l3lab/ntp-mathlib huggingface dataset:
/-- The truncated and non-truncated versions of `Xᵢ` have the same asymptotic behavior, as they
almost surely coincide at all but finitely many steps. This follows from a probability computation
and Borel-Cantelli. -/
theorem strong_law_aux5 :
    ∀ᵐ ω, (fun n : ℕ => ∑ i in range n, truncation (X i) i ω - ∑ i in range n, X i ω) =o[atTop]
    fun n : ℕ => (n : ℝ) := by
  have A : (∑' j : ℕ, ℙ {ω | X j ω ∈ Set.Ioi (j : ℝ)}) < ∞ := by
    convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2
    ext1 j
    exact (hident j).measure_mem_eq measurableSet_Ioi
  have B : ∀ᵐ ω, Tendsto (fun n : ℕ => truncation (X n) n ω - X n ω) atTop (𝓝 0) := by
    filter_upwards [ae_eventually_not_mem A.ne] with ω hω
    apply tendsto_const_nhds.congr' _
    filter_upwards [hω, Ioi_mem_atTop 0] with n hn npos
    simp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply]
    

Proof state:
case h
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ↑↑ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ⊤
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, X n ω ∉ Set.Ioi ↑n
n : ℕ
hn : X n ω ∉ Set.Ioi ↑n
npos : n ∈ Set.Ioi 0
⊢ 0 = (if -↑n < X n ω ∧ X n ω ≤ ↑n then X n ω else 0) - X n ω

Next tactic:
split_ifs with h
/-- The truncated and non-truncated versions of `Xᵢ` have the same asymptotic behavior, as they
almost surely coincide at all but finitely many steps. This follows from a probability computation
and Borel-Cantelli. -/
theorem strong_law_aux5 :
    ∀ᵐ ω, (fun n : ℕ => ∑ i in range n, truncation (X i) i ω - ∑ i in range n, X i ω) =o[atTop]
    fun n : ℕ => (n : ℝ) := by
  have A : (∑' j : ℕ, ℙ {ω | X j ω ∈ Set.Ioi (j : ℝ)}) < ∞ := by
    convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2
    ext1 j
    exact (hident j).measure_mem_eq measurableSet_Ioi
  have B : ∀ᵐ ω, Tendsto (fun n : ℕ => truncation (X n) n ω - X n ω) atTop (𝓝 0) := by
    filter_upwards [ae_eventually_not_mem A.ne] with ω hω
    apply tendsto_const_nhds.congr' _
    filter_upwards [hω, Ioi_mem_atTop 0] with n hn npos
    simp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply]
    split_ifs with h
    · 

Proof state:
case pos
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ↑↑ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ⊤
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, X n ω ∉ Set.Ioi ↑n
n : ℕ
hn : X n ω ∉ Set.Ioi ↑n
npos : n ∈ Set.Ioi 0
h : -↑n < X n ω ∧ X n ω ≤ ↑n
⊢ 0 = X n ω - X n ω

Next tactic:
exact (sub_self _).symm
/-- The truncated and non-truncated versions of `Xᵢ` have the same asymptotic behavior, as they
almost surely coincide at all but finitely many steps. This follows from a probability computation
and Borel-Cantelli. -/
theorem strong_law_aux5 :
    ∀ᵐ ω, (fun n : ℕ => ∑ i in range n, truncation (X i) i ω - ∑ i in range n, X i ω) =o[atTop]
    fun n : ℕ => (n : ℝ) := by
  have A : (∑' j : ℕ, ℙ {ω | X j ω ∈ Set.Ioi (j : ℝ)}) < ∞ := by
    convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2
    ext1 j
    exact (hident j).measure_mem_eq measurableSet_Ioi
  have B : ∀ᵐ ω, Tendsto (fun n : ℕ => truncation (X n) n ω - X n ω) atTop (𝓝 0) := by
    filter_upwards [ae_eventually_not_mem A.ne] with ω hω
    apply tendsto_const_nhds.congr' _
    filter_upwards [hω, Ioi_mem_atTop 0] with n hn npos
    simp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply]
    split_ifs with h
    · exact (sub_self _).symm
    · 

Proof state:
case neg
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ↑↑ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ⊤
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, X n ω ∉ Set.Ioi ↑n
n : ℕ
hn : X n ω ∉ Set.Ioi ↑n
npos : n ∈ Set.Ioi 0
h : ¬(-↑n < X n ω ∧ X n ω ≤ ↑n)
⊢ 0 = 0 - X n ω

Next tactic:
have : -(n : ℝ) < X n ω := by
        apply lt_of_lt_of_le _ (hnonneg n ω)
        simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos
/-- The truncated and non-truncated versions of `Xᵢ` have the same asymptotic behavior, as they
almost surely coincide at all but finitely many steps. This follows from a probability computation
and Borel-Cantelli. -/
theorem strong_law_aux5 :
    ∀ᵐ ω, (fun n : ℕ => ∑ i in range n, truncation (X i) i ω - ∑ i in range n, X i ω) =o[atTop]
    fun n : ℕ => (n : ℝ) := by
  have A : (∑' j : ℕ, ℙ {ω | X j ω ∈ Set.Ioi (j : ℝ)}) < ∞ := by
    convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2
    ext1 j
    exact (hident j).measure_mem_eq measurableSet_Ioi
  have B : ∀ᵐ ω, Tendsto (fun n : ℕ => truncation (X n) n ω - X n ω) atTop (𝓝 0) := by
    filter_upwards [ae_eventually_not_mem A.ne] with ω hω
    apply tendsto_const_nhds.congr' _
    filter_upwards [hω, Ioi_mem_atTop 0] with n hn npos
    simp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply]
    split_ifs with h
    · exact (sub_self _).symm
    · have : -(n : ℝ) < X n ω := by
        

Proof state:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ↑↑ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ⊤
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, X n ω ∉ Set.Ioi ↑n
n : ℕ
hn : X n ω ∉ Set.Ioi ↑n
npos : n ∈ Set.Ioi 0
h : ¬(-↑n < X n ω ∧ X n ω ≤ ↑n)
⊢ -↑n < X n ω

Next tactic:
apply lt_of_lt_of_le _ (hnonneg n ω)
/-- The truncated and non-truncated versions of `Xᵢ` have the same asymptotic behavior, as they
almost surely coincide at all but finitely many steps. This follows from a probability computation
and Borel-Cantelli. -/
theorem strong_law_aux5 :
    ∀ᵐ ω, (fun n : ℕ => ∑ i in range n, truncation (X i) i ω - ∑ i in range n, X i ω) =o[atTop]
    fun n : ℕ => (n : ℝ) := by
  have A : (∑' j : ℕ, ℙ {ω | X j ω ∈ Set.Ioi (j : ℝ)}) < ∞ := by
    convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2
    ext1 j
    exact (hident j).measure_mem_eq measurableSet_Ioi
  have B : ∀ᵐ ω, Tendsto (fun n : ℕ => truncation (X n) n ω - X n ω) atTop (𝓝 0) := by
    filter_upwards [ae_eventually_not_mem A.ne] with ω hω
    apply tendsto_const_nhds.congr' _
    filter_upwards [hω, Ioi_mem_atTop 0] with n hn npos
    simp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply]
    split_ifs with h
    · exact (sub_self _).symm
    · have : -(n : ℝ) < X n ω := by
        apply lt_of_lt_of_le _ (hnonneg n ω)
        

Proof state:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ↑↑ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ⊤
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, X n ω ∉ Set.Ioi ↑n
n : ℕ
hn : X n ω ∉ Set.Ioi ↑n
npos : n ∈ Set.Ioi 0
h : ¬(-↑n < X n ω ∧ X n ω ≤ ↑n)
⊢ -↑n < 0

Next tactic:
simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos
/-- The truncated and non-truncated versions of `Xᵢ` have the same asymptotic behavior, as they
almost surely coincide at all but finitely many steps. This follows from a probability computation
and Borel-Cantelli. -/
theorem strong_law_aux5 :
    ∀ᵐ ω, (fun n : ℕ => ∑ i in range n, truncation (X i) i ω - ∑ i in range n, X i ω) =o[atTop]
    fun n : ℕ => (n : ℝ) := by
  have A : (∑' j : ℕ, ℙ {ω | X j ω ∈ Set.Ioi (j : ℝ)}) < ∞ := by
    convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2
    ext1 j
    exact (hident j).measure_mem_eq measurableSet_Ioi
  have B : ∀ᵐ ω, Tendsto (fun n : ℕ => truncation (X n) n ω - X n ω) atTop (𝓝 0) := by
    filter_upwards [ae_eventually_not_mem A.ne] with ω hω
    apply tendsto_const_nhds.congr' _
    filter_upwards [hω, Ioi_mem_atTop 0] with n hn npos
    simp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply]
    split_ifs with h
    · exact (sub_self _).symm
    · have : -(n : ℝ) < X n ω := by
        apply lt_of_lt_of_le _ (hnonneg n ω)
        simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos
      

Proof state:
case neg
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ↑↑ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ⊤
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, X n ω ∉ Set.Ioi ↑n
n : ℕ
hn : X n ω ∉ Set.Ioi ↑n
npos : n ∈ Set.Ioi 0
h : ¬(-↑n < X n ω ∧ X n ω ≤ ↑n)
this : -↑n < X n ω
⊢ 0 = 0 - X n ω

Next tactic:
simp only [this, true_and_iff, not_le] at h
/-- The truncated and non-truncated versions of `Xᵢ` have the same asymptotic behavior, as they
almost surely coincide at all but finitely many steps. This follows from a probability computation
and Borel-Cantelli. -/
theorem strong_law_aux5 :
    ∀ᵐ ω, (fun n : ℕ => ∑ i in range n, truncation (X i) i ω - ∑ i in range n, X i ω) =o[atTop]
    fun n : ℕ => (n : ℝ) := by
  have A : (∑' j : ℕ, ℙ {ω | X j ω ∈ Set.Ioi (j : ℝ)}) < ∞ := by
    convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2
    ext1 j
    exact (hident j).measure_mem_eq measurableSet_Ioi
  have B : ∀ᵐ ω, Tendsto (fun n : ℕ => truncation (X n) n ω - X n ω) atTop (𝓝 0) := by
    filter_upwards [ae_eventually_not_mem A.ne] with ω hω
    apply tendsto_const_nhds.congr' _
    filter_upwards [hω, Ioi_mem_atTop 0] with n hn npos
    simp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply]
    split_ifs with h
    · exact (sub_self _).symm
    · have : -(n : ℝ) < X n ω := by
        apply lt_of_lt_of_le _ (hnonneg n ω)
        simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos
      simp only [this, true_and_iff, not_le] at h
      

Proof state:
case neg
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ↑↑ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ⊤
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, X n ω ∉ Set.Ioi ↑n
n : ℕ
hn : X n ω ∉ Set.Ioi ↑n
npos : n ∈ Set.Ioi 0
this : -↑n < X n ω
h : ↑n < X n ω
⊢ 0 = 0 - X n ω

Next tactic:
exact (hn h).elim
/-- The truncated and non-truncated versions of `Xᵢ` have the same asymptotic behavior, as they
almost surely coincide at all but finitely many steps. This follows from a probability computation
and Borel-Cantelli. -/
theorem strong_law_aux5 :
    ∀ᵐ ω, (fun n : ℕ => ∑ i in range n, truncation (X i) i ω - ∑ i in range n, X i ω) =o[atTop]
    fun n : ℕ => (n : ℝ) := by
  have A : (∑' j : ℕ, ℙ {ω | X j ω ∈ Set.Ioi (j : ℝ)}) < ∞ := by
    convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2
    ext1 j
    exact (hident j).measure_mem_eq measurableSet_Ioi
  have B : ∀ᵐ ω, Tendsto (fun n : ℕ => truncation (X n) n ω - X n ω) atTop (𝓝 0) := by
    filter_upwards [ae_eventually_not_mem A.ne] with ω hω
    apply tendsto_const_nhds.congr' _
    filter_upwards [hω, Ioi_mem_atTop 0] with n hn npos
    simp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply]
    split_ifs with h
    · exact (sub_self _).symm
    · have : -(n : ℝ) < X n ω := by
        apply lt_of_lt_of_le _ (hnonneg n ω)
        simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos
      simp only [this, true_and_iff, not_le] at h
      exact (hn h).elim
  

Proof state:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ↑↑ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ⊤
B : ∀ᵐ (ω : Ω), Tendsto (fun n => truncation (X n) (↑n) ω - X n ω) atTop (𝓝 0)
⊢ ∀ᵐ (ω : Ω), (fun n => ∑ i in range n, truncation (X i) (↑i) ω - ∑ i in range n, X i ω) =o[atTop] fun n => ↑n

Next tactic:
filter_upwards [B] with ω hω
/-- The truncated and non-truncated versions of `Xᵢ` have the same asymptotic behavior, as they
almost surely coincide at all but finitely many steps. This follows from a probability computation
and Borel-Cantelli. -/
theorem strong_law_aux5 :
    ∀ᵐ ω, (fun n : ℕ => ∑ i in range n, truncation (X i) i ω - ∑ i in range n, X i ω) =o[atTop]
    fun n : ℕ => (n : ℝ) := by
  have A : (∑' j : ℕ, ℙ {ω | X j ω ∈ Set.Ioi (j : ℝ)}) < ∞ := by
    convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2
    ext1 j
    exact (hident j).measure_mem_eq measurableSet_Ioi
  have B : ∀ᵐ ω, Tendsto (fun n : ℕ => truncation (X n) n ω - X n ω) atTop (𝓝 0) := by
    filter_upwards [ae_eventually_not_mem A.ne] with ω hω
    apply tendsto_const_nhds.congr' _
    filter_upwards [hω, Ioi_mem_atTop 0] with n hn npos
    simp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply]
    split_ifs with h
    · exact (sub_self _).symm
    · have : -(n : ℝ) < X n ω := by
        apply lt_of_lt_of_le _ (hnonneg n ω)
        simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos
      simp only [this, true_and_iff, not_le] at h
      exact (hn h).elim
  filter_upwards [B] with ω hω
  

Proof state:
case h
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ↑↑ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ⊤
B : ∀ᵐ (ω : Ω), Tendsto (fun n => truncation (X n) (↑n) ω - X n ω) atTop (𝓝 0)
ω : Ω
hω : Tendsto (fun n => truncation (X n) (↑n) ω - X n ω) atTop (𝓝 0)
⊢ (fun n => ∑ i in range n, truncation (X i) (↑i) ω - ∑ i in range n, X i ω) =o[atTop] fun n => ↑n

Next tactic:
convert isLittleO_sum_range_of_tendsto_zero hω using 1
/-- The truncated and non-truncated versions of `Xᵢ` have the same asymptotic behavior, as they
almost surely coincide at all but finitely many steps. This follows from a probability computation
and Borel-Cantelli. -/
theorem strong_law_aux5 :
    ∀ᵐ ω, (fun n : ℕ => ∑ i in range n, truncation (X i) i ω - ∑ i in range n, X i ω) =o[atTop]
    fun n : ℕ => (n : ℝ) := by
  have A : (∑' j : ℕ, ℙ {ω | X j ω ∈ Set.Ioi (j : ℝ)}) < ∞ := by
    convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2
    ext1 j
    exact (hident j).measure_mem_eq measurableSet_Ioi
  have B : ∀ᵐ ω, Tendsto (fun n : ℕ => truncation (X n) n ω - X n ω) atTop (𝓝 0) := by
    filter_upwards [ae_eventually_not_mem A.ne] with ω hω
    apply tendsto_const_nhds.congr' _
    filter_upwards [hω, Ioi_mem_atTop 0] with n hn npos
    simp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply]
    split_ifs with h
    · exact (sub_self _).symm
    · have : -(n : ℝ) < X n ω := by
        apply lt_of_lt_of_le _ (hnonneg n ω)
        simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos
      simp only [this, true_and_iff, not_le] at h
      exact (hn h).elim
  filter_upwards [B] with ω hω
  convert isLittleO_sum_range_of_tendsto_zero hω using 1
  

Proof state:
case h.e'_7
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ↑↑ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ⊤
B : ∀ᵐ (ω : Ω), Tendsto (fun n => truncation (X n) (↑n) ω - X n ω) atTop (𝓝 0)
ω : Ω
hω : Tendsto (fun n => truncation (X n) (↑n) ω - X n ω) atTop (𝓝 0)
⊢ (fun n => ∑ i in range n, truncation (X i) (↑i) ω - ∑ i in range n, X i ω) = fun n =>
    ∑ i in range n, (truncation (X i) (↑i) ω - X i ω)

Next tactic:
ext n
/-- The truncated and non-truncated versions of `Xᵢ` have the same asymptotic behavior, as they
almost surely coincide at all but finitely many steps. This follows from a probability computation
and Borel-Cantelli. -/
theorem strong_law_aux5 :
    ∀ᵐ ω, (fun n : ℕ => ∑ i in range n, truncation (X i) i ω - ∑ i in range n, X i ω) =o[atTop]
    fun n : ℕ => (n : ℝ) := by
  have A : (∑' j : ℕ, ℙ {ω | X j ω ∈ Set.Ioi (j : ℝ)}) < ∞ := by
    convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2
    ext1 j
    exact (hident j).measure_mem_eq measurableSet_Ioi
  have B : ∀ᵐ ω, Tendsto (fun n : ℕ => truncation (X n) n ω - X n ω) atTop (𝓝 0) := by
    filter_upwards [ae_eventually_not_mem A.ne] with ω hω
    apply tendsto_const_nhds.congr' _
    filter_upwards [hω, Ioi_mem_atTop 0] with n hn npos
    simp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply]
    split_ifs with h
    · exact (sub_self _).symm
    · have : -(n : ℝ) < X n ω := by
        apply lt_of_lt_of_le _ (hnonneg n ω)
        simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos
      simp only [this, true_and_iff, not_le] at h
      exact (hn h).elim
  filter_upwards [B] with ω hω
  convert isLittleO_sum_range_of_tendsto_zero hω using 1
  ext n
  

Proof state:
case h.e'_7.h
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ↑↑ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ⊤
B : ∀ᵐ (ω : Ω), Tendsto (fun n => truncation (X n) (↑n) ω - X n ω) atTop (𝓝 0)
ω : Ω
hω : Tendsto (fun n => truncation (X n) (↑n) ω - X n ω) atTop (𝓝 0)
n : ℕ
⊢ ∑ i in range n, truncation (X i) (↑i) ω - ∑ i in range n, X i ω = ∑ i in range n, (truncation (X i) (↑i) ω - X i ω)

Next tactic:
rw [sum_sub_distrib]
theorem strong_law_aux6 {c : ℝ} (c_one : 1 < c) :
    ∀ᵐ ω, Tendsto (fun n : ℕ => (∑ i in range ⌊c ^ n⌋₊, X i ω) / ⌊c ^ n⌋₊) atTop (𝓝 𝔼[X 0]) := by
  

Proof state:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
⊢ ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range ⌊c ^ n⌋₊, X i ω) / ↑⌊c ^ n⌋₊) atTop (𝓝 (∫ (a : Ω), X 0 a))

Next tactic:
have H : ∀ n : ℕ, (0 : ℝ) < ⌊c ^ n⌋₊ := by
    intro n
    refine' zero_lt_one.trans_le _
    simp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le c_one.le n]
theorem strong_law_aux6 {c : ℝ} (c_one : 1 < c) :
    ∀ᵐ ω, Tendsto (fun n : ℕ => (∑ i in range ⌊c ^ n⌋₊, X i ω) / ⌊c ^ n⌋₊) atTop (𝓝 𝔼[X 0]) := by
  have H : ∀ n : ℕ, (0 : ℝ) < ⌊c ^ n⌋₊ := by
    

Proof state:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
⊢ ∀ (n : ℕ), 0 < ↑⌊c ^ n⌋₊

Next tactic:
intro n
theorem strong_law_aux6 {c : ℝ} (c_one : 1 < c) :
    ∀ᵐ ω, Tendsto (fun n : ℕ => (∑ i in range ⌊c ^ n⌋₊, X i ω) / ⌊c ^ n⌋₊) atTop (𝓝 𝔼[X 0]) := by
  have H : ∀ n : ℕ, (0 : ℝ) < ⌊c ^ n⌋₊ := by
    intro n
    

Proof state:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
n : ℕ
⊢ 0 < ↑⌊c ^ n⌋₊

Next tactic:
refine' zero_lt_one.trans_le _
theorem strong_law_aux6 {c : ℝ} (c_one : 1 < c) :
    ∀ᵐ ω, Tendsto (fun n : ℕ => (∑ i in range ⌊c ^ n⌋₊, X i ω) / ⌊c ^ n⌋₊) atTop (𝓝 𝔼[X 0]) := by
  have H : ∀ n : ℕ, (0 : ℝ) < ⌊c ^ n⌋₊ := by
    intro n
    refine' zero_lt_one.trans_le _
    

Proof state:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
n : ℕ
⊢ 1 ≤ ↑⌊c ^ n⌋₊

Next tactic:
simp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le c_one.le n]
theorem strong_law_aux6 {c : ℝ} (c_one : 1 < c) :
    ∀ᵐ ω, Tendsto (fun n : ℕ => (∑ i in range ⌊c ^ n⌋₊, X i ω) / ⌊c ^ n⌋₊) atTop (𝓝 𝔼[X 0]) := by
  have H : ∀ n : ℕ, (0 : ℝ) < ⌊c ^ n⌋₊ := by
    intro n
    refine' zero_lt_one.trans_le _
    simp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le c_one.le n]
  

Proof state:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
H : ∀ (n : ℕ), 0 < ↑⌊c ^ n⌋₊
⊢ ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range ⌊c ^ n⌋₊, X i ω) / ↑⌊c ^ n⌋₊) atTop (𝓝 (∫ (a : Ω), X 0 a))

Next tactic:
filter_upwards [strong_law_aux4 X hint hindep hident hnonneg c_one,
    strong_law_aux5 X hint hident hnonneg] with ω hω h'ω
theorem strong_law_aux6 {c : ℝ} (c_one : 1 < c) :
    ∀ᵐ ω, Tendsto (fun n : ℕ => (∑ i in range ⌊c ^ n⌋₊, X i ω) / ⌊c ^ n⌋₊) atTop (𝓝 𝔼[X 0]) := by
  have H : ∀ n : ℕ, (0 : ℝ) < ⌊c ^ n⌋₊ := by
    intro n
    refine' zero_lt_one.trans_le _
    simp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le c_one.le n]
  filter_upwards [strong_law_aux4 X hint hindep hident hnonneg c_one,
    strong_law_aux5 X hint hident hnonneg] with ω hω h'ω
  

Proof state:
case h
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
H : ∀ (n : ℕ), 0 < ↑⌊c ^ n⌋₊
ω : Ω
hω :
  (fun n => ∑ i in range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ↑⌊c ^ n⌋₊ * ∫ (a : Ω), X 0 a) =o[atTop] fun n => ↑⌊c ^ n⌋₊
h'ω : (fun n => ∑ i in range n, truncation (X i) (↑i) ω - ∑ i in range n, X i ω) =o[atTop] fun n => ↑n
⊢ Tendsto (fun n => (∑ i in range ⌊c ^ n⌋₊, X i ω) / ↑⌊c ^ n⌋₊) atTop (𝓝 (∫ (a : Ω), X 0 a))

Next tactic:
rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ]
theorem strong_law_aux6 {c : ℝ} (c_one : 1 < c) :
    ∀ᵐ ω, Tendsto (fun n : ℕ => (∑ i in range ⌊c ^ n⌋₊, X i ω) / ⌊c ^ n⌋₊) atTop (𝓝 𝔼[X 0]) := by
  have H : ∀ n : ℕ, (0 : ℝ) < ⌊c ^ n⌋₊ := by
    intro n
    refine' zero_lt_one.trans_le _
    simp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le c_one.le n]
  filter_upwards [strong_law_aux4 X hint hindep hident hnonneg c_one,
    strong_law_aux5 X hint hident hnonneg] with ω hω h'ω
  rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ]
  

Proof state:
case h
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
H : ∀ (n : ℕ), 0 < ↑⌊c ^ n⌋₊
ω : Ω
hω :
  (fun n => ∑ i in range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ↑⌊c ^ n⌋₊ * ∫ (a : Ω), X 0 a) =o[atTop] fun n => ↑⌊c ^ n⌋₊
h'ω : (fun n => ∑ i in range n, truncation (X i) (↑i) ω - ∑ i in range n, X i ω) =o[atTop] fun n => ↑n
⊢ (fun x => (∑ i in range ⌊c ^ x⌋₊, X i ω) / ↑⌊c ^ x⌋₊ - ∫ (a : Ω), X 0 a) =o[atTop] fun _x => 1

Next tactic:
have L : (fun n : ℕ => ∑ i in range ⌊c ^ n⌋₊, X i ω - ⌊c ^ n⌋₊ * 𝔼[X 0]) =o[atTop] fun n =>
      (⌊c ^ n⌋₊ : ℝ) := by
    have A : Tendsto (fun n : ℕ => ⌊c ^ n⌋₊) atTop atTop :=
      tendsto_nat_floor_atTop.comp (tendsto_pow_atTop_atTop_of_one_lt c_one)
    convert hω.sub (h'ω.comp_tendsto A) using 1
    ext1 n
    simp only [Function.comp_apply, sub_sub_sub_cancel_left]