mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Martingale.OptionalSampling

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
ι : Type u_3
inst✝¹⁰ : LinearOrder ι
inst✝⁹ : LocallyFiniteOrder ι
inst✝⁸ : OrderBot ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : DiscreteTopology ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
ℱ : Filtration ι m
τ σ : Ω → WithTop ι
f : ι → Ω → E
inst✝ : SigmaFiniteFiltration μ ℱ
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hσ : IsStoppingTime ℱ σ
n : ι
hτ_le : ∀ (x : Ω), τ x ≤ ↑n
h_sf_min : SigmaFinite (μ.trim ⋯)
⊢ μ[stoppedValue f τ | ⋯.measurableSpace] =ᵐ[μ.restrict {x | τ x ≤ σ x}] μ[stoppedValue f τ | ⋯.measurableSpace]

Tactic:
IsStoppingTime.measurableSpace_min hσ hτ,

Goal state after:
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
ι : Type u_3
inst✝¹⁰ : LinearOrder ι
inst✝⁹ : LocallyFiniteOrder ι
inst✝⁸ : OrderBot ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : DiscreteTopology ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
ℱ : Filtration ι m
τ σ : Ω → WithTop ι
f : ι → Ω → E
inst✝ : SigmaFiniteFiltration μ ℱ
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hσ : IsStoppingTime ℱ σ
n : ι
hτ_le : ∀ (x : Ω), τ x ≤ ↑n
h_sf_min : SigmaFinite (μ.trim ⋯)
⊢ μ[stoppedValue f τ | hσ.measurableSpace ⊓ hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x ≤ σ x}]
    μ[stoppedValue f τ | ⋯.measurableSpace]
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: case right
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
⊢ eLpNorm (fun ω ↦ rexp (t * X ω)) (↑p) (κ ω') < ∞

Tactic:
by_cases hp : p = 0

Goal state after:
case pos
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
⊢ eLpNorm (fun ω ↦ rexp (t * X ω)) (↑p) (κ ω') < ∞

case neg
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
⊢ eLpNorm (fun ω ↦ rexp (t * X ω)) (↑p) (κ ω') < ∞
Module: Mathlib.Probability.Moments.Tilted

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
t : ℝ
ht : t ∈ interior (integrableExpSet X μ)
p : ℝ≥0
hX : AEMeasurable X μ
⊢ MemLp X (↑p) (μ.tilted fun x ↦ t * X x)

Tactic:
by_cases hp : p = 0

Goal state after:
case pos
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
t : ℝ
ht : t ∈ interior (integrableExpSet X μ)
p : ℝ≥0
hX : AEMeasurable X μ
hp : p = 0
⊢ MemLp X (↑p) (μ.tilted fun x ↦ t * X x)

case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
t : ℝ
ht : t ∈ interior (integrableExpSet X μ)
p : ℝ≥0
hX : AEMeasurable X μ
hp : ¬p = 0
⊢ MemLp X (↑p) (μ.tilted fun x ↦ t * X x)
Module: Mathlib.Probability.Moments.Tilted

Goal state before:
pp: case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
t : ℝ
ht : t ∈ interior (integrableExpSet X μ)
p : ℝ≥0
hX : AEMeasurable X μ
hp : ¬p = 0
⊢ (∫⁻ (a : Ω), ENNReal.ofReal (|X a| ^ ↑p) ∂μ.tilted fun x ↦ t * X x) < ∞

Tactic:
refine Integrable.lintegral_lt_top ?_

Goal state after:
case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
t : ℝ
ht : t ∈ interior (integrableExpSet X μ)
p : ℝ≥0
hX : AEMeasurable X μ
hp : ¬p = 0
⊢ Integrable (fun a ↦ |X a| ^ ↑p) (μ.tilted fun x ↦ t * X x)
Module: Mathlib.Probability.ProbabilityMassFunction.Binomial

Goal state before:
pp: p : ℝ≥0
h : p ≤ 1
n : ℕ
⊢ ∑ a, ↑(p ^ ↑a * (1 - p) ^ (↑(Fin.last n) - ↑a) * ↑(n.choose ↑a)) = 1

Tactic:
norm_cast

Goal state after:
p : ℝ≥0
h : p ≤ 1
n : ℕ
⊢ ∑ x, p ^ ↑x * (1 - p) ^ (n - ↑x) * ↑(n.choose ↑x) = 1
Module: Mathlib.Probability.ProbabilityMassFunction.Binomial

Goal state before:
pp: case e'_2
p : ℝ≥0
h : p ≤ 1
n : ℕ
⊢ ∀ x ∈ Finset.range (n + 1),
    (if h : x < n + 1 then p ^ ↑⟨x, h⟩ * (1 - p) ^ (n - ↑⟨x, h⟩) * ↑(n.choose ↑⟨x, h⟩) else 0) =
      p ^ x * (1 - p) ^ (n - x) * ↑(n.choose x)

Tactic:
intro i

Goal state after:
case e'_2
p : ℝ≥0
h : p ≤ 1
n i : ℕ
⊢ i ∈ Finset.range (n + 1) →
    (if h : i < n + 1 then p ^ ↑⟨i, h⟩ * (1 - p) ^ (n - ↑⟨i, h⟩) * ↑(n.choose ↑⟨i, h⟩) else 0) =
      p ^ i * (1 - p) ^ (n - i) * ↑(n.choose i)
Module: Mathlib.Probability.ProbabilityMassFunction.Binomial

Goal state before:
pp: k b : ℕ
hb : k ≤ b
x : ℝ≥0
h : x ≤ 1
eq0 : k % (b + 1) = k
⊢ 1 - ↑x = ENNReal.ofReal (1 - ↑x)

Tactic:
norm_cast

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
t : ℝ
⊢ (↑2).HolderTriple (↑2) 1

Tactic:
norm_cast

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
t : ℝ
⊢ 2.HolderTriple 2 1
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case pos
α : Type u_1
f : α → ℝ
A : ℝ
x : α
h : f x ∈ Set.Ioc (-A) A
⊢ |f x| ≤ |A|

Tactic:
exact abs_le_abs h.2 (neg_le.2 h.1.le)

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: α : Type u_1
f : α → ℝ
A : ℝ
x : α
⊢ |truncation f A x| ≤ |f x|

Tactic:
simp only [truncation, indicator, id, Function.comp_apply]
  split_ifs
  · exact le_rfl
  · simp [abs_nonneg]

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: α : Type u_1
f : α → ℝ
A : ℝ
h : ∀ (x : α), 0 ≤ f x
⊢ truncation f A = (Set.Ioc 0 A).indicator id ∘ f

Tactic:
by
  ext x
  rcases (h x).lt_or_eq with (hx | hx)
  · simp only [truncation, indicator, hx, Set.mem_Ioc, id, Function.comp_apply]
    by_cases h'x : f x ≤ A
    · have : -A < f x := by linarith [h x]
      simp only [this, true_and]
    · simp only [h'x, and_false]
  · simp only [truncation, indicator, hx, id, Function.comp_apply, ite_self]

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
ρ : Measure ℝ := Measure.map X ℙ
this : IsProbabilityMeasure ρ
⊢ ∫ (x : ℝ) in ↑0..↑N, x + 1 ∂ρ = ∫ (x : ℝ) in 0..↑N, x + 1 ∂ρ

Tactic:
norm_cast

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case e_f.e_a
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
j : ℕ
⊢ ∫ (x : ℝ) in 0..↑j, x ^ 2 ∂ρ = ∫ (x : ℝ) in ↑0..↑j, x ^ 2 ∂ρ

Tactic:
norm_cast

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
⊢ 2 * ∫ (x : ℝ) in ↑0..↑K, x ∂ρ = 2 * ∫ (x : ℝ) in 0..↑K, x ∂ρ

Tactic:
norm_cast

Goal state after:
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
⊢ (↑2).HolderTriple (↑2) 1

Tactic:
norm_cast

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
⊢ 2.HolderTriple 2 1
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: 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 i : ℕ
x✝ : i ∈ range n
⊢ Integrable (truncation (X i) ↑i) ℙ

Tactic:
exact ((hident i).symm.integrable_snd hint).1.integrable_truncation

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
X : ℕ → Ω → ℝ
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
mΩ : MeasureSpace Ω := { toMeasurableSpace := m, volume := μ }
h : ∀ᵐ (ω : Ω), X 0 ω = 0
⊢ ∀ (i : ℕ), ∀ᵐ (a : Ω), X i a = 0

Tactic:
intro i

Goal state after:
Ω : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
X : ℕ → Ω → ℝ
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
mΩ : MeasureSpace Ω := { toMeasurableSpace := m, volume := μ }
h : ∀ᵐ (ω : Ω), X 0 ω = 0
i : ℕ
⊢ ∀ᵐ (a : Ω), X i a = 0
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : IsProbabilityMeasure μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSpace E
X : ℕ → Ω → E
h' : Measurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
φ : SimpleFunc E E
c : E
s : Set E
hs : MeasurableSet s
F : E → ℝ := s.indicator 1
F_meas : Measurable F
Y : ℕ → Ω → ℝ := fun n ↦ F ∘ X n
⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y i ω) atTop (𝓝 (∫ (x : Ω), Y 0 x ∂μ))

Tactic:
simp only [smul_eq_mul, ← div_eq_inv_mul]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : IsProbabilityMeasure μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSpace E
X : ℕ → Ω → E
h' : Measurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
φ : SimpleFunc E E
c : E
s : Set E
hs : MeasurableSet s
F : E → ℝ := s.indicator 1
F_meas : Measurable F
Y : ℕ → Ω → ℝ := fun n ↦ F ∘ X n
⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (∑ i ∈ Finset.range n, Y i ω) / ↑n) atTop (𝓝 (∫ (x : Ω), Y 0 x ∂μ))
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case hindep
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁵ : IsProbabilityMeasure μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
h' : StronglyMeasurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
s : Set E := Set.range (X 0) ∪ {0}
zero_s : 0 ∈ s
this✝ : SeparableSpace ↑s
this : Nonempty ↑s
φ : ℕ → SimpleFunc E E := SimpleFunc.nearestPt fun k ↦ Nat.casesOn k 0 (Subtype.val ∘ denseSeq ↑s)
Y : ℕ → ℕ → Ω → E := fun k i ↦ ⇑(φ k) ∘ X i
A : ∀ᵐ (ω : Ω) ∂μ, ∀ (k : ℕ), Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y k i ω) atTop (𝓝 (∫ (x : Ω), Y k 0 x ∂μ))
k : ℕ
G : ℕ → E → ℝ := fun k x ↦ ‖x - (φ k) x‖
G_meas : ∀ (k : ℕ), Measurable (G k)
I : ∀ (k i : ℕ), (fun ω ↦ ‖(X i - Y k i) ω‖) = G k ∘ X i
⊢ Pairwise fun x y ↦ (G k ∘ X x) ⟂ᵢ[μ] G k ∘ X y

Tactic:
intro i

Goal state after:
case hindep
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁵ : IsProbabilityMeasure μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
h' : StronglyMeasurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
s : Set E := Set.range (X 0) ∪ {0}
zero_s : 0 ∈ s
this✝ : SeparableSpace ↑s
this : Nonempty ↑s
φ : ℕ → SimpleFunc E E := SimpleFunc.nearestPt fun k ↦ Nat.casesOn k 0 (Subtype.val ∘ denseSeq ↑s)
Y : ℕ → ℕ → Ω → E := fun k i ↦ ⇑(φ k) ∘ X i
A : ∀ᵐ (ω : Ω) ∂μ, ∀ (k : ℕ), Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y k i ω) atTop (𝓝 (∫ (x : Ω), Y k 0 x ∂μ))
k : ℕ
G : ℕ → E → ℝ := fun k x ↦ ‖x - (φ k) x‖
G_meas : ∀ (k : ℕ), Measurable (G k)
I : ∀ (k i : ℕ), (fun ω ↦ ‖(X i - Y k i) ω‖) = G k ∘ X i
i : ℕ
⊢ ∀ ⦃j : ℕ⦄, i ≠ j → (fun x y ↦ (G k ∘ X x) ⟂ᵢ[μ] G k ∘ X y) i j
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case hident
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁵ : IsProbabilityMeasure μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
h' : StronglyMeasurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
s : Set E := ⋯
zero_s : 0 ∈ s
this✝ : SeparableSpace ↑s
this : Nonempty ↑s
φ : ℕ → SimpleFunc E E := ⋯
Y : ℕ → ℕ → Ω → E := ⋯
A : ∀ᵐ (ω : Ω) ∂μ, ∀ (k : ℕ), Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y k i ω) atTop (𝓝 (∫ (x : Ω), Y k 0 x ∂μ))
k : ℕ
G : ℕ → E → ℝ := ⋯
G_meas : ∀ (k : ℕ), Measurable (G k)
I : ∀ (k i : ℕ), (fun ω ↦ ‖(X i - Y k i) ω‖) = G k ∘ X i
⊢ ∀ (i : ℕ), IdentDistrib (fun ω ↦ ‖(X i - Y k i) ω‖) (fun ω ↦ ‖(X 0 - Y k 0) ω‖) μ μ

Tactic:
intro i

Goal state after:
case hident
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁵ : IsProbabilityMeasure μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
h' : StronglyMeasurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
s : Set E := Set.range (X 0) ∪ {0}
zero_s : 0 ∈ s
this✝ : SeparableSpace ↑s
this : Nonempty ↑s
φ : ℕ → SimpleFunc E E := SimpleFunc.nearestPt fun k ↦ Nat.casesOn k 0 (Subtype.val ∘ denseSeq ↑s)
Y : ℕ → ℕ → Ω → E := fun k i ↦ ⇑(φ k) ∘ X i
A : ∀ᵐ (ω : Ω) ∂μ, ∀ (k : ℕ), Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y k i ω) atTop (𝓝 (∫ (x : Ω), Y k 0 x ∂μ))
k : ℕ
G : ℕ → E → ℝ := fun k x ↦ ‖x - (φ k) x‖
G_meas : ∀ (k : ℕ), Measurable (G k)
I : ∀ (k i : ℕ), (fun ω ↦ ‖(X i - Y k i) ω‖) = G k ∘ X i
i : ℕ
⊢ IdentDistrib (fun ω ↦ ‖(X i - Y k i) ω‖) (fun ω ↦ ‖(X 0 - Y k 0) ω‖) μ μ
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
h : ∀ᵐ (ω : Ω) ∂μ, X 0 ω = 0
⊢ ∀ (i : ℕ), ∀ᵐ (a : Ω) ∂μ, X i a = 0

Tactic:
intro i

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
h : ∀ᵐ (ω : Ω) ∂μ, X 0 ω = 0
i : ℕ
⊢ ∀ᵐ (a : Ω) ∂μ, X i a = 0
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
X : ℕ → Ω → E
hℒp : MemLp (X 0) p μ
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
h : ∀ᵐ (ω : Ω) ∂μ, X 0 ω = 0
⊢ ∀ (i : ℕ), ∀ᵐ (a : Ω) ∂μ, X i a = 0

Tactic:
intro i

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
X : ℕ → Ω → E
hℒp : MemLp (X 0) p μ
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
h : ∀ᵐ (ω : Ω) ∂μ, X 0 ω = 0
i : ℕ
⊢ ∀ᵐ (a : Ω) ∂μ, X i a = 0
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : CommRing k
inst✝ : Monoid G
X Y : Rep k G
⊢ Hom.hom ((β_ X Y).hom ≫ (β_ Y X).hom) = Hom.hom (𝟙 (X ⊗ Y))

Tactic:
simp [← comm_symm Y.ρ X.ρ, ← toIntertwiningMap_trans,
    trans_symm, toIntertwiningMap_refl]

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

Goal state before:
pp: k : Type u
G H K : Type v
inst✝¹⁰ : Ring k
inst✝⁹ : TopologicalSpace k
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : IsTopologicalGroup H
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : IsTopologicalGroup K
X : TopRep k G
Y : TopRep k H
Z : TopRep k K
φ : H →ₜ* G
ψ : K →ₜ* H
f : res (↑φ) X ⟶ Y
g : res (↑ψ) Y ⟶ Z
i : ℕ
v : ↑(X.homogeneousCochains.X i).toModuleCat
x : K
⊢ ↑((ConcreteCategory.hom ((cochainsMap (φ.comp ψ) ((resFunctor ↑ψ).map f ≫ g)).f i)) v) x =
    ↑((ConcreteCategory.hom ((cochainsMap φ f ≫ cochainsMap ψ g).f i)) v) x

Tactic:
exact congr($(resolutionMap_comp φ ψ f g (i + 1)).hom v.1 x)

Goal state after:
Module: Mathlib.RepresentationTheory.FiniteIndex

Goal state before:
pp: k : Type u
G : Type v
inst✝³ : CommRing k
inst✝² : Group G
S : Subgroup G
inst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
inst✝ : S.FiniteIndex
g : G
a : ↑A
⊢ ((AlgebraTensorModule.curry
            ((A.coindToInd ∘ₗ A.indToCoind) ∘ₗ
              Coinvariants.mk (tprod (MonoidHom.comp (Representation.leftRegular k G) S.subtype) A.ρ)) ∘ₗ
          MonoidAlgebra.lsingle g)
        1)
      a =
    ((AlgebraTensorModule.curry
            (LinearMap.id ∘ₗ Coinvariants.mk (tprod (MonoidHom.comp (Representation.leftRegular k G) S.subtype) A.ρ)) ∘ₗ
          MonoidAlgebra.lsingle g)
        1)
      a

Tactic:
simp only [LinearMap.comp_apply, AlgebraTensorModule.curry_apply,
    TensorProduct.curry_apply, LinearMap.coe_restrictScalars, LinearMap.id_apply]

Goal state after:
k : Type u
G : Type v
inst✝³ : CommRing k
inst✝² : Group G
S : Subgroup G
inst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
inst✝ : S.FiniteIndex
g : G
a : ↑A
⊢ A.coindToInd
      (A.indToCoind
        ((Coinvariants.mk (tprod (MonoidHom.comp (Representation.leftRegular k G) S.subtype) A.ρ))
          ((MonoidAlgebra.lsingle g) 1 ⊗ₜ[k] a))) =
    (Coinvariants.mk (tprod (MonoidHom.comp (Representation.leftRegular k G) S.subtype) A.ρ))
      ((MonoidAlgebra.lsingle g) 1 ⊗ₜ[k] a)
Module: Mathlib.RepresentationTheory.Homological.FiniteCyclic

Goal state before:
pp: case pos
k : Type u_1
G : Type u_2
inst✝² : CommRing k
inst✝¹ : Group G
inst✝ : Finite G
this : Fintype G
x : MonoidAlgebra k G
hx : x ∈ ((linearCombination k fun x ↦ 1) ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k)).ker
g : G
hg : g = 1
⊢ x.coeff g = (x.coeff.sum fun g r ↦ MonoidAlgebra.single g r - MonoidAlgebra.single 1 r).coeff g

pp: case neg
k : Type u_1
G : Type u_2
inst✝² : CommRing k
inst✝¹ : Group G
inst✝ : Finite G
this : Fintype G
x : MonoidAlgebra k G
hx : x ∈ ((linearCombination k fun x ↦ 1) ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k)).ker
g : G
hg : ¬g = 1
⊢ x.coeff g = (x.coeff.sum fun g r ↦ MonoidAlgebra.single g r - MonoidAlgebra.single 1 r).coeff g

Tactic:
· simp_all [linearCombination, sum_apply', MonoidAlgebra.coeff_finsuppSum]

Goal state after:
case neg
k : Type u_1
G : Type u_2
inst✝² : CommRing k
inst✝¹ : Group G
inst✝ : Finite G
this : Fintype G
x : MonoidAlgebra k G
hx : x ∈ ((linearCombination k fun x ↦ 1) ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k)).ker
g : G
hg : ¬g = 1
⊢ x.coeff g = (x.coeff.sum fun g r ↦ MonoidAlgebra.single g r - MonoidAlgebra.single 1 r).coeff g
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.FiniteCyclic

Goal state before:
pp: case neg
k G : Type u
inst✝² : CommRing k
inst✝¹ : CommGroup G
inst✝ : Fintype G
A : Rep k G
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
i : ℕ
x : leftRegular k G ⟶ A
hi : ¬Even i
⊢ (ModuleCat.Hom.hom (A.leftRegularHomEquiv.toModuleIso.hom ≫ (moduleCatCochainComplex A g).d i (i + 1))) x =
    (ModuleCat.Hom.hom
        (((resolution k g hg).complex.linearYonedaObj k A).d i (i + 1) ≫ A.leftRegularHomEquiv.toModuleIso.hom))
      x

Tactic:
· simp [hi, Nat.even_add_one.2 hi, Representation.norm, homEquiv, Rep.norm, ← hom_comm_apply x]

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.Resolution

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Monoid G
⊢ (standardComplex k G).d 1 0 ≫ ε k G = 0

Tactic:
ext : 3

Goal state after:
k G : Type u
inst✝¹ : CommRing k
inst✝ : Monoid G
x✝ : ↑((standardComplex k G).X 1)
⊢ (Hom.hom ((standardComplex k G).d 1 0 ≫ ε k G)).toLinearMap x✝ = (Hom.hom 0).toLinearMap x✝
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.Hilbert90

Goal state before:
pp: K L : Type
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
g : Gal(L/K)
hg : ∀ (x : Gal(L/K)), x ∈ Subgroup.zpowers g
x✝ : L
hx✝ : (Algebra.norm K) x✝ = 1
x : L
hx : (Algebra.norm K) x = 1
⊢ ∃ y, g ↑y / ↑y = x

Tactic:
let xu : Lˣ := (Algebra.norm_ne_zero_iff.1 <| hx ▸ zero_ne_one.symm).isUnit.unit

Goal state after:
K L : Type
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
g : Gal(L/K)
hg : ∀ (x : Gal(L/K)), x ∈ Subgroup.zpowers g
x✝ : L
hx✝ : (Algebra.norm K) x✝ = 1
x : L
hx : (Algebra.norm K) x = 1
xu : Lˣ := ⋯.unit
⊢ ∃ y, g ↑y / ↑y = x
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Basic

Goal state before:
pp: k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
n : ℕ
inst✝ : DecidableEq G
⊢ d A n =
    (A.coinvariantsTensorFreeLEquiv (Fin (n + 1) → G)).toModuleIso.inv ≫
      (HomologicalComplex.coinvariantsTensorObj A (barComplex k G)).d (n + 1) n ≫
        (A.coinvariantsTensorFreeLEquiv (Fin n → G)).toModuleIso.hom

Tactic:
ext : 3

Goal state after:
k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
n : ℕ
inst✝ : DecidableEq G
a✝ : Fin (n + 1) → G
x✝ : ↑A
⊢ (ModuleCat.Hom.hom (d A n) ∘ₗ lsingle a✝) x✝ =
    (ModuleCat.Hom.hom
          ((A.coinvariantsTensorFreeLEquiv (Fin (n + 1) → G)).toModuleIso.inv ≫
            (HomologicalComplex.coinvariantsTensorObj A (barComplex k G)).d (n + 1) n ≫
              (A.coinvariantsTensorFreeLEquiv (Fin n → G)).toModuleIso.hom) ∘ₗ
        lsingle a✝)
      x✝
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Basic

Goal state before:
pp: case refine_1
k✝ G✝ : Type u
inst✝⁴ : CommRing k✝
inst✝³ : Group G✝
k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
n : ℕ
inst✝ : DecidableEq G
⊢ (i : ℕ) → (inhomogeneousChains A).X i ≅ (HomologicalComplex.coinvariantsTensorObj A (barComplex k G)).X i

Tactic:
intro i

Goal state after:
case refine_1
k✝ G✝ : Type u
inst✝⁴ : CommRing k✝
inst✝³ : Group G✝
k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
n : ℕ
inst✝ : DecidableEq G
i : ℕ
⊢ (inhomogeneousChains A).X i ≅ (HomologicalComplex.coinvariantsTensorObj A (barComplex k G)).X i
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree

Goal state before:
pp: G : Type u_1
M : Type u_2
inst✝² : Monoid G
inst✝¹ : CommGroup M
inst✝ : MulAction G M
f : G × G → M
hf : IsMulCocycle₂ f
g : G
⊢ f (1, g) = f (1, 1)

Tactic:
simpa only [one_smul, one_mul, mul_one, mul_right_inj] using (hf 1 1 g).symm

Goal state after: