mathlib-initiative/mathlib-tactics huggingface dataset:
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
Ω'' : Type u_3
mΩ'' : MeasurableSpace Ω''
Y : Ω'' → ℝ
cY : ℝ≥0
inst✝¹ : SFinite ν
η : Kernel (Ω' × Ω) Ω''
inst✝ : IsZeroOrMarkovKernel η
hX : HasSubgaussianMGF X c κ ν
hY : HasSubgaussianMGF Y cY η (ν ⊗ₘ κ)
hκ : IsSFiniteKernel κ
q : ℚ
ω' : Ω'
hX_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
hX_int : Integrable (fun y ↦ rexp (↑q * X y)) (κ ω')
hY_mgf : ∀ᵐ (b : Ω) ∂κ ω', ∀ (t : ℝ), mgf Y (η (ω', b)) t ≤ rexp (↑cY * t ^ 2 / 2)
h_int_mul : Integrable (fun ω ↦ rexp (↑q * (X ω.1 + Y ω.2))) ((κ ⊗ₖ η) ω')
⊢ (∫ (a : Ω), rexp (↑q * X a) ∂κ ω') * rexp (↑cY * ↑q ^ 2 / 2) ≤ rexp (↑c * ↑q ^ 2 / 2) * rexp (↑cY * ↑q ^ 2 / 2)

Tactic:
]

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
Ω'' : Type u_3
mΩ'' : MeasurableSpace Ω''
Y : Ω'' → ℝ
cY : ℝ≥0
inst✝¹ : SFinite ν
η : Kernel (Ω' × Ω) Ω''
inst✝ : IsZeroOrMarkovKernel η
hX : HasSubgaussianMGF X c κ ν
hY : HasSubgaussianMGF Y cY η (ν ⊗ₘ κ)
hκ : IsSFiniteKernel κ
q : ℚ
ω' : Ω'
hX_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
hX_int : Integrable (fun y ↦ rexp (↑q * X y)) (κ ω')
hY_mgf : ∀ᵐ (b : Ω) ∂κ ω', ∀ (t : ℝ), mgf Y (η (ω', b)) t ≤ rexp (↑cY * t ^ 2 / 2)
h_int_mul : Integrable (fun ω ↦ rexp (↑q * (X ω.1 + Y ω.2))) ((κ ⊗ₖ η) ω')
⊢ (∫ (a : Ω), rexp (↑q * X a) ∂κ ω') * rexp (↑cY * ↑q ^ 2 / 2) ≤ rexp (↑c * ↑q ^ 2 / 2) * rexp (↑cY * ↑q ^ 2 / 2)
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
a✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
x : ℝ
hx : x ∈ Set.Ioc ↑k ↑(k + 1)
⊢ 2 / (↑k + 1) * x ^ 2 = x / (↑k + 1) * (2 * x)

Tactic:
by ring

Goal state after:
Module: Mathlib.RepresentationTheory.Action

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Monoid G
inst✝ : Semiring k
X : Type w
g : G
x : X
⊢ MonoidAlgebra.single ((ConcreteCategory.hom ((Action.trivial G X).ρ g)) x) 1 = MonoidAlgebra.single x 1

Tactic:
]

Goal state after:
k : Type u
G : Type v
inst✝¹ : Monoid G
inst✝ : Semiring k
X : Type w
g : G
x : X
⊢ MonoidAlgebra.single ((ConcreteCategory.hom ((Action.trivial G X).ρ g)) x) 1 = MonoidAlgebra.single x 1
Module: Mathlib.RepresentationTheory.Action

Goal state before:
pp: k : Type u
G : Type v
V : Type u'
W : Type v'
inst✝⁵ : Monoid G
inst✝⁴ : Semiring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddCommGroup W
inst✝ : Module k W
σ : Representation k G V
ρ : Representation k G W
X✝ Y Z : Action (Type w) G
X : Type w
g : G
⊢ ↑(LinearEquiv.refl k k[(Action.trivial G X).V]) = ↑(LinearEquiv.refl k k[(Action.trivial G X).V])

Tactic:
]

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
a✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
x : ℝ
hx : x ∈ Set.Ioc ↑k ↑(k + 1)
⊢ 2 * x = 2 * x

Tactic:
]

Goal state after:
Module: Mathlib.RepresentationTheory.AlgebraRepresentation.Basic

Goal state before:
pp: case inr
A : Type u_1
V : Type u_2
k : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : Ring A
inst✝⁸ : Algebra k A
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : Module A V
inst✝⁴ : IsScalarTower k A V
inst✝³ : IsSimpleModule A V
inst✝² : FiniteDimensional k V
inst✝¹ : IsAlgClosed k
inst✝ : IsMulCommutative A
this : Nontrivial V
v : V
v_nz : v ≠ 0
U : Submodule A V :=
  let __src := k ∙ v;
  { toAddSubmonoid := __src.toAddSubmonoid, smul_mem' := ⋯ }
hU : U = ⊤
⊢ k ∙ v = ⊤

Tactic:
]

Goal state after:
case inr
A : Type u_1
V : Type u_2
k : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : Ring A
inst✝⁸ : Algebra k A
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : Module A V
inst✝⁴ : IsScalarTower k A V
inst✝³ : IsSimpleModule A V
inst✝² : FiniteDimensional k V
inst✝¹ : IsAlgClosed k
inst✝ : IsMulCommutative A
this : Nontrivial V
v : V
v_nz : v ≠ 0
U : Submodule A V :=
  let __src := k ∙ v;
  { toAddSubmonoid := __src.toAddSubmonoid, smul_mem' := ⋯ }
hU : U = ⊤
⊢ k ∙ v = ⊤
Module: Mathlib.RepresentationTheory.AlgebraRepresentation.Basic

Goal state before:
pp: case inr
A : Type u_1
V : Type u_2
k : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : Ring A
inst✝⁸ : Algebra k A
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : Module A V
inst✝⁴ : IsScalarTower k A V
inst✝³ : IsSimpleModule A V
inst✝² : FiniteDimensional k V
inst✝¹ : IsAlgClosed k
inst✝ : IsMulCommutative A
this : Nontrivial V
v : V
v_nz : v ≠ 0
U : Submodule A V :=
  let __src := k ∙ v;
  { toAddSubmonoid := __src.toAddSubmonoid, smul_mem' := ⋯ }
hU : ⊤ ≤ U
⊢ ⊤ ≤ k ∙ v

Tactic:
]

Goal state after:
case inr
A : Type u_1
V : Type u_2
k : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : Ring A
inst✝⁸ : Algebra k A
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : Module A V
inst✝⁴ : IsScalarTower k A V
inst✝³ : IsSimpleModule A V
inst✝² : FiniteDimensional k V
inst✝¹ : IsAlgClosed k
inst✝ : IsMulCommutative A
this : Nontrivial V
v : V
v_nz : v ≠ 0
U : Submodule A V :=
  let __src := k ∙ v;
  { toAddSubmonoid := __src.toAddSubmonoid, smul_mem' := ⋯ }
hU : ⊤ ≤ U
⊢ ⊤ ≤ k ∙ v
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
⊢ ∫ (x : ℝ) in ↑0..↑K, 2 * x ∂ρ = 2 * ∫ (x : ℝ) in 0..↑K, x ∂ρ

pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
x✝ : k < K
⊢ IntervalIntegrable (HMul.hMul 2) ρ ↑k ↑(k + 1)

Tactic:
]

Goal state after:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
⊢ ∫ (x : ℝ) in ↑0..↑K, 2 * x ∂ρ = 2 * ∫ (x : ℝ) in 0..↑K, x ∂ρ

Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
x✝ : k < K
⊢ IntervalIntegrable (HMul.hMul 2) ρ ↑k ↑(k + 1)
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
m mΩ : MeasurableSpace Ω
hm : m ≤ mΩ
inst✝¹ : StandardBorelSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
c : ℝ≥0
h : HasCondSubgaussianMGF m hm X c μ
t : ℝ
h_eq✝ : μ[fun ω ↦ rexp (t * X ω) | m] =ᵐ[μ.trim hm] fun ω ↦ ∫ (ω : Ω), rexp (t * X ω) ∂(condExpKernel μ m) ω
ω' : Ω
h_mgf : ∀ (t : ℝ), mgf X ((condExpKernel μ m) ω') t ≤ rexp (↑c * t ^ 2 / 2)
h_eq : μ[fun ω ↦ rexp (t * X ω) | m] ω' = ∫ (ω : Ω), rexp (t * X ω) ∂(condExpKernel μ m) ω'
⊢ ∫ (ω : Ω), rexp (t * X ω) ∂(condExpKernel μ m) ω' ≤ rexp (↑c * t ^ 2 / 2)

Tactic:
]

Goal state after:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
hm : m ≤ mΩ
inst✝¹ : StandardBorelSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
c : ℝ≥0
h : HasCondSubgaussianMGF m hm X c μ
t : ℝ
h_eq✝ : μ[fun ω ↦ rexp (t * X ω) | m] =ᵐ[μ.trim hm] fun ω ↦ ∫ (ω : Ω), rexp (t * X ω) ∂(condExpKernel μ m) ω
ω' : Ω
h_mgf : ∀ (t : ℝ), mgf X ((condExpKernel μ m) ω') t ≤ rexp (↑c * t ^ 2 / 2)
h_eq : μ[fun ω ↦ rexp (t * X ω) | m] ω' = ∫ (ω : Ω), rexp (t * X ω) ∂(condExpKernel μ m) ω'
⊢ ∫ (ω : Ω), rexp (t * X ω) ∂(condExpKernel μ m) ω' ≤ rexp (↑c * t ^ 2 / 2)
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
⊢ 2 * ∫ (x : ℝ) in ↑0..↑K, x ∂ρ = 2 * ∫ (x : ℝ) in 0..↑K, x ∂ρ

Tactic:
]

Goal state after:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
⊢ 2 * ∫ (x : ℝ) in ↑0..↑K, x ∂ρ = 2 * ∫ (x : ℝ) in 0..↑K, x ∂ρ
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
h : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())
Y : Ω → ℝ
h' : X =ᵐ[μ] Y
⊢ Kernel.HasSubgaussianMGF Y c (Kernel.const Unit μ) (Measure.dirac ())

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
h : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())
Y : Ω → ℝ
h' : X =ᵐ[μ] Y
⊢ Kernel.HasSubgaussianMGF Y c (Kernel.const Unit μ) (Measure.dirac ())
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
h : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())
t : ℝ
p : ℝ≥0
⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) μ

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
h : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())
t : ℝ
p : ℝ≥0
⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) μ
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
h : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())
t : ℝ
⊢ cgf X μ t ≤ ↑c * t ^ 2 / 2

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
h : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())
t : ℝ
⊢ cgf X μ t ≤ ↑c * t ^ 2 / 2
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
c : ℝ≥0
Ω' : Type u_2
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Y : Ω' → Ω
X : Ω → ℝ
hY : AEMeasurable Y μ
h : HasSubgaussianMGF X c (Measure.map Y μ)
t : ℝ
h1 : Integrable ((fun ω ↦ rexp (t * X ω)) ∘ Y) μ
⊢ Integrable (fun ω ↦ rexp (t * (X ∘ Y) ω)) μ

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
c : ℝ≥0
Ω' : Type u_2
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Y : Ω' → Ω
X : Ω → ℝ
hY : AEMeasurable Y μ
h : HasSubgaussianMGF X c (Measure.map Y μ)
t : ℝ
h1 : Integrable ((fun ω ↦ rexp (t * X ω)) ∘ Y) μ
⊢ Integrable (fun ω ↦ rexp (t * (X ∘ Y) ω)) μ
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
⊢ ∫ (x : Ω), truncation X (↑K) x ≤ ∫ (a : Ω), X a

Tactic:
]

Goal state after:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
⊢ ∫ (x : Ω), truncation X (↑K) x ≤ ∫ (a : Ω), X a
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
c : ℝ≥0
Ω' : Type u_2
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Y : Ω' → Ω
X : Ω → ℝ
hY : AEMeasurable Y μ
h : HasSubgaussianMGF X c (Measure.map Y μ)
t : ℝ
⊢ mgf (X ∘ Y) μ t = mgf (X ∘ Y) μ t

Tactic:
]

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

Goal state before:
pp: case refine_1
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
hX : AEMeasurable X μ
h : HasSubgaussianMGF id c (Measure.map X μ)
⊢ HasSubgaussianMGF (id ∘ X) c μ

Tactic:
]

Goal state after:
case refine_1
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
hX : AEMeasurable X μ
h : HasSubgaussianMGF id c (Measure.map X μ)
⊢ HasSubgaussianMGF (id ∘ X) c μ
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: case refine_2
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
hX : AEMeasurable X μ
h : HasSubgaussianMGF X c μ
t : ℝ
⊢ Integrable ((fun ω ↦ rexp (t * id ω)) ∘ X) μ

Tactic:
]

Goal state after:
case refine_2
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
hX : AEMeasurable X μ
h : HasSubgaussianMGF X c μ
t : ℝ
⊢ Integrable ((fun ω ↦ rexp (t * id ω)) ∘ X) μ
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: case refine_3
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
hX : AEMeasurable X μ
h : HasSubgaussianMGF X c μ
t : ℝ
⊢ mgf X μ t ≤ rexp (↑c * t ^ 2 / 2)

Tactic:
]

Goal state after:
case refine_3
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
hX : AEMeasurable X μ
h : HasSubgaussianMGF X c μ
t : ℝ
⊢ mgf X μ t ≤ rexp (↑c * t ^ 2 / 2)
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
Ω' : Type u_2
mΩ' : MeasurableSpace Ω'
μ' : Measure Ω'
Y : Ω' → ℝ
hX : HasSubgaussianMGF id c (Measure.map X μ)
hXY : IdentDistrib X Y μ μ'
⊢ HasSubgaussianMGF Y c μ'

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
Ω' : Type u_2
mΩ' : MeasurableSpace Ω'
μ' : Measure Ω'
Y : Ω' → ℝ
hX : HasSubgaussianMGF id c (Measure.map X μ)
hXY : IdentDistrib X Y μ μ'
⊢ HasSubgaussianMGF Y c μ'
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
Ω' : Type u_2
mΩ' : MeasurableSpace Ω'
μ' : Measure Ω'
Y : Ω' → ℝ
hX : HasSubgaussianMGF id c (Measure.map X μ)
hXY : IdentDistrib X Y μ μ'
⊢ HasSubgaussianMGF id c (Measure.map X μ)

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
Ω' : Type u_2
mΩ' : MeasurableSpace Ω'
μ' : Measure Ω'
Y : Ω' → ℝ
hX : HasSubgaussianMGF id c (Measure.map X μ)
hXY : IdentDistrib X Y μ μ'
⊢ HasSubgaussianMGF id c (Measure.map X μ)
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
hm : m ≤ mΩ
hXm : Measurable X
hX : HasSubgaussianMGF X c μ
t : ℝ
⊢ ∫ (x : Ω), rexp (t * X x) ∂μ ≤ rexp (↑c * t ^ 2 / 2)

pp: case hf
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
hm : m ≤ mΩ
hXm : Measurable X
hX : HasSubgaussianMGF X c μ
t : ℝ
⊢ StronglyMeasurable fun ω ↦ rexp (t * X ω)

Tactic:
]

Goal state after:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
hm : m ≤ mΩ
hXm : Measurable X
hX : HasSubgaussianMGF X c μ
t : ℝ
⊢ ∫ (x : Ω), rexp (t * X x) ∂μ ≤ rexp (↑c * t ^ 2 / 2)

case hf
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
hm : m ≤ mΩ
hXm : Measurable X
hX : HasSubgaussianMGF X c μ
t : ℝ
⊢ StronglyMeasurable fun ω ↦ rexp (t * X ω)
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
h : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())
r : ℝ
⊢ Kernel.HasSubgaussianMGF (fun ω ↦ r * X ω) (⟨r ^ 2, ⋯⟩ * c) (Kernel.const Unit μ) (Measure.dirac ())

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
h : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())
r : ℝ
⊢ Kernel.HasSubgaussianMGF (fun ω ↦ r * X ω) (⟨r ^ 2, ⋯⟩ * c) (Kernel.const Unit μ) (Measure.dirac ())
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
h : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())
ε : ℝ
hε : 0 ≤ ε
⊢ μ.real {ω | ε ≤ X ω} ≤ rexp (-ε ^ 2 / (2 * ↑c))

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
h : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())
ε : ℝ
hε : 0 ≤ ε
⊢ μ.real {ω | ε ≤ X ω} ≤ rexp (-ε ^ 2 / (2 * ↑c))
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case hbc
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n ↦ truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n ↦ ∑ i ∈ range n, Y i
hS : S = fun n ↦ ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n ↦ ⌊c ^ n⌋₊
u_mono : Monotone u
K j : ℕ
a✝ : j ∈ range K
⊢ Var[truncation (X 0) ↑j; ℙ] ≤ ∫ (a : Ω), (truncation (X 0) ↑j ^ 2) a

Tactic:
]

Goal state after:
case hbc
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n ↦ truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n ↦ ∑ i ∈ range n, Y i
hS : S = fun n ↦ ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n ↦ ⌊c ^ n⌋₊
u_mono : Monotone u
K j : ℕ
a✝ : j ∈ range K
⊢ Var[truncation (X 0) ↑j; ℙ] ≤ ∫ (a : Ω), (truncation (X 0) ↑j ^ 2) a
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF X cX μ
hY : HasSubgaussianMGF Y cY μ
hindep : X ⟂ᵢ[μ] Y
t : ℝ
⊢ rexp (↑cX * t ^ 2 / 2 + ↑cY * t ^ 2 / 2) = rexp ((↑cX + ↑cY) * t ^ 2 / 2)

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF X cX μ
hY : HasSubgaussianMGF Y cY μ
hindep : X ⟂ᵢ[μ] Y
t : ℝ
⊢ rexp (↑cX * t ^ 2 / 2 + ↑cY * t ^ 2 / 2) = rexp ((↑cX + ↑cY) * t ^ 2 / 2)
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : Semiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
f : ρ.IntertwiningMap σ
g : G
l : U →ₗ[A] V
⊢ σ g ∘ₗ f.toLinearMap ∘ₗ l = σ g ∘ₗ f.toLinearMap ∘ₗ l

Tactic:
]

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : Semiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f✝ : V →ₗ[A] W
f : ρ.IntertwiningMap σ
n : ℕ
⊢ n • ⇑f = n • ⇑f

Tactic:
]

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
h_indep : iIndepFun X μ
c : ι → ℝ≥0
h_meas : ∀ (i : ι), AEMeasurable (X i) μ
this : IsProbabilityMeasure μ
i : ι
s : Finset ι
his : i ∉ s
h : (∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ) → HasSubgaussianMGF (fun ω ↦ ∑ i ∈ s, X i ω) (∑ i ∈ s, c i) μ
h_subG : ∀ i_1 ∈ insert i s, HasSubgaussianMGF (X i_1) (c i_1) μ
h_indep' : X i ⟂ᵢ[μ] ∑ j ∈ s, X j
x✝ : Ω
⊢ ∑ c ∈ s, X c x✝ = ∑ c ∈ s, X c x✝

Tactic:
]

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
h_indep : iIndepFun X μ
c : ι → ℝ≥0
s : Finset ι
h_subG : ∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ
this : HasSubgaussianMGF (fun ω ↦ ∑ i, X (↑i) ω) (∑ i ∈ s, c i) μ
⊢ HasSubgaussianMGF (fun ω ↦ ∑ i ∈ s, X i ω) (∑ i ∈ s, c i) μ

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
h_indep : iIndepFun X μ
c : ι → ℝ≥0
s : Finset ι
h_subG : ∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ
this : HasSubgaussianMGF (fun ω ↦ ∑ i, X (↑i) ω) (∑ i ∈ s, c i) μ
⊢ HasSubgaussianMGF (fun ω ↦ ∑ i ∈ s, X i ω) (∑ i ∈ s, c i) μ
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
h_indep : iIndepFun X μ
c : ι → ℝ≥0
s : Finset ι
h_subG : ∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ
⊢ HasSubgaussianMGF (fun ω ↦ ∑ i ∈ s, X i ω) (∑ i ∈ s, c i) μ

Tactic:
have : HasSubgaussianMGF (fun ω ↦ ∑ (i : s), X i ω) (∑ (i : s), c i) μ := by
    apply sum_of_iIndepFun_of_forall_aemeasurable
    · exact h_indep.precomp Subtype.val_injective
    · exact fun i ↦ (h_subG i i.2).aemeasurable
    · exact fun i _ ↦ h_subG i i.2
  rw [Finset.sum_coe_sort] at this
  exact this.congr (ae_of_all _ fun ω ↦ Finset.sum_attach s (fun i ↦ X i ω))

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

Goal state before:
pp: case refine_1
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) cX μ
hY : HasSubgaussianMGF (fun ω ↦ Y ω - ∫ (x : Ω), Y x ∂μ) cY μ
hindep : X ⟂ᵢ[μ] Y
h_le : ∫ (x : Ω), Y x ∂μ ≤ ∫ (x : Ω), X x ∂μ
⊢ HasSubgaussianMGF (fun ω ↦ Y ω - ∫ (x : Ω), Y x ∂μ - (X ω - ∫ (x : Ω), X x ∂μ)) (cY + cX) μ

Tactic:
]

Goal state after:
case refine_1
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) cX μ
hY : HasSubgaussianMGF (fun ω ↦ Y ω - ∫ (x : Ω), Y x ∂μ) cY μ
hindep : X ⟂ᵢ[μ] Y
h_le : ∫ (x : Ω), Y x ∂μ ≤ ∫ (x : Ω), X x ∂μ
⊢ HasSubgaussianMGF (fun ω ↦ Y ω - ∫ (x : Ω), Y x ∂μ - (X ω - ∫ (x : Ω), X x ∂μ)) (cY + cX) μ
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : Semiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : V →ₗ[A] W
φ✝ φ : ρ.Equiv σ
g : G
⊢ σ g = σ g

Tactic:
]

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : Monoid G
inst✝³ : AddCommMonoid V
inst✝² : AddCommMonoid W
inst✝¹ : Module A V
inst✝ : Module A W
ρ : Representation A G V
σ : Representation A G W
e : V ≃ₗ[A] W
he : ∀ (g : G), ↑e ∘ₗ ρ g = σ g ∘ₗ ↑e
g : G
⊢ σ g = σ g

Tactic:
]

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : Semiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : V →ₗ[A] W
φ✝ φ : ρ.Equiv σ
ψ : σ.Equiv τ
g : G
⊢ τ g ∘ₗ (↑ψ).toLinearMap ∘ₗ (↑φ).toLinearMap = τ g ∘ₗ ↑ψ.toLinearEquiv ∘ₗ ↑φ.toLinearEquiv

Tactic:
]

Goal state after: