-- Mathlib/Probability/Martingale/Centering.lean
theorem martingalePart_add_ae_eq [SigmaFiniteFiltration μ ℱ] {f g : ℕ → Ω → E}
    (hf : Martingale f ℱ μ) (hg : Adapted ℱ fun n => g (n + 1)) (hg0 : g 0 = 0)
    (hgint : ∀ n, Integrable (g n) μ) (n : ℕ) : martingalePart (f + g) ℱ μ n =ᵐ[μ] f n

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 : Filtration ℕ m0
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
h : ℕ → Ω → E := f - martingalePart (f + g) ℱ μ
hhdef : h = f - martingalePart (f + g) ℱ μ
hh : h = predictablePart (f + g) ℱ μ - g
⊢ Adapted ℱ fun n => h (n + 1)

Tactic:
rw [hh]

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 : Filtration ℕ m0
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
h : ℕ → Ω → E := f - martingalePart (f + g) ℱ μ
hhdef : h = f - martingalePart (f + g) ℱ μ
hh : h = predictablePart (f + g) ℱ μ - g
⊢ Adapted ℱ fun n => (predictablePart (f + g) ℱ μ - g) (n + 1)

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 : Filtration ℕ m0
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
h : ℕ → Ω → E := f - martingalePart (f + g) ℱ μ
hhdef : h = f - martingalePart (f + g) ℱ μ
hh : h = predictablePart (f + g) ℱ μ - g
hhpred : Adapted ℱ fun n => h (n + 1)
hhmgle : Martingale h ℱ μ
⊢ martingalePart (f + g) ℱ μ n =ᶠ[ae μ] f n

Tactic:
refine (eventuallyEq_iff_sub.2 ?_).symm

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 : Filtration ℕ m0
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
h : ℕ → Ω → E := f - martingalePart (f + g) ℱ μ
hhdef : h = f - martingalePart (f + g) ℱ μ
hh : h = predictablePart (f + g) ℱ μ - g
hhpred : Adapted ℱ fun n => h (n + 1)
hhmgle : Martingale h ℱ μ
⊢ f n - martingalePart (f + g) ℱ μ n =ᶠ[ae μ] 0

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 : Filtration ℕ m0
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
⊢ martingalePart (f + g) ℱ μ n =ᶠ[ae μ] f n

Tactic:
set h := f - martingalePart (f + g) ℱ μ with hhdef

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 : Filtration ℕ m0
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
h : ℕ → Ω → E := f - martingalePart (f + g) ℱ μ
hhdef : h = f - martingalePart (f + g) ℱ μ
⊢ martingalePart (f + g) ℱ μ n =ᶠ[ae μ] f n

Proof state:
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 : Filtration ℕ m0
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
h : ℕ → Ω → E := f - martingalePart (f + g) ℱ μ
hhdef : h = f - martingalePart (f + g) ℱ μ
hh : h = predictablePart (f + g) ℱ μ - g
hhpred : Adapted ℱ fun n => h (n + 1)
hhmgle : Martingale h ℱ μ
ω : Ω
hω : h n ω = h 0 ω
⊢ (f n - martingalePart (f + g) ℱ μ n) ω = 0 ω

Tactic:
unfold h at hω

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 : Filtration ℕ m0
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
h : ℕ → Ω → E := f - martingalePart (f + g) ℱ μ
hhdef : h = f - martingalePart (f + g) ℱ μ
hh : h = predictablePart (f + g) ℱ μ - g
hhpred : Adapted ℱ fun n => h (n + 1)
hhmgle : Martingale h ℱ μ
ω : Ω
hω : (f - martingalePart (f + g) ℱ μ) n ω = (f - martingalePart (f + g) ℱ μ) 0 ω
⊢ (f n - martingalePart (f + g) ℱ μ n) ω = 0 ω

Proof state:
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 : Filtration ℕ m0
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
h : ℕ → Ω → E := f - martingalePart (f + g) ℱ μ
hhdef : h = f - martingalePart (f + g) ℱ μ
hh : h = predictablePart (f + g) ℱ μ - g
hhpred : Adapted ℱ fun n => h (n + 1)
hhmgle : Martingale h ℱ μ
ω : Ω
hω : (f n - martingalePart (f + g) ℱ μ n) ω = (f - martingalePart (f + g) ℱ μ) 0 ω
⊢ (f n - martingalePart (f + g) ℱ μ n) ω = 0 ω

Tactic:
rw [hω, Pi.sub_apply, martingalePart]

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 : Filtration ℕ m0
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
h : ℕ → Ω → E := f - martingalePart (f + g) ℱ μ
hhdef : h = f - martingalePart (f + g) ℱ μ
hh : h = predictablePart (f + g) ℱ μ - g
hhpred : Adapted ℱ fun n => h (n + 1)
hhmgle : Martingale h ℱ μ
ω : Ω
hω : (f n - martingalePart (f + g) ℱ μ n) ω = (f - martingalePart (f + g) ℱ μ) 0 ω
⊢ (f 0 - ((f + g) 0 - predictablePart (f + g) ℱ μ 0)) ω = 0 ω

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 : Filtration ℕ m0
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
h : ℕ → Ω → E := f - martingalePart (f + g) ℱ μ
hhdef : h = f - martingalePart (f + g) ℱ μ
⊢ h = predictablePart (f + g) ℱ μ - g

Tactic:
rw [hhdef, sub_eq_sub_iff_add_eq_add, add_comm (predictablePart (f + g) ℱ μ),
  martingalePart_add_predictablePart]

Result:
no goals

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 : Filtration ℕ m0
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
h : ℕ → Ω → E := f - martingalePart (f + g) ℱ μ
hhdef : h = f - martingalePart (f + g) ℱ μ
hh : h = predictablePart (f + g) ℱ μ - g
⊢ Adapted ℱ fun n => (predictablePart (f + g) ℱ μ - g) (n + 1)

Tactic:
exact adapted_predictablePart.sub hg

Result:
no goals