l3lab/ntp-mathlib huggingface dataset:
theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ ∞) {ε : ℝ≥0}
    (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.innerContent U ≤ μ K + ε := by
  have h'ε := ENNReal.coe_ne_zero.2 hε
  rcases le_or_lt (μ.innerContent U) ε with h | h
  · exact ⟨⊥, empty_subset _, le_add_left h⟩
  have h₂ := ENNReal.sub_lt_self hU h.ne_bot h'ε
  conv at h₂ => rhs; rw [innerContent]
  

Proof state:
case inr
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U : Opens G
hU : innerContent μ U ≠ ⊤
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < innerContent μ U
h₂ : innerContent μ U - ↑ε < ⨆ K, ⨆ (_ : ↑K ⊆ ↑U), (fun s => ↑(toFun μ s)) K
⊢ ∃ K, ↑K ⊆ ↑U ∧ innerContent μ U ≤ (fun s => ↑(toFun μ s)) K + ↑ε

Next tactic:
simp only [lt_iSup_iff] at h₂
theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ ∞) {ε : ℝ≥0}
    (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.innerContent U ≤ μ K + ε := by
  have h'ε := ENNReal.coe_ne_zero.2 hε
  rcases le_or_lt (μ.innerContent U) ε with h | h
  · exact ⟨⊥, empty_subset _, le_add_left h⟩
  have h₂ := ENNReal.sub_lt_self hU h.ne_bot h'ε
  conv at h₂ => rhs; rw [innerContent]
  simp only [lt_iSup_iff] at h₂
  

Proof state:
case inr
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U : Opens G
hU : innerContent μ U ≠ ⊤
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < innerContent μ U
h₂ : ∃ i, ∃ (_ : ↑i ⊆ ↑U), innerContent μ U - ↑ε < ↑(toFun μ i)
⊢ ∃ K, ↑K ⊆ ↑U ∧ innerContent μ U ≤ (fun s => ↑(toFun μ s)) K + ↑ε

Next tactic:
rcases h₂ with ⟨U, h1U, h2U⟩
theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ ∞) {ε : ℝ≥0}
    (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.innerContent U ≤ μ K + ε := by
  have h'ε := ENNReal.coe_ne_zero.2 hε
  rcases le_or_lt (μ.innerContent U) ε with h | h
  · exact ⟨⊥, empty_subset _, le_add_left h⟩
  have h₂ := ENNReal.sub_lt_self hU h.ne_bot h'ε
  conv at h₂ => rhs; rw [innerContent]
  simp only [lt_iSup_iff] at h₂
  rcases h₂ with ⟨U, h1U, h2U⟩; 

Proof state:
case inr.intro.intro
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U✝ : Opens G
hU : innerContent μ U✝ ≠ ⊤
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < innerContent μ U✝
U : Compacts G
h1U : ↑U ⊆ ↑U✝
h2U : innerContent μ U✝ - ↑ε < ↑(toFun μ U)
⊢ ∃ K, ↑K ⊆ ↑U✝ ∧ innerContent μ U✝ ≤ (fun s => ↑(toFun μ s)) K + ↑ε

Next tactic:
refine' ⟨U, h1U, _⟩
theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ ∞) {ε : ℝ≥0}
    (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.innerContent U ≤ μ K + ε := by
  have h'ε := ENNReal.coe_ne_zero.2 hε
  rcases le_or_lt (μ.innerContent U) ε with h | h
  · exact ⟨⊥, empty_subset _, le_add_left h⟩
  have h₂ := ENNReal.sub_lt_self hU h.ne_bot h'ε
  conv at h₂ => rhs; rw [innerContent]
  simp only [lt_iSup_iff] at h₂
  rcases h₂ with ⟨U, h1U, h2U⟩; refine' ⟨U, h1U, _⟩
  

Proof state:
case inr.intro.intro
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U✝ : Opens G
hU : innerContent μ U✝ ≠ ⊤
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < innerContent μ U✝
U : Compacts G
h1U : ↑U ⊆ ↑U✝
h2U : innerContent μ U✝ - ↑ε < ↑(toFun μ U)
⊢ innerContent μ U✝ ≤ (fun s => ↑(toFun μ s)) U + ↑ε

Next tactic:
rw [← tsub_le_iff_right]
theorem innerContent_exists_compact {U : Opens G} (hU : μ.innerContent U ≠ ∞) {ε : ℝ≥0}
    (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.innerContent U ≤ μ K + ε := by
  have h'ε := ENNReal.coe_ne_zero.2 hε
  rcases le_or_lt (μ.innerContent U) ε with h | h
  · exact ⟨⊥, empty_subset _, le_add_left h⟩
  have h₂ := ENNReal.sub_lt_self hU h.ne_bot h'ε
  conv at h₂ => rhs; rw [innerContent]
  simp only [lt_iSup_iff] at h₂
  rcases h₂ with ⟨U, h1U, h2U⟩; refine' ⟨U, h1U, _⟩
  rw [← tsub_le_iff_right]; 

Proof state:
case inr.intro.intro
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U✝ : Opens G
hU : innerContent μ U✝ ≠ ⊤
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < innerContent μ U✝
U : Compacts G
h1U : ↑U ⊆ ↑U✝
h2U : innerContent μ U✝ - ↑ε < ↑(toFun μ U)
⊢ innerContent μ U✝ - ↑ε ≤ (fun s => ↑(toFun μ s)) U

Next tactic:
exact le_of_lt h2U
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  

Proof state:
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
⊢ innerContent μ (⨆ i, U i) ≤ ∑' (i : ℕ), innerContent μ (U i)

Next tactic:
have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      rw [Finset.sup_insert, Finset.sum_insert hn]
      exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    

Proof state:
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
⊢ ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)

Next tactic:
intro t K
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    

Proof state:
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)

Next tactic:
refine' Finset.induction_on t _ _
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · 

Proof state:
case refine'_1
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ (fun s => ↑(toFun μ s)) (Finset.sup ∅ K) ≤ Finset.sum ∅ fun i => (fun s => ↑(toFun μ s)) (K i)

Next tactic:
simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · 

Proof state:
case refine'_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ ∀ ⦃a : ℕ⦄ {s : Finset ℕ},
    a ∉ s →
      ((fun s => ↑(toFun μ s)) (Finset.sup s K) ≤ Finset.sum s fun i => (fun s => ↑(toFun μ s)) (K i)) →
        (fun s => ↑(toFun μ s)) (Finset.sup (insert a s) K) ≤
          Finset.sum (insert a s) fun i => (fun s => ↑(toFun μ s)) (K i)

Next tactic:
intro n s hn ih
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      

Proof state:
case refine'_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
n : ℕ
s : Finset ℕ
hn : n ∉ s
ih : (fun s => ↑(toFun μ s)) (Finset.sup s K) ≤ Finset.sum s fun i => (fun s => ↑(toFun μ s)) (K i)
⊢ (fun s => ↑(toFun μ s)) (Finset.sup (insert n s) K) ≤ Finset.sum (insert n s) fun i => (fun s => ↑(toFun μ s)) (K i)

Next tactic:
rw [Finset.sup_insert, Finset.sum_insert hn]
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      rw [Finset.sup_insert, Finset.sum_insert hn]
      

Proof state:
case refine'_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
n : ℕ
s : Finset ℕ
hn : n ∉ s
ih : (fun s => ↑(toFun μ s)) (Finset.sup s K) ≤ Finset.sum s fun i => (fun s => ↑(toFun μ s)) (K i)
⊢ (fun s => ↑(toFun μ s)) (K n ⊔ Finset.sup s K) ≤
    (fun s => ↑(toFun μ s)) (K n) + Finset.sum s fun x => (fun s => ↑(toFun μ s)) (K x)

Next tactic:
exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      rw [Finset.sup_insert, Finset.sum_insert hn]
      exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
  

Proof state:
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)
⊢ innerContent μ (⨆ i, U i) ≤ ∑' (i : ℕ), innerContent μ (U i)

Next tactic:
refine' iSup₂_le fun K hK => _
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      rw [Finset.sup_insert, Finset.sum_insert hn]
      exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
  refine' iSup₂_le fun K hK => _
  

Proof state:
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
⊢ (fun s => ↑(toFun μ s)) K ≤ ∑' (i : ℕ), innerContent μ (U i)

Next tactic:
obtain ⟨t, ht⟩ :=
    K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by rwa [← Opens.coe_iSup])
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      rw [Finset.sup_insert, Finset.sum_insert hn]
      exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
  refine' iSup₂_le fun K hK => _
  obtain ⟨t, ht⟩ :=
    K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by 

Proof state:
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
⊢ ↑K ⊆ ⋃ i, ↑(U i)

Next tactic:
rwa [← Opens.coe_iSup]
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      rw [Finset.sup_insert, Finset.sum_insert hn]
      exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
  refine' iSup₂_le fun K hK => _
  obtain ⟨t, ht⟩ :=
    K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by rwa [← Opens.coe_iSup])
  

Proof state:
case intro
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
⊢ (fun s => ↑(toFun μ s)) K ≤ ∑' (i : ℕ), innerContent μ (U i)

Next tactic:
rcases K.isCompact.finite_compact_cover t (SetLike.coe ∘ U) (fun i _ => (U i).isOpen) ht with
    ⟨K', h1K', h2K', h3K'⟩
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      rw [Finset.sup_insert, Finset.sum_insert hn]
      exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
  refine' iSup₂_le fun K hK => _
  obtain ⟨t, ht⟩ :=
    K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by rwa [← Opens.coe_iSup])
  rcases K.isCompact.finite_compact_cover t (SetLike.coe ∘ U) (fun i _ => (U i).isOpen) ht with
    ⟨K', h1K', h2K', h3K'⟩
  

Proof state:
case intro.intro.intro.intro
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
⊢ (fun s => ↑(toFun μ s)) K ≤ ∑' (i : ℕ), innerContent μ (U i)

Next tactic:
let L : ℕ → Compacts G := fun n => ⟨K' n, h1K' n⟩
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      rw [Finset.sup_insert, Finset.sum_insert hn]
      exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
  refine' iSup₂_le fun K hK => _
  obtain ⟨t, ht⟩ :=
    K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by rwa [← Opens.coe_iSup])
  rcases K.isCompact.finite_compact_cover t (SetLike.coe ∘ U) (fun i _ => (U i).isOpen) ht with
    ⟨K', h1K', h2K', h3K'⟩
  let L : ℕ → Compacts G := fun n => ⟨K' n, h1K' n⟩
  

Proof state:
case intro.intro.intro.intro
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }
⊢ (fun s => ↑(toFun μ s)) K ≤ ∑' (i : ℕ), innerContent μ (U i)

Next tactic:
convert le_trans (h3 t L) _
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      rw [Finset.sup_insert, Finset.sum_insert hn]
      exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
  refine' iSup₂_le fun K hK => _
  obtain ⟨t, ht⟩ :=
    K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by rwa [← Opens.coe_iSup])
  rcases K.isCompact.finite_compact_cover t (SetLike.coe ∘ U) (fun i _ => (U i).isOpen) ht with
    ⟨K', h1K', h2K', h3K'⟩
  let L : ℕ → Compacts G := fun n => ⟨K' n, h1K' n⟩
  convert le_trans (h3 t L) _
  · 

Proof state:
case h.e'_3.h.e'_1
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }
⊢ K = Finset.sup t L

Next tactic:
ext1
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      rw [Finset.sup_insert, Finset.sum_insert hn]
      exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
  refine' iSup₂_le fun K hK => _
  obtain ⟨t, ht⟩ :=
    K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by rwa [← Opens.coe_iSup])
  rcases K.isCompact.finite_compact_cover t (SetLike.coe ∘ U) (fun i _ => (U i).isOpen) ht with
    ⟨K', h1K', h2K', h3K'⟩
  let L : ℕ → Compacts G := fun n => ⟨K' n, h1K' n⟩
  convert le_trans (h3 t L) _
  · ext1
    

Proof state:
case h.e'_3.h.e'_1.h
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }
⊢ ↑K = ↑(Finset.sup t L)

Next tactic:
rw [Compacts.coe_finset_sup, Finset.sup_eq_iSup]
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      rw [Finset.sup_insert, Finset.sum_insert hn]
      exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
  refine' iSup₂_le fun K hK => _
  obtain ⟨t, ht⟩ :=
    K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by rwa [← Opens.coe_iSup])
  rcases K.isCompact.finite_compact_cover t (SetLike.coe ∘ U) (fun i _ => (U i).isOpen) ht with
    ⟨K', h1K', h2K', h3K'⟩
  let L : ℕ → Compacts G := fun n => ⟨K' n, h1K' n⟩
  convert le_trans (h3 t L) _
  · ext1
    rw [Compacts.coe_finset_sup, Finset.sup_eq_iSup]
    

Proof state:
case h.e'_3.h.e'_1.h
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }
⊢ ↑K = ⨆ a ∈ t, ↑(L a)

Next tactic:
exact h3K'
/-- The inner content of a supremum of opens is at most the sum of the individual inner contents. -/
theorem innerContent_iSup_nat [T2Space G] (U : ℕ → Opens G) :
    μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) := by
  have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine' Finset.induction_on t _ _
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      rw [Finset.sup_insert, Finset.sum_insert hn]
      exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
  refine' iSup₂_le fun K hK => _
  obtain ⟨t, ht⟩ :=
    K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by rwa [← Opens.coe_iSup])
  rcases K.isCompact.finite_compact_cover t (SetLike.coe ∘ U) (fun i _ => (U i).isOpen) ht with
    ⟨K', h1K', h2K', h3K'⟩
  let L : ℕ → Compacts G := fun n => ⟨K' n, h1K' n⟩
  convert le_trans (h3 t L) _
  · ext1
    rw [Compacts.coe_finset_sup, Finset.sup_eq_iSup]
    exact h3K'
  

Proof state:
case intro.intro.intro.intro.convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : T2Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    (fun s => ↑(toFun μ s)) (Finset.sup t K) ≤ Finset.sum t fun i => (fun s => ↑(toFun μ s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) }
⊢ (Finset.sum t fun i => (fun s => ↑(toFun μ s)) (L i)) ≤ ∑' (i : ℕ), innerContent μ (U i)

Next tactic:
refine' le_trans (Finset.sum_le_sum _) (ENNReal.sum_le_tsum t)