l3lab/ntp-mathlib huggingface dataset:
/-- Given any function `m` assigning measures to sets satisying `m ∅ = 0`, there is
  a unique maximal outer measure `μ` satisfying `μ s ≤ m s` for all `s : Set α`. -/
protected def ofFunction : OuterMeasure α :=
  let μ s := ⨅ (f : ℕ → Set α) (_ : s ⊆ ⋃ i, f i), ∑' i, m (f i)
  { measureOf := μ
    empty :=
      le_antisymm
        ((iInf_le_of_le fun _ => ∅) <| iInf_le_of_le (empty_subset _) <| by simp [m_empty])
        (zero_le _)
    mono := fun {s₁ s₂} hs => iInf_mono fun f => iInf_mono' fun hb => ⟨hs.trans hb, le_rfl⟩
    iUnion_nat := fun s =>
      ENNReal.le_of_forall_pos_le_add <| by
        intro ε hε (hb : (∑' i, μ (s i)) < ∞)
        rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 hε).ne' ℕ with ⟨ε', hε', hl⟩
        refine' le_trans _ (add_le_add_left (le_of_lt hl) _)
        rw [← ENNReal.tsum_add]
        choose f hf using
          show ∀ i, ∃ f : ℕ → Set α, (s i ⊆ ⋃ i, f i) ∧ (∑' i, m (f i)) < μ (s i) + ε' i by
            

Proof state:
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
μ : Set α → ℝ≥0∞ := fun s => ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)
s : ℕ → Set α
ε : ℝ≥0
hε : 0 < ε
hb : ∑' (i : ℕ), μ (s i) < ⊤
ε' : ℕ → ℝ≥0
hε' : ∀ (i : ℕ), 0 < ε' i
hl : ∑' (i : ℕ), ↑(ε' i) < ↑ε
⊢ ∀ (i : ℕ), ∃ f, s i ⊆ ⋃ i, f i ∧ ∑' (i : ℕ), m (f i) < μ (s i) + ↑(ε' i)

Next tactic:
intro i
/-- Given any function `m` assigning measures to sets satisying `m ∅ = 0`, there is
  a unique maximal outer measure `μ` satisfying `μ s ≤ m s` for all `s : Set α`. -/
protected def ofFunction : OuterMeasure α :=
  let μ s := ⨅ (f : ℕ → Set α) (_ : s ⊆ ⋃ i, f i), ∑' i, m (f i)
  { measureOf := μ
    empty :=
      le_antisymm
        ((iInf_le_of_le fun _ => ∅) <| iInf_le_of_le (empty_subset _) <| by simp [m_empty])
        (zero_le _)
    mono := fun {s₁ s₂} hs => iInf_mono fun f => iInf_mono' fun hb => ⟨hs.trans hb, le_rfl⟩
    iUnion_nat := fun s =>
      ENNReal.le_of_forall_pos_le_add <| by
        intro ε hε (hb : (∑' i, μ (s i)) < ∞)
        rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 hε).ne' ℕ with ⟨ε', hε', hl⟩
        refine' le_trans _ (add_le_add_left (le_of_lt hl) _)
        rw [← ENNReal.tsum_add]
        choose f hf using
          show ∀ i, ∃ f : ℕ → Set α, (s i ⊆ ⋃ i, f i) ∧ (∑' i, m (f i)) < μ (s i) + ε' i by
            intro i
            

Proof state:
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
μ : Set α → ℝ≥0∞ := fun s => ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)
s : ℕ → Set α
ε : ℝ≥0
hε : 0 < ε
hb : ∑' (i : ℕ), μ (s i) < ⊤
ε' : ℕ → ℝ≥0
hε' : ∀ (i : ℕ), 0 < ε' i
hl : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
⊢ ∃ f, s i ⊆ ⋃ i, f i ∧ ∑' (i : ℕ), m (f i) < μ (s i) + ↑(ε' i)

Next tactic:
have : μ (s i) < μ (s i) + ε' i :=
              ENNReal.lt_add_right (ne_top_of_le_ne_top hb.ne <| ENNReal.le_tsum _)
                (by simpa using (hε' i).ne')
/-- Given any function `m` assigning measures to sets satisying `m ∅ = 0`, there is
  a unique maximal outer measure `μ` satisfying `μ s ≤ m s` for all `s : Set α`. -/
protected def ofFunction : OuterMeasure α :=
  let μ s := ⨅ (f : ℕ → Set α) (_ : s ⊆ ⋃ i, f i), ∑' i, m (f i)
  { measureOf := μ
    empty :=
      le_antisymm
        ((iInf_le_of_le fun _ => ∅) <| iInf_le_of_le (empty_subset _) <| by simp [m_empty])
        (zero_le _)
    mono := fun {s₁ s₂} hs => iInf_mono fun f => iInf_mono' fun hb => ⟨hs.trans hb, le_rfl⟩
    iUnion_nat := fun s =>
      ENNReal.le_of_forall_pos_le_add <| by
        intro ε hε (hb : (∑' i, μ (s i)) < ∞)
        rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 hε).ne' ℕ with ⟨ε', hε', hl⟩
        refine' le_trans _ (add_le_add_left (le_of_lt hl) _)
        rw [← ENNReal.tsum_add]
        choose f hf using
          show ∀ i, ∃ f : ℕ → Set α, (s i ⊆ ⋃ i, f i) ∧ (∑' i, m (f i)) < μ (s i) + ε' i by
            intro i
            have : μ (s i) < μ (s i) + ε' i :=
              ENNReal.lt_add_right (ne_top_of_le_ne_top hb.ne <| ENNReal.le_tsum _)
                (by 

Proof state:
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
μ : Set α → ℝ≥0∞ := fun s => ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)
s : ℕ → Set α
ε : ℝ≥0
hε : 0 < ε
hb : ∑' (i : ℕ), μ (s i) < ⊤
ε' : ℕ → ℝ≥0
hε' : ∀ (i : ℕ), 0 < ε' i
hl : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
⊢ ↑(ε' i) ≠ 0

Next tactic:
simpa using (hε' i).ne'
/-- Given any function `m` assigning measures to sets satisying `m ∅ = 0`, there is
  a unique maximal outer measure `μ` satisfying `μ s ≤ m s` for all `s : Set α`. -/
protected def ofFunction : OuterMeasure α :=
  let μ s := ⨅ (f : ℕ → Set α) (_ : s ⊆ ⋃ i, f i), ∑' i, m (f i)
  { measureOf := μ
    empty :=
      le_antisymm
        ((iInf_le_of_le fun _ => ∅) <| iInf_le_of_le (empty_subset _) <| by simp [m_empty])
        (zero_le _)
    mono := fun {s₁ s₂} hs => iInf_mono fun f => iInf_mono' fun hb => ⟨hs.trans hb, le_rfl⟩
    iUnion_nat := fun s =>
      ENNReal.le_of_forall_pos_le_add <| by
        intro ε hε (hb : (∑' i, μ (s i)) < ∞)
        rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 hε).ne' ℕ with ⟨ε', hε', hl⟩
        refine' le_trans _ (add_le_add_left (le_of_lt hl) _)
        rw [← ENNReal.tsum_add]
        choose f hf using
          show ∀ i, ∃ f : ℕ → Set α, (s i ⊆ ⋃ i, f i) ∧ (∑' i, m (f i)) < μ (s i) + ε' i by
            intro i
            have : μ (s i) < μ (s i) + ε' i :=
              ENNReal.lt_add_right (ne_top_of_le_ne_top hb.ne <| ENNReal.le_tsum _)
                (by simpa using (hε' i).ne')
            

Proof state:
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
μ : Set α → ℝ≥0∞ := fun s => ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)
s : ℕ → Set α
ε : ℝ≥0
hε : 0 < ε
hb : ∑' (i : ℕ), μ (s i) < ⊤
ε' : ℕ → ℝ≥0
hε' : ∀ (i : ℕ), 0 < ε' i
hl : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
this : μ (s i) < μ (s i) + ↑(ε' i)
⊢ ∃ f, s i ⊆ ⋃ i, f i ∧ ∑' (i : ℕ), m (f i) < μ (s i) + ↑(ε' i)

Next tactic:
rcases iInf_lt_iff.mp this with ⟨t, ht⟩
/-- Given any function `m` assigning measures to sets satisying `m ∅ = 0`, there is
  a unique maximal outer measure `μ` satisfying `μ s ≤ m s` for all `s : Set α`. -/
protected def ofFunction : OuterMeasure α :=
  let μ s := ⨅ (f : ℕ → Set α) (_ : s ⊆ ⋃ i, f i), ∑' i, m (f i)
  { measureOf := μ
    empty :=
      le_antisymm
        ((iInf_le_of_le fun _ => ∅) <| iInf_le_of_le (empty_subset _) <| by simp [m_empty])
        (zero_le _)
    mono := fun {s₁ s₂} hs => iInf_mono fun f => iInf_mono' fun hb => ⟨hs.trans hb, le_rfl⟩
    iUnion_nat := fun s =>
      ENNReal.le_of_forall_pos_le_add <| by
        intro ε hε (hb : (∑' i, μ (s i)) < ∞)
        rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 hε).ne' ℕ with ⟨ε', hε', hl⟩
        refine' le_trans _ (add_le_add_left (le_of_lt hl) _)
        rw [← ENNReal.tsum_add]
        choose f hf using
          show ∀ i, ∃ f : ℕ → Set α, (s i ⊆ ⋃ i, f i) ∧ (∑' i, m (f i)) < μ (s i) + ε' i by
            intro i
            have : μ (s i) < μ (s i) + ε' i :=
              ENNReal.lt_add_right (ne_top_of_le_ne_top hb.ne <| ENNReal.le_tsum _)
                (by simpa using (hε' i).ne')
            rcases iInf_lt_iff.mp this with ⟨t, ht⟩
            

Proof state:
case intro
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
μ : Set α → ℝ≥0∞ := fun s => ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)
s : ℕ → Set α
ε : ℝ≥0
hε : 0 < ε
hb : ∑' (i : ℕ), μ (s i) < ⊤
ε' : ℕ → ℝ≥0
hε' : ∀ (i : ℕ), 0 < ε' i
hl : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
this : μ (s i) < μ (s i) + ↑(ε' i)
t : ℕ → Set α
ht : ⨅ (_ : s i ⊆ ⋃ i, t i), ∑' (i : ℕ), m (t i) < μ (s i) + ↑(ε' i)
⊢ ∃ f, s i ⊆ ⋃ i, f i ∧ ∑' (i : ℕ), m (f i) < μ (s i) + ↑(ε' i)

Next tactic:
exists t
/-- Given any function `m` assigning measures to sets satisying `m ∅ = 0`, there is
  a unique maximal outer measure `μ` satisfying `μ s ≤ m s` for all `s : Set α`. -/
protected def ofFunction : OuterMeasure α :=
  let μ s := ⨅ (f : ℕ → Set α) (_ : s ⊆ ⋃ i, f i), ∑' i, m (f i)
  { measureOf := μ
    empty :=
      le_antisymm
        ((iInf_le_of_le fun _ => ∅) <| iInf_le_of_le (empty_subset _) <| by simp [m_empty])
        (zero_le _)
    mono := fun {s₁ s₂} hs => iInf_mono fun f => iInf_mono' fun hb => ⟨hs.trans hb, le_rfl⟩
    iUnion_nat := fun s =>
      ENNReal.le_of_forall_pos_le_add <| by
        intro ε hε (hb : (∑' i, μ (s i)) < ∞)
        rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 hε).ne' ℕ with ⟨ε', hε', hl⟩
        refine' le_trans _ (add_le_add_left (le_of_lt hl) _)
        rw [← ENNReal.tsum_add]
        choose f hf using
          show ∀ i, ∃ f : ℕ → Set α, (s i ⊆ ⋃ i, f i) ∧ (∑' i, m (f i)) < μ (s i) + ε' i by
            intro i
            have : μ (s i) < μ (s i) + ε' i :=
              ENNReal.lt_add_right (ne_top_of_le_ne_top hb.ne <| ENNReal.le_tsum _)
                (by simpa using (hε' i).ne')
            rcases iInf_lt_iff.mp this with ⟨t, ht⟩
            exists t
            

Proof state:
case intro
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
μ : Set α → ℝ≥0∞ := fun s => ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)
s : ℕ → Set α
ε : ℝ≥0
hε : 0 < ε
hb : ∑' (i : ℕ), μ (s i) < ⊤
ε' : ℕ → ℝ≥0
hε' : ∀ (i : ℕ), 0 < ε' i
hl : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
this : μ (s i) < μ (s i) + ↑(ε' i)
t : ℕ → Set α
ht : ⨅ (_ : s i ⊆ ⋃ i, t i), ∑' (i : ℕ), m (t i) < μ (s i) + ↑(ε' i)
⊢ s i ⊆ ⋃ i, t i ∧ ∑' (i : ℕ), m (t i) < μ (s i) + ↑(ε' i)

Next tactic:
contrapose! ht
/-- Given any function `m` assigning measures to sets satisying `m ∅ = 0`, there is
  a unique maximal outer measure `μ` satisfying `μ s ≤ m s` for all `s : Set α`. -/
protected def ofFunction : OuterMeasure α :=
  let μ s := ⨅ (f : ℕ → Set α) (_ : s ⊆ ⋃ i, f i), ∑' i, m (f i)
  { measureOf := μ
    empty :=
      le_antisymm
        ((iInf_le_of_le fun _ => ∅) <| iInf_le_of_le (empty_subset _) <| by simp [m_empty])
        (zero_le _)
    mono := fun {s₁ s₂} hs => iInf_mono fun f => iInf_mono' fun hb => ⟨hs.trans hb, le_rfl⟩
    iUnion_nat := fun s =>
      ENNReal.le_of_forall_pos_le_add <| by
        intro ε hε (hb : (∑' i, μ (s i)) < ∞)
        rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 hε).ne' ℕ with ⟨ε', hε', hl⟩
        refine' le_trans _ (add_le_add_left (le_of_lt hl) _)
        rw [← ENNReal.tsum_add]
        choose f hf using
          show ∀ i, ∃ f : ℕ → Set α, (s i ⊆ ⋃ i, f i) ∧ (∑' i, m (f i)) < μ (s i) + ε' i by
            intro i
            have : μ (s i) < μ (s i) + ε' i :=
              ENNReal.lt_add_right (ne_top_of_le_ne_top hb.ne <| ENNReal.le_tsum _)
                (by simpa using (hε' i).ne')
            rcases iInf_lt_iff.mp this with ⟨t, ht⟩
            exists t
            contrapose! ht
            

Proof state:
case intro
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
μ : Set α → ℝ≥0∞ := fun s => ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)
s : ℕ → Set α
ε : ℝ≥0
hε : 0 < ε
hb : ∑' (i : ℕ), μ (s i) < ⊤
ε' : ℕ → ℝ≥0
hε' : ∀ (i : ℕ), 0 < ε' i
hl : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
this : μ (s i) < μ (s i) + ↑(ε' i)
t : ℕ → Set α
ht : s i ⊆ ⋃ i, t i → μ (s i) + ↑(ε' i) ≤ ∑' (i : ℕ), m (t i)
⊢ μ (s i) + ↑(ε' i) ≤ ⨅ (_ : s i ⊆ ⋃ i, t i), ∑' (i : ℕ), m (t i)

Next tactic:
exact le_iInf ht
/-- Given any function `m` assigning measures to sets satisying `m ∅ = 0`, there is
  a unique maximal outer measure `μ` satisfying `μ s ≤ m s` for all `s : Set α`. -/
protected def ofFunction : OuterMeasure α :=
  let μ s := ⨅ (f : ℕ → Set α) (_ : s ⊆ ⋃ i, f i), ∑' i, m (f i)
  { measureOf := μ
    empty :=
      le_antisymm
        ((iInf_le_of_le fun _ => ∅) <| iInf_le_of_le (empty_subset _) <| by simp [m_empty])
        (zero_le _)
    mono := fun {s₁ s₂} hs => iInf_mono fun f => iInf_mono' fun hb => ⟨hs.trans hb, le_rfl⟩
    iUnion_nat := fun s =>
      ENNReal.le_of_forall_pos_le_add <| by
        intro ε hε (hb : (∑' i, μ (s i)) < ∞)
        rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 hε).ne' ℕ with ⟨ε', hε', hl⟩
        refine' le_trans _ (add_le_add_left (le_of_lt hl) _)
        rw [← ENNReal.tsum_add]
        choose f hf using
          show ∀ i, ∃ f : ℕ → Set α, (s i ⊆ ⋃ i, f i) ∧ (∑' i, m (f i)) < μ (s i) + ε' i by
            intro i
            have : μ (s i) < μ (s i) + ε' i :=
              ENNReal.lt_add_right (ne_top_of_le_ne_top hb.ne <| ENNReal.le_tsum _)
                (by simpa using (hε' i).ne')
            rcases iInf_lt_iff.mp this with ⟨t, ht⟩
            exists t
            contrapose! ht
            exact le_iInf ht
        

Proof state:
case intro.intro
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
μ : Set α → ℝ≥0∞ := fun s => ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)
s : ℕ → Set α
ε : ℝ≥0
hε : 0 < ε
hb : ∑' (i : ℕ), μ (s i) < ⊤
ε' : ℕ → ℝ≥0
hε' : ∀ (i : ℕ), 0 < ε' i
hl : ∑' (i : ℕ), ↑(ε' i) < ↑ε
f : ℕ → ℕ → Set α
hf : ∀ (i : ℕ), s i ⊆ ⋃ i_1, f i i_1 ∧ ∑' (i_1 : ℕ), m (f i i_1) < μ (s i) + ↑(ε' i)
⊢ μ (⋃ i, s i) ≤ ∑' (a : ℕ), (μ (s a) + ↑(ε' a))

Next tactic:
refine' le_trans _ (ENNReal.tsum_le_tsum fun i => le_of_lt (hf i).2)
/-- Given any function `m` assigning measures to sets satisying `m ∅ = 0`, there is
  a unique maximal outer measure `μ` satisfying `μ s ≤ m s` for all `s : Set α`. -/
protected def ofFunction : OuterMeasure α :=
  let μ s := ⨅ (f : ℕ → Set α) (_ : s ⊆ ⋃ i, f i), ∑' i, m (f i)
  { measureOf := μ
    empty :=
      le_antisymm
        ((iInf_le_of_le fun _ => ∅) <| iInf_le_of_le (empty_subset _) <| by simp [m_empty])
        (zero_le _)
    mono := fun {s₁ s₂} hs => iInf_mono fun f => iInf_mono' fun hb => ⟨hs.trans hb, le_rfl⟩
    iUnion_nat := fun s =>
      ENNReal.le_of_forall_pos_le_add <| by
        intro ε hε (hb : (∑' i, μ (s i)) < ∞)
        rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 hε).ne' ℕ with ⟨ε', hε', hl⟩
        refine' le_trans _ (add_le_add_left (le_of_lt hl) _)
        rw [← ENNReal.tsum_add]
        choose f hf using
          show ∀ i, ∃ f : ℕ → Set α, (s i ⊆ ⋃ i, f i) ∧ (∑' i, m (f i)) < μ (s i) + ε' i by
            intro i
            have : μ (s i) < μ (s i) + ε' i :=
              ENNReal.lt_add_right (ne_top_of_le_ne_top hb.ne <| ENNReal.le_tsum _)
                (by simpa using (hε' i).ne')
            rcases iInf_lt_iff.mp this with ⟨t, ht⟩
            exists t
            contrapose! ht
            exact le_iInf ht
        refine' le_trans _ (ENNReal.tsum_le_tsum fun i => le_of_lt (hf i).2)
        

Proof state:
case intro.intro
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
μ : Set α → ℝ≥0∞ := fun s => ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)
s : ℕ → Set α
ε : ℝ≥0
hε : 0 < ε
hb : ∑' (i : ℕ), μ (s i) < ⊤
ε' : ℕ → ℝ≥0
hε' : ∀ (i : ℕ), 0 < ε' i
hl : ∑' (i : ℕ), ↑(ε' i) < ↑ε
f : ℕ → ℕ → Set α
hf : ∀ (i : ℕ), s i ⊆ ⋃ i_1, f i i_1 ∧ ∑' (i_1 : ℕ), m (f i i_1) < μ (s i) + ↑(ε' i)
⊢ μ (⋃ i, s i) ≤ ∑' (a : ℕ) (i : ℕ), m (f a i)

Next tactic:
rw [← ENNReal.tsum_prod, ← Nat.pairEquiv.symm.tsum_eq]
/-- Given any function `m` assigning measures to sets satisying `m ∅ = 0`, there is
  a unique maximal outer measure `μ` satisfying `μ s ≤ m s` for all `s : Set α`. -/
protected def ofFunction : OuterMeasure α :=
  let μ s := ⨅ (f : ℕ → Set α) (_ : s ⊆ ⋃ i, f i), ∑' i, m (f i)
  { measureOf := μ
    empty :=
      le_antisymm
        ((iInf_le_of_le fun _ => ∅) <| iInf_le_of_le (empty_subset _) <| by simp [m_empty])
        (zero_le _)
    mono := fun {s₁ s₂} hs => iInf_mono fun f => iInf_mono' fun hb => ⟨hs.trans hb, le_rfl⟩
    iUnion_nat := fun s =>
      ENNReal.le_of_forall_pos_le_add <| by
        intro ε hε (hb : (∑' i, μ (s i)) < ∞)
        rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 hε).ne' ℕ with ⟨ε', hε', hl⟩
        refine' le_trans _ (add_le_add_left (le_of_lt hl) _)
        rw [← ENNReal.tsum_add]
        choose f hf using
          show ∀ i, ∃ f : ℕ → Set α, (s i ⊆ ⋃ i, f i) ∧ (∑' i, m (f i)) < μ (s i) + ε' i by
            intro i
            have : μ (s i) < μ (s i) + ε' i :=
              ENNReal.lt_add_right (ne_top_of_le_ne_top hb.ne <| ENNReal.le_tsum _)
                (by simpa using (hε' i).ne')
            rcases iInf_lt_iff.mp this with ⟨t, ht⟩
            exists t
            contrapose! ht
            exact le_iInf ht
        refine' le_trans _ (ENNReal.tsum_le_tsum fun i => le_of_lt (hf i).2)
        rw [← ENNReal.tsum_prod, ← Nat.pairEquiv.symm.tsum_eq]
        

Proof state:
case intro.intro
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
μ : Set α → ℝ≥0∞ := fun s => ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)
s : ℕ → Set α
ε : ℝ≥0
hε : 0 < ε
hb : ∑' (i : ℕ), μ (s i) < ⊤
ε' : ℕ → ℝ≥0
hε' : ∀ (i : ℕ), 0 < ε' i
hl : ∑' (i : ℕ), ↑(ε' i) < ↑ε
f : ℕ → ℕ → Set α
hf : ∀ (i : ℕ), s i ⊆ ⋃ i_1, f i i_1 ∧ ∑' (i_1 : ℕ), m (f i i_1) < μ (s i) + ↑(ε' i)
⊢ μ (⋃ i, s i) ≤ ∑' (c : ℕ), m (f (Nat.pairEquiv.symm c).1 (Nat.pairEquiv.symm c).2)

Next tactic:
refine' iInf_le_of_le _ (iInf_le _ _)
/-- Given any function `m` assigning measures to sets satisying `m ∅ = 0`, there is
  a unique maximal outer measure `μ` satisfying `μ s ≤ m s` for all `s : Set α`. -/
protected def ofFunction : OuterMeasure α :=
  let μ s := ⨅ (f : ℕ → Set α) (_ : s ⊆ ⋃ i, f i), ∑' i, m (f i)
  { measureOf := μ
    empty :=
      le_antisymm
        ((iInf_le_of_le fun _ => ∅) <| iInf_le_of_le (empty_subset _) <| by simp [m_empty])
        (zero_le _)
    mono := fun {s₁ s₂} hs => iInf_mono fun f => iInf_mono' fun hb => ⟨hs.trans hb, le_rfl⟩
    iUnion_nat := fun s =>
      ENNReal.le_of_forall_pos_le_add <| by
        intro ε hε (hb : (∑' i, μ (s i)) < ∞)
        rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 hε).ne' ℕ with ⟨ε', hε', hl⟩
        refine' le_trans _ (add_le_add_left (le_of_lt hl) _)
        rw [← ENNReal.tsum_add]
        choose f hf using
          show ∀ i, ∃ f : ℕ → Set α, (s i ⊆ ⋃ i, f i) ∧ (∑' i, m (f i)) < μ (s i) + ε' i by
            intro i
            have : μ (s i) < μ (s i) + ε' i :=
              ENNReal.lt_add_right (ne_top_of_le_ne_top hb.ne <| ENNReal.le_tsum _)
                (by simpa using (hε' i).ne')
            rcases iInf_lt_iff.mp this with ⟨t, ht⟩
            exists t
            contrapose! ht
            exact le_iInf ht
        refine' le_trans _ (ENNReal.tsum_le_tsum fun i => le_of_lt (hf i).2)
        rw [← ENNReal.tsum_prod, ← Nat.pairEquiv.symm.tsum_eq]
        refine' iInf_le_of_le _ (iInf_le _ _)
        

Proof state:
case intro.intro
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
μ : Set α → ℝ≥0∞ := fun s => ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)
s : ℕ → Set α
ε : ℝ≥0
hε : 0 < ε
hb : ∑' (i : ℕ), μ (s i) < ⊤
ε' : ℕ → ℝ≥0
hε' : ∀ (i : ℕ), 0 < ε' i
hl : ∑' (i : ℕ), ↑(ε' i) < ↑ε
f : ℕ → ℕ → Set α
hf : ∀ (i : ℕ), s i ⊆ ⋃ i_1, f i i_1 ∧ ∑' (i_1 : ℕ), m (f i i_1) < μ (s i) + ↑(ε' i)
⊢ ⋃ i, s i ⊆ ⋃ i, f (Nat.pairEquiv.symm i).1 (Nat.pairEquiv.symm i).2

Next tactic:
apply iUnion_subset
/-- Given any function `m` assigning measures to sets satisying `m ∅ = 0`, there is
  a unique maximal outer measure `μ` satisfying `μ s ≤ m s` for all `s : Set α`. -/
protected def ofFunction : OuterMeasure α :=
  let μ s := ⨅ (f : ℕ → Set α) (_ : s ⊆ ⋃ i, f i), ∑' i, m (f i)
  { measureOf := μ
    empty :=
      le_antisymm
        ((iInf_le_of_le fun _ => ∅) <| iInf_le_of_le (empty_subset _) <| by simp [m_empty])
        (zero_le _)
    mono := fun {s₁ s₂} hs => iInf_mono fun f => iInf_mono' fun hb => ⟨hs.trans hb, le_rfl⟩
    iUnion_nat := fun s =>
      ENNReal.le_of_forall_pos_le_add <| by
        intro ε hε (hb : (∑' i, μ (s i)) < ∞)
        rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 hε).ne' ℕ with ⟨ε', hε', hl⟩
        refine' le_trans _ (add_le_add_left (le_of_lt hl) _)
        rw [← ENNReal.tsum_add]
        choose f hf using
          show ∀ i, ∃ f : ℕ → Set α, (s i ⊆ ⋃ i, f i) ∧ (∑' i, m (f i)) < μ (s i) + ε' i by
            intro i
            have : μ (s i) < μ (s i) + ε' i :=
              ENNReal.lt_add_right (ne_top_of_le_ne_top hb.ne <| ENNReal.le_tsum _)
                (by simpa using (hε' i).ne')
            rcases iInf_lt_iff.mp this with ⟨t, ht⟩
            exists t
            contrapose! ht
            exact le_iInf ht
        refine' le_trans _ (ENNReal.tsum_le_tsum fun i => le_of_lt (hf i).2)
        rw [← ENNReal.tsum_prod, ← Nat.pairEquiv.symm.tsum_eq]
        refine' iInf_le_of_le _ (iInf_le _ _)
        apply iUnion_subset
        

Proof state:
case intro.intro.h
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
μ : Set α → ℝ≥0∞ := fun s => ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)
s : ℕ → Set α
ε : ℝ≥0
hε : 0 < ε
hb : ∑' (i : ℕ), μ (s i) < ⊤
ε' : ℕ → ℝ≥0
hε' : ∀ (i : ℕ), 0 < ε' i
hl : ∑' (i : ℕ), ↑(ε' i) < ↑ε
f : ℕ → ℕ → Set α
hf : ∀ (i : ℕ), s i ⊆ ⋃ i_1, f i i_1 ∧ ∑' (i_1 : ℕ), m (f i i_1) < μ (s i) + ↑(ε' i)
⊢ ∀ (i : ℕ), s i ⊆ ⋃ i, f (Nat.pairEquiv.symm i).1 (Nat.pairEquiv.symm i).2

Next tactic:
intro i
/-- Given any function `m` assigning measures to sets satisying `m ∅ = 0`, there is
  a unique maximal outer measure `μ` satisfying `μ s ≤ m s` for all `s : Set α`. -/
protected def ofFunction : OuterMeasure α :=
  let μ s := ⨅ (f : ℕ → Set α) (_ : s ⊆ ⋃ i, f i), ∑' i, m (f i)
  { measureOf := μ
    empty :=
      le_antisymm
        ((iInf_le_of_le fun _ => ∅) <| iInf_le_of_le (empty_subset _) <| by simp [m_empty])
        (zero_le _)
    mono := fun {s₁ s₂} hs => iInf_mono fun f => iInf_mono' fun hb => ⟨hs.trans hb, le_rfl⟩
    iUnion_nat := fun s =>
      ENNReal.le_of_forall_pos_le_add <| by
        intro ε hε (hb : (∑' i, μ (s i)) < ∞)
        rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 hε).ne' ℕ with ⟨ε', hε', hl⟩
        refine' le_trans _ (add_le_add_left (le_of_lt hl) _)
        rw [← ENNReal.tsum_add]
        choose f hf using
          show ∀ i, ∃ f : ℕ → Set α, (s i ⊆ ⋃ i, f i) ∧ (∑' i, m (f i)) < μ (s i) + ε' i by
            intro i
            have : μ (s i) < μ (s i) + ε' i :=
              ENNReal.lt_add_right (ne_top_of_le_ne_top hb.ne <| ENNReal.le_tsum _)
                (by simpa using (hε' i).ne')
            rcases iInf_lt_iff.mp this with ⟨t, ht⟩
            exists t
            contrapose! ht
            exact le_iInf ht
        refine' le_trans _ (ENNReal.tsum_le_tsum fun i => le_of_lt (hf i).2)
        rw [← ENNReal.tsum_prod, ← Nat.pairEquiv.symm.tsum_eq]
        refine' iInf_le_of_le _ (iInf_le _ _)
        apply iUnion_subset
        intro i
        

Proof state:
case intro.intro.h
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
μ : Set α → ℝ≥0∞ := fun s => ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)
s : ℕ → Set α
ε : ℝ≥0
hε : 0 < ε
hb : ∑' (i : ℕ), μ (s i) < ⊤
ε' : ℕ → ℝ≥0
hε' : ∀ (i : ℕ), 0 < ε' i
hl : ∑' (i : ℕ), ↑(ε' i) < ↑ε
f : ℕ → ℕ → Set α
hf : ∀ (i : ℕ), s i ⊆ ⋃ i_1, f i i_1 ∧ ∑' (i_1 : ℕ), m (f i i_1) < μ (s i) + ↑(ε' i)
i : ℕ
⊢ s i ⊆ ⋃ i, f (Nat.pairEquiv.symm i).1 (Nat.pairEquiv.symm i).2

Next tactic:
apply Subset.trans (hf i).1