mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Process.Adapted

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁴ : Preorder ι
f : Filtration ι m
β : Type u_3
inst✝³ : TopologicalSpace β
inst✝² : MeasurableSpace ι
γ : Type u_4
inst✝¹ : CommMonoid β
inst✝ : ContinuousMul β
U : γ → ι → Ω → β
s : Finset γ
h : ∀ c ∈ s, IsStronglyProgressive f (U c)
⊢ IsStronglyProgressive f fun i a ↦ ∏ c ∈ s, U c i a

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Adapted

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : Preorder ι
f : Filtration ι m
β : Type u_3
inst✝⁴ : TopologicalSpace β
u : ι → Ω → β
γ : Type u_4
inst✝³ : MeasurableSpace ι
inst✝² : PseudoMetrizableSpace β
fltr : Filter γ
inst✝¹ : fltr.NeBot
inst✝ : fltr.IsCountablyGenerated
U : γ → ι → Ω → β
h : ∀ (l : γ), IsStronglyProgressive f (U l)
h_tendsto : Tendsto U fltr (𝓝 u)
⊢ IsStronglyProgressive f u

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
⊢ predictablePart f ℱ μ 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
n : ℕ
⊢ predictablePart f ℱ μ (n + 1) = predictablePart f ℱ μ n + μ[f (n + 1) - f n | ↑ℱ n]

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : CompleteSpace E
c : ℝ
n : ℕ
⊢ predictablePart (c • f) ℱ μ n =ᵐ[μ] c • predictablePart f ℱ μ n

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝¹ : LinearOrder ι
inst✝ : OrderBot ι
𝓕 : Filtration ι m
i j : ι
s : Set Ω
hs : MeasurableSet s
hij : i ≤ j
⊢ (Set.Ioi i \ Set.Ioi j) ×ˢ s = Set.Ioi i ×ˢ (s \ s) ∪ (Set.Ioi i \ Set.Ioi j) ×ˢ s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝¹ : LinearOrder ι
inst✝ : OrderBot ι
𝓕 : Filtration ι m
i j : ι
s : Set Ω
hs : MeasurableSet s
⊢ MeasurableSet (Set.Ioc i j ×ˢ s)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
𝓕 : Filtration ℕ m
n : ℕ
s : Set Ω
hs : MeasurableSet s
⊢ MeasurableSet ({n + 1} ×ˢ s)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
𝓕 : Filtration ℕ m
s : Set (ℕ × Ω)
hs : MeasurableSet s
n : ℕ
⊢ {ω | (n + 1, ω) ∈ s} = Prod.mk ⟨n + 1, ⋯⟩ ⁻¹' (fun p ↦ (↑p.1, p.2)) ⁻¹' s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f g : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : CompleteSpace E
hfint : ∀ (n : ℕ), Integrable (f n) μ
hgint : ∀ (n : ℕ), Integrable (g n) μ
n i : ℕ
hi : i ∈ Finset.range n
⊢ μ[(f + g) (i + 1) - (f + g) i | ↑ℱ i] =ᵐ[μ] μ[f (i + 1) - f i + (g (i + 1) - g i) | ↑ℱ i]

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f g : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : CompleteSpace E
hfint : ∀ (n : ℕ), Integrable (f n) μ
hgint : ∀ (n : ℕ), Integrable (g n) μ
n i : ℕ
hi : i ∈ Finset.range n
⊢ μ[f (i + 1) - f i + (g (i + 1) - g i) | ↑ℱ i] =ᵐ[μ] μ[f (i + 1) - f i | ↑ℱ i] + μ[g (i + 1) - g i | ↑ℱ i]

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f g : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : CompleteSpace E
hfint : ∀ (n : ℕ), Integrable (f n) μ
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
⊢ predictablePart (f + g) ℱ μ n =ᵐ[μ] predictablePart f ℱ μ n + predictablePart g ℱ μ n

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : CompleteSpace E
hf : Martingale f ℱ μ
n i : ℕ
hi : i ∈ Finset.range n
⊢ μ[f (i + 1) - f i | ↑ℱ i] =ᵐ[μ] μ[f (i + 1) | ↑ℱ i] - μ[f i | ↑ℱ i]

Tactic:
simp [condExp_sub (hf.integrable (i + 1)) (hf.integrable i) (ℱ i)]

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : CompleteSpace E
hf : Martingale f ℱ μ
n i : ℕ
hi : i ∈ Finset.range n
⊢ μ[f (i + 1) - f i | ↑ℱ i] =ᵐ[μ] μ[f (i + 1) | ↑ℱ i] - μ[f i | ↑ℱ i]

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : CompleteSpace E
hf : Martingale f ℱ μ
n i : ℕ
hi : i ∈ Finset.range n
⊢ f i - f i =ᵐ[μ] 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : CompleteSpace E
hf : Martingale f ℱ μ
n : ℕ
⊢ predictablePart f ℱ μ n =ᵐ[μ] 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
𝓕 : Filtration ℕ m
s : Set (ℕ × Ω)
hs : MeasurableSet s
n i : ℕ
A : Set Ω
hA : MeasurableSet A
hin : i ≤ n
p : ↑(Set.singleton (n + 1)) × Ω
hp2 : p.2 ∈ A
x✝ : i < n + 1
⊢ p.1 = ⟨n + 1, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
𝓕 : Filtration ℕ m
s : Set (ℕ × Ω)
hs : MeasurableSet s
n : ℕ
⊢ MeasurableSet {ω | (n + 1, ω) ∈ s}

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
inst✝ : OrderBot ι
𝓕 : Filtration ι m
m' : MeasurableSpace (ι × Ω)
hm'bot : ∀ (A : Set Ω), MeasurableSet A → MeasurableSet ({⊥} ×ˢ A)
hm' : ∀ (i : ι) (A : Set Ω), MeasurableSet A → MeasurableSet (Set.Ioi i ×ˢ A)
⊢ 𝓕.predictable ≤ m'

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i : ι
A : Set Ω
hA : MeasurableSet A
⊢ (fun p ↦ (↑p.1, p.2)) ⁻¹' {⊥} ×ˢ A = {⊥} ×ˢ A

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝² : CompleteSpace E
inst✝¹ : PartialOrder E
inst✝ : IsOrderedAddMonoid E
hf : Submartingale f ℱ μ
this : ∀ᵐ (a : Ω) ∂μ, ∀ (i : ℕ), 0 a ≤ μ[f i.succ - f i | ↑ℱ i] a
ω : Ω
h : ∀ (i : ℕ), 0 ω ≤ μ[f i.succ - f i | ↑ℱ i] ω
⊢ Monotone fun x ↦ predictablePart f ℱ μ x ω

Tactic:
simp only [Pi.zero_apply, Nat.succ_eq_add_one, ← ge_iff_le] at h

Goal state after:
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝² : CompleteSpace E
inst✝¹ : PartialOrder E
inst✝ : IsOrderedAddMonoid E
hf : Submartingale f ℱ μ
this : ∀ᵐ (a : Ω) ∂μ, ∀ (i : ℕ), 0 a ≤ μ[f i.succ - f i | ↑ℱ i] a
ω : Ω
h : ∀ (i : ℕ), μ[f (i + 1) - f i | ↑ℱ i] ω ≥ 0
⊢ Monotone fun x ↦ predictablePart f ℱ μ x ω
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i j : ι
A : Set Ω
hA : MeasurableSet A
hji : j ≤ i
⊢ (fun p ↦ (↑p.1, p.2)) ⁻¹' Set.Ioi j ×ˢ A = (Subtype.val ⁻¹' Set.Ioc j i) ×ˢ A

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i j : ι
A : Set Ω
hA : MeasurableSet A
hij : i ≤ j
⊢ (fun p ↦ (↑p.1, p.2)) ⁻¹' Set.Ioi j ×ˢ A = ∅

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
i : ι
⊢ Measurable fun x ↦ (↑x.1, x.2)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝⁶ : TopologicalSpace E
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m✝
u : ι → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
i : ι
this✝ : MeasurableSpace (ι × Ω) := 𝓕.predictable
this : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑𝓕 i)
X : ℕ → ↑(Set.Iic i) × Ω → E := fun m x ↦ (StronglyMeasurable.approx h𝓕 m) (↑x.1, x.2)
m : ℕ
⊢ Set.range (X m) ⊆ Set.range ⇑(StronglyMeasurable.approx h𝓕 m)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝² : CompleteSpace E
inst✝¹ : PartialOrder E
inst✝ : IsOrderedAddMonoid E
hf : Submartingale f ℱ μ
⊢ ∀ᵐ (ω : Ω) ∂μ, Monotone fun x ↦ predictablePart f ℱ μ x ω

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
E : Type u_3
inst✝⁶ : TopologicalSpace E
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
u : ι → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
i : ι
this✝ : MeasurableSpace (ι × Ω) := 𝓕.predictable
this : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑𝓕 i)
X : ℕ → ↑(Set.Iic i) × Ω → E := fun m_1 x ↦ (StronglyMeasurable.approx h𝓕 m_1) (↑x.1, x.2)
n : { x // x ∈ Set.Iic i } × Ω
⊢ Tendsto (fun n_1 ↦ ((fun m_1 ↦ { toFun := X m_1, measurableSet_fiber' := ⋯, finite_range' := ⋯ }) n_1) n) atTop
    (𝓝 ((fun p ↦ u (↑p.1) p.2) n))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
E : Type u_3
inst✝⁶ : TopologicalSpace E
inst✝⁵ : LinearOrder ι
inst✝⁴ : OrderBot ι
inst✝³ : MeasurableSpace ι
inst✝² : TopologicalSpace ι
inst✝¹ : OpensMeasurableSpace ι
inst✝ : OrderClosedTopology ι
𝓕 : Filtration ι m
u : ι → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
⊢ IsStronglyProgressive 𝓕 u

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
n : ℕ
this✝ : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
this : MeasurableSpace Ω := ↑𝓕 n
X : ℕ → Ω → E := fun m ↦ Function.curry (⇑(StronglyMeasurable.approx h𝓕 m)) (n + 1)
m : ℕ
s : E
⊢ X m ⁻¹' {s} = {ω | (n + 1, ω) ∈ ⇑(StronglyMeasurable.approx h𝓕 m) ⁻¹' {s}}

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝² : CompleteSpace E
inst✝¹ : PartialOrder E
inst✝ : IsOrderedAddMonoid E
hf : Submartingale f ℱ μ
⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), 0 ≤ predictablePart f ℱ μ n ω

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : SigmaFiniteFiltration μ ℱ
hf : IsStronglyPredictable ℱ f
hfint : ∀ (n : ℕ), Integrable (f n) μ
n : ℕ
⊢ predictablePart f ℱ μ n =ᵐ[μ] f n - f 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝² : SecondCountableTopology E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ StronglyMeasurable (predictablePart f ℱ μ 0)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m
u : ℕ → Ω → E
h𝓕 : IsStronglyPredictable 𝓕 u
n : ℕ
⊢ StronglyMeasurable (u (n + 1))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
⊢ martingalePart f ℱ μ 0 = f 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : CompleteSpace E
c : ℝ
n : ℕ
⊢ martingalePart (c • f) ℱ μ n =ᵐ[μ] c • martingalePart f ℱ μ n

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f g : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : CompleteSpace E
hfint : ∀ (n : ℕ), Integrable (f n) μ
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
⊢ martingalePart (f + g) ℱ μ n =ᵐ[μ] martingalePart f ℱ μ n + martingalePart g ℱ μ n

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : CompleteSpace E
hf : Martingale f ℱ μ
n : ℕ
⊢ martingalePart f ℱ μ n =ᵐ[μ] f n

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
⊢ Y m ⁻¹' {s} = ⋃ n, {n} ×ˢ (Function.curry (Y m) n ⁻¹' {s})

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : SigmaFiniteFiltration μ ℱ
hf : IsStronglyPredictable ℱ f
hfint : ∀ (n : ℕ), Integrable (f n) μ
n : ℕ
⊢ martingalePart f ℱ μ n =ᵐ[μ] f 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Basic

Goal state before:
pp: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : Preorder ι
m0 : MeasurableSpace Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
ℱ : Filtration ι m0
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
x : E
i j : ι
x✝ : i ≤ j
⊢ μ[(fun x_1 x_2 ↦ x) j | ↑ℱ i] =ᵐ[μ] (fun x_1 x_2 ↦ x) i

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Basic

Goal state before:
pp: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : OrderBot ι
ℱ : Filtration ι m0
μ : Measure Ω
inst✝ : SigmaFiniteFiltration μ ℱ
f : Ω → E
hf : StronglyMeasurable f
hfint : Integrable f μ
⊢ Martingale (fun x ↦ f) ℱ μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Basic

Goal state before:
pp: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : Preorder ι
m0 : MeasurableSpace Ω
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
ℱ : Filtration ι m0
μ : Measure Ω
i j : ι
x✝ : i ≤ j
⊢ μ[0 j | ↑ℱ i] =ᵐ[μ] 0 i

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: case refine_1.zero
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := ⋯
X : ℕ → ℕ × Ω → E := ⋯
Y : ℕ → ℕ × Ω → E := ⋯
m : ℕ
s : E
⊢ MeasurableSet (Function.curry (Y m) 0 ⁻¹' {s})

Tactic:
let : MeasurableSpace Ω := 𝓕 0

Goal state after:
case refine_1.zero
Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this✝ : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
this : MeasurableSpace Ω := ↑𝓕 0
⊢ MeasurableSet (Function.curry (Y m) 0 ⁻¹' {s})
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
⊢ martingalePart f ℱ μ = fun n ↦ f 0 + ∑ i ∈ Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i | ↑ℱ i])

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
n : ℕ
hmk : n + 1 ≤ m
⊢ Function.curry (Y m) (n + 1) = Function.curry (X m) (n + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Basic

Goal state before:
pp: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : ι → Ω → E
ℱ : Filtration ι m0
inst✝¹ : CompleteSpace E
inst✝ : SigmaFiniteFiltration μ ℱ
hf : Martingale f ℱ μ
i j : ι
hij : i ≤ j
s : Set Ω
hs : MeasurableSet s
⊢ ∫ (ω : Ω) in s, f i ω ∂μ = ∫ (ω : Ω) in s, f j ω ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Basic

Goal state before:
pp: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f g : ι → Ω → E
ℱ : Filtration ι m0
inst✝ : CompleteSpace E
hf : Martingale f ℱ μ
hg : StronglyAdapted ℱ g
h_eq : ∀ (t : ι), f t =ᵐ[μ] g t
⊢ Martingale g ℱ μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
inst✝ : CompleteSpace E
hf_int : ∀ (n : ℕ), Integrable (f n) μ
n : ℕ
⊢ Integrable (martingalePart f ℱ μ n) μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Basic

Goal state before:
pp: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f g : ι → Ω → E
ℱ : Filtration ι m0
inst✝ : CompleteSpace E
hf : Martingale f ℱ μ
hg : Martingale g ℱ μ
⊢ Martingale (f + g) ℱ μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Basic

Goal state before:
pp: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f g : ι → Ω → E
ℱ : Filtration ι m0
inst✝ : CompleteSpace E
hf : Martingale f ℱ μ
hg : Martingale g ℱ μ
⊢ Martingale (f - g) ℱ μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Basic

Goal state before:
pp: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ι → Ω → E
ℱ : Filtration ι m0
inst✝ : CompleteSpace E
c : ℝ
hf : Martingale f ℱ μ
⊢ Martingale (c • f) ℱ μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
n : ℕ
hmk : m < n + 1
⊢ Function.curry (Y m) (n + 1) = Function.curry (X m) 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Centering

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

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.Basic

Goal state before:
pp: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁸ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
ℱ : Filtration ι m0
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ ℱ
f : ι → Ω → E
hf : Supermartingale f ℱ μ
i j : ι
hij : i ≤ j
s : Set Ω
hs : MeasurableSet s
⊢ ∫ (ω : Ω) in s, f j ω ∂μ ≤ ∫ (ω : Ω) in s, f i ω ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Process.Predictable

Goal state before:
pp: Ω : Type u_1
m✝ : MeasurableSpace Ω
E : Type u_3
inst✝ : TopologicalSpace E
𝓕 : Filtration ℕ m✝
u : ℕ → Ω → E
h₀ : StronglyMeasurable (u 0)
h : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))
this : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
X : ℕ → ℕ × Ω → E :=
  fun m x ↦
    match x.1 with
    | 0 => (h₀.approx m) x.2
    | n.succ => (⋯.approx m) x.2
Y : ℕ → ℕ × Ω → E := fun m x ↦ if x.1 ≤ m then X m x else X m (0, x.2)
m : ℕ
s : E
n : ℕ
hmk : m < n + 1
⊢ 0 ≤ n

Tactic:
by

Goal state after: