l3lab/ntp-mathlib huggingface dataset:
instance (priority := 50) [BorelSpace α] [ClosableCompactSubsetOpenSpace α]
    [h : InnerRegularCompactLTTop μ] [IsFiniteMeasure μ] : Regular μ := by
  constructor
  

Proof state:
case innerRegular
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ : Measure α
inst✝² : BorelSpace α
inst✝¹ : ClosableCompactSubsetOpenSpace α
h : InnerRegularCompactLTTop μ
inst✝ : IsFiniteMeasure μ
⊢ InnerRegularWRT μ IsCompact IsOpen

Next tactic:
apply InnerRegularWRT.trans h.innerRegular
instance (priority := 50) [BorelSpace α] [ClosableCompactSubsetOpenSpace α]
    [h : InnerRegularCompactLTTop μ] [IsFiniteMeasure μ] : Regular μ := by
  constructor
  apply InnerRegularWRT.trans h.innerRegular
  

Proof state:
case innerRegular
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ : Measure α
inst✝² : BorelSpace α
inst✝¹ : ClosableCompactSubsetOpenSpace α
h : InnerRegularCompactLTTop μ
inst✝ : IsFiniteMeasure μ
⊢ InnerRegularWRT μ (fun s => MeasurableSet s ∧ ↑↑μ s ≠ ⊤) IsOpen

Next tactic:
exact InnerRegularWRT.of_imp (fun U hU ↦ ⟨hU.measurableSet, measure_ne_top μ U⟩)
/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
compact space, then any compact set can be approximated from outside by open sets. -/
protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
  

Proof state:
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
⊢ ↑↑μ K = ⨅ U, ⨅ (_ : K ⊆ U), ⨅ (_ : IsOpen U), ↑↑μ U

Next tactic:
apply le_antisymm
/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
compact space, then any compact set can be approximated from outside by open sets. -/
protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
  apply le_antisymm
  · 

Proof state:
case a
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
⊢ ↑↑μ K ≤ ⨅ U, ⨅ (_ : K ⊆ U), ⨅ (_ : IsOpen U), ↑↑μ U

Next tactic:
simp only [le_iInf_iff]
/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
compact space, then any compact set can be approximated from outside by open sets. -/
protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
  apply le_antisymm
  · simp only [le_iInf_iff]
    

Proof state:
case a
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
⊢ ∀ (i : Set α), K ⊆ i → IsOpen i → ↑↑μ K ≤ ↑↑μ i

Next tactic:
rintro U KU -
/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
compact space, then any compact set can be approximated from outside by open sets. -/
protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
  apply le_antisymm
  · simp only [le_iInf_iff]
    rintro U KU -
    

Proof state:
case a
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
U : Set α
KU : K ⊆ U
⊢ ↑↑μ K ≤ ↑↑μ U

Next tactic:
exact measure_mono KU
/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
compact space, then any compact set can be approximated from outside by open sets. -/
protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
  apply le_antisymm
  · simp only [le_iInf_iff]
    rintro U KU -
    exact measure_mono KU
  

Proof state:
case a
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
⊢ ⨅ U, ⨅ (_ : K ⊆ U), ⨅ (_ : IsOpen U), ↑↑μ U ≤ ↑↑μ K

Next tactic:
apply le_of_forall_lt' (fun r hr ↦ ?_)
/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
compact space, then any compact set can be approximated from outside by open sets. -/
protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
  apply le_antisymm
  · simp only [le_iInf_iff]
    rintro U KU -
    exact measure_mono KU
  apply le_of_forall_lt' (fun r hr ↦ ?_)
  

Proof state:
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : ↑↑μ K < r
⊢ ⨅ U, ⨅ (_ : K ⊆ U), ⨅ (_ : IsOpen U), ↑↑μ U < r

Next tactic:
simp only [iInf_lt_iff, exists_prop, exists_and_left]
/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
compact space, then any compact set can be approximated from outside by open sets. -/
protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
  apply le_antisymm
  · simp only [le_iInf_iff]
    rintro U KU -
    exact measure_mono KU
  apply le_of_forall_lt' (fun r hr ↦ ?_)
  simp only [iInf_lt_iff, exists_prop, exists_and_left]
  

Proof state:
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : ↑↑μ K < r
⊢ ∃ i, K ⊆ i ∧ IsOpen i ∧ ↑↑μ i < r

Next tactic:
obtain ⟨L, L_comp, KL, -⟩ : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ univ :=
    exists_compact_between hK isOpen_univ (subset_univ _)
/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
compact space, then any compact set can be approximated from outside by open sets. -/
protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
  apply le_antisymm
  · simp only [le_iInf_iff]
    rintro U KU -
    exact measure_mono KU
  apply le_of_forall_lt' (fun r hr ↦ ?_)
  simp only [iInf_lt_iff, exists_prop, exists_and_left]
  obtain ⟨L, L_comp, KL, -⟩ : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ univ :=
    exists_compact_between hK isOpen_univ (subset_univ _)
  

Proof state:
case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : ↑↑μ K < r
L : Set α
L_comp : IsCompact L
KL : K ⊆ interior L
⊢ ∃ i, K ⊆ i ∧ IsOpen i ∧ ↑↑μ i < r

Next tactic:
have : Fact (μ (interior L) < ∞) :=
    ⟨(measure_mono interior_subset).trans_lt L_comp.measure_lt_top⟩
/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
compact space, then any compact set can be approximated from outside by open sets. -/
protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
  apply le_antisymm
  · simp only [le_iInf_iff]
    rintro U KU -
    exact measure_mono KU
  apply le_of_forall_lt' (fun r hr ↦ ?_)
  simp only [iInf_lt_iff, exists_prop, exists_and_left]
  obtain ⟨L, L_comp, KL, -⟩ : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ univ :=
    exists_compact_between hK isOpen_univ (subset_univ _)
  have : Fact (μ (interior L) < ∞) :=
    ⟨(measure_mono interior_subset).trans_lt L_comp.measure_lt_top⟩
  

Proof state:
case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : ↑↑μ K < r
L : Set α
L_comp : IsCompact L
KL : K ⊆ interior L
this : Fact (↑↑μ (interior L) < ⊤)
⊢ ∃ i, K ⊆ i ∧ IsOpen i ∧ ↑↑μ i < r

Next tactic:
obtain ⟨U, KU, U_open, hU⟩ : ∃ U, K ⊆ U ∧ IsOpen U ∧ μ.restrict (interior L) U < r := by
    apply exists_isOpen_lt_of_lt K r
    exact (restrict_apply_le _ _).trans_lt hr
/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
compact space, then any compact set can be approximated from outside by open sets. -/
protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
  apply le_antisymm
  · simp only [le_iInf_iff]
    rintro U KU -
    exact measure_mono KU
  apply le_of_forall_lt' (fun r hr ↦ ?_)
  simp only [iInf_lt_iff, exists_prop, exists_and_left]
  obtain ⟨L, L_comp, KL, -⟩ : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ univ :=
    exists_compact_between hK isOpen_univ (subset_univ _)
  have : Fact (μ (interior L) < ∞) :=
    ⟨(measure_mono interior_subset).trans_lt L_comp.measure_lt_top⟩
  obtain ⟨U, KU, U_open, hU⟩ : ∃ U, K ⊆ U ∧ IsOpen U ∧ μ.restrict (interior L) U < r := by
    

Proof state:
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : ↑↑μ K < r
L : Set α
L_comp : IsCompact L
KL : K ⊆ interior L
this : Fact (↑↑μ (interior L) < ⊤)
⊢ ∃ U, K ⊆ U ∧ IsOpen U ∧ ↑↑(Measure.restrict μ (interior L)) U < r

Next tactic:
apply exists_isOpen_lt_of_lt K r
/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
compact space, then any compact set can be approximated from outside by open sets. -/
protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
  apply le_antisymm
  · simp only [le_iInf_iff]
    rintro U KU -
    exact measure_mono KU
  apply le_of_forall_lt' (fun r hr ↦ ?_)
  simp only [iInf_lt_iff, exists_prop, exists_and_left]
  obtain ⟨L, L_comp, KL, -⟩ : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ univ :=
    exists_compact_between hK isOpen_univ (subset_univ _)
  have : Fact (μ (interior L) < ∞) :=
    ⟨(measure_mono interior_subset).trans_lt L_comp.measure_lt_top⟩
  obtain ⟨U, KU, U_open, hU⟩ : ∃ U, K ⊆ U ∧ IsOpen U ∧ μ.restrict (interior L) U < r := by
    apply exists_isOpen_lt_of_lt K r
    

Proof state:
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : ↑↑μ K < r
L : Set α
L_comp : IsCompact L
KL : K ⊆ interior L
this : Fact (↑↑μ (interior L) < ⊤)
⊢ ↑↑(Measure.restrict μ (interior L)) K < r

Next tactic:
exact (restrict_apply_le _ _).trans_lt hr
/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
compact space, then any compact set can be approximated from outside by open sets. -/
protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
  apply le_antisymm
  · simp only [le_iInf_iff]
    rintro U KU -
    exact measure_mono KU
  apply le_of_forall_lt' (fun r hr ↦ ?_)
  simp only [iInf_lt_iff, exists_prop, exists_and_left]
  obtain ⟨L, L_comp, KL, -⟩ : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ univ :=
    exists_compact_between hK isOpen_univ (subset_univ _)
  have : Fact (μ (interior L) < ∞) :=
    ⟨(measure_mono interior_subset).trans_lt L_comp.measure_lt_top⟩
  obtain ⟨U, KU, U_open, hU⟩ : ∃ U, K ⊆ U ∧ IsOpen U ∧ μ.restrict (interior L) U < r := by
    apply exists_isOpen_lt_of_lt K r
    exact (restrict_apply_le _ _).trans_lt hr
  

Proof state:
case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : ↑↑μ K < r
L : Set α
L_comp : IsCompact L
KL : K ⊆ interior L
this : Fact (↑↑μ (interior L) < ⊤)
U : Set α
KU : K ⊆ U
U_open : IsOpen U
hU : ↑↑(Measure.restrict μ (interior L)) U < r
⊢ ∃ i, K ⊆ i ∧ IsOpen i ∧ ↑↑μ i < r

Next tactic:
refine ⟨U ∩ interior L, subset_inter KU KL, U_open.inter isOpen_interior, ?_⟩
/-- I`μ` is inner regular for finite measure sets with respect to compact sets in a regular locally
compact space, then any compact set can be approximated from outside by open sets. -/
protected lemma _root_.IsCompact.measure_eq_infi_isOpen [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) :
    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U := by
  apply le_antisymm
  · simp only [le_iInf_iff]
    rintro U KU -
    exact measure_mono KU
  apply le_of_forall_lt' (fun r hr ↦ ?_)
  simp only [iInf_lt_iff, exists_prop, exists_and_left]
  obtain ⟨L, L_comp, KL, -⟩ : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ univ :=
    exists_compact_between hK isOpen_univ (subset_univ _)
  have : Fact (μ (interior L) < ∞) :=
    ⟨(measure_mono interior_subset).trans_lt L_comp.measure_lt_top⟩
  obtain ⟨U, KU, U_open, hU⟩ : ∃ U, K ⊆ U ∧ IsOpen U ∧ μ.restrict (interior L) U < r := by
    apply exists_isOpen_lt_of_lt K r
    exact (restrict_apply_le _ _).trans_lt hr
  refine ⟨U ∩ interior L, subset_inter KU KL, U_open.inter isOpen_interior, ?_⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : ↑↑μ K < r
L : Set α
L_comp : IsCompact L
KL : K ⊆ interior L
this : Fact (↑↑μ (interior L) < ⊤)
U : Set α
KU : K ⊆ U
U_open : IsOpen U
hU : ↑↑(Measure.restrict μ (interior L)) U < r
⊢ ↑↑μ (U ∩ interior L) < r

Next tactic:
rwa [restrict_apply U_open.measurableSet] at hU
protected lemma _root_.IsCompact.exists_isOpen_lt_of_lt [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) (r : ℝ≥0∞) (hr : μ K < r) :
    ∃ U, K ⊆ U ∧ IsOpen U ∧ μ U < r := by
  

Proof state:
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : ↑↑μ K < r
⊢ ∃ U, K ⊆ U ∧ IsOpen U ∧ ↑↑μ U < r

Next tactic:
have : ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U < r := by
    rwa [hK.measure_eq_infi_isOpen] at hr
protected lemma _root_.IsCompact.exists_isOpen_lt_of_lt [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) (r : ℝ≥0∞) (hr : μ K < r) :
    ∃ U, K ⊆ U ∧ IsOpen U ∧ μ U < r := by
  have : ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U < r := by
    

Proof state:
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : ↑↑μ K < r
⊢ ⨅ U, ⨅ (_ : K ⊆ U), ⨅ (_ : IsOpen U), ↑↑μ U < r

Next tactic:
rwa [hK.measure_eq_infi_isOpen] at hr
protected lemma _root_.IsCompact.exists_isOpen_lt_of_lt [InnerRegularCompactLTTop μ]
    [IsFiniteMeasureOnCompacts μ] [LocallyCompactSpace α] [RegularSpace α]
    [BorelSpace α] {K : Set α} (hK : IsCompact K) (r : ℝ≥0∞) (hr : μ K < r) :
    ∃ U, K ⊆ U ∧ IsOpen U ∧ μ U < r := by
  have : ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U < r := by
    rwa [hK.measure_eq_infi_isOpen] at hr
  

Proof state:
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
μ : Measure α
inst✝⁴ : InnerRegularCompactLTTop μ
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : LocallyCompactSpace α
inst✝¹ : RegularSpace α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : ↑↑μ K < r
this : ⨅ U, ⨅ (_ : K ⊆ U), ⨅ (_ : IsOpen U), ↑↑μ U < r
⊢ ∃ U, K ⊆ U ∧ IsOpen U ∧ ↑↑μ U < r

Next tactic:
simpa only [iInf_lt_iff, exists_prop, exists_and_left]
instance smul [h : InnerRegularCompactLTTop μ] (c : ℝ≥0∞) : InnerRegularCompactLTTop (c • μ) := by
  

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ : Measure α
h : InnerRegularCompactLTTop μ
c : ℝ≥0∞
⊢ InnerRegularCompactLTTop (c • μ)

Next tactic:
by_cases hc : c = 0
instance smul [h : InnerRegularCompactLTTop μ] (c : ℝ≥0∞) : InnerRegularCompactLTTop (c • μ) := by
  by_cases hc : c = 0
  · 

Proof state:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ : Measure α
h : InnerRegularCompactLTTop μ
c : ℝ≥0∞
hc : c = 0
⊢ InnerRegularCompactLTTop (c • μ)

Next tactic:
simp only [hc, zero_smul]
instance smul [h : InnerRegularCompactLTTop μ] (c : ℝ≥0∞) : InnerRegularCompactLTTop (c • μ) := by
  by_cases hc : c = 0
  · simp only [hc, zero_smul]
    

Proof state:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ : Measure α
h : InnerRegularCompactLTTop μ
c : ℝ≥0∞
hc : c = 0
⊢ InnerRegularCompactLTTop 0

Next tactic:
infer_instance
instance smul [h : InnerRegularCompactLTTop μ] (c : ℝ≥0∞) : InnerRegularCompactLTTop (c • μ) := by
  by_cases hc : c = 0
  · simp only [hc, zero_smul]
    infer_instance
  

Proof state:
case neg
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ : Measure α
h : InnerRegularCompactLTTop μ
c : ℝ≥0∞
hc : ¬c = 0
⊢ InnerRegularCompactLTTop (c • μ)

Next tactic:
by_cases h'c : c = ∞
instance smul [h : InnerRegularCompactLTTop μ] (c : ℝ≥0∞) : InnerRegularCompactLTTop (c • μ) := by
  by_cases hc : c = 0
  · simp only [hc, zero_smul]
    infer_instance
  by_cases h'c : c = ∞
  · 

Proof state:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ : Measure α
h : InnerRegularCompactLTTop μ
c : ℝ≥0∞
hc : ¬c = 0
h'c : c = ⊤
⊢ InnerRegularCompactLTTop (c • μ)

Next tactic:
constructor
instance smul [h : InnerRegularCompactLTTop μ] (c : ℝ≥0∞) : InnerRegularCompactLTTop (c • μ) := by
  by_cases hc : c = 0
  · simp only [hc, zero_smul]
    infer_instance
  by_cases h'c : c = ∞
  · constructor
    

Proof state:
case pos.innerRegular
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ : Measure α
h : InnerRegularCompactLTTop μ
c : ℝ≥0∞
hc : ¬c = 0
h'c : c = ⊤
⊢ InnerRegularWRT (c • μ) IsCompact fun s => MeasurableSet s ∧ ↑↑(c • μ) s ≠ ⊤

Next tactic:
intro s hs r hr
instance smul [h : InnerRegularCompactLTTop μ] (c : ℝ≥0∞) : InnerRegularCompactLTTop (c • μ) := by
  by_cases hc : c = 0
  · simp only [hc, zero_smul]
    infer_instance
  by_cases h'c : c = ∞
  · constructor
    intro s hs r hr
    

Proof state:
case pos.innerRegular
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ : Measure α
h : InnerRegularCompactLTTop μ
c : ℝ≥0∞
hc : ¬c = 0
h'c : c = ⊤
s : Set α
hs : MeasurableSet s ∧ ↑↑(c • μ) s ≠ ⊤
r : ℝ≥0∞
hr : r < ↑↑(c • μ) s
⊢ ∃ K ⊆ s, IsCompact K ∧ r < ↑↑(c • μ) K

Next tactic:
simp only [h'c, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, smul_eq_mul] at hr
instance smul [h : InnerRegularCompactLTTop μ] (c : ℝ≥0∞) : InnerRegularCompactLTTop (c • μ) := by
  by_cases hc : c = 0
  · simp only [hc, zero_smul]
    infer_instance
  by_cases h'c : c = ∞
  · constructor
    intro s hs r hr
    simp only [h'c, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, smul_eq_mul] at hr
    

Proof state:
case pos.innerRegular
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ : Measure α
h : InnerRegularCompactLTTop μ
c : ℝ≥0∞
hc : ¬c = 0
h'c : c = ⊤
s : Set α
hs : MeasurableSet s ∧ ↑↑(c • μ) s ≠ ⊤
r : ℝ≥0∞
hr : r < ⊤ * ↑↑μ s
⊢ ∃ K ⊆ s, IsCompact K ∧ r < ↑↑(c • μ) K

Next tactic:
by_cases h's : μ s = 0
instance smul [h : InnerRegularCompactLTTop μ] (c : ℝ≥0∞) : InnerRegularCompactLTTop (c • μ) := by
  by_cases hc : c = 0
  · simp only [hc, zero_smul]
    infer_instance
  by_cases h'c : c = ∞
  · constructor
    intro s hs r hr
    simp only [h'c, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, smul_eq_mul] at hr
    by_cases h's : μ s = 0
    · 

Proof state:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ : Measure α
h : InnerRegularCompactLTTop μ
c : ℝ≥0∞
hc : ¬c = 0
h'c : c = ⊤
s : Set α
hs : MeasurableSet s ∧ ↑↑(c • μ) s ≠ ⊤
r : ℝ≥0∞
hr : r < ⊤ * ↑↑μ s
h's : ↑↑μ s = 0
⊢ ∃ K ⊆ s, IsCompact K ∧ r < ↑↑(c • μ) K

Next tactic:
simp [h's] at hr