Module: Mathlib.Probability.Martingale.BorelCantelli

Goal state before:
pp: case mp
Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hfmono : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), f n ω ≤ f (n + 1) ω
hf : StronglyAdapted ℱ f
hint : ∀ (n : ℕ), Integrable (f n) μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), ‖f (n + 1) ω - f n ω‖ ≤ ↑R
h₀ : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |martingalePart f ℱ μ (i + 1) ω - martingalePart f ℱ μ i ω| ≤ ↑(2 * R)
h₁ : ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n ↦ martingalePart f ℱ μ n ω) atTop atTop
h₂ : ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n ↦ martingalePart f ℱ μ n ω) atTop atBot
h₃ : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), 0 ≤ μ[f (n + 1) - f n | ↑ℱ n] ω
ω : Ω
hω₁ : ¬Tendsto (fun n ↦ martingalePart f ℱ μ n ω) atTop atTop
hω₂ : ¬Tendsto (fun n ↦ martingalePart f ℱ μ n ω) atTop atBot
hω₃ : ∀ (n : ℕ), 0 ≤ μ[f (n + 1) - f n | ↑ℱ n] ω
hω₄ : ∀ (n : ℕ), f n ω ≤ f (n + 1) ω
ht : Tendsto (fun n ↦ f n ω) atTop atTop
⊢ Tendsto (fun n ↦ predictablePart f ℱ μ n ω) atTop atTop

pp: case mpr
Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hfmono : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), f n ω ≤ f (n + 1) ω
hf : StronglyAdapted ℱ f
hint : ∀ (n : ℕ), Integrable (f n) μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), ‖f (n + 1) ω - f n ω‖ ≤ ↑R
h₀ : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |martingalePart f ℱ μ (i + 1) ω - martingalePart f ℱ μ i ω| ≤ ↑(2 * R)
h₁ : ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n ↦ martingalePart f ℱ μ n ω) atTop atTop
h₂ : ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n ↦ martingalePart f ℱ μ n ω) atTop atBot
h₃ : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), 0 ≤ μ[f (n + 1) - f n | ↑ℱ n] ω
ω : Ω
hω₁ : ¬Tendsto (fun n ↦ martingalePart f ℱ μ n ω) atTop atTop
hω₂ : ¬Tendsto (fun n ↦ martingalePart f ℱ μ n ω) atTop atBot
hω₃ : ∀ (n : ℕ), 0 ≤ μ[f (n + 1) - f n | ↑ℱ n] ω
hω₄ : ∀ (n : ℕ), f n ω ≤ f (n + 1) ω
ht : Tendsto (fun n ↦ predictablePart f ℱ μ n ω) atTop atTop
⊢ Tendsto (fun n ↦ f n ω) atTop atTop

Tactic:
· refine tendsto_atTop_atTop_of_monotone' ?_ ?_
    · intro n m hnm
      simp only [predictablePart, Finset.sum_apply]
      exact Finset.sum_mono_set_of_nonneg hω₃ (Finset.range_mono hnm)
    rintro ⟨b, hbdd⟩
    rw [← tendsto_neg_atBot_iff] at ht
    simp only [martingalePart, sub_eq_add_neg] at hω₁
    exact hω₁ (tendsto_atTop_add_right_of_le _ (-b) (tendsto_neg_atBot_iff.1 ht) fun n =>
      neg_le_neg (hbdd ⟨n, rfl⟩))

Goal state after:
case mpr
Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hfmono : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), f n ω ≤ f (n + 1) ω
hf : StronglyAdapted ℱ f
hint : ∀ (n : ℕ), Integrable (f n) μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), ‖f (n + 1) ω - f n ω‖ ≤ ↑R
h₀ : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |martingalePart f ℱ μ (i + 1) ω - martingalePart f ℱ μ i ω| ≤ ↑(2 * R)
h₁ : ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n ↦ martingalePart f ℱ μ n ω) atTop atTop
h₂ : ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n ↦ martingalePart f ℱ μ n ω) atTop atBot
h₃ : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), 0 ≤ μ[f (n + 1) - f n | ↑ℱ n] ω
ω : Ω
hω₁ : ¬Tendsto (fun n ↦ martingalePart f ℱ μ n ω) atTop atTop
hω₂ : ¬Tendsto (fun n ↦ martingalePart f ℱ μ n ω) atTop atBot
hω₃ : ∀ (n : ℕ), 0 ≤ μ[f (n + 1) - f n | ↑ℱ n] ω
hω₄ : ∀ (n : ℕ), f n ω ≤ f (n + 1) ω
ht : Tendsto (fun n ↦ predictablePart f ℱ μ n ω) atTop atTop
⊢ Tendsto (fun n ↦ f n ω) atTop atTop