assms obtain x where "x∈s" "∀y∈s. dist a x ≤ dist a y"

using `s ≠ {}` distance_attains_inf [of s a] by blast

with `x∈s` show ?thesis using dist_pos_lt[of a x] and`a ∉ s`

by blast

qed

lemma separate_compact_closed:

fixes s t :: "'a::heine_borel set"

assumes "compact s"

and t: "closed t" "s ∩ t = {}"

shows "∃d>0. ∀x∈s. ∀y∈t. d ≤ dist x y"

proof cases

assume "s ≠ {} ∧ t ≠ {}"

then have "s ≠ {}" "t ≠ {}" by auto

let ?inf = "λx. infdist x t"

have "continuous_on s ?inf"

by (auto intro!: continuous_at_imp_continuous_on continuous_infdist continuous_at_id)

then obtain x where x: "x ∈ s" "∀y∈s. ?inf x ≤ ?inf y"

using continuous_attains_inf[OF `compact s` `s ≠ {}`] by auto

then have "0 < ?inf x"

using t `t ≠ {}` in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)

moreover have "∀x'∈s. ∀y∈t. ?inf x ≤ dist x' y"

using x by (auto intro: order_trans infdist_le)

ultimately show ?thesis by auto

qed (auto intro!: exI[of _ 1])

lemma separate_closed_compact:

fixes s t :: "'a::heine_borel set"

assumes "closed s"

and "compact t"

and "s ∩ t = {}"

shows "∃d>0. ∀x∈s. ∀y∈t. d ≤ dist x y"

proof -

have *: "t ∩ s = {}"

using assms(3) by auto

show ?thesis

using separate_compact_closed[OF assms(2,1) *]

apply auto

apply (rule_tac x=d in exI)

apply auto

apply (erule_tac x=y in ballE)

apply (auto simp add: dist_commute)

done

qed

subsection {* Intervals *}

lemma interval:

fixes a :: "'a::ordered_euclidean_space"

shows "{a <..< b} = {x::'a. ∀i∈Basis. a•i < x•i ∧ x•i < b•i}"

and "{a .. b} = {x::'a. ∀i∈Basis. a•i ≤ x•i ∧ x•i ≤ b•i}"

by (auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

lemma mem_interval:

fixes a :: "'a::ordered_euclidean_space"

shows "x ∈ {a<..<b} <-> (∀i∈Basis. a•i < x•i ∧ x•i < b•i)"

and "x ∈ {a .. b} <-> (∀i∈Basis. a•i ≤ x•i ∧ x•i ≤ b•i)"

using interval[of a b]

by (auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

lemma interval_eq_empty:

fixes a :: "'a::ordered_euclidean_space"

shows "({a <..< b} = {} <-> (∃i∈Basis. b•i ≤ a•i))" (is ?th1)

and "({a .. b} = {} <-> (∃i∈Basis. b•i < a•i))" (is ?th2)

proof -

{

fix i x

assume i: "i∈Basis" and as:"b•i ≤ a•i" and x:"x∈{a <..< b}"

then have "a • i < x • i ∧ x • i < b • i"

unfolding mem_interval by auto

then have "a•i < b•i" by auto

then have False using as by auto

}

moreover

{

assume as: "∀i∈Basis. ¬ (b•i ≤ a•i)"

let ?x = "(1/2) *⇩_{R}(a + b)"

{

fix i :: 'a

assume i: "i ∈ Basis"

have "a•i < b•i"

using as[THEN bspec[where x=i]] i by auto

then have "a•i < ((1/2) *⇩_{R}(a+b)) • i" "((1/2) *⇩_{R}(a+b)) • i < b•i"

by (auto simp: inner_add_left)

}

then have "{a <..< b} ≠ {}"

using mem_interval(1)[of "?x" a b] by auto

}

ultimately show ?th1 by blast

{

fix i x

assume i: "i ∈ Basis" and as:"b•i < a•i" and x:"x∈{a .. b}"

then have "a • i ≤ x • i ∧ x • i ≤ b • i"

unfolding mem_interval by auto

then have "a•i ≤ b•i" by auto

then have False using as by auto

}

moreover

{

assume as:"∀i∈Basis. ¬ (b•i < a•i)"

let ?x = "(1/2) *⇩_{R}(a + b)"

{

fix i :: 'a

assume i:"i ∈ Basis"

have "a•i ≤ b•i"

using as[THEN bspec[where x=i]] i by auto

then have "a•i ≤ ((1/2) *⇩_{R}(a+b)) • i" "((1/2) *⇩_{R}(a+b)) • i ≤ b•i"

by (auto simp: inner_add_left)

}

then have "{a .. b} ≠ {}"

using mem_interval(2)[of "?x" a b] by auto

}

ultimately show ?th2 by 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<..<a} = {}"

unfolding set_eq_iff mem_interval eq_iff [symmetric]

by (auto intro: euclidean_eqI simp: ex_in_conv)

lemma subset_interval_imp:

fixes a :: "'a::ordered_euclidean_space"

shows "(∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i) ==> {c .. d} ⊆ {a .. b}"

and "(∀i∈Basis. a•i < c•i ∧ d•i < b•i) ==> {c .. d} ⊆ {a<..<b}"

and "(∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i) ==> {c<..<d} ⊆ {a .. b}"

and "(∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i) ==> {c<..<d} ⊆ {a<..<b}"

unfolding subset_eq[unfolded Ball_def] unfolding mem_interval

by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

lemma interval_open_subset_closed:

fixes a :: "'a::ordered_euclidean_space"

shows "{a<..<b} ⊆ {a .. b}"

unfolding subset_eq [unfolded Ball_def] mem_interval

by (fast intro: less_imp_le)

lemma subset_interval:

fixes a :: "'a::ordered_euclidean_space"

shows "{c .. d} ⊆ {a .. b} <-> (∀i∈Basis. c•i ≤ d•i) --> (∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i)" (is ?th1)

and "{c .. d} ⊆ {a<..<b} <-> (∀i∈Basis. c•i ≤ d•i) --> (∀i∈Basis. a•i < c•i ∧ d•i < b•i)" (is ?th2)

and "{c<..<d} ⊆ {a .. b} <-> (∀i∈Basis. c•i < d•i) --> (∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i)" (is ?th3)

and "{c<..<d} ⊆ {a<..<b} <-> (∀i∈Basis. c•i < d•i) --> (∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i)" (is ?th4)

proof -

show ?th1

unfolding subset_eq and Ball_def and mem_interval

by (auto intro: order_trans)

show ?th2

unfolding subset_eq and Ball_def and mem_interval

by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)

{

assume as: "{c<..<d} ⊆ {a .. b}" "∀i∈Basis. c•i < d•i"

then have "{c<..<d} ≠ {}"

unfolding interval_eq_empty by auto

fix i :: 'a

assume i: "i ∈ Basis"

(** TODO combine the following two parts as done in the HOL_light version. **)

{

let ?x = "(∑j∈Basis. (if j=i then ((min (a•j) (d•j))+c•j)/2 else (c•j+d•j)/2) *⇩_{R}j)::'a"

assume as2: "a•i > c•i"

{

fix j :: 'a

assume j: "j ∈ Basis"

then have "c • j < ?x • j ∧ ?x • j < d • j"

apply (cases "j = i")

using as(2)[THEN bspec[where x=j]] i

apply (auto simp add: as2)

done

}

then have "?x∈{c<..<d}"

using i unfolding mem_interval by auto

moreover

have "?x ∉ {a .. b}"

unfolding mem_interval

apply auto

apply (rule_tac x=i in bexI)

using as(2)[THEN bspec[where x=i]] and as2 i

apply auto

done

ultimately have False using as by auto

}

then have "a•i ≤ c•i" by (rule ccontr) auto

moreover

{

let ?x = "(∑j∈Basis. (if j=i then ((max (b•j) (c•j))+d•j)/2 else (c•j+d•j)/2) *⇩_{R}j)::'a"

assume as2: "b•i < d•i"

{

fix j :: 'a

assume "j∈Basis"

then have "d • j > ?x • j ∧ ?x • j > c • j"

apply (cases "j = i")

using as(2)[THEN bspec[where x=j]]

apply (auto simp add: as2)

done

}

then have "?x∈{c<..<d}"

unfolding mem_interval by auto

moreover

have "?x∉{a .. b}"

unfolding mem_interval

apply auto

apply (rule_tac x=i in bexI)

using as(2)[THEN bspec[where x=i]] and as2 using i

apply auto

done

ultimately have False using as by auto

}

then have "b•i ≥ d•i" by (rule ccontr) auto

ultimately

have "a•i ≤ c•i ∧ d•i ≤ b•i" by auto

} note part1 = this

show ?th3

unfolding subset_eq and Ball_def and mem_interval

apply (rule, rule, rule, rule)

apply (rule part1)

unfolding subset_eq and Ball_def and mem_interval

prefer 4

apply auto

apply (erule_tac x=xa in allE, erule_tac x=xa in allE, fastforce)+

done

{

assume as: "{c<..<d} ⊆ {a<..<b}" "∀i∈Basis. c•i < d•i"

fix i :: 'a

assume i:"i∈Basis"

from as(1) have "{c<..<d} ⊆ {a..b}"

using interval_open_subset_closed[of a b] by auto

then have "a•i ≤ c•i ∧ d•i ≤ b•i"

using part1 and as(2) using i by auto

} note * = this

show ?th4

unfolding subset_eq and Ball_def and mem_interval

apply (rule, rule, rule, rule)

apply (rule *)

unfolding subset_eq and Ball_def and mem_interval

prefer 4

apply auto

apply (erule_tac x=xa in allE, simp)+

done

qed

lemma inter_interval:

fixes a :: "'a::ordered_euclidean_space"

shows "{a .. b} ∩ {c .. d} =

{(∑i∈Basis. max (a•i) (c•i) *⇩_{R}i) .. (∑i∈Basis. min (b•i) (d•i) *⇩_{R}i)}"

unfolding set_eq_iff and Int_iff and mem_interval

by 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<..<d} = {} <-> (∃i∈Basis. (b•i < a•i ∨ d•i ≤ c•i ∨ b•i ≤ c•i ∨ d•i ≤ a•i))" (is ?th2)

and "{a<..<b} ∩ {c .. d} = {} <-> (∃i∈Basis. (b•i ≤ a•i ∨ d•i < c•i ∨ b•i ≤ c•i ∨ d•i ≤ a•i))" (is ?th3)

and "{a<..<b} ∩ {c<..<d} = {} <-> (∃i∈Basis. (b•i ≤ a•i ∨ d•i ≤ c•i ∨ b•i ≤ c•i ∨ d•i ≤ a•i))" (is ?th4)

proof -

let ?z = "(∑i∈Basis. (((max (a•i) (c•i)) + (min (b•i) (d•i))) / 2) *⇩_{R}i)::'a"

have **: "!!P Q. (!!i :: 'a. i ∈ Basis ==> Q ?z i ==> P i) ==>

(!!i x :: 'a. i ∈ Basis ==> P i ==> Q x i) ==> (∀x. ∃i∈Basis. Q x i) <-> (∃i∈Basis. P i)"

by blast

note * = set_eq_iff Int_iff empty_iff mem_interval ball_conj_distrib[symmetric] eq_False ball_simps(10)

show ?th1 unfolding * by (intro **) auto

show ?th2 unfolding * by (intro **) auto

show ?th3 unfolding * by (intro **) auto

show ?th4 unfolding * by (intro **) auto

qed

(* Moved interval_open_subset_closed a bit upwards *)

lemma open_interval[intro]:

fixes a b :: "'a::ordered_euclidean_space"

shows "open {a<..<b}"

proof -

have "open (\<Inter>i∈Basis. (λx. x•i) -` {a•i<..<b•i})"

by (intro open_INT finite_lessThan ballI continuous_open_vimage allI

linear_continuous_at open_real_greaterThanLessThan finite_Basis bounded_linear_inner_left)

also have "(\<Inter>i∈Basis. (λx. x•i) -` {a•i<..<b•i}) = {a<..<b}"

by (auto simp add: eucl_less [where 'a='a])

finally show "open {a<..<b}" .

qed

lemma closed_interval[intro]:

fixes a b :: "'a::ordered_euclidean_space"

shows "closed {a .. b}"

proof -

have "closed (\<Inter>i∈Basis. (λx. x•i) -` {a•i .. b•i})"

by (intro closed_INT ballI continuous_closed_vimage allI

linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)

also have "(\<Inter>i∈Basis. (λx. x•i) -` {a•i .. b•i}) = {a .. b}"

by (auto simp add: eucl_le [where 'a='a])

finally show "closed {a .. b}" .

qed

lemma interior_closed_interval [intro]:

fixes a b :: "'a::ordered_euclidean_space"

shows "interior {a..b} = {a<..<b}" (is "?L = ?R")

proof(rule subset_antisym)

show "?R ⊆ ?L"

using interval_open_subset_closed open_interval

by (rule interior_maximal)

{

fix x

assume "x ∈ interior {a..b}"

then obtain s where s: "open s" "x ∈ s" "s ⊆ {a..b}" ..

then obtain e where "e>0" and e:"∀x'. dist x' x < e --> x' ∈ {a..b}"

unfolding open_dist and subset_eq by auto

{

fix i :: 'a

assume i: "i ∈ Basis"

have "dist (x - (e / 2) *⇩_{R}i) x < e"

and "dist (x + (e / 2) *⇩_{R}i) x < e"

unfolding dist_norm

apply auto

unfolding norm_minus_cancel

using norm_Basis[OF i] `e>0`

apply auto

done

then have "a • i ≤ (x - (e / 2) *⇩_{R}i) • i" and "(x + (e / 2) *⇩_{R}i) • i ≤ b • i"

using e[THEN spec[where x="x - (e/2) *⇩_{R}i"]]

and e[THEN spec[where x="x + (e/2) *⇩_{R}i"]]

unfolding mem_interval

using i

by blast+

then have "a • i < x • i" and "x • i < b • i"

using `e>0` i

by (auto simp: inner_diff_left inner_Basis inner_add_left)

}

then have "x ∈ {a<..<b}"

unfolding mem_interval by auto

}

then show "?L ⊆ ?R" ..

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"

then have "¦x•i¦ ≤ ¦a•i¦ + ¦b•i¦"

using x[THEN bspec[where x=i]] by auto

}

then have "(∑i∈Basis. ¦x • i¦) ≤ ?b"

apply -

apply (rule setsum_mono)

apply auto

done

then have "norm x ≤ ?b"

using norm_le_l1[of x] by auto

}

then show ?thesis

unfolding interval and bounded_iff by auto

qed

lemma bounded_interval:

fixes a :: "'a::ordered_euclidean_space"

shows "bounded {a .. b} ∧ bounded {a<..<b}"

using bounded_closed_interval[of a b]

using interval_open_subset_closed[of a b]

using bounded_subset[of "{a..b}" "{a<..<b}"]

by simp

lemma not_interval_univ:

fixes a :: "'a::ordered_euclidean_space"

shows "{a .. b} ≠ UNIV ∧ {a<..<b} ≠ 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<..<b} ≠ {}"

shows "((1/2) *⇩_{R}(a + b)) ∈ {a<..<b}"

proof -

{

fix i :: 'a

assume "i ∈ Basis"

then have "a • i < ((1 / 2) *⇩_{R}(a + b)) • i ∧ ((1 / 2) *⇩_{R}(a + b)) • i < b • i"

using assms[unfolded interval_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left)

}

then show ?thesis unfolding mem_interval by auto

qed

lemma open_closed_interval_convex:

fixes x :: "'a::ordered_euclidean_space"

assumes x: "x ∈ {a<..<b}"

and y: "y ∈ {a .. b}"

and e: "0 < e" "e ≤ 1"

shows "(e *⇩_{R}x + (1 - e) *⇩_{R}y) ∈ {a<..<b}"

proof -

{

fix i :: 'a

assume i: "i ∈ Basis"

have "a • i = e * (a • i) + (1 - e) * (a • i)"

unfolding left_diff_distrib by simp

also have "… < e * (x • i) + (1 - e) * (y • i)"

apply (rule add_less_le_mono)

using e unfolding mult_less_cancel_left and mult_le_cancel_left

apply simp_all

using x unfolding mem_interval using i

apply simp

using y unfolding mem_interval using i

apply simp

done

finally have "a • i < (e *⇩_{R}x + (1 - e) *⇩_{R}y) • i"

unfolding inner_simps by auto

moreover

{

have "b • i = e * (b•i) + (1 - e) * (b•i)"

unfolding left_diff_distrib by simp

also have "… > e * (x • i) + (1 - e) * (y • i)"

apply (rule add_less_le_mono)

using e unfolding mult_less_cancel_left and mult_le_cancel_left

apply simp_all

using x

unfolding mem_interval

using i

apply simp

using y

unfolding mem_interval

using i

apply simp

done

finally have "(e *⇩_{R}x + (1 - e) *⇩_{R}y) • i < b • i"

unfolding inner_simps by auto

}

ultimately have "a • i < (e *⇩_{R}x + (1 - e) *⇩_{R}y) • i ∧ (e *⇩_{R}x + (1 - e) *⇩_{R}y) • i < b • i"

by auto

}

then show ?thesis

unfolding mem_interval by auto

qed

lemma closure_open_interval:

fixes a :: "'a::ordered_euclidean_space"

assumes "{a<..<b} ≠ {}"

shows "closure {a<..<b} = {a .. b}"

proof -

have ab: "a < b"

using assms[unfolded interval_ne_empty]

apply (subst eucl_less)

apply auto

done

let ?c = "(1 / 2) *⇩_{R}(a + b)"

{

fix x

assume as:"x ∈ {a .. b}"

def f ≡ "λn::nat. x + (inverse (real n + 1)) *⇩_{R}(?c - x)"

{

fix n

assume fn: "f n < b --> a < f n --> f n = x" and xc: "x ≠ ?c"

have *: "0 < inverse (real n + 1)" "inverse (real n + 1) ≤ 1"

unfolding inverse_le_1_iff by auto

have "(inverse (real n + 1)) *⇩_{R}((1 / 2) *⇩_{R}(a + b)) + (1 - inverse (real n + 1)) *⇩_{R}x =

x + (inverse (real n + 1)) *⇩_{R}(((1 / 2) *⇩_{R}(a + b)) - x)"

by (auto simp add: algebra_simps)

then have "f n < b" and "a < f n"

using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *]

unfolding f_def by auto

then have False

using fn unfolding f_def using xc by auto

}

moreover

{

assume "¬ (f ---> x) sequentially"

{

fix e :: real

assume "e > 0"

then have "∃N::nat. inverse (real (N + 1)) < e"

using real_arch_inv[of e]

apply (auto simp add: Suc_pred')

apply (rule_tac x="n - 1" in exI)

apply auto

done

then obtain N :: nat where "inverse (real (N + 1)) < e"

by auto

then have "∀n≥N. inverse (real n + 1) < e"

apply auto

apply (metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans

real_of_nat_Suc real_of_nat_Suc_gt_zero)

done

then have "∃N::nat. ∀n≥N. inverse (real n + 1) < e" by auto

}

then have "((λn. inverse (real n + 1)) ---> 0) sequentially"

unfolding LIMSEQ_def by(auto simp add: dist_norm)

then have "(f ---> x) sequentially"

unfolding f_def

using tendsto_add[OF tendsto_const, of "λn::nat. (inverse (real n + 1)) *⇩_{R}((1 / 2) *⇩_{R}(a + b) - x)" 0 sequentially x]

using tendsto_scaleR [OF _ tendsto_const, of "λn::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *⇩_{R}(a + b) - x)"]

by auto

}

ultimately have "x ∈ closure {a<..<b}"

using as and open_interval_midpoint[OF assms]

unfolding closure_def

unfolding islimpt_sequential

by (cases "x=?c") auto

}

then show ?thesis

using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast

qed

lemma bounded_subset_open_interval_symmetric:

fixes s::"('a::ordered_euclidean_space) set"

assumes "bounded s"

shows "∃a. s ⊆ {-a<..<a}"

proof -

obtain b where "b>0" and b: "∀x∈s. norm x ≤ b"

using assms[unfolded bounded_pos] by auto

def a ≡ "(∑i∈Basis. (b + 1) *⇩_{R}i)::'a"

{

fix x

assume "x ∈ s"

fix i :: 'a

assume i: "i ∈ Basis"

then have "(-a)•i < x•i" and "x•i < a•i"

using b[THEN bspec[where x=x], OF `x∈s`]

using Basis_le_norm[OF i, of x]

unfolding inner_simps and a_def

by auto

}

then show ?thesis

by (auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])

qed

lemma bounded_subset_open_interval:

fixes s :: "('a::ordered_euclidean_space) set"

shows "bounded s ==> (∃a b. s ⊆ {a<..<b})"

by (auto dest!: bounded_subset_open_interval_symmetric)

lemma bounded_subset_closed_interval_symmetric:

fixes s :: "('a::ordered_euclidean_space) set"

assumes "bounded s"

shows "∃a. s ⊆ {-a .. a}"

proof -

obtain a where "s ⊆ {- a<..<a}"

using bounded_subset_open_interval_symmetric[OF assms] by auto

then show ?thesis

using interval_open_subset_closed[of "-a" a] by auto

qed

lemma bounded_subset_closed_interval:

fixes s :: "('a::ordered_euclidean_space) set"

shows "bounded s ==> ∃a b. s ⊆ {a .. b}"

using bounded_subset_closed_interval_symmetric[of s] by auto

lemma frontier_closed_interval:

fixes a b :: "'a::ordered_euclidean_space"

shows "frontier {a .. b} = {a .. b} - {a<..<b}"

unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..

lemma frontier_open_interval:

fixes a b :: "'a::ordered_euclidean_space"

shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"

proof (cases "{a<..<b} = {}")

case True

then show ?thesis

using frontier_empty by auto

next

case False

then show ?thesis

unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval]

by auto

qed

lemma inter_interval_mixed_eq_empty:

fixes a :: "'a::ordered_euclidean_space"

assumes "{c<..<d} ≠ {}"

shows "{a<..<b} ∩ {c .. d} = {} <-> {a<..<b} ∩ {c<..<d} = {}"

unfolding closure_open_interval[OF assms, symmetric]

unfolding open_inter_closure_eq_empty[OF open_interval] ..

lemma open_box: "open (box a b)"

proof -

have "open (\<Inter>i∈Basis. (op • i) -` {a • i <..< b • i})"

by (auto intro!: continuous_open_vimage continuous_inner continuous_at_id continuous_const)

also have "(\<Inter>i∈Basis. (op • i) -` {a • i <..< b • i}) = box a b"

by (auto simp add: box_def inner_commute)

finally show ?thesis .

qed

instance euclidean_space ⊆ second_countable_topology

proof

def a ≡ "λf :: 'a => (real × real). ∑i∈Basis. fst (f i) *⇩_{R}i"

then have a: "!!f. (∑i∈Basis. fst (f i) *⇩_{R}i) = a f"

by simp

def b ≡ "λf :: 'a => (real × real). ∑i∈Basis. snd (f i) *⇩_{R}i"

then have b: "!!f. (∑i∈Basis. snd (f i) *⇩_{R}i) = b f"

by simp

def B ≡ "(λf. box (a f) (b f)) ` (Basis ->⇩_{E}(\<rat> × \<rat>))"

have "Ball B open" by (simp add: B_def open_box)

moreover have "(∀A. open A --> (∃B'⊆B. \<Union>B' = A))"

proof safe

fix A::"'a set"

assume "open A"

show "∃B'⊆B. \<Union>B' = A"

apply (rule exI[of _ "{b∈B. b ⊆ A}"])

apply (subst (3) open_UNION_box[OF `open A`])

apply (auto simp add: a b B_def)

done

qed

ultimately

have "topological_basis B"

unfolding topological_basis_def by blast

moreover

have "countable B"

unfolding B_def

by (intro countable_image countable_PiE finite_Basis countable_SIGMA countable_rat)

ultimately show "∃B::'a set set. countable B ∧ open = generate_topology B"

by (blast intro: topological_basis_imp_subbasis)

qed

instance euclidean_space ⊆ polish_space ..

text {* Intervals in general, including infinite and mixtures of open and closed. *}

definition "is_interval (s::('a::euclidean_space) set) <->

(∀a∈s. ∀b∈s. ∀x. (∀i∈Basis. ((a•i ≤ x•i ∧ x•i ≤ b•i) ∨ (b•i ≤ x•i ∧ x•i ≤ a•i))) --> x ∈ s)"

lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)

"is_interval {a<..<b}" (is ?th2) proof -

show ?th1 ?th2 unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff

by(meson order_trans le_less_trans less_le_trans less_trans)+ qed

lemma is_interval_empty:

"is_interval {}"

unfolding is_interval_def

by simp

lemma is_interval_univ:

"is_interval UNIV"

unfolding is_interval_def

by simp

subsection {* Closure of halfspaces and hyperplanes *}

lemma isCont_open_vimage:

assumes "!!x. isCont f x"

and "open s"

shows "open (f -` s)"

proof -

from assms(1) have "continuous_on UNIV f"

unfolding isCont_def continuous_on_def by simp

then have "open {x ∈ UNIV. f x ∈ s}"

using open_UNIV `open s` by (rule continuous_open_preimage)

then show "open (f -` s)"

by (simp add: vimage_def)

qed

lemma isCont_closed_vimage:

assumes "!!x. isCont f x"

and "closed s"

shows "closed (f -` s)"

using assms unfolding closed_def vimage_Compl [symmetric]

by (rule isCont_open_vimage)

lemma open_Collect_less:

fixes f g :: "'a::t2_space => real"

assumes f: "!!x. isCont f x"

and g: "!!x. isCont g x"

shows "open {x. f x < g x}"

proof -

have "open ((λx. g x - f x) -` {0<..})"

using isCont_diff [OF g f] open_real_greaterThan

by (rule isCont_open_vimage)

also have "((λx. g x - f x) -` {0<..}) = {x. f x < g x}"

by auto

finally show ?thesis .

qed

lemma closed_Collect_le:

fixes f g :: "'a::t2_space => real"

assumes f: "!!x. isCont f x"

and g: "!!x. isCont g x"

shows "closed {x. f x ≤ g x}"

proof -

have "closed ((λx. g x - f x) -` {0..})"

using isCont_diff [OF g f] closed_real_atLeast

by (rule isCont_closed_vimage)

also have "((λx. g x - f x) -` {0..}) = {x. f x ≤ g x}"

by auto

finally show ?thesis .

qed

lemma closed_Collect_eq:

fixes f g :: "'a::t2_space => 'b::t2_space"

assumes f: "!!x. isCont f x"

and g: "!!x. isCont g x"

shows "closed {x. f x = g x}"

proof -

have "open {(x::'b, y::'b). x ≠ y}"

unfolding open_prod_def by (auto dest!: hausdorff)

then have "closed {(x::'b, y::'b). x = y}"

unfolding closed_def split_def Collect_neg_eq .

with isCont_Pair [OF f g]

have "closed ((λx. (f x, g x)) -` {(x, y). x = y})"

by (rule isCont_closed_vimage)

also have "… = {x. f x = g x}" by auto

finally show ?thesis .

qed

lemma continuous_at_inner: "continuous (at x) (inner a)"

unfolding continuous_at by (intro tendsto_intros)

lemma closed_halfspace_le: "closed {x. inner a x ≤ b}"

by (simp add: closed_Collect_le)

lemma closed_halfspace_ge: "closed {x. inner a x ≥ b}"

by (simp add: closed_Collect_le)

lemma closed_hyperplane: "closed {x. inner a x = b}"

by (simp add: closed_Collect_eq)

lemma closed_halfspace_component_le: "closed {x::'a::euclidean_space. x•i ≤ a}"

by (simp add: closed_Collect_le)

lemma closed_halfspace_component_ge: "closed {x::'a::euclidean_space. x•i ≥ a}"

by (simp add: closed_Collect_le)

lemma closed_interval_left:

fixes b :: "'a::euclidean_space"

shows "closed {x::'a. ∀i∈Basis. x•i ≤ b•i}"

by (simp add: Collect_ball_eq closed_INT closed_Collect_le)

lemma closed_interval_right:

fixes a :: "'a::euclidean_space"

shows "closed {x::'a. ∀i∈Basis. a•i ≤ x•i}"

by (simp add: Collect_ball_eq closed_INT closed_Collect_le)

text {* Openness of halfspaces. *}

lemma open_halfspace_lt: "open {x. inner a x < b}"

by (simp add: open_Collect_less)

lemma open_halfspace_gt: "open {x. inner a x > b}"

by (simp add: open_Collect_less)

lemma open_halfspace_component_lt: "open {x::'a::euclidean_space. x•i < a}"

by (simp add: open_Collect_less)

lemma open_halfspace_component_gt: "open {x::'a::euclidean_space. x•i > a}"

by (simp add: open_Collect_less)

text{* Instantiation for intervals on @{text ordered_euclidean_space} *}

lemma eucl_lessThan_eq_halfspaces:

fixes a :: "'a::ordered_euclidean_space"

shows "{..<a} = (\<Inter>i∈Basis. {x. x • i < a • i})"

by (auto simp: eucl_less[where 'a='a])

lemma eucl_greaterThan_eq_halfspaces:

fixes a :: "'a::ordered_euclidean_space"

shows "{a<..} = (\<Inter>i∈Basis. {x. a • i < x • i})"

by (auto simp: eucl_less[where 'a='a])

lemma eucl_atMost_eq_halfspaces:

fixes a :: "'a::ordered_euclidean_space"

shows "{.. a} = (\<Inter>i∈Basis. {x. x • i ≤ a • i})"

by (auto simp: eucl_le[where 'a='a])

lemma eucl_atLeast_eq_halfspaces:

fixes a :: "'a::ordered_euclidean_space"

shows "{a ..} = (\<Inter>i∈Basis. {x. a • i ≤ x • i})"

by (auto simp: eucl_le[where 'a='a])

lemma open_eucl_lessThan[simp, intro]:

fixes a :: "'a::ordered_euclidean_space"

shows "open {..< a}"

by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)

lemma open_eucl_greaterThan[simp, intro]:

fixes a :: "'a::ordered_euclidean_space"

shows "open {a <..}"

by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)

lemma closed_eucl_atMost[simp, intro]:

fixes a :: "'a::ordered_euclidean_space"

shows "closed {.. a}"

unfolding eucl_atMost_eq_halfspaces

by (simp add: closed_INT closed_Collect_le)

lemma closed_eucl_atLeast[simp, intro]:

fixes a :: "'a::ordered_euclidean_space"

shows "closed {a ..}"

unfolding eucl_atLeast_eq_halfspaces

by (simp add: closed_INT closed_Collect_le)

text {* This gives a simple derivation of limit component bounds. *}

lemma Lim_component_le:

fixes f :: "'a => 'b::euclidean_space"

assumes "(f ---> l) net"

and "¬ (trivial_limit net)"

and "eventually (λx. f(x)•i ≤ b) net"

shows "l•i ≤ b"

by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)])

lemma Lim_component_ge:

fixes f :: "'a => 'b::euclidean_space"

assumes "(f ---> l) net"

and "¬ (trivial_limit net)"

and "eventually (λx. b ≤ (f x)•i) net"

shows "b ≤ l•i"

by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)])

lemma Lim_component_eq:

fixes f :: "'a => 'b::euclidean_space"

assumes net: "(f ---> l) net" "¬ trivial_limit net"

and ev:"eventually (λx. f(x)•i = b) net"

shows "l•i = b"

using ev[unfolded order_eq_iff eventually_conj_iff]

using Lim_component_ge[OF net, of b i]

using Lim_component_le[OF net, of i b]

by auto

text {* Limits relative to a union. *}

lemma eventually_within_Un:

"eventually P (at x within (s ∪ t)) <->

eventually P (at x within s) ∧ eventually P (at x within t)"

unfolding eventually_at_filter

by (auto elim!: eventually_rev_mp)

lemma Lim_within_union:

"(f ---> l) (at x within (s ∪ t)) <->

(f ---> l) (at x within s) ∧ (f ---> l) (at x within t)"

unfolding tendsto_def

by (auto simp add: eventually_within_Un)

lemma Lim_topological:

"(f ---> l) net <->

trivial_limit net ∨ (∀S. open S --> l ∈ S --> eventually (λx. f x ∈ S) net)"

unfolding tendsto_def trivial_limit_eq by auto

text{* Some more convenient intermediate-value theorem formulations. *}

lemma connected_ivt_hyperplane:

assumes "connected s"

and "x ∈ s"

and "y ∈ s"

and "inner a x ≤ b"

and "b ≤ inner a y"

shows "∃z ∈ s. inner a z = b"

proof (rule ccontr)

assume as:"¬ (∃z∈s. inner a z = b)"

let ?A = "{x. inner a x < b}"

let ?B = "{x. inner a x > b}"

have "open ?A" "open ?B"

using open_halfspace_lt and open_halfspace_gt by auto

moreover

have "?A ∩ ?B = {}" by auto

moreover

have "s ⊆ ?A ∪ ?B" using as by auto

ultimately

show False

using assms(1)[unfolded connected_def not_ex,

THEN spec[where x="?A"], THEN spec[where x="?B"]]

using assms(2-5)

by auto

qed

lemma connected_ivt_component:

fixes x::"'a::euclidean_space"

shows "connected s ==>

x ∈ s ==> y ∈ s ==>

x•k ≤ a ==> a ≤ y•k ==> (∃z∈s. z•k = a)"

using connected_ivt_hyperplane[of s x y "k::'a" a]

by (auto simp: inner_commute)

subsection {* Homeomorphisms *}

definition "homeomorphism s t f g <->

(∀x∈s. (g(f x) = x)) ∧ (f ` s = t) ∧ continuous_on s f ∧

(∀y∈t. (f(g y) = y)) ∧ (g ` t = s) ∧ continuous_on t g"

definition homeomorphic :: "'a::topological_space set => 'b::topological_space set => bool"

(infixr "homeomorphic" 60)

where "s homeomorphic t ≡ (∃f g. homeomorphism s t f g)"

lemma homeomorphic_refl: "s homeomorphic s"

unfolding homeomorphic_def

unfolding homeomorphism_def

using continuous_on_id

apply (rule_tac x = "(λx. x)" in exI)

apply (rule_tac x = "(λx. x)" in exI)

apply blast

done

lemma homeomorphic_sym: "s homeomorphic t <-> t homeomorphic s"

unfolding homeomorphic_def

unfolding homeomorphism_def

by blast

lemma homeomorphic_trans:

assumes "s homeomorphic t"

and "t homeomorphic u"

shows "s homeomorphic u"

proof -

obtain f1 g1 where fg1: "∀x∈s. g1 (f1 x) = x" "f1 ` s = t"

"continuous_on s f1" "∀y∈t. f1 (g1 y) = y" "g1 ` t = s" "continuous_on t g1"

using assms(1) unfolding homeomorphic_def homeomorphism_def by auto

obtain f2 g2 where fg2: "∀x∈t. g2 (f2 x) = x" "f2 ` t = u" "continuous_on t f2"

"∀y∈u. f2 (g2 y) = y" "g2 ` u = t" "continuous_on u g2"

using assms(2) unfolding homeomorphic_def homeomorphism_def by auto

{

fix x

assume "x∈s"

then have "(g1 o g2) ((f2 o f1) x) = x"

using fg1(1)[THEN bspec[where x=x]] and fg2(1)[THEN bspec[where x="f1 x"]] and fg1(2)

by auto

}

moreover have "(f2 o f1) ` s = u"

using fg1(2) fg2(2) by auto

moreover have "continuous_on s (f2 o f1)"

using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto

moreover

{

fix y

assume "y∈u"

then have "(f2 o f1) ((g1 o g2) y) = y"

using fg2(4)[THEN bspec[where x=y]] and fg1(4)[THEN bspec[where x="g2 y"]] and fg2(5)

by auto

}

moreover have "(g1 o g2) ` u = s" using fg1(5) fg2(5) by auto

moreover have "continuous_on u (g1 o g2)"

using continuous_on_compose[OF fg2(6)] and fg1(6)

unfolding fg2(5)

by auto

ultimately show ?thesis

unfolding homeomorphic_def homeomorphism_def

apply (rule_tac x="f2 o f1" in exI)

apply (rule_tac x="g1 o g2" in exI)

apply auto

done

qed

lemma homeomorphic_minimal:

"s homeomorphic t <->

(∃f g. (∀x∈s. f(x) ∈ t ∧ (g(f(x)) = x)) ∧

(∀y∈t. g(y) ∈ s ∧ (f(g(y)) = y)) ∧

continuous_on s f ∧ continuous_on t g)"

unfolding homeomorphic_def homeomorphism_def

apply auto

apply (rule_tac x=f in exI)

apply (rule_tac x=g in exI)

apply auto

apply (rule_tac x=f in exI)

apply (rule_tac x=g in exI)

apply auto

unfolding image_iff

apply (erule_tac x="g x" in ballE)

apply (erule_tac x="x" in ballE)

apply auto

apply (rule_tac x="g x" in bexI)

apply auto

apply (erule_tac x="f x" in ballE)

apply (erule_tac x="x" in ballE)

apply auto

apply (rule_tac x="f x" in bexI)

apply auto

done

text {* Relatively weak hypotheses if a set is compact. *}

lemma homeomorphism_compact:

fixes f :: "'a::topological_space => 'b::t2_space"

assumes "compact s" "continuous_on s f" "f ` s = t" "inj_on f s"

shows "∃g. homeomorphism s t f g"

proof -

def g ≡ "λx. SOME y. y∈s ∧ f y = x"

have g: "∀x∈s. g (f x) = x"

using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto

{

fix y

assume "y ∈ t"

then obtain x where x:"f x = y" "x∈s"

using assms(3) by auto

then have "g (f x) = x" using g by auto

then have "f (g y) = y" unfolding x(1)[symmetric] by auto

}

then have g':"∀x∈t. f (g x) = x" by auto

moreover

{

fix x

have "x∈s ==> x ∈ g ` t"

using g[THEN bspec[where x=x]]

unfolding image_iff

using assms(3)

by (auto intro!: bexI[where x="f x"])

moreover

{

assume "x∈g ` t"

then obtain y where y:"y∈t" "g y = x" by auto

then obtain x' where x':"x'∈s" "f x' = y"

using assms(3) by auto

then have "x ∈ s"

unfolding g_def

using someI2[of "λb. b∈s ∧ f b = y" x' "λx. x∈s"]

unfolding y(2)[symmetric] and g_def

by auto

}

ultimately have "x∈s <-> x ∈ g ` t" ..

}

then have "g ` t = s" by auto

ultimately show ?thesis

unfolding homeomorphism_def homeomorphic_def

apply (rule_tac x=g in exI)

using g and assms(3) and continuous_on_inv[OF assms(2,1), of g, unfolded assms(3)] and assms(2)

apply auto

done

qed

lemma homeomorphic_compact:

fixes f :: "'a::topological_space => 'b::t2_space"

shows "compact s ==> continuous_on s f ==> (f ` s = t) ==> inj_on f s ==> s homeomorphic t"

unfolding homeomorphic_def by (metis homeomorphism_compact)

text{* Preservation of topological properties. *}

lemma homeomorphic_compactness: "s homeomorphic t ==> (compact s <-> compact t)"

unfolding homeomorphic_def homeomorphism_def

by (metis compact_continuous_image)

text{* Results on translation, scaling etc. *}

lemma homeomorphic_scaling:

fixes s :: "'a::real_normed_vector set"

assumes "c ≠ 0"

shows "s homeomorphic ((λx. c *⇩_{R}x) ` s)"

unfolding homeomorphic_minimal

apply (rule_tac x="λx. c *⇩_{R}x" in exI)

apply (rule_tac x="λx. (1 / c) *⇩_{R}x" in exI)

using assms

apply (auto simp add: continuous_on_intros)

done

lemma homeomorphic_translation:

fixes s :: "'a::real_normed_vector set"

shows "s homeomorphic ((λx. a + x) ` s)"

unfolding homeomorphic_minimal

apply (rule_tac x="λx. a + x" in exI)

apply (rule_tac x="λx. -a + x" in exI)

using continuous_on_add[OF continuous_on_const continuous_on_id]

apply auto

done

lemma homeomorphic_affinity:

fixes s :: "'a::real_normed_vector set"

assumes "c ≠ 0"

shows "s homeomorphic ((λx. a + c *⇩_{R}x) ` s)"

proof -

have *: "op + a ` op *⇩_{R}c ` s = (λx. a + c *⇩_{R}x) ` s" by auto

show ?thesis

using homeomorphic_trans

using homeomorphic_scaling[OF assms, of s]

using homeomorphic_translation[of "(λx. c *⇩_{R}x) ` s" a]

unfolding *

by auto

qed

lemma homeomorphic_balls:

fixes a b ::"'a::real_normed_vector"

assumes "0 < d" "0 < e"

shows "(ball a d) homeomorphic (ball b e)" (is ?th)

and "(cball a d) homeomorphic (cball b e)" (is ?cth)

proof -

show ?th unfolding homeomorphic_minimal

apply(rule_tac x="λx. b + (e/d) *⇩_{R}(x - a)" in exI)

apply(rule_tac x="λx. a + (d/e) *⇩_{R}(x - b)" in exI)

using assms

apply (auto intro!: continuous_on_intros

simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)

done

show ?cth unfolding homeomorphic_minimal

apply(rule_tac x="λx. b + (e/d) *⇩_{R}(x - a)" in exI)

apply(rule_tac x="λx. a + (d/e) *⇩_{R}(x - b)" in exI)

using assms

apply (auto intro!: continuous_on_intros

simp: dist_commute dist_norm pos_divide_le_eq mult_strict_left_mono)

done

qed

text{* "Isometry" (up to constant bounds) of injective linear map etc. *}

lemma cauchy_isometric:

fixes x :: "nat => 'a::euclidean_space"

assumes e: "e > 0"

and s: "subspace s"

and f: "bounded_linear f"

and normf: "∀x∈s. norm (f x) ≥ e * norm x"

and xs: "∀n. x n ∈ s"

and cf: "Cauchy (f o x)"

shows "Cauchy x"

proof -

interpret f: bounded_linear f by fact

{

fix d :: real

assume "d > 0"

then obtain N where N:"∀n≥N. norm (f (x n) - f (x N)) < e * d"

using cf[unfolded cauchy o_def dist_norm, THEN spec[where x="e*d"]]

and e and mult_pos_pos[of e d]

by auto

{

fix n

assume "n≥N"

have "e * norm (x n - x N) ≤ norm (f (x n - x N))"

using subspace_sub[OF s, of "x n" "x N"]

using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]

using normf[THEN bspec[where x="x n - x N"]]

by auto

also have "norm (f (x n - x N)) < e * d"

using `N ≤ n` N unfolding f.diff[symmetric] by auto

finally have "norm (x n - x N) < d" using `e>0` by simp

}

then have "∃N. ∀n≥N. norm (x n - x N) < d" by auto

}

then show ?thesis unfolding cauchy and dist_norm by 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 "λ n xa. xa ∈ s ∧ g n = f xa"]

by auto

then have x:"∀n. x n ∈ s" "∀n. g n = f (x n)"

by auto

then have "f o x = g"

unfolding fun_eq_iff

by auto

then obtain l where "l∈s" and l:"(x ---> l) sequentially"

using cs[unfolded complete_def, THEN spec[where x="x"]]

using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1)

by auto

then have "∃l∈f ` s. (g ---> l) sequentially"

using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]

unfolding `f o x = g`

by auto

}

then show ?thesis

unfolding complete_def by auto

qed

lemma injective_imp_isometric:

fixes f :: "'a::euclidean_space => 'b::euclidean_space"

assumes s: "closed s" "subspace s"

and f: "bounded_linear f" "∀x∈s. f x = 0 --> x = 0"

shows "∃e>0. ∀x∈s. norm (f x) ≥ e * norm x"

proof (cases "s ⊆ {0::'a}")

case True

{

fix x

assume "x ∈ s"

then have "x = 0" using True by auto

then have "norm x ≤ norm (f x)" by auto

}

then show ?thesis by (auto intro!: exI[where x=1])

next

interpret f: bounded_linear f by fact

case False

then obtain a where a: "a ≠ 0" "a ∈ s"

by auto

from False have "s ≠ {}"

by auto

let ?S = "{f x| x. (x ∈ s ∧ norm x = norm a)}"

let ?S' = "{x::'a. x∈s ∧ norm x = norm a}"

let ?S'' = "{x::'a. norm x = norm a}"

have "?S'' = frontier(cball 0 (norm a))"

unfolding frontier_cball and dist_norm by auto

then have "compact ?S''"

using compact_frontier[OF compact_cball, of 0 "norm a"] by auto

moreover have "?S' = s ∩ ?S''" by auto

ultimately have "compact ?S'"

using closed_inter_compact[of s ?S''] using s(1) by auto

moreover have *:"f ` ?S' = ?S" by auto

ultimately have "compact ?S"

using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto

then have "closed ?S" using compact_imp_closed by auto

moreover have "?S ≠ {}" using a by auto

ultimately obtain b' where "b'∈?S" "∀y∈?S. norm b' ≤ norm y"

using distance_attains_inf[of ?S 0] unfolding dist_0_norm by auto

then obtain b where "b∈s"

and ba: "norm b = norm a"

and b: "∀x∈{x ∈ s. norm x = norm a}. norm (f b) ≤ norm (f x)"

unfolding *[symmetric] unfolding image_iff by auto

let ?e = "norm (f b) / norm b"

have "norm b > 0" using ba and a and norm_ge_zero by auto

moreover have "norm (f b) > 0"

using f(2)[THEN bspec[where x=b], OF `b∈s`]

using `norm b >0`

unfolding zero_less_norm_iff

by auto

ultimately have "0 < norm (f b) / norm b"

by (simp only: divide_pos_pos)

moreover

{

fix x

assume "x∈s"

then have "norm (f b) / norm b * norm x ≤ norm (f x)"

proof (cases "x=0")

case True

then show "norm (f b) / norm b * norm x ≤ norm (f x)" by auto

next

case False

then have *: "0 < norm a / norm x"

using `a≠0`

unfolding zero_less_norm_iff[symmetric]

by (simp only: divide_pos_pos)

have "∀c. ∀x∈s. c *⇩_{R}x ∈ s"

using s[unfolded subspace_def] by auto

then have "(norm a / norm x) *⇩_{R}x ∈ {x ∈ s. norm x = norm a}"

using `x∈s` and `x≠0` by auto

then show "norm (f b) / norm b * norm x ≤ norm (f x)"

using b[THEN bspec[where x="(norm a / norm x) *⇩_{R}x"]]

unfolding f.scaleR and ba using `x≠0` `a≠0`

by (auto simp add: mult_commute pos_le_divide_eq pos_divide_le_eq)

qed

}

ultimately show ?thesis by auto

qed

lemma closed_injective_image_subspace:

fixes f :: "'a::euclidean_space => 'b::euclidean_space"

assumes "subspace s" "bounded_linear f" "∀x∈s. f x = 0 --> x = 0" "closed s"

shows "closed(f ` s)"

proof -

obtain e where "e > 0" and e: "∀x∈s. e * norm x ≤ norm (f x)"

using injective_imp_isometric[OF assms(4,1,2,3)] by auto

show ?thesis

using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4)

unfolding complete_eq_closed[symmetric] by auto

qed

subsection {* Some properties of a canonical subspace *}

lemma subspace_substandard:

"subspace {x::'a::euclidean_space. (∀i∈Basis. P i --> x•i = 0)}"

unfolding subspace_def by (auto simp: inner_add_left)

lemma closed_substandard:

"closed {x::'a::euclidean_space. ∀i∈Basis. P i --> x•i = 0}" (is "closed ?A")

proof -

let ?D = "{i∈Basis. P i}"

have "closed (\<Inter>i∈?D. {x::'a. x•i = 0})"

by (simp add: closed_INT closed_Collect_eq)

also have "(\<Inter>i∈?D. {x::'a. x•i = 0}) = ?A"

by auto

finally show "closed ?A" .

qed

lemma dim_substandard:

assumes d: "d ⊆ Basis"

shows "dim {x::'a::euclidean_space. ∀i∈Basis. i ∉ d --> x•i = 0} = card d" (is "dim ?A = _")

proof (rule dim_unique)

show "d ⊆ ?A"

using d by (auto simp: inner_Basis)

show "independent d"

using independent_mono [OF independent_Basis d] .

show "?A ⊆ span d"

proof (clarify)

fix x assume x: "∀i∈Basis. i ∉ d --> x • i = 0"

have "finite d"

using finite_subset [OF d finite_Basis] .

then have "(∑i∈d. (x • i) *⇩_{R}i) ∈ span d"

by (simp add: span_setsum span_clauses)

also have "(∑i∈d. (x • i) *⇩_{R}i) = (∑i∈Basis. (x • i) *⇩_{R}i)"

by (rule setsum_mono_zero_cong_left [OF finite_Basis d]) (auto simp add: x)

finally show "x ∈ span d"

unfolding euclidean_representation .

qed

qed simp

text{* Hence closure and completeness of all subspaces. *}

lemma ex_card:

assumes "n ≤ card A"

shows "∃S⊆A. card S = n"

proof cases

assume "finite A"

from ex_bij_betw_nat_finite[OF this] guess f .. note f = this

moreover from f `n ≤ card A` have "{..< n} ⊆ {..< card A}" "inj_on f {..< n}"

by (auto simp: bij_betw_def intro: subset_inj_on)

ultimately have "f ` {..< n} ⊆ A" "card (f ` {..< n}) = n"

by (auto simp: bij_betw_def card_image)

then show ?thesis by blast

next

assume "¬ finite A"

with `n ≤ card A` show ?thesis by force

qed

lemma closed_subspace:

fixes s :: "'a::euclidean_space set"

assumes "subspace s"

shows "closed s"

proof -

have "dim s ≤ card (Basis :: 'a set)"

using dim_subset_UNIV by auto

with ex_card[OF this] obtain d :: "'a set" where t: "card d = dim s" and d: "d ⊆ Basis"

by auto

let ?t = "{x::'a. ∀i∈Basis. i ∉ d --> x•i = 0}"

have "∃f. linear f ∧ f ` {x::'a. ∀i∈Basis. i ∉ d --> x • i = 0} = s ∧

inj_on f {x::'a. ∀i∈Basis. i ∉ d --> x • i = 0}"

using dim_substandard[of d] t d assms

by (intro subspace_isomorphism[OF subspace_substandard[of "λi. i ∉ d"]]) (auto simp: inner_Basis)

then 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

then show ?thesis using dim_subset[OF closure_subset, of s]

by auto

qed

subsection {* Affine transformations of intervals *}

lemma real_affinity_le:

"0 < (m::'a::linordered_field) ==> (m * x + c ≤ y <-> x ≤ inverse(m) * y + -(c / m))"

by (simp add: field_simps inverse_eq_divide)

lemma real_le_affinity:

"0 < (m::'a::linordered_field) ==> (y ≤ m * x + c <-> inverse(m) * y + -(c / m) ≤ x)"

by (simp add: field_simps inverse_eq_divide)

lemma real_affinity_lt:

"0 < (m::'a::linordered_field) ==> (m * x + c < y <-> x < inverse(m) * y + -(c / m))"

by (simp add: field_simps inverse_eq_divide)

lemma real_lt_affinity:

"0 < (m::'a::linordered_field) ==> (y < m * x + c <-> inverse(m) * y + -(c / m) < x)"

by (simp add: field_simps inverse_eq_divide)

lemma real_affinity_eq:

"(m::'a::linordered_field) ≠ 0 ==> (m * x + c = y <-> x = inverse(m) * y + -(c / m))"

by (simp add: field_simps inverse_eq_divide)

lemma real_eq_affinity:

"(m::'a::linordered_field) ≠ 0 ==> (y = m * x + c <-> inverse(m) * y + -(c / m) = x)"

by (simp add: field_simps inverse_eq_divide)

lemma image_affinity_interval: fixes m::real

fixes a b c :: "'a::ordered_euclidean_space"

shows "(λx. m *⇩_{R}x + c) ` {a .. b} =

(if {a .. b} = {} then {}

else (if 0 ≤ m then {m *⇩_{R}a + c .. m *⇩_{R}b + c}

else {m *⇩_{R}b + c .. m *⇩_{R}a + c}))"

proof (cases "m = 0")

case True

{

fix x

assume "x ≤ c" "c ≤ x"

then have "x = c"

unfolding eucl_le[where 'a='a]

apply -

apply (subst euclidean_eq_iff)

apply (auto intro: order_antisym)

done

}

moreover have "c ∈ {m *⇩_{R}a + c..m *⇩_{R}b + c}"

unfolding True by (auto simp add: eucl_le[where 'a='a])

ultimately show ?thesis using True by auto

next

case False

{

fix y

assume "a ≤ y" "y ≤ b" "m > 0"

then have "m *⇩_{R}a + c ≤ m *⇩_{R}y + c" and "m *⇩_{R}y + c ≤ m *⇩_{R}b + c"

unfolding eucl_le[where 'a='a] by (auto simp: inner_simps)

}

moreover

{

fix y

assume "a ≤ y" "y ≤ b" "m < 0"

then have "m *⇩_{R}b + c ≤ m *⇩_{R}y + c" and "m *⇩_{R}y + c ≤ m *⇩_{R}a + c"

unfolding eucl_le[where 'a='a] by (auto simp add: mult_left_mono_neg inner_simps)

}

moreover

{

fix y

assume "m > 0" and "m *⇩_{R}a + c ≤ y" and "y ≤ m *⇩_{R}b + c"

then have "y ∈ (λx. m *⇩_{R}x + c) ` {a..b}"

unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]

apply (intro exI[where x="(1 / m) *⇩_{R}(y - c)"])

apply (auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff inner_simps)

done

}

moreover

{

fix y

assume "m *⇩_{R}b + c ≤ y" "y ≤ m *⇩_{R}a + c" "m < 0"

then have "y ∈ (λx. m *⇩_{R}x + c) ` {a..b}"

unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]

apply (intro exI[where x="(1 / m) *⇩_{R}(y - c)"])

apply (auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff inner_simps)

done

}

ultimately show ?thesis using False by auto

qed

lemma image_smult_interval:"(λx. m *⇩_{R}(x::_::ordered_euclidean_space)) ` {a..b} =

(if {a..b} = {} then {} else if 0 ≤ m then {m *⇩_{R}a..m *⇩_{R}b} else {m *⇩_{R}b..m *⇩_{R}a})"

using image_affinity_interval[of m 0 a b] by 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 ≡ "λn. (f ^^ n) z0"

{

fix n :: nat

have "z n ∈ s" unfolding z_def

proof (induct n)

case 0

then show ?case using `z0 ∈ s` by auto

next

case Suc

then show ?case using f by auto qed

} note z_in_s = this

def d ≡ "dist (z 0) (z 1)"

have fzn:"!!n. f (z n) = z (Suc n)" unfolding z_def by auto

{

fix n :: nat

have "dist (z n) (z (Suc n)) ≤ (c ^ n) * d"

proof (induct n)

case 0

then show ?case

unfolding d_def by auto

next

case (Suc m)

then have "c * dist (z m) (z (Suc m)) ≤ c ^ Suc m * d"

using `0 ≤ c`

using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c]

by auto

then show ?case

using lipschitz[THEN bspec[where x="z m"], OF z_in_s, THEN bspec[where x="z (Suc m)"], OF z_in_s]

unfolding fzn and mult_le_cancel_left

by auto

qed

} note cf_z = this

{

fix n m :: nat

have "(1 - c) * dist (z m) (z (m+n)) ≤ (c ^ m) * d * (1 - c ^ n)"

proof (induct n)

case 0

show ?case by auto

next

case (Suc k)

have "(1 - c) * dist (z m) (z (m + Suc k)) ≤

(1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))"

using dist_triangle and c by (auto simp add: dist_triangle)

also have "… ≤ (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"

using cf_z[of "m + k"] and c by auto

also have "… ≤ c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"

using Suc by (auto simp add: field_simps)

also have "… = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"

unfolding power_add by (auto simp add: field_simps)

also have "… ≤ (c ^ m) * d * (1 - c ^ Suc k)"

using c by (auto simp add: field_simps)

finally show ?case by auto

qed

} note cf_z2 = this

{

fix e :: real

assume "e > 0"

then have "∃N. ∀m n. N ≤ m ∧ N ≤ n --> dist (z m) (z n) < e"

proof (cases "d = 0")

case True

have *: "!!x. ((1 - c) * x ≤ 0) = (x ≤ 0)" using `1 - c > 0`

by (metis mult_zero_left mult_commute real_mult_le_cancel_iff1)

from True have "!!n. z n = z0" using cf_z2[of 0] and c unfolding z_def

by (simp add: *)

then show ?thesis using `e>0` by auto

next

case False

then have "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"]

by (metis False d_def less_le)

then have "0 < e * (1 - c) / d"

using `e>0` and `1-c>0`

using divide_pos_pos[of "e * (1 - c)" d] and mult_pos_pos[of e "1 - c"]

by auto

then obtain N where N:"c ^ N < e * (1 - c) / d"

using real_arch_pow_inv[of "e * (1 - c) / d" c] and c by auto

{

fix m n::nat

assume "m>n" and as:"m≥N" "n≥N"

have *:"c ^ n ≤ c ^ N" using `n≥N` and c

using power_decreasing[OF `n≥N`, of c] by auto

have "1 - c ^ (m - n) > 0"

using c and power_strict_mono[of c 1 "m - n"] using `m>n` by auto

then have **: "d * (1 - c ^ (m - n)) / (1 - c) > 0"

using mult_pos_pos[OF `d>0`, of "1 - c ^ (m - n)"]

using divide_pos_pos[of "d * (1 - c ^ (m - n))" "1 - c"]

using `0 < 1 - c`

by auto

have "dist (z m) (z n) ≤ c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"

using cf_z2[of n "m - n"] and `m>n`

unfolding pos_le_divide_eq[OF `1-c>0`]

by (auto simp add: mult_commute dist_commute)

also have "… ≤ c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"

using mult_right_mono[OF * order_less_imp_le[OF **]]

unfolding mult_assoc by auto

also have "… < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"

using mult_strict_right_mono[OF N **] unfolding mult_assoc by auto

also have "… = e * (1 - c ^ (m - n))"

using c and `d>0` and `1 - c > 0` by auto

also have "… ≤ e" using c and `1 - c ^ (m - n) > 0` and `e>0`

using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto

finally have "dist (z m) (z n) < e" by auto

} note * = this

{

fix m n :: nat

assume as: "N ≤ m" "N ≤ n"

then have "dist (z n) (z m) < e"

proof (cases "n = m")

case True

then show ?thesis using `e>0` by auto

next

case False

then show ?thesis using as and *[of n m] *[of m n]

unfolding nat_neq_iff by (auto simp add: dist_commute)

qed

}

then show ?thesis by auto

qed

}

then have "Cauchy z"

unfolding cauchy_def by auto

then obtain x where "x∈s" and x:"(z ---> x) sequentially"

using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto

def e ≡ "dist (f x) x"

have "e = 0"

proof (rule ccontr)

assume "e ≠ 0"

then have "e > 0"

unfolding e_def using zero_le_dist[of "f x" x]

by (metis dist_eq_0_iff dist_nz e_def)

then obtain N where N:"∀n≥N. dist (z n) x < e / 2"

using x[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] by auto

then have N':"dist (z N) x < e / 2" by auto

have *: "c * dist (z N) x ≤ dist (z N) x"

unfolding mult_le_cancel_right2

using zero_le_dist[of "z N" x] and c

by (metis dist_eq_0_iff dist_nz order_less_asym less_le)

have "dist (f (z N)) (f x) ≤ c * dist (z N) x"

using lipschitz[THEN bspec[where x="z N"], THEN bspec[where x=x]]

using z_in_s[of N] `x∈s`

using c

by auto

also have "… < e / 2"

using N' and c using * by auto

finally show False

unfolding fzn

using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]

unfolding e_def

by auto

qed

then have "f x = x" unfolding e_def by auto

moreover

{

fix y

assume "f y = y" "y∈s"

then have "dist x y ≤ c * dist x y"

using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]]

using `x∈s` and `f x = x`

by auto

then have "dist x y = 0"

unfolding mult_le_cancel_right1

using c and zero_le_dist[of x y]

by auto

then have "y = x" by auto

}

ultimately show ?thesis using `x∈s` by blast+

qed

subsection {* Edelstein fixed point theorem *}

lemma edelstein_fix:

fixes s :: "'a::metric_space set"

assumes s: "compact s" "s ≠ {}"

and gs: "(g ` s) ⊆ s"

and dist: "∀x∈s. ∀y∈s. x ≠ y --> dist (g x) (g y) < dist x y"

shows "∃!x∈s. g x = x"

proof -

let ?D = "(λx. (x, x)) ` s"

have D: "compact ?D" "?D ≠ {}"

by (rule compact_continuous_image)

(auto intro!: s continuous_Pair continuous_within_id simp: continuous_on_eq_continuous_within)

have "!!x y e. x ∈ s ==> y ∈ s ==> 0 < e ==> dist y x < e ==> dist (g y) (g x) < e"

using dist by fastforce

then have "continuous_on s g"

unfolding continuous_on_iff by auto

then have cont: "continuous_on ?D (λx. dist ((g o fst) x) (snd x))"

unfolding continuous_on_eq_continuous_within

by (intro continuous_dist ballI continuous_within_compose)

(auto intro!: continuous_fst continuous_snd continuous_within_id simp: image_image)

obtain a where "a ∈ s" and le: "!!x. x ∈ s ==> dist (g a) a ≤ dist (g x) x"

using continuous_attains_inf[OF D cont] by auto

have "g a = a"

proof (rule ccontr)

assume "g a ≠ a"

with `a ∈ s` gs have "dist (g (g a)) (g a) < dist (g a) a"

by (intro dist[rule_format]) auto

moreover have "dist (g a) a ≤ dist (g (g a)) (g a)"

using `a ∈ s` gs by (intro le) auto

ultimately show False by auto

qed

moreover have "!!x. x ∈ s ==> g x = x ==> x = a"

using dist[THEN bspec[where x=a]] `g a = a` and `a∈s` by auto

ultimately show "∃!x∈s. g x = x" using `a ∈ s` by blast

qed

declare tendsto_const [intro] (* FIXME: move *)

end