mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: case mp
Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
s : ι → Set Ω
hπ : ∀ (i : ι), MeasurableSet (s i)
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ (∀ (s_1 : Finset ι) {f : ι → Set Ω},
      (∀ i ∈ s_1, f i = s i) →
        μ[(⋂ i ∈ s_1, s i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ x ∈ s_1, μ[(s x).indicator fun ω ↦ 1 | m']) →
    ∀ (S : Finset ι), μ[(⋂ i ∈ S, s i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ i ∈ S, μ[(s i).indicator fun ω ↦ 1 | m']

Tactic:
intros
      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
H : Type u_7
inst✝⁹ : (s : S) → TopologicalSpace (E s)
inst✝⁸ : (s : S) → MeasurableSpace (E s)
inst✝⁷ : ∀ (s : S), BorelSpace (E s)
inst✝⁶ : ∀ (s : S), HasOuterApproxClosed (E s)
inst✝⁵ : TopologicalSpace H
inst✝⁴ : MeasurableSpace H
inst✝³ : BorelSpace H
inst✝² : HasOuterApproxClosed H
X : (s : S) → Ω → E s
U : Ω → H
inst✝¹ : IsFiniteMeasure P
inst✝ : Finite S
mX : ∀ (s : S), AEMeasurable (X s) P
mU : AEMeasurable U P
h :
  ∀ (f : ((s : S) → E s) →ᵇ ℝ) (g : H →ᵇ ℝ),
    ∫ (x : Ω), (fun ω ↦ (f fun x ↦ X x ω) * g (U ω)) x ∂P =
      (∫ (x : Ω), (fun ω ↦ f fun x ↦ X x ω) x ∂P) * ∫ (x : Ω), (⇑g ∘ U) x ∂P
⊢ (fun ω s ↦ X s ω) ⟂ᵢ[P] U

Tactic:
have := Fintype.ofFinite S

Goal state after:
Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
H : Type u_7
inst✝⁹ : (s : S) → TopologicalSpace (E s)
inst✝⁸ : (s : S) → MeasurableSpace (E s)
inst✝⁷ : ∀ (s : S), BorelSpace (E s)
inst✝⁶ : ∀ (s : S), HasOuterApproxClosed (E s)
inst✝⁵ : TopologicalSpace H
inst✝⁴ : MeasurableSpace H
inst✝³ : BorelSpace H
inst✝² : HasOuterApproxClosed H
X : (s : S) → Ω → E s
U : Ω → H
inst✝¹ : IsFiniteMeasure P
inst✝ : Finite S
mX : ∀ (s : S), AEMeasurable (X s) P
mU : AEMeasurable U P
h :
  ∀ (f : ((s : S) → E s) →ᵇ ℝ) (g : H →ᵇ ℝ),
    ∫ (x : Ω), (fun ω ↦ (f fun x ↦ X x ω) * g (U ω)) x ∂P =
      (∫ (x : Ω), (fun ω ↦ f fun x ↦ X x ω) x ∂P) * ∫ (x : Ω), (⇑g ∘ U) x ∂P
this : Fintype S
⊢ (fun ω s ↦ X s ω) ⟂ᵢ[P] U
Module: Mathlib.Probability.Kernel.Representation

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
κ : Kernel X ↑I
inst✝ : IsMarkovKernel κ
f : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}
a : ↑I
⊢ MeasurableSet (uncurry f ⁻¹' Ioi a)

Tactic:
simp only [preimage, uncurry, mem_Ioi]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
κ : Kernel X ↑I
inst✝ : IsMarkovKernel κ
f : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}
a : ↑I
⊢ MeasurableSet {x | a < f x.1 x.2}
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: case e'_2
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 fun ω ↦ (f ω, g ω)) = Measure.map (fun ω ↦ (ω, f ω, g ω)) μ

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:
· rw [Measure.compProd_map (by fun_prop), compProd_trim_condExpKernel]
    exact Measure.map_map (by fun_prop) ((measurable_id.mono le_rfl hm').prodMk measurable_id)

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
⊢ 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'
Module: Mathlib.Probability.Martingale.OptionalSampling

Goal state before:
pp: case neg
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → WithTop ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ ↑n
inst✝ : SigmaFinite (μ.trim ⋯)
i : ι
hin : ¬i ≤ n
x : Ω
⊢ ¬τ x = ↑i

Tactic:
contrapose hin

Goal state after:
case neg
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → WithTop ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ ↑n
inst✝ : SigmaFinite (μ.trim ⋯)
i : ι
x : Ω
hin : τ x = ↑i
⊢ i ≤ n
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
⊢ A.indicator 1 ⟂ᵢ[P] fun ω s ↦ X s ω

Tactic:
have := Fintype.ofFinite S

Goal state after:
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
⊢ A.indicator 1 ⟂ᵢ[P] fun ω s ↦ X s ω
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)})
⊢ False

Tactic:
refine (by simp : ¬ (1 / 2 : ℝ≥0∞) ^ n ≤ 0) <| this.trans <| nonpos_iff_eq_zero.2 ?_
    rw [measure_eq_zero_iff_ae_notMem]
    filter_upwards [(hσ n).tendsto_top] with ω hTop hmem
    simp_rw [WithTop.tendsto_nhds_top_iff, eventually_atTop] at hTop
    simp only [Set.mem_iInter, Set.mem_ofPred_eq] at hmem
    obtain ⟨N, hN⟩ := hTop (T n)
    specialize hN N le_rfl
    specialize hmem N
    grind

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
⊢ Tendsto (fun A ↦ ∫ (x : α), truncation f A x ∂μ) atTop (𝓝 (∫ (x : α), f x ∂μ))

Tactic:
by
  refine tendsto_integral_filter_of_dominated_convergence (fun x => abs (f x)) ?_ ?_ ?_ ?_
  · exact Eventually.of_forall fun A ↦ hf.aestronglyMeasurable.truncation
  · filter_upwards with A
    filter_upwards with x
    rw [Real.norm_eq_abs]
    exact abs_truncation_le_abs_self _ _ _
  · exact hf.abs
  · filter_upwards with x
    apply tendsto_const_nhds.congr' _
    filter_upwards [Ioi_mem_atTop (abs (f x))] with A hA
    exact (truncation_eq_self hA).symm

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✝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
h : ∀ (n : ℕ), ∃ k, P {ω | σ n k ω < min (τ n ω) ↑(T n)} ≤ (1 / 2) ^ n
⊢ ∃ nk T, StrictMono nk ∧ Tendsto T atTop atTop ∧ ∀ (n : ℕ), P {ω | σ n (nk n) ω < min (τ n ω) ↑(T n)} ≤ (1 / 2) ^ n

Tactic:
choose nk hnk using h

Goal state after:
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
nk : ℕ → ℕ
hnk : ∀ (n : ℕ), P {ω | σ n (nk n) ω < min (τ n ω) ↑(T n)} ≤ (1 / 2) ^ n
⊢ ∃ nk T, StrictMono nk ∧ Tendsto T atTop atTop ∧ ∀ (n : ℕ), P {ω | σ n (nk n) ω < min (τ n ω) ↑(T n)} ≤ (1 / 2) ^ n
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
ρ : Measure ℝ := Measure.map X ℙ
this : IsProbabilityMeasure ρ
A : ∑ j ∈ range K, ∫ (x : ℝ) in ↑j..↑N, 1 ∂ρ ≤ (∫ (a : Ω), X a) + 1
a b : ℝ
⊢ ℙ {ω | X ω ∈ Set.Ioc a b} = ρ (Set.Ioc a b)

Tactic:
Measure.map_apply_of_aemeasurable hint.aemeasurable measurableSet_Ioc

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
this : IsProbabilityMeasure ρ
A : ∑ j ∈ range K, ∫ (x : ℝ) in ↑j..↑N, 1 ∂ρ ≤ (∫ (a : Ω), X a) + 1
a b : ℝ
⊢ ℙ {ω | X ω ∈ Set.Ioc a b} = ℙ (X ⁻¹' Set.Ioc a b)
Module: Mathlib.RepresentationTheory.Action

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Monoid G
inst✝ : Semiring k
⊢ (ε k G) 1 = MonoidAlgebra.single PUnit.unit 1

Tactic:
simp [← toLinearMap_apply, types_tensorUnit_def]

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 : PUnit.{u + 1}
⊢ (η k G) (MonoidAlgebra.single x 1) = 1

Tactic:
simp [← toLinearMap_apply, types_tensorUnit_def]

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

Goal state before:
pp: case hbc
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')
ω : Ω
hY_mgf : mgf Y (η (ω', ω)) ↑q ≤ rexp (↑cY * ↑q ^ 2 / 2)
⊢ ∫ (y : Ω''), rexp (↑q * Y y) ∂η (ω', ω) ≤ rexp (↑cY * ↑q ^ 2 / 2)

Tactic:
exact hY_mgf

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
f g : ρ.IntertwiningMap σ
h : f.toLinearMap = g.toLinearMap
⊢ f = g

Tactic:
by
  cases f; cases g
  simpa using h

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
⊢ Function.Injective fun f ↦ f.toFun

Tactic:
intro f

Goal state after:
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
f : ρ.IntertwiningMap σ
⊢ ∀ ⦃a₂ : ρ.IntertwiningMap σ⦄, (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) a₂ → f = a₂
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 : ρ.IntertwiningMap σ) (n : ℕ), ⇑(n • x) = n • ⇑x

Tactic:
intro f

Goal state after:
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
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 (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
⊢ μ.real {ω | ∫ (x : Ω), X x ∂μ - ∫ (x : Ω), Y x ∂μ ≤ Y ω - ∫ (x : Ω), Y x ∂μ - (X ω - ∫ (x : Ω), X x ∂μ)} ≤
    rexp (-(∫ (x : Ω), Y x ∂μ - ∫ (x : Ω), X x ∂μ) ^ 2 / (2 * (↑cX + ↑cY)))

Tactic:
refine (measure_ge_le (X := fun ω ↦ (Y ω - μ[Y]) - (X ω - μ[X])) (c := cX + cY) ?_ ?_).trans_eq
      ?_

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)) (cX + cY) μ

case refine_2
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
⊢ 0 ≤ ∫ (x : Ω), X x ∂μ - ∫ (x : Ω), Y x ∂μ

case refine_3
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
⊢ rexp (-(∫ (x : Ω), X x ∂μ - ∫ (x : Ω), Y x ∂μ) ^ 2 / (2 * ↑(cX + cY))) =
    rexp (-(∫ (x : Ω), Y x ∂μ - ∫ (x : Ω), X x ∂μ) ^ 2 / (2 * (↑cX + ↑cY)))
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
⊢ Var[X; μ.tilted fun x ↦ u * X x] ≤ ((b - a) / 2) ^ 2

Tactic:
convert! variance_le_sq_of_bounded ((tilted_absolutelyContinuous μ (u * X ·)) hb) _

Goal state after:
case convert_1
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
⊢ IsProbabilityMeasure (μ.tilted fun x ↦ u * X x)

case convert_2
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
⊢ AEMeasurable X (μ.tilted fun x ↦ u * X x)
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
⊢ ∀ (x : ρ.IntertwiningMap ρ) (n : ℕ), (x ^ n).toLinearMap = x.toLinearMap ^ n

Tactic:
intro f

Goal state after:
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.IntertwiningMap ρ
⊢ ∀ (n : ℕ), (f ^ n).toLinearMap = f.toLinearMap ^ n
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
⊢ ∀ (x y : ρ.IntertwiningMap ρ),
    (equivLinearMapAsModule ρ ρ) (x * y) = (equivLinearMapAsModule ρ ρ) x * (equivLinearMapAsModule ρ ρ) y

Tactic:
intro f

Goal state after:
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.IntertwiningMap ρ
⊢ ∀ (y : ρ.IntertwiningMap ρ),
    (equivLinearMapAsModule ρ ρ) (f * y) = (equivLinearMapAsModule ρ ρ) f * (equivLinearMapAsModule ρ ρ) y
Module: Mathlib.RepresentationTheory.FDRep

Goal state before:
pp: k : Type u
G : Type v
V : Type u
inst✝⁴ : Field k
inst✝³ : Group G
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : FiniteDimensional k V
ρV : Representation k G V
W : FDRep k G
⊢ of ρV.dual ⊗ W ≅ of (ρV.linHom W.ρ)

Tactic:
refine Action.mkIso (dualTensorIsoLinHomAux ρV W) (fun g => ?_)
  ext : 1
  exact dualTensorHom_comm ρV W.ρ g

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✝ : Ring k
M : Rep k G
⊢ Subsingleton ↑(res f M) ↔ Subsingleton ↑M

Tactic:
Rep.res_obj_V

Goal state after:
G : Type v1
H : Type v2
inst✝² : Monoid G
inst✝¹ : Monoid H
f : H →* G
k : Type u
inst✝ : Ring k
M : Rep k G
⊢ Subsingleton ↑M ↔ Subsingleton ↑M
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
⊢ x✝¹ ∈ ↑(Submodule.comap (ρ g) (ker (MonoidHom.comp ρ S.subtype)))

Tactic:
by
    simpa [← hs] using mem_ker_of_eq
      ⟨g * s * g⁻¹, Subgroup.Normal.conj_mem ‹_› s.1 s.2 g⟩ (ρ g x) _ <| 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✝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
x : V
⊢ (ρ.finsuppToCoinvariants α) (single a ((Coinvariants.mk ρ) x)) = (Coinvariants.mk (ρ.finsupp α)) (single a x)

Tactic:
simp [finsuppToCoinvariants]

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.ContCohomology.Basic

Goal state before:
pp: case succ
k : Type u_1
G : Type u_2
inst✝⁴ : Ring k
inst✝³ : Group G
inst✝² : TopologicalSpace k
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
X : TopRep k G
n : ℕ
ih : X.d n ≫ X.d (n + 1) = 0
⊢ X.d (n + 1) ≫ ofHom (X.resolutionX (n + 1 + 1)).ρ.coind₁ι -
      (ofHom (X.resolutionX (n + 1)).ρ.coind₁ι ≫ (coind₁Functor k G).map (X.d (n + 1)) -
        (coind₁Functor k G).map (X.d n) ≫ (coind₁Functor k G).map (X.d (n + 1))) =
    0

Tactic:
← Functor.map_comp,

Goal state after:
case succ
k : Type u_1
G : Type u_2
inst✝⁴ : Ring k
inst✝³ : Group G
inst✝² : TopologicalSpace k
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
X : TopRep k G
n : ℕ
ih : X.d n ≫ X.d (n + 1) = 0
⊢ X.d (n + 1) ≫ ofHom (X.resolutionX (n + 1 + 1)).ρ.coind₁ι -
      (ofHom (X.resolutionX (n + 1)).ρ.coind₁ι ≫ (coind₁Functor k G).map (X.d (n + 1)) -
        (coind₁Functor k G).map (X.d n ≫ X.d (n + 1))) =
    0
Module: Mathlib.RepresentationTheory.FiniteIndex

Goal state before:
pp: k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
g : G
a : ↑A
⊢ (A.indToCoindAux g) a g = a

Tactic:
indToCoindAux,

Goal state after:
k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
g : G
a : ↑A
⊢ (LinearMap.pi fun g₁ ↦ if h : (QuotientGroup.rightRel S) g₁ g then A.ρ ⟨g₁ * g⁻¹, ⋯⟩ else 0) a g = a
Module: Mathlib.RepresentationTheory.FiniteIndex

Goal state before:
pp: case inl
k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
g₁ g₂ g₃ : G
a : ↑A
s : ↥S
hs : (fun m ↦ m • g₁) s = g₂ * g₃⁻¹
⊢ (A.indToCoindAux g₁) a (g₂ * g₃⁻¹) = (A.indToCoindAux (g₁ * g₃)) a g₂

pp: case inr
k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
g₁ g₂ g₃ : G
a : ↑A
h : ¬(QuotientGroup.rightRel S) (g₂ * g₃⁻¹) g₁
⊢ (A.indToCoindAux g₁) a (g₂ * g₃⁻¹) = (A.indToCoindAux (g₁ * g₃)) a g₂

Tactic:
· simp [S.1.smul_def, mul_assoc, ← eq_mul_inv_iff_mul_eq.1 hs]

Goal state after:
case inr
k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
g₁ g₂ g₃ : G
a : ↑A
h : ¬(QuotientGroup.rightRel S) (g₂ * g₃⁻¹) g₁
⊢ (A.indToCoindAux g₁) a (g₂ * g₃⁻¹) = (A.indToCoindAux (g₁ * g₃)) a g₂