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
h : HasSubgaussianMGF X c κ ν
r : ℝ
ω : Ω'
hω : ∀ (t : ℝ), mgf X (κ ω) t ≤ rexp (↑c * t ^ 2 / 2)
⊢ ∀ (t : ℝ), mgf (fun ω ↦ r * X ω) (κ ω) t ≤ rexp (↑(NNReal.mk (r ^ 2) ⋯ * 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
r : ℝ
ω : Ω'
hω : ∀ (t : ℝ), mgf X (κ ω) t ≤ rexp (↑c * t ^ 2 / 2)
t : ℝ
⊢ mgf (fun ω ↦ r * X ω) (κ ω) t ≤ rexp (↑(NNReal.mk (r ^ 2) ⋯ * c) * t ^ 2 / 2)
Module: Mathlib.Probability.Process.LocalProperty

Goal state before:
pp: ι : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : Preorder ι
inst✝¹ : TopologicalSpace ι
inst✝ : OrderTopology ι
𝓕 : Filtration ι mΩ
P : Measure Ω
x✝³ : Ω
x✝² x✝¹ : ℕ
x✝ : x✝² ≤ x✝¹
⊢ (fun x ↦ ⊤) x✝² ≤ (fun x ↦ ⊤) x✝¹

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
ε : ℝ
ω' : Ω'
h1 : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
h2 : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
a✝ : IsFiniteMeasure (κ ω')
⊢ ∀ (t : ℝ), 0 ≤ t → (κ ω').real {ω | ε ≤ X ω} ≤ rexp (-t * ε + ↑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
ε : ℝ
ω' : Ω'
h1 : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
h2 : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
a✝ : IsFiniteMeasure (κ ω')
t : ℝ
⊢ 0 ≤ t → (κ ω').real {ω | ε ≤ X ω} ≤ rexp (-t * ε + ↑c * t ^ 2 / 2)
Module: Mathlib.RepresentationTheory.Subrepresentation

Goal state before:
pp: A : Type u_1
G : Type u_2
W : Type u_3
M : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : Monoid G
inst✝³ : AddCommMonoid W
inst✝² : Module A W
ρ : Representation A G W
inst✝¹ : AddCommMonoid M
inst✝ : Module A[G] M
⊢ ∀ (g : G) ⦃v : W⦄, v ∈ ⊤ → (ρ g) v ∈ ⊤

Tactic:
by simp

Goal state after:
Module: Mathlib.RepresentationTheory.Subrepresentation

Goal state before:
pp: A : Type u_1
G : Type u_2
W : Type u_3
M : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : Monoid G
inst✝³ : AddCommMonoid W
inst✝² : Module A W
ρ : Representation A G W
inst✝¹ : AddCommMonoid M
inst✝ : Module A[G] M
⊢ ∀ (g : G) ⦃v : W⦄, v ∈ ⊥ → (ρ g) v ∈ ⊥

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 Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF X cX κ ν
hY : HasSubgaussianMGF Y cY κ ν
hX0 : ¬cX = 0
hY0 : ¬cY = 0
p : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX
q : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cY
ω' : Ω'
hmX : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑cX * t ^ 2 / 2)
hmY : ∀ (t : ℝ), mgf Y (κ ω') t ≤ rexp (↑cY * t ^ 2 / 2)
hlX : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (κ ω')
hlY : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * Y ω)) (↑q) (κ ω')
⊢ ∀ (t : ℝ), mgf (fun ω ↦ X ω + Y ω) (κ ω') t ≤ rexp (↑((NNReal.sqrt cX + NNReal.sqrt cY) ^ 2) * 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 Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF X cX κ ν
hY : HasSubgaussianMGF Y cY κ ν
hX0 : ¬cX = 0
hY0 : ¬cY = 0
p : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX
q : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cY
ω' : Ω'
hmX : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑cX * t ^ 2 / 2)
hmY : ∀ (t : ℝ), mgf Y (κ ω') t ≤ rexp (↑cY * t ^ 2 / 2)
hlX : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (κ ω')
hlY : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * Y ω)) (↑q) (κ ω')
t : ℝ
⊢ mgf (fun ω ↦ X ω + Y ω) (κ ω') t ≤ rexp (↑((NNReal.sqrt cX + NNReal.sqrt cY) ^ 2) * t ^ 2 / 2)
Module: Mathlib.Probability.Process.LocalProperty

Goal state before:
pp: ι : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁶ : ConditionallyCompleteLinearOrderBot ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
𝓕 : Filtration ι mΩ
inst✝³ : DenselyOrdered ι
inst✝² : FirstCountableTopology ι
inst✝¹ : NoMaxOrder ι
τ : ℕ → Ω → WithTop ι
inst✝ : 𝓕.IsRightContinuous
hτ : IsPreLocalizingSequence 𝓕 τ P
ω : Ω
hω : Tendsto (fun x ↦ τ x ω) atTop (𝓝 ⊤)
⊢ Tendsto (fun x ↦ ⨅ j, ⨅ (_ : j ≥ x), τ j ω) atTop (𝓝 ⊤)

Tactic:
replace hω := hω.liminf_eq

Goal state after:
ι : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁶ : ConditionallyCompleteLinearOrderBot ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
𝓕 : Filtration ι mΩ
inst✝³ : DenselyOrdered ι
inst✝² : FirstCountableTopology ι
inst✝¹ : NoMaxOrder ι
τ : ℕ → Ω → WithTop ι
inst✝ : 𝓕.IsRightContinuous
hτ : IsPreLocalizingSequence 𝓕 τ P
ω : Ω
hω : liminf (fun x ↦ τ x ω) atTop = ⊤
⊢ Tendsto (fun x ↦ ⨅ j, ⨅ (_ : j ≥ x), τ j ω) atTop (𝓝 ⊤)
Module: Mathlib.Probability.Process.LocalProperty

Goal state before:
pp: ι : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁴ : ConditionallyCompleteLinearOrderBot ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
𝓕 : Filtration ι mΩ
inst✝¹ : SecondCountableTopology ι
inst✝ : IsFiniteMeasure P
τ : ℕ → Ω → WithTop ι
σ : ℕ → ℕ → Ω → WithTop ι
hτ : IsLocalizingSequence 𝓕 τ P
hσ : ∀ (n : ℕ), IsLocalizingSequence 𝓕 (σ n) P
T : ℕ → ι
hT : Tendsto T atTop atTop
n : ℕ
hn : ∀ (k : ℕ), (1 / 2) ^ n < P {ω | σ n k ω < min (τ n ω) ↑(T n)}
this : (1 / 2) ^ n ≤ P (⋂ k, {ω | σ n k ω < min (τ n ω) ↑(T n)})
⊢ ¬(1 / 2) ^ n ≤ 0

Tactic:
by simp

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 : ℕ
hKN : K ≤ N
⊢ ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N} ≤ ENNReal.ofReal ((∫ (a : Ω), X a) + 1)

Tactic:
let ρ : Measure ℝ := Measure.map X ℙ

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 : ℕ
hKN : K ≤ N
ρ : Measure ℝ := Measure.map X ℙ
⊢ ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N} ≤ ENNReal.ofReal ((∫ (a : Ω), X a) + 1)
Module: Mathlib.Probability.Process.LocalProperty

Goal state before:
pp: ι : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : ConditionallyCompleteLinearOrderBot ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
𝓕 : Filtration ι mΩ
inst✝² : SecondCountableTopology ι
inst✝¹ : IsFiniteMeasure P
inst✝ : NoMaxOrder ι
τ : ℕ → Ω → WithTop ι
σ : ℕ → ℕ → Ω → WithTop ι
hτ : IsLocalizingSequence 𝓕 τ P
hσ : ∀ (n : ℕ), IsLocalizingSequence 𝓕 (σ n) P
nk : ℕ → ℕ
T : ℕ → ι
hnk : StrictMono nk
hT : Tendsto T atTop atTop
hP : ∀ (n : ℕ), P {ω | σ n (nk n) ω < min (τ n ω) ↑(T n)} ≤ (1 / 2) ^ n
⊢ 1 / 2 < 1

Tactic:
by simp

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

Goal state before:
pp: ι : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : ConditionallyCompleteLinearOrderBot ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
𝓕 : Filtration ι mΩ
inst✝² : SecondCountableTopology ι
inst✝¹ : IsFiniteMeasure P
inst✝ : NoMaxOrder ι
τ : ℕ → Ω → WithTop ι
σ : ℕ → ℕ → Ω → WithTop ι
hτ : IsLocalizingSequence 𝓕 τ P
hσ : ∀ (n : ℕ), IsLocalizingSequence 𝓕 (σ n) P
nk : ℕ → ℕ
T : ℕ → ι
hnk : StrictMono nk
hT : Tendsto T atTop atTop
hP : ∀ (n : ℕ), P {ω | σ n (nk n) ω < min (τ n ω) ↑(T n)} ≤ (1 / 2) ^ n
this : ∑' (n : ℕ), P {ω | σ n (nk n) ω < min (τ n ω) ↑(T n)} < ∞
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, ¬σ n (nk n) ω < min (τ n ω) ↑(T n)
hωτ : Tendsto (fun x ↦ τ x ω) atTop (𝓝 ⊤)
⊢ ∀ᶠ (b : ℕ) in atTop, σ b (nk b) ω ≤ min ⊤ ⊤

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
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) * rexp (↑q * Y ω.2)) ((κ ⊗ₖ η) ω')
⊢ ∫ (x : Ω × Ω''), rexp (↑q * X x.1) * rexp (↑q * Y x.2) ∂(κ ⊗ₖ η) ω' =
    ∫ (x : Ω), rexp (↑q * X x) * ∫ (y : Ω''), rexp (↑q * Y y) ∂η (ω', x) ∂κ ω'

Tactic:
integral_compProd h_int_mul,

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) * rexp (↑q * Y ω.2)) ((κ ⊗ₖ η) ω')
⊢ ∫ (x : Ω), ∫ (y : Ω''), rexp (↑q * X x) * rexp (↑q * Y y) ∂η (ω', x) ∂κ ω' =
    ∫ (x : Ω), rexp (↑q * X x) * ∫ (y : Ω''), rexp (↑q * Y y) ∂η (ω', x) ∂κ ω'
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
⊢ ∑ j ∈ range K, (↑j ^ 2)⁻¹ * ∫ (a : Ω), (truncation X ↑j ^ 2) a ≤ 2 * ∫ (a : Ω), X a

Tactic:
let ρ : Measure ℝ := Measure.map X ℙ

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
⊢ ∑ j ∈ range K, (↑j ^ 2)⁻¹ * ∫ (a : Ω), (truncation X ↑j ^ 2) a ≤ 2 * ∫ (a : Ω), X a
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)
⊢ 1 * (2 * x) = 2 * x

Tactic:
rw [one_mul]

Goal state after:
Module: Mathlib.RepresentationTheory.Equiv

Goal state before:
pp: k✝ : Type u
inst✝¹¹ : Semiring k✝
G : Type v
inst✝¹⁰ : Monoid G
V : Type v'
inst✝⁹ : AddCommMonoid V
inst✝⁸ : Module k✝ V
W : Type w'
inst✝⁷ : AddCommMonoid W
inst✝⁶ : Module k✝ W
H : Type w
inst✝⁵ : Subsingleton H
inst✝⁴ : MulOneClass H
inst✝³ : MulAction G H
k : Type u
inst✝² : CommSemiring k
inst✝¹ : Module k V
inst✝ : Module k W
σ : Representation k G V
ρ : Representation k G W
v : V
⊢ (fun f ↦ (Finsupp.llift V k k G).symm (f.toLinearMap ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k).symm) 1)
      ((fun v ↦
          { toLinearMap := ((Finsupp.llift V k k G) fun g ↦ (σ g) v) ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k),
            isIntertwining' := ⋯ })
        v) =
    v

Tactic:
by simp

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
⊢ ∀ (g : G), 0 ∘ₗ ρ g = σ g ∘ₗ 0

Tactic:
by simp

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
⊢ ∀ (g : G), LinearMap.id ∘ₗ ρ g = ρ g ∘ₗ LinearMap.id

Tactic:
by simp

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
x✝ : G
⊢ ∀ (x : V × W), (LinearMap.fst A V W ∘ₗ (ρ.prod σ) x✝) x = (ρ x✝ ∘ₗ LinearMap.fst A V W) x

Tactic:
by simp

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
x✝ : G
⊢ ∀ (x : V × W), (LinearMap.snd A V W ∘ₗ (ρ.prod σ) x✝) x = (σ x✝ ∘ₗ LinearMap.snd A V W) x

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case e_f.e_a.h
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
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
N i k : ℕ
a✝ : k ∈ ↑(range (u i))
⊢ ∀ ⦃y : ℕ⦄, y ∈ ↑(range (u i)) → k ≠ y → (fun i j ↦ Y i ⟂ᵢ Y j) k y

Tactic:
l

Goal state after:
case e_f.e_a.h
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
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
N i k : ℕ
a✝ : k ∈ ↑(range (u i))
l : ℕ
⊢ l ∈ ↑(range (u i)) → k ≠ l → (fun i j ↦ Y i ⟂ᵢ Y j) k l
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 : ρ.Equiv σ
g : G
⊢ ↑(LinearEquiv.refl A V) ∘ₗ ρ g = ρ g ∘ₗ ↑(LinearEquiv.refl A V)

Tactic:
by simp

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
inst✝ : IsProbabilityMeasure μ
a b t : ℝ
hm : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hc : ∫ (x : Ω), X x ∂μ = 0
ht : 0 < t
hi : ∀ (u : ℝ), Integrable (fun ω ↦ rexp (u * X ω)) μ
hs : Set.Icc 0 t ⊆ interior (integrableExpSet X μ)
u : ℝ
h1 : u ∈ Set.Ioo 0 t
h2 : cgf X μ t = iteratedDeriv 2 (cgf X μ) u * t ^ 2 / 2
⊢ rexp (cgf X μ t) ≤ rexp ((↑‖b - a‖₊ / 2) ^ 2 * t ^ 2 / 2)

Tactic:
exp_le_exp,

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
inst✝ : IsProbabilityMeasure μ
a b t : ℝ
hm : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hc : ∫ (x : Ω), X x ∂μ = 0
ht : 0 < t
hi : ∀ (u : ℝ), Integrable (fun ω ↦ rexp (u * X ω)) μ
hs : Set.Icc 0 t ⊆ interior (integrableExpSet X μ)
u : ℝ
h1 : u ∈ Set.Ioo 0 t
h2 : cgf X μ t = iteratedDeriv 2 (cgf X μ) u * t ^ 2 / 2
⊢ cgf X μ t ≤ (↑‖b - a‖₊ / 2) ^ 2 * 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 : CommSemiring 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.asModule →ₗ[A[G]] σ.asModule
a : A
v : V
⊢ f.toFun (a • v) = (RingHom.id A) a • f.toFun v

Tactic:
by simp

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Res

Goal state before:
pp: G : Type v1
H : Type v2
inst✝² : Monoid G
inst✝¹ : Monoid H
f : H →* G
k : Type u
inst✝ : CommRing k
S : ShortComplex (Rep k G)
⊢ (S.map (resFunctor f)).Exact ↔ S.Exact

Tactic:
ShortComplex.exact_map_iff_of_faithful

Goal state after:
G : Type v1
H : Type v2
inst✝² : Monoid G
inst✝¹ : Monoid H
f : H →* G
k : Type u
inst✝ : CommRing k
S : ShortComplex (Rep k G)
⊢ S.Exact ↔ S.Exact
Module: Mathlib.RepresentationTheory.Rep.Res

Goal state before:
pp: case mp
G : Type v1
H : Type v2
inst✝² : Monoid G
inst✝¹ : Monoid H
k : Type u
inst✝ : CommRing k
φ : H →* G
S : ShortComplex (Rep k G)
h : (S.map (resFunctor φ)).ShortExact
h₁ : (S.map (resFunctor φ)).Exact
h₂ : Mono (S.map (resFunctor φ)).f
h₃ : Epi (S.map (resFunctor φ)).g
⊢ S.ShortExact

Tactic:
ShortComplex.exact_map_iff_of_faithful

Goal state after:
case mp
G : Type v1
H : Type v2
inst✝² : Monoid G
inst✝¹ : Monoid H
k : Type u
inst✝ : CommRing k
φ : H →* G
S : ShortComplex (Rep k G)
h : (S.map (resFunctor φ)).ShortExact
h₁ : S.Exact
h₂ : Mono (S.map (resFunctor φ)).f
h₃ : Epi (S.map (resFunctor φ)).g
⊢ S.ShortExact
Module: Mathlib.RepresentationTheory.Rep.Res

Goal state before:
pp: G : Type v1
H : Type v2
inst✝² : Monoid G
inst✝¹ : Monoid H
k : Type u
inst✝ : CommRing k
φ : H →* G
S : ShortComplex (Rep k G)
exact✝ : S.Exact
mono_f : Mono S.f
epi_g : Epi S.g
⊢ (S.map (resFunctor φ)).Exact

Tactic:
ShortComplex.exact_map_iff_of_faithful

Goal state after:
G : Type v1
H : Type v2
inst✝² : Monoid G
inst✝¹ : Monoid H
k : Type u
inst✝ : CommRing k
φ : H →* G
S : ShortComplex (Rep k G)
exact✝ : S.Exact
mono_f : Mono S.f
epi_g : Epi S.g
⊢ S.Exact
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case e'_7
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
A : Tendsto (fun i ↦ ∫ (a : Ω), truncation (X i) (↑i) a) atTop (𝓝 (∫ (a : Ω), X 0 a))
n : ℕ
⊢ (∫ (a : Ω), (∑ i ∈ range n, truncation (X i) ↑i) a) - ↑n * ∫ (a : Ω), X 0 a =
    ∑ i ∈ range n, ((∫ (a : Ω), truncation (X i) (↑i) a) - ∫ (a : Ω), X 0 a)

Tactic:
simp only [sum_sub_distrib, sum_const, card_range, nsmul_eq_mul, Finset.sum_apply, sub_left_inj]

Goal state after:
case e'_7
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
A : Tendsto (fun i ↦ ∫ (a : Ω), truncation (X i) (↑i) a) atTop (𝓝 (∫ (a : Ω), X 0 a))
n : ℕ
⊢ ∫ (a : Ω), ∑ c ∈ range n, truncation (X c) (↑c) a = ∑ x ∈ range n, ∫ (a : Ω), truncation (X x) (↑x) a
Module: Mathlib.RepresentationTheory.Coinduced

Goal state before:
pp: k : Type u_1
G : Type u_2
H : Type u_3
inst✝⁶ : Semiring k
inst✝⁵ : Monoid G
inst✝⁴ : Monoid H
φ : G →* H
A : Type u_4
B : Type u_5
inst✝³ : AddCommMonoid A
inst✝² : Module k A
inst✝¹ : AddCommMonoid B
inst✝ : Module k B
σ : Representation k G A
ρ : Representation k G B
⊢ 0 ∈ {f | ∀ (g : G) (h : H), f (φ g * h) = (σ g) (f h)}

Tactic:
by simp

Goal state after:
Module: Mathlib.RepresentationTheory.Coinduced

Goal state before:
pp: k : Type u
G✝ : Type v
H : Type w
inst✝⁴ : CommRing k
inst✝³ : Monoid G✝
inst✝² : Monoid H
φ : G✝ →* H
A : Rep k G✝
G : Type v'
inst✝¹ : Group G
S : Subgroup G
X Y : Rep k ↥S
f : X ⟶ Y
inst✝ : Epi f
y : ↑((coindFunctor k S.subtype).obj Y)
this : Setoid G := QuotientGroup.rightRel S
s✝ : ↑Y → ↑X
hs : ∀ (b : ↑Y), (Hom.hom f) (s✝ b) = b
i : Quotient this → G
hi : ∀ (b : Quotient this), Quotient.mk' (i b) = b
γ : G → ↥S := fun g ↦ ⟨g * (i (Quotient.mk' g))⁻¹, ⋯⟩
s : ↥S
g : G
⊢ g * (↑s * g)⁻¹ ∈ S

Tactic:
by simp

Goal state after:
Module: Mathlib.RepresentationTheory.Coinvariants

Goal state before:
pp: k : Type u_6
G : Type u_7
V : Type u_8
inst✝⁴ : CommRing k
inst✝³ : Group G
inst✝² : AddCommGroup V
inst✝¹ : Module k V
ρ : Representation k G V
S : Subgroup G
inst✝ : S.Normal
g : G
x✝¹ : V
x✝ : x✝¹ ∈ Set.range fun gv ↦ ((MonoidHom.comp ρ S.subtype) gv.1) gv.2 - gv.2
s : ↥S
x : V
hs : (fun gv ↦ ((MonoidHom.comp ρ S.subtype) gv.1) gv.2 - gv.2) (s, x) = x✝¹
⊢ ((MonoidHom.comp ρ S.subtype) ⟨g * ↑s * g⁻¹, ⋯⟩) ((ρ g) x) - (ρ g) x = (ρ g) ((ρ ↑s) x) - (ρ g) x

Tactic:
by simp

Goal state after:
Module: Mathlib.RepresentationTheory.Coinduced

Goal state before:
pp: k : Type u
G : Type v
H : Type w
inst✝² : CommRing k
inst✝¹ : Monoid G
inst✝ : Monoid H
φ : G →* H
A : Rep k G
x : ↥(Representation.coindV φ A.ρ)
⊢ (fun f ↦ ⟨fun h ↦ (Hom.hom f).toLinearMap (MonoidAlgebra.single h 1), ⋯⟩)
      ((fun f ↦
          ofHom { toLinearMap := linearCombination k ↑f ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k), isIntertwining' := ⋯ })
        x) =
    x

Tactic:
by simp

Goal state after:
Module: Mathlib.RepresentationTheory.Coinduced

Goal state before:
pp: k : Type u
G : Type v
H : Type w
inst✝² : CommRing k
inst✝¹ : Monoid G
inst✝ : Monoid H
φ : G →* H
A : Rep k G
x : res φ (leftRegular k H) ⟶ A
x✝ : H
⊢ (Hom.hom
          ((fun f ↦
              ofHom
                { toLinearMap := linearCombination k ↑f ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k), isIntertwining' := ⋯ })
            ((fun f ↦ ⟨fun h ↦ (Hom.hom f).toLinearMap (MonoidAlgebra.single h 1), ⋯⟩) x))).toLinearMap
      (MonoidAlgebra.single x✝ 1) =
    (Hom.hom x).toLinearMap (MonoidAlgebra.single x✝ 1)

Tactic:
by simp

Goal state after:
Module: Mathlib.RepresentationTheory.Coinvariants

Goal state before:
pp: k✝ : Type u_1
G✝ : Type u_2
V✝ : Type u_3
W : Type u_4
X : Type u_5
inst✝¹¹ : CommRing k✝
inst✝¹⁰ : Monoid G✝
inst✝⁹ : AddCommGroup V✝
inst✝⁸ : Module k✝ V✝
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module k✝ W
inst✝⁵ : AddCommGroup X
inst✝⁴ : Module k✝ X
ρ✝ : Representation k✝ G✝ V✝
τ : Representation k✝ G✝ W
υ : Representation k✝ G✝ X
k : Type u_6
G : Type u_7
V : Type u_8
inst✝³ : CommRing k
inst✝² : Group G
inst✝¹ : AddCommGroup V
inst✝ : Module k V
ρ : Representation k G V
α : Type u_9
a : α
g : G
x : V
⊢ ((ρ.finsupp α) g) (single a x) - single a x = (lsingle a) ((ρ g) x) - (lsingle a) x

Tactic:
by simp

Goal state after:
Module: Mathlib.RepresentationTheory.Coinvariants

Goal state before:
pp: k : Type u_6
G : Type u_7
V : Type u_8
W : Type u_9
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddCommGroup W
inst✝ : Module k W
ρ : Representation k G V
τ : Representation k G W
x : V
y : W
g : G
⊢ ((ρ.tprod τ) g⁻¹) (x ⊗ₜ[k] (τ g) y) - x ⊗ₜ[k] (τ g) y = (ρ g⁻¹) x ⊗ₜ[k] y - x ⊗ₜ[k] (τ g) y

Tactic:
by simp

Goal state after:
Module: Mathlib.RepresentationTheory.Coinvariants

Goal state before:
pp: k : Type u_6
G : Type u_7
V : Type u_8
W : Type u_9
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddCommGroup W
inst✝ : Module k W
ρ : Representation k G V
τ : Representation k G W
x : V
y : W
g : G
⊢ ((ρ.tprod τ) g⁻¹) ((ρ g) x ⊗ₜ[k] y) - (ρ g) x ⊗ₜ[k] y = x ⊗ₜ[k] (τ g⁻¹) y - (ρ g) x ⊗ₜ[k] y

Tactic:
by simp

Goal state after: