mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: X : ℕ → Type u_1
inst✝⁴ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝³ : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
a b c : ℕ
hab : a ≤ b
hbc : b ≤ c
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
⊢ ((traj κ a) x₀)[f | ↑piLE b] =ᵐ[(traj κ a) x₀] fun x ↦
    ∫ (y : (i : ↥(Iic c)) → X ↑i),
      ((traj κ a) x₀)[f | ↑piLE c] (updateFinset x (Iic c) y) ∂(partialTraj κ b c) (frestrictLe b x)

Tactic:
have i_cf : Integrable ((traj κ a x₀)[f | piLE c]) (traj κ a x₀) :=
    integrable_condExp
  have mcf : StronglyMeasurable ((traj κ a x₀)[f | piLE c]) :=
    stronglyMeasurable_condExp.mono (piLE.le c)
  filter_upwards [piLE.condExp_condExp f hbc, condExp_traj hab i_cf] with x h1 h2
  rw [← h1, h2, ← traj_map_frestrictLe, Kernel.map_apply, integral_map]
  · congr with y
    apply stronglyMeasurable_condExp.dependsOn_of_piLE
    simp only [Set.mem_Iic, updateFinset, mem_Iic, frestrictLe_apply, dite_eq_ite]
    exact fun i hi ↦ (if_pos hi).symm
  any_goals fun_prop
  exact (mcf.comp_measurable measurable_updateFinset).aestronglyMeasurable

Goal state after:
Module: Mathlib.Probability.Distributions.Poisson.Basic

Goal state before:
pp: r : ℝ≥0
n : ℕ
hr : 0 < r
⊢ 0 < Po(r).real {n}

Tactic:
rw [poissonMeasure_real_singleton]

Goal state after:
r : ℝ≥0
n : ℕ
hr : 0 < r
⊢ 0 < rexp (-↑r) * ↑r ^ n / ↑n !
Module: Mathlib.Probability.Distributions.Poisson.Basic

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
r : ℝ≥0
f : ℕ → E
⊢ ∀ (i : ℕ), ENNReal.ofReal (rexp (-↑r) * ↑r ^ i / ↑i !) ≠ ⊤

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.Distributions.Poisson.Basic

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
r : ℝ≥0
f : ℕ → E
hf : Integrable f Po(r)
this : (fun n ↦ (rexp (-↑r) * ↑r ^ n / ↑n !) • f n) = fun n ↦ (ENNReal.ofReal (rexp (-↑r) * ↑r ^ n / ↑n !)).toReal • f n
⊢ ∀ (i : ℕ), ENNReal.ofReal (rexp (-↑r) * ↑r ^ i / ↑i !) ≠ ⊤

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Monad

Goal state before:
pp: α : Type u_1
a a' : α
⊢ a' ∈ (pure a).support ↔ a' = a

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Monad

Goal state before:
pp: case pos
α : Type u_1
a : α
s : Set α
ha : a ∈ s
⊢ ∑' (x : α), s.indicator (⇑(pure a)) x = 1

Tactic:
refine (tsum_congr fun b => ?_).trans (tsum_ite_eq a 1)
    exact ite_eq_left_iff.2 fun hb =>
      symm (ite_eq_right_iff.2 fun h => (hb <| h.symm ▸ ha).elim)

Goal state after:
Module: Mathlib.Probability.Distributions.Poisson.Basic

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
r : ℝ≥0
f : ℕ → E
hf : Integrable f Po(r)
⊢ HasSum (fun n ↦ (rexp (-↑r) * ↑r ^ n / ↑n !) • f n) (∫ (n : ℕ), f n ∂Po(r))

Tactic:
have : (fun n ↦ (exp (-r) * r ^ n / (n)!) • f n) =
      fun n ↦ (ENNReal.ofReal (exp (-r) * r ^ n / (n)!)).toReal • f n := by
    ext; rw [ENNReal.toReal_ofReal (by positivity)]
  rw [this]
  apply hasSum_integral_sum_dirac (by simp)
  convert! integrable_poissonMeasure_iff.1 hf
  rw [ENNReal.toReal_ofReal (by positivity)]

Goal state after:
Module: Mathlib.Probability.Distributions.Poisson.PoissonLimitThm

Goal state before:
pp: p : ℕ → ℝ
r : ℝ
k : ℕ
hr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)
hp_lt_half : ∀ᶠ (n : ℕ) in atTop, p n < 1 / 2
hEq : (fun n ↦ (1 - p n) ^ (n - k)) =ᶠ[atTop] fun n ↦ (1 - p n) ^ n * ((1 - p n) ^ k)⁻¹
this : Real.exp (-r) = Real.exp (-r) * (1 ^ k)⁻¹
⊢ 1 ^ k ≠ 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.Distributions.Poisson.Basic

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
r : ℝ≥0
f : ℕ → E
⊢ ∀ (i : ℕ), ENNReal.ofReal (rexp (-↑r) * ↑r ^ i / ↑i !) ≠ ⊤

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Constructions

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
p : PMF α
b : β
⊢ b ∈ (map f p).support ↔ ∃ a ∈ p.support, f a = b

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Constructions

Goal state before:
pp: α : Type u_1
β : Type u_2
q : PMF (α → β)
p : PMF α
b : β
⊢ b ∈ (q.seq p).support ↔ ∃ f ∈ q.support, b ∈ f '' p.support

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Constructions

Goal state before:
pp: α : Type u_1
f : α → ℝ≥0∞
hf0 : tsum f ≠ 0
hf : tsum f ≠ ∞
a : α
⊢ a ∈ (normalize f hf0 hf).support ↔ f a ≠ 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.Distributions.TwoValued

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
hXmeas : AEMeasurable X μ
hX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1
⊢ μ.real {ω | 1 - X ω = 1} = μ.real {ω | X ω = 0}

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Monad

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
p : PMF α
q : PMF β
f : (a : α) → a ∈ p.support → (b : β) → b ∈ q.support → PMF γ
c : γ
b : β
a : α
⊢ (p a * if h : p a = 0 then 0 else q b * if h : q b = 0 then 0 else (f a h b h) c) =
    q b * if h : q b = 0 then 0 else p a * if h : p a = 0 then 0 else (f a h b h) c

Tactic:
split_ifs with h1 h2 h2 <;> ring

Goal state after:
Module: Mathlib.Probability.Distributions.TwoValued

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
hXmeas : AEMeasurable X μ
hX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1
⊢ ∫ (ω : Ω), 1 - X ω ∂μ = μ.real {ω | X ω = 0}

Tactic:
calc
    _ = μ.real {ω | 1 - X ω = 1} :=
      integral_of_ae_eq_zero_or_one (aemeasurable_const (b := 1).sub hXmeas)
        (by simpa [sub_eq_zero, or_comm, eq_comm (a := (1 : ℝ))] using hX)
    _ = μ.real {ω | X ω = 0} := by simp

Goal state after:
Module: Mathlib.Probability.Distributions.Uniform

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
a : α
⊢ a ∈ (uniformOfFintype α).support

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.Independence.BoundedContinuousFunction

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
𝓧 : Type u_4
mX : MeasurableSpace 𝓧
A : Set Ω
X : Ω → 𝓧
h : A.indicator 1 ⟂ᵢ[P] X
s : Set Ω
hs : s ∈ {A}
t : Set 𝓧
ht : MeasurableSet t
hA' : A = A.indicator 1 ⁻¹' {1}
⊢ MeasurableSet {1}

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.Independence.ZeroOne

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
β : Type u_4
p : Set ι → Prop
f : Filter ι
ns : β → Set ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
hf : ∀ (t : Set ι), p t → tᶜ ∈ f
hns : Directed (fun x1 x2 ↦ x1 ⊆ x2) ns
hnsp : ∀ (a : β), p (ns a)
hns_univ : ∀ (n : ι), ∃ a, n ∈ ns a
t : Set Ω
ht_tail : MeasurableSet t
⊢ ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1

Tactic:
apply measure_eq_zero_or_one_of_indepSet_self' ?_
    ((indep_limsup_self h_le h_indep hf hns hnsp hns_univ).indepSet_of_measurableSet ht_tail
      ht_tail)

Goal state after:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
β : Type u_4
p : Set ι → Prop
f : Filter ι
ns : β → Set ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
hf : ∀ (t : Set ι), p t → tᶜ ∈ f
hns : Directed (fun x1 x2 ↦ x1 ⊆ x2) ns
hnsp : ∀ (a : β), p (ns a)
hns_univ : ∀ (n : ι), ∃ a, n ∈ ns a
t : Set Ω
ht_tail : MeasurableSet t
⊢ ∀ᵐ (a : α) ∂μα, IsFiniteMeasure (κ a)
Module: Mathlib.Probability.Kernel.Category.Stoch

Goal state before:
pp: X Y : Stoch
κ : X ⟶ Y
inst✝ : Deterministic κ
this : copy Y.obj.carrier ∘ₖ κ.hom.hom = Kernel.id ∥ₖ κ.hom.hom ∘ₖ (κ.hom.hom ∥ₖ Kernel.id) ∘ₖ copy X.obj.carrier
⊢ κ.hom.hom ∥ₖ κ.hom.hom ∘ₖ copy X.obj.carrier = copy Y.obj.carrier ∘ₖ κ.hom.hom

Tactic:
id_parallelComp_comp_parallelComp_id

Goal state after:
X Y : Stoch
κ : X ⟶ Y
inst✝ : Deterministic κ
this : copy Y.obj.carrier ∘ₖ κ.hom.hom = κ.hom.hom ∥ₖ κ.hom.hom ∘ₖ copy X.obj.carrier
⊢ κ.hom.hom ∥ₖ κ.hom.hom ∘ₖ copy X.obj.carrier = copy Y.obj.carrier ∘ₖ κ.hom.hom
Module: Mathlib.Probability.Kernel.Invariance

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
κ : Kernel α α
inst✝ : IsMarkovKernel κ
π : Measure α
h_rev : κ.IsReversible π
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (x : α) in s, (κ x) Set.univ ∂π = π s

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X₁✝ X₂✝ X₃✝ Y₁✝ Y₂✝ Y₃✝ : SFinKer
κ₁ : X₁✝ ⟶ Y₁✝
κ₂ : X₂✝ ⟶ Y₂✝
η : X₃✝ ⟶ Y₃✝
a✝ : (X₁✝.carrier × X₂✝.carrier) × X₃✝.carrier
s✝ : Set (Y₁✝.carrier × Y₂✝.carrier × Y₃✝.carrier)
hs : MeasurableSet s✝
⊢ (((κ₁.hom ∥ₖ κ₂.hom ∥ₖ η.hom).map ⇑prodAssoc) a✝) s✝ = (((κ₁.hom ∥ₖ (κ₂.hom ∥ₖ η.hom)).comap ⇑prodAssoc ⋯) a✝) s✝

Tactic:
rw [Kernel.map_apply' _ (by fun_prop) _ hs, Kernel.comap_apply' _ (by fun_prop)]

Goal state after:
X₁✝ X₂✝ X₃✝ Y₁✝ Y₂✝ Y₃✝ : SFinKer
κ₁ : X₁✝ ⟶ Y₁✝
κ₂ : X₂✝ ⟶ Y₂✝
η : X₃✝ ⟶ Y₃✝
a✝ : (X₁✝.carrier × X₂✝.carrier) × X₃✝.carrier
s✝ : Set (Y₁✝.carrier × Y₂✝.carrier × Y₃✝.carrier)
hs : MeasurableSet s✝
⊢ ((κ₁.hom ∥ₖ κ₂.hom ∥ₖ η.hom) a✝) (⇑prodAssoc ⁻¹' s✝) = ((κ₁.hom ∥ₖ (κ₂.hom ∥ₖ η.hom)) (prodAssoc a✝)) s✝
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: case e'_3
Ω : Type u_1
β : Type u_3
β' : Type u_4
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
hf : Measurable f
hg : Measurable g
⊢ μ.trim hm' ⊗ₘ ((condExpKernel μ m').map f ×ₖ (condExpKernel μ m').map g) =
    ⇑(Kernel.id ×ₖ ((condExpKernel μ m').map f ×ₖ (condExpKernel μ m').map g)) ∘ₘ μ.trim hm'

Tactic:
Measure.compProd_eq_comp_prod

Goal state after:
case e'_3
Ω : Type u_1
β : Type u_3
β' : Type u_4
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
hf : Measurable f
hg : Measurable g
⊢ ⇑(Kernel.id ×ₖ ((condExpKernel μ m').map f ×ₖ (condExpKernel μ m').map g)) ∘ₘ μ.trim hm' =
    ⇑(Kernel.id ×ₖ ((condExpKernel μ m').map f ×ₖ (condExpKernel μ m').map g)) ∘ₘ μ.trim hm'
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: Ω : Type u_1
β : Type u_3
β' : Type u_4
mΩ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
f : Ω → β
g : Ω → β'
γ : Type u_5
mγ : MeasurableSpace γ
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝³ : StandardBorelSpace β
inst✝² : Nonempty β
inst✝¹ : StandardBorelSpace β'
inst✝ : Nonempty β'
hf : Measurable f
hg : Measurable g
k : Ω → γ
hk : Measurable k
⊢ Measure.map (fun ω ↦ (k ω, g ω, f ω)) μ =
      ⇑(Kernel.id ×ₖ (condDistrib g k μ ×ₖ condDistrib f k μ)) ∘ₘ Measure.map k μ ↔
    Measure.map (fun x ↦ ((k x, g x), f x)) μ =
      Measure.map (fun ω ↦ (k ω, g ω)) μ ⊗ₘ Kernel.prodMkRight β' (condDistrib f k μ)

Tactic:
Measure.compProd_eq_comp_prod

Goal state after:
Ω : Type u_1
β : Type u_3
β' : Type u_4
mΩ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
f : Ω → β
g : Ω → β'
γ : Type u_5
mγ : MeasurableSpace γ
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝³ : StandardBorelSpace β
inst✝² : Nonempty β
inst✝¹ : StandardBorelSpace β'
inst✝ : Nonempty β'
hf : Measurable f
hg : Measurable g
k : Ω → γ
hk : Measurable k
⊢ Measure.map (fun ω ↦ (k ω, g ω, f ω)) μ =
      ⇑(Kernel.id ×ₖ (condDistrib g k μ ×ₖ condDistrib f k μ)) ∘ₘ Measure.map k μ ↔
    Measure.map (fun x ↦ ((k x, g x), f x)) μ =
      ⇑(Kernel.id ×ₖ Kernel.prodMkRight β' (condDistrib f k μ)) ∘ₘ Measure.map (fun ω ↦ (k ω, g ω)) μ
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h : HasSubgaussianMGF X c κ ν
h_int✝ : ∀ᵐ (a : Ω') ∂ν, ∀ (i : ℤ), Integrable (fun ω ↦ rexp (↑i * X ω)) (κ a)
ω' : Ω'
h_int : ∀ (i : ℤ), Integrable (fun ω ↦ rexp (↑i * X ω)) (κ ω')
⊢ ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')

Tactic:
t

Goal state after:
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h : HasSubgaussianMGF X c κ ν
h_int✝ : ∀ᵐ (a : Ω') ∂ν, ∀ (i : ℤ), Integrable (fun ω ↦ rexp (↑i * X ω)) (κ a)
ω' : Ω'
h_int : ∀ (i : ℤ), Integrable (fun ω ↦ rexp (↑i * X ω)) (κ ω')
t : ℝ
⊢ Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h : HasSubgaussianMGF X c κ ν
p : ℝ≥0
ω' : Ω'
hi : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
⊢ ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (κ ω')

Tactic:
t

Goal state after:
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h : HasSubgaussianMGF X c κ ν
p : ℝ≥0
ω' : Ω'
hi : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
t : ℝ
⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (κ ω')
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h : HasSubgaussianMGF X c κ ν
p : ℝ≥0
ω' : Ω'
hi : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
t : ℝ
hp : ¬p = 0
⊢ ↑p ≠ ∞

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.Independence.BoundedContinuousFunction

Goal state before:
pp: Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
inst✝⁵ : (s : S) → TopologicalSpace (E s)
inst✝⁴ : (s : S) → MeasurableSpace (E s)
inst✝³ : ∀ (s : S), BorelSpace (E s)
inst✝² : ∀ (s : S), HasOuterApproxClosed (E s)
X : (s : S) → Ω → E s
inst✝¹ : IsProbabilityMeasure P
inst✝ : Finite S
A : Set Ω
mA : NullMeasurableSet A P
mX : ∀ (s : S), AEMeasurable (X s) P
h : ∀ (f : ((s : S) → E s) →ᵇ ℝ), ∫ (ω : Ω) in A, f fun x ↦ X x ω ∂P = P.real A * ∫ (ω : Ω), f fun x ↦ X x ω ∂P
this : Fintype S
f : (s : S) → E s →ᵇ ℝ
⊢ ∫ (ω : Ω) in A, ∏ s, (f s) (X s ω) ∂P = P.real A * ∫ (ω : Ω), ∏ s, (f s) (X s ω) ∂P

Tactic:
convert! h (∏ s, (f s).compContinuous ⟨Function.eval s, by fun_prop⟩)

Goal state after:
case e'_2
Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
inst✝⁵ : (s : S) → TopologicalSpace (E s)
inst✝⁴ : (s : S) → MeasurableSpace (E s)
inst✝³ : ∀ (s : S), BorelSpace (E s)
inst✝² : ∀ (s : S), HasOuterApproxClosed (E s)
X : (s : S) → Ω → E s
inst✝¹ : IsProbabilityMeasure P
inst✝ : Finite S
A : Set Ω
mA : NullMeasurableSet A P
mX : ∀ (s : S), AEMeasurable (X s) P
h : ∀ (f : ((s : S) → E s) →ᵇ ℝ), ∫ (ω : Ω) in A, f fun x ↦ X x ω ∂P = P.real A * ∫ (ω : Ω), f fun x ↦ X x ω ∂P
this : Fintype S
f : (s : S) → E s →ᵇ ℝ
x✝ : Ω
⊢ ∏ s, (f s) (X s x✝) = (∏ s, (f s).compContinuous { toFun := Function.eval s, continuous_toFun := ⋯ }) fun x ↦ X x x✝

case e'_3
Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
inst✝⁵ : (s : S) → TopologicalSpace (E s)
inst✝⁴ : (s : S) → MeasurableSpace (E s)
inst✝³ : ∀ (s : S), BorelSpace (E s)
inst✝² : ∀ (s : S), HasOuterApproxClosed (E s)
X : (s : S) → Ω → E s
inst✝¹ : IsProbabilityMeasure P
inst✝ : Finite S
A : Set Ω
mA : NullMeasurableSet A P
mX : ∀ (s : S), AEMeasurable (X s) P
h : ∀ (f : ((s : S) → E s) →ᵇ ℝ), ∫ (ω : Ω) in A, f fun x ↦ X x ω ∂P = P.real A * ∫ (ω : Ω), f fun x ↦ X x ω ∂P
this : Fintype S
f : (s : S) → E s →ᵇ ℝ
x✝ : Ω
⊢ ∏ s, (f s) (X s x✝) = (∏ s, (f s).compContinuous { toFun := Function.eval s, continuous_toFun := ⋯ }) fun x ↦ X x x✝
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h : HasSubgaussianMGF X c κ ν
t : ℝ
p : ℝ≥0
hp0 : ¬p = 0
⊢ ↑p ≠ ∞

Tactic:
by simp

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

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h✝ : HasSubgaussianMGF X c κ ν
ω' : Ω'
h : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
h_int : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
⊢ ∀ (t : ℝ), cgf X (κ ω') t ≤ ↑c * t ^ 2 / 2

Tactic:
t

Goal state after:
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h✝ : HasSubgaussianMGF X c κ ν
ω' : Ω'
h : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
h_int : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
t : ℝ
⊢ cgf X (κ ω') t ≤ ↑c * t ^ 2 / 2
Module: Mathlib.Probability.Process.Kolmogorov

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
inst✝¹ : PseudoEMetricSpace T
mΩ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace E
p q : ℝ
M : ℝ≥0
P : Measure Ω
X : T → Ω → E
hX : IsKolmogorovProcess X P p q M
⊢ ∀ (t : T), X t =ᵐ[P] X t

Tactic:
by simp

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

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
inst✝⁴ : PseudoEMetricSpace T
mΩ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace E
p q : ℝ
M : ℝ≥0
P : Measure Ω
X : T → Ω → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : SecondCountableTopology E
h_meas : ∀ (s : T), Measurable (X s)
h_kol : ∀ (s t : T), ∫⁻ (ω : Ω), edist (X s ω) (X t ω) ^ p ∂P ≤ ↑M * edist s t ^ q
hp : 0 < p
hq : 0 < q
s t : T
⊢ Measurable fun ω ↦ (X s ω, X t ω)

Tactic:
by
    suffices Measurable (fun ω ↦ (X s ω, X t ω)) by
      rwa [Prod.borelSpace.measurable_eq] at this
    fun_prop

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

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h_int✝ : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑κ ∘ₘ ν)
h_mgf : ∀ᵐ (a : Ω') ∂ν, ∀ (i : ℚ), mgf X (κ a) ↑i ≤ rexp (↑c * ↑i ^ 2 / 2)
h_int' : ∀ᵐ (a : Ω') ∂ν, ∀ (i : ℤ), Integrable (fun ω ↦ rexp (↑i * X ω)) (κ a)
ω' : Ω'
h_int : ∀ (i : ℤ), Integrable (fun ω ↦ rexp (↑i * X ω)) (κ ω')
⊢ ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')

Tactic:
t

Goal state after:
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h_int✝ : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑κ ∘ₘ ν)
h_mgf : ∀ᵐ (a : Ω') ∂ν, ∀ (i : ℚ), mgf X (κ a) ↑i ≤ rexp (↑c * ↑i ^ 2 / 2)
h_int' : ∀ᵐ (a : Ω') ∂ν, ∀ (i : ℤ), Integrable (fun ω ↦ rexp (↑i * X ω)) (κ a)
ω' : Ω'
h_int : ∀ (i : ℤ), Integrable (fun ω ↦ rexp (↑i * X ω)) (κ ω')
t : ℝ
⊢ Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
Module: Mathlib.Probability.Process.Kolmogorov

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
inst✝¹ : PseudoEMetricSpace T
mΩ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace E
p q : ℝ
P : Measure Ω
X : T → Ω → E
hX : IsAEKolmogorovProcess X P p q 0
s t : T
⊢ 0 * edist s t ^ q = 0

Tactic:
by simp

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

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h_int✝¹ : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑κ ∘ₘ ν)
h_mgf✝ : ∀ᵐ (a : Ω') ∂ν, ∀ (i : ℚ), mgf X (κ a) ↑i ≤ rexp (↑c * ↑i ^ 2 / 2)
h_int✝ : ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
ω' : Ω'
h_mgf : ∀ (i : ℚ), mgf X (κ ω') ↑i ≤ rexp (↑c * ↑i ^ 2 / 2)
h_int : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
⊢ ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)

Tactic:
t

Goal state after:
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h_int✝¹ : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑κ ∘ₘ ν)
h_mgf✝ : ∀ᵐ (a : Ω') ∂ν, ∀ (i : ℚ), mgf X (κ a) ↑i ≤ rexp (↑c * ↑i ^ 2 / 2)
h_int✝ : ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
ω' : Ω'
h_mgf : ∀ (i : ℚ), mgf X (κ ω') ↑i ≤ rexp (↑c * ↑i ^ 2 / 2)
h_int : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
t : ℝ
⊢ mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsZeroOrMarkovKernel κ
⊢ ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * 0)) (⇑κ ∘ₘ ν)

Tactic:
by simp

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

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsZeroOrMarkovKernel κ
⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), mgf (fun x ↦ 0) (κ ω') t ≤ rexp (↑0 * t ^ 2 / 2)

Tactic:
by simp

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

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
⊢ ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑κ ∘ₘ 0)

Tactic:
by simp

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

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
⊢ ∀ᵐ (ω' : Ω') ∂0, ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)

Tactic:
by simp

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

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h : HasSubgaussianMGF X c κ ν
ω' : Ω'
hm : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
⊢ ∀ (t : ℝ), mgf (-X) (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)

Tactic:
t

Goal state after:
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h : HasSubgaussianMGF X c κ ν
ω' : Ω'
hm : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
t : ℝ
⊢ mgf (-X) (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
Y : Ω → ℝ
h : HasSubgaussianMGF X c κ ν
h'✝ : X =ᵐ[⇑κ ∘ₘ ν] Y
h'' : ∀ᵐ (ω' : Ω') ∂ν, ∀ᵐ (ω : Ω) ∂κ ω', X ω = Y ω
ω' : Ω'
h_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
h' : ∀ᵐ (ω : Ω) ∂κ ω', X ω = Y ω
⊢ ∀ (t : ℝ), mgf Y (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)

Tactic:
t

Goal state after:
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
Y : Ω → ℝ
h : HasSubgaussianMGF X c κ ν
h'✝ : X =ᵐ[⇑κ ∘ₘ ν] Y
h'' : ∀ᵐ (ω' : Ω') ∂ν, ∀ᵐ (ω : Ω) ∂κ ω', X ω = Y ω
ω' : Ω'
h_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
h' : ∀ᵐ (ω : Ω) ∂κ ω', X ω = Y ω
t : ℝ
⊢ mgf Y (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
c : ℝ≥0
Ω'' : Type u_3
mΩ'' : MeasurableSpace Ω''
κ : Kernel Ω' Ω''
Y : Ω'' → Ω
X : Ω → ℝ
hY : Measurable Y
h : HasSubgaussianMGF X c (κ.map Y) ν
ω' : Ω'
h_int : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) ((κ.map Y) ω')
h_mgf : ∀ (t : ℝ), mgf X ((κ.map Y) ω') t ≤ rexp (↑c * t ^ 2 / 2)
⊢ ∀ (t : ℝ), mgf (X ∘ Y) (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)

Tactic:
t

Goal state after:
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
c : ℝ≥0
Ω'' : Type u_3
mΩ'' : MeasurableSpace Ω''
κ : Kernel Ω' Ω''
Y : Ω'' → Ω
X : Ω → ℝ
hY : Measurable Y
h : HasSubgaussianMGF X c (κ.map Y) ν
ω' : Ω'
h_int : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) ((κ.map Y) ω')
h_mgf : ∀ (t : ℝ), mgf X ((κ.map Y) ω') t ≤ rexp (↑c * t ^ 2 / 2)
t : ℝ
⊢ mgf (X ∘ Y) (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)