-- Mathlib/Probability/Martingale/Centering.lean
theorem martingale_martingalePart (hf : Adapted ℱ f) (hf_int : ∀ n, Integrable (f n) μ)
    [SigmaFiniteFiltration μ ℱ] : Martingale (martingalePart f ℱ μ) ℱ μ

Proof state:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
⊢ μ[f 0|↑ℱ i] + μ[∑ i ∈ Finset.range j, (f (i + 1) - f i - μ[f (i + 1) - f i|↑ℱ i])|↑ℱ i] =ᶠ[ae μ]
    f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])

Tactic:
refine (EventuallyEq.rfl.add (condExp_finset_sum (fun i _ => by fun_prop) _)).trans ?_

Result:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
⊢ (fun x =>
      (μ[f 0|↑ℱ i]) x +
        (∑ i_1 ∈ Finset.range j, μ[f (i_1 + 1) - f i_1 - μ[f (i_1 + 1) - f i_1|↑ℱ i_1]|↑ℱ i]) x) =ᶠ[ae μ]
    f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])

Proof state:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum :
  μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ]
    f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
⊢ f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i]) =ᶠ[ae μ]
    martingalePart f ℱ μ i

Tactic:
have h_ge : ∀ k, i ≤ k → μ[f (k + 1) - f k|ℱ i] - μ[μ[f (k + 1) - f k|ℱ k]|ℱ i] =ᵐ[μ] 0 := by
  intro k hk
  have : μ[μ[f (k + 1) - f k|ℱ k]|ℱ i] =ᵐ[μ] μ[f (k + 1) - f k|ℱ i] :=
    condExp_condExp_of_le (ℱ.mono hk) (ℱ.le k)
  filter_upwards [this] with x hx
  rw [Pi.sub_apply, Pi.zero_apply, hx, sub_self]

Result:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum :
  μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ]
    f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
h_ge : ∀ (k : ℕ), i ≤ k → μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] 0
⊢ f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i]) =ᶠ[ae μ]
    martingalePart f ℱ μ i

Proof state:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum :
  μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ]
    f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
h_ge : ∀ (k : ℕ), i ≤ k → μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] 0
h_lt :
  ∀ k < i, μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] f (k + 1) - f k - μ[f (k + 1) - f k|↑ℱ k]
⊢ f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i]) =ᶠ[ae μ]
    martingalePart f ℱ μ i

Tactic:
rw [martingalePart_eq_sum]

Result:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum :
  μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ]
    f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
h_ge : ∀ (k : ℕ), i ≤ k → μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] 0
h_lt :
  ∀ k < i, μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] f (k + 1) - f k - μ[f (k + 1) - f k|↑ℱ k]
⊢ f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i]) =ᶠ[ae μ]
    (fun n => f 0 + ∑ i ∈ Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i|↑ℱ i])) i

Proof state:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum :
  μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ]
    f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
k : ℕ
hk : i ≤ k
this : μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] μ[f (k + 1) - f k|↑ℱ i]
⊢ μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] 0

Tactic:
filter_upwards [this] with x hx

Result:
case h
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum :
  μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ]
    f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
k : ℕ
hk : i ≤ k
this : μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] μ[f (k + 1) - f k|↑ℱ i]
x : Ω
hx : (μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i]) x = (μ[f (k + 1) - f k|↑ℱ i]) x
⊢ (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i]) x = 0 x

Proof state:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
⊢ μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ]
    f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])

Tactic:
rw [martingalePart_eq_sum]

Result:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
⊢ μ[(fun n => f 0 + ∑ i ∈ Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i|↑ℱ i])) j|↑ℱ i] =ᶠ[ae μ]
    f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])

Proof state:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
k : ℕ
x✝ : k ∈ Finset.range j
⊢ Integrable (f (k + 1) - f k) μ

Tactic:
fun_prop

Result:
no goals