mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Moments.Basic

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsZeroOrProbabilityMeasure μ
⊢ cgf X μ 0 = 0

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
hX : ¬Integrable (fun ω ↦ rexp (t * X ω)) μ
⊢ mgf X μ t = 0

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
hX : ¬Integrable (fun ω ↦ rexp (t * X ω)) μ
⊢ cgf X μ t = 0

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝⁶ : LinearOrder ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : SecondCountableTopology ι
inst✝¹ : BorelSpace ι
inst✝ : TopologicalSpace β
f : Filtration ι m
h : IsStronglyProgressive f u
hτ : IsStoppingTime f τ
n : ι
hτ_le : ∀ (ω : Ω), τ ω ≤ ↑n
⊢ StronglyMeasurable (stoppedValue u τ)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
⊢ 0 ≤ mgf X μ t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
hμ : μ ≠ 0
h_int_X : Integrable (fun ω ↦ rexp (t * X ω)) μ
⊢ ∫ (x : Ω), rexp (t * X x) ∂μ = ∫ (x : Ω) in Set.univ, rexp (t * X x) ∂μ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹⁰ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝⁹ : LinearOrder ι
inst✝⁸ : MeasurableSpace ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : BorelSpace ι
inst✝³ : TopologicalSpace β
f : Filtration ι m
inst✝² : PseudoMetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
hf_prog : IsStronglyProgressive f u
hτ : IsStoppingTime f τ
t : Set β
ht : MeasurableSet t
i : ι
h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω ↦ min (τ ω) ↑i)
this : stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ ↑i} = (stoppedValue u fun ω ↦ min (τ ω) ↑i) ⁻¹' t ∩ {ω | τ ω ≤ ↑i}
⊢ MeasurableSet (stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ ↑i})

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
hμ : μ ≠ 0
h_int_X : Integrable (fun ω ↦ rexp (t * X ω)) μ
this : ∫ (x : Ω), rexp (t * X x) ∂μ = ∫ (x : Ω) in Set.univ, rexp (t * X x) ∂μ
⊢ (Function.support fun x ↦ rexp (t * X x)) = Set.univ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
hμ : μ ≠ 0
h_int_X : Integrable (fun ω ↦ rexp (t * X ω)) μ
⊢ 0 < mgf X μ t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
hμ : NeZero μ
⊢ 0 < mgf X μ t ↔ Integrable (fun ω ↦ rexp (t * X ω)) μ

Tactic:
by

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

Goal state before:
pp: case e'_2
Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
g : ℕ → Ω → ℝ := ⋯
hg : Submartingale g ℱ μ
hg0 : g 0 = 0
hgbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |g (i + 1) ω - g i ω| ≤ ↑R
ω : Ω
hω : BddAbove (Set.range fun n ↦ g n ω) ↔ ∃ c, Tendsto (fun n ↦ g n ω) atTop (𝓝 c)
⊢ (∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n ↦ g n ω) atTop (𝓝 c)

Tactic:
· refine ⟨fun h => ?_, fun h => ?_⟩ <;> obtain ⟨c, hc⟩ := h
    · exact ⟨c - f 0 ω, hc.sub_const _⟩
    · refine ⟨c + f 0 ω, ?_⟩
      have := hc.add_const (f 0 ω)
      simpa only [g, sub_add_cancel]

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
hμ : NeZero μ
hX : Integrable (fun ω ↦ rexp (t * X ω)) μ
⊢ rexp (cgf X μ t) = mgf X μ t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ f n ω) ↔ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Y : Ω' → Ω
X : Ω → ℝ
hY : AEMeasurable Y μ
t : ℝ
hX : AEStronglyMeasurable (fun ω ↦ rexp (t * X ω)) (Measure.map Y μ)
⊢ mgf X (Measure.map Y μ) t = mgf (X ∘ Y) μ t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹⁰ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝⁹ : LinearOrder ι
inst✝⁸ : MeasurableSpace ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : BorelSpace ι
inst✝³ : TopologicalSpace β
f : Filtration ι m
inst✝² : PseudoMetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
hf_prog : IsStronglyProgressive f u
hτ : IsStoppingTime f τ
t : Set β
ht : MeasurableSet t
i : ι
⊢ MeasurableSet (stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ ↑i})

Tactic:
by

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

Goal state before:
pp: Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BorelCantelli

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
s : ℕ → Set Ω
hsm : ∀ (n : ℕ), MeasurableSet (s n)
hs : iIndepSet s μ
hs' : ∑' (n : ℕ), μ (s n) = ∞
this✝ : IsProbabilityMeasure μ
this :
  ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), μ[(s (n + 1)).indicator 1 | ↑(filtrationOfSet hsm) n] ω = (fun x ↦ μ.real (s (n + 1))) ω
ω : Ω
hω : ∀ (n : ℕ), μ[(s (n + 1)).indicator 1 | ↑(filtrationOfSet hsm) n] ω = μ.real (s (n + 1))
htends : ∀ (x : NNReal), ∀ᶠ (a : ℕ) in atTop, ↑x < ∑ k ∈ Finset.range a, μ (s (k + 1))
B : ℝ
hB : ∀ x ∈ Set.range fun n ↦ ∑ x ∈ Finset.range n, μ.real (s (x + 1)), x ≤ B
n : ℕ
⊢ ∀ a ∈ Finset.range n, μ (s (a + 1)) ≠ ∞

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
hX : AEMeasurable X μ
⊢ mgf id (Measure.map X μ) = mgf X μ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
Y : Ω → ℝ
h : X =ᵐ[μ] Y
ω : Ω
hω : X ω = Y ω
⊢ rexp (t * X ω) = rexp (t * Y ω)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
Y : Ω → ℝ
h : X =ᵐ[μ] Y
⊢ (fun ω ↦ rexp (t * X ω)) =ᵐ[μ] fun ω ↦ rexp (t * Y ω)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ' : Measure Ω'
Y : Ω' → ℝ
h : IdentDistrib X Y μ μ'
⊢ mgf X μ = mgf Y μ'

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
⊢ mgf (-X) μ t = mgf X μ (-t)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
⊢ cgf (-X) μ t = cgf X μ (-t)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t α : ℝ
⊢ mgf (α • X) μ t = mgf X μ (α * t)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹⁰ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝⁹ : LinearOrder ι
inst✝⁸ : MeasurableSpace ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : BorelSpace ι
inst✝³ : TopologicalSpace β
f : Filtration ι m
inst✝² : PseudoMetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
hf_prog : IsStronglyProgressive f u
hτ : IsStoppingTime f τ
h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω ↦ min (τ ω) ↑i)
t : Set β
ht : MeasurableSet t
seq : ℕ → ι
h_seq_tendsto : Tendsto seq atTop atTop
⊢ stoppedValue u τ ⁻¹' t = (⋃ n, stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ ↑(seq n)}) ∪ stoppedValue u τ ⁻¹' t ∩ {ω | τ ω = ⊤}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t α : ℝ
⊢ mgf (fun ω ↦ α + X ω) μ t = rexp (t * α) * mgf X μ t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t α : ℝ
⊢ mgf (fun ω ↦ X ω + α) μ t = mgf X μ t * rexp (t * α)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
ν : Measure Ω
hμ : Integrable (fun ω ↦ rexp (t * X ω)) μ
hν : Integrable (fun ω ↦ rexp (t * X ω)) ν
⊢ mgf X (μ + ν) t = mgf X μ t + mgf X ν t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
t : ℝ
ι : Type u_3
μ : ι → Measure Ω
hμ : Integrable (fun ω ↦ rexp (t * X ω)) (Measure.sum μ)
⊢ mgf X (Measure.sum μ) t = ∑' (i : ι), mgf X (μ i) t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
c : ℝ≥0∞
⊢ mgf X (c • μ) t = c.toReal * mgf X μ t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹⁰ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝⁹ : LinearOrder ι
inst✝⁸ : MeasurableSpace ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : BorelSpace ι
inst✝³ : TopologicalSpace β
f : Filtration ι m
inst✝² : PseudoMetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
hf_prog : IsStronglyProgressive f u
hτ : IsStoppingTime f τ
h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω ↦ min (τ ω) ↑i)
t : Set β
ht : MeasurableSet t
seq : ℕ → ι
h_seq_tendsto : Tendsto seq atTop atTop
this :
  stoppedValue u τ ⁻¹' t = (⋃ n, stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ ↑(seq n)}) ∪ stoppedValue u τ ⁻¹' t ∩ {ω | τ ω = ⊤}
⊢ stoppedValue u τ ⁻¹' t ∩ {ω | τ ω = ⊤} = (fun ω ↦ u (Classical.arbitrary ι) ω) ⁻¹' t ∩ {ω | τ ω = ⊤}

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BorelCantelli

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
s : ℕ → Set Ω
hsm : ∀ (n : ℕ), MeasurableSet (s n)
hs : iIndepSet s μ
hs' : ∑' (n : ℕ), μ (s n) = ∞
this✝ : IsProbabilityMeasure μ
this :
  ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), μ[(s (n + 1)).indicator 1 | ↑(filtrationOfSet hsm) n] ω = (fun x ↦ μ.real (s (n + 1))) ω
ω : Ω
hω : ∀ (n : ℕ), μ[(s (n + 1)).indicator 1 | ↑(filtrationOfSet hsm) n] ω = μ.real (s (n + 1))
htends : ∀ (x : NNReal), ∀ᶠ (a : ℕ) in atTop, ↑x < ∑ k ∈ Finset.range a, μ (s (k + 1))
B : ℝ
n : ℕ
hB : (∑ k ∈ Finset.range n, μ (s (k + 1))).toReal ≤ B
⊢ 0 ≤ (∑ k ∈ Finset.range n, μ (s (k + 1))).toReal

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
Y : Ω → ℝ
hXY : X ≤ᵐ[μ] Y
ht : 0 ≤ t
htY : Integrable (fun ω ↦ rexp (t * Y ω)) μ
htX : Integrable (fun ω ↦ rexp (t * X ω)) μ
ω : Ω
hω : X ω ≤ Y ω
⊢ rexp (t * X ω) ≤ rexp (t * Y ω)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹⁰ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝⁹ : LinearOrder ι
inst✝⁸ : MeasurableSpace ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : BorelSpace ι
inst✝³ : TopologicalSpace β
f : Filtration ι m
inst✝² : PseudoMetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
hf_prog : IsStronglyProgressive f u
hτ : IsStoppingTime f τ
⊢ Measurable (stoppedValue u τ)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BorelCantelli

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
s : ℕ → Set Ω
hsm : ∀ (n : ℕ), MeasurableSet (s n)
hs : iIndepSet s μ
hs' : ∑' (n : ℕ), μ (s n) = ∞
⊢ μ (limsup s atTop) = 1

Tactic:
by

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

Goal state before:
pp: Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R
hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ f n ω) ↔ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)
hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ (-f) n ω) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω) atTop (𝓝 c)
ω : Ω
hω₁ : BddAbove (Set.range fun n ↦ f n ω) ↔ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)
hω₂ : BddAbove (Set.range fun n ↦ (-f) n ω) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω) atTop (𝓝 c)
⊢ (∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω) atTop (𝓝 c)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
Y : Ω → ℝ
hXY : X ≤ᵐ[μ] Y
ht : 0 ≤ t
htY : Integrable (fun ω ↦ rexp (t * Y ω)) μ
⊢ mgf X μ t ≤ mgf Y μ t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
inst✝¹ : Nonempty ι
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝ : AddCommMonoid E
s : Finset ι
hbdd : ∀ (ω : Ω), τ ω ∈ WithTop.some '' ↑s
y : Ω
this : {i ∈ s | y ∈ {ω | τ ω = ↑i}} = {(τ y).untopA}
⊢ u (τ y).untopA y = ∑ i ∈ s with y ∈ {ω | τ ω = ↑i}, u i y

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
inst✝¹ : Nonempty ι
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝ : AddCommMonoid E
s : Finset ι
y : Ω
hbdd : τ y ∈ WithTop.some '' ↑s
ω : ι
h : ω = (τ y).untopA
h_contra : τ y = ⊤
⊢ False

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
Y : Ω → ℝ
hXY : X ≤ᵐ[μ] Y
ht : t ≤ 0
htX : Integrable (fun ω ↦ rexp (t * X ω)) μ
⊢ mgf Y μ t ≤ mgf X μ t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
inst✝¹ : Nonempty ι
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝ : AddCommMonoid E
s : Finset ι
hbdd : ∀ (ω : Ω), τ ω ∈ WithTop.some '' ↑s
⊢ stoppedValue u τ = ∑ i ∈ s, {ω | τ ω = ↑i}.indicator (u i)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
inst✝³ : Nonempty ι
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝² : Preorder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : AddCommMonoid E
N : ι
ω : Ω
hbdd : τ ω ≤ ↑N
h_contra : τ ω = ⊤
⊢ False

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
h_indep : X ⟂ᵢ[μ] Y
s t : ℝ
⊢ (fun ω ↦ rexp (s * X ω)) ⟂ᵢ[μ] fun ω ↦ rexp (t * Y ω)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
inst✝³ : Nonempty ι
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝² : Preorder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : AddCommMonoid E
N : ι
hbdd : ∀ (ω : Ω), τ ω ≤ ↑N
⊢ stoppedValue u τ = ∑ i ≤ N, {ω | τ ω = ↑i}.indicator (u i)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
X Y : Ω → ℝ
h_indep : X ⟂ᵢ[μ] Y
hX : AEStronglyMeasurable (fun ω ↦ rexp (t * X ω)) μ
hY : AEStronglyMeasurable (fun ω ↦ rexp (t * Y ω)) μ
⊢ mgf (X + Y) μ t = mgf X μ t * mgf Y μ t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
X Y : Ω → ℝ
h_indep : X ⟂ᵢ[μ] Y
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
⊢ Continuous fun x ↦ rexp (t * x)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
X Y : Ω → ℝ
h_indep : X ⟂ᵢ[μ] Y
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
⊢ mgf (X + Y) μ t = mgf X μ t * mgf Y μ t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
X Y : Ω → ℝ
h_indep : X ⟂ᵢ[μ] Y
h_int_X : Integrable (fun ω ↦ rexp (t * X ω)) μ
h_int_Y : Integrable (fun ω ↦ rexp (t * Y ω)) μ
⊢ cgf (X + Y) μ t = cgf X μ t + cgf Y μ t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
X Y : Ω → ℝ
h_int_X : AEStronglyMeasurable (fun ω ↦ rexp (t * X ω)) μ
h_int_Y : AEStronglyMeasurable (fun ω ↦ rexp (t * Y ω)) μ
⊢ AEStronglyMeasurable (fun ω ↦ rexp (t * (X + Y) ω)) μ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
inst✝² : Nonempty ι
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝¹ : LinearOrder ι
inst✝ : AddCommMonoid E
s : Finset ι
n : ι
hbdd : ∀ (ω : Ω), τ ω < ↑n → τ ω ∈ WithTop.some '' ↑s
ω : Ω
h : τ ω < ↑n
h_contra : τ ω = ⊤
⊢ False

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
X : ι → Ω → ℝ
s : Finset ι
h_int : ∀ i ∈ s, AEStronglyMeasurable (fun ω ↦ rexp (t * X i ω)) μ
⊢ AEStronglyMeasurable (fun ω ↦ rexp (t * (∑ i ∈ s, X i) ω)) μ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
X Y : Ω → ℝ
h_indep : X ⟂ᵢ[μ] Y
h_int_X : Integrable (fun ω ↦ rexp (t * X ω)) μ
h_int_Y : Integrable (fun ω ↦ rexp (t * Y ω)) μ
⊢ Integrable (fun ω ↦ rexp (t * (X + Y) ω)) μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Moments.IntegrableExpMul

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t a b : ℝ
ha : Integrable (fun ω ↦ rexp (a * X ω)) μ
hb : Integrable (fun ω ↦ rexp (b * X ω)) μ
hat : a ≤ t
htb : t ≤ b
ω : Ω
⊢ 0 ≤ rexp (a * X ω) + rexp (b * X ω)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Moments.IntegrableExpMul

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t a b : ℝ
ha : Integrable (fun ω ↦ rexp (a * X ω)) μ
hb : Integrable (fun ω ↦ rexp (b * X ω)) μ
hat : a ≤ t
htb : t ≤ b
⊢ Integrable (fun ω ↦ rexp (t * X ω)) μ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun X μ
h_meas : ∀ (i : ι), Measurable (X i)
s : Finset ι
h_int : ∀ i ∈ s, Integrable (fun ω ↦ rexp (t * X i ω)) μ
⊢ Integrable (fun ω ↦ rexp (t * (∑ i ∈ s, X i) ω)) μ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ f n ω) ↔ BddBelow (Set.range fun n ↦ f n ω)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
X : ι → Ω → ℝ
h_indep : iIndepFun X μ
h_meas : ∀ (i : ι), AEMeasurable (X i) μ
s : Finset ι
⊢ mgf (∑ i ∈ s, X i) μ t = ∏ i ∈ s, mgf (X i) μ t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
⊢ ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n ↦ f n ω) atTop atTop

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
X : ι → Ω → ℝ
h_indep : iIndepFun X μ
h_meas : ∀ (i : ι), Measurable (X i)
s : Finset ι
⊢ ∀ (i : ι), AEMeasurable (X i) μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Moments.IntegrableExpMul

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t u : ℝ
hu_int_pos : Integrable (fun ω ↦ rexp (u * X ω)) μ
hu_int_neg : Integrable (fun ω ↦ rexp (-u * X ω)) μ
htu : |t| ≤ |u|
⊢ Integrable (fun ω ↦ rexp (t * X ω)) μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Moments.IntegrableExpMul

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t u : ℝ
inst✝ : IsFiniteMeasure μ
hu : Integrable (fun ω ↦ rexp (u * X ω)) μ
h_nonneg : 0 ≤ t
htu : t ≤ u
⊢ Integrable (fun ω ↦ rexp (0 * X ω)) μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Moments.IntegrableExpMul

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t u : ℝ
inst✝ : IsFiniteMeasure μ
hu : Integrable (fun ω ↦ rexp (u * X ω)) μ
h_nonpos : t ≤ 0
htu : u ≤ t
⊢ Integrable (fun ω ↦ rexp (0 * X ω)) μ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
⊢ ∀ᵐ (ω : Ω) ∂μ, ¬Tendsto (fun n ↦ f n ω) atTop atBot

Tactic:
by

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

Goal state before:
pp: Ω : Type u_2
s : ℕ → Set Ω
⊢ process s 0 = 0

Tactic:
by

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

Goal state before:
pp: Ω : Type u_2
m0 : MeasurableSpace Ω
ℱ : Filtration ℕ m0
μ : Measure Ω
s : ℕ → Set Ω
n : ℕ
⊢ martingalePart (process s) ℱ μ n = ∑ k ∈ Finset.range n, ((s (k + 1)).indicator 1 - μ[(s (k + 1)).indicator 1 | ↑ℱ k])

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
X : ι → Ω → ℝ
h_indep : iIndepFun X μ
h_meas : ∀ (i : ι), AEMeasurable (X i) μ
s : Finset ι
h_int : ∀ i ∈ s, Integrable (fun ω ↦ rexp (t * X i ω)) μ
⊢ cgf (∑ i ∈ s, X i) μ t = ∑ i ∈ s, cgf (X i) μ t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
inst✝² : Nonempty ι
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝¹ : LinearOrder ι
inst✝ : AddCommMonoid E
s : Finset ι
n : ι
hbdd : ∀ (ω : Ω), τ ω < ↑n → τ ω ∈ WithTop.some '' ↑s
⊢ stoppedProcess u τ n = {a | ↑n ≤ τ a}.indicator (u n) + ∑ i ∈ s with i < n, {ω | τ ω = ↑i}.indicator (u i)

Tactic:
by

Goal state after: