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

Goal state before:
pp: X : ℕ → Type u_1
inst✝³ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝² : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℕ
hab : a ≤ b
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
hf : Integrable f (⇑(traj κ b) ∘ₘ (partialTraj κ a b) x₀)
A : Set ((i : ↥(Iic b)) → X ↑i)
hA : MeasurableSet A
⊢ Integrable (uncurry fun x ↦ f) ((partialTraj κ a b) x₀ ⊗ₘ traj κ b)

Tactic:
← Measure.snd_compProd

Goal state after:
X : ℕ → Type u_1
inst✝³ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝² : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℕ
hab : a ≤ b
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
hf : Integrable f ((partialTraj κ a b) x₀ ⊗ₘ traj κ b).snd
A : Set ((i : ↥(Iic b)) → X ↑i)
hA : MeasurableSet A
⊢ Integrable (uncurry fun x ↦ f) ((partialTraj κ a b) x₀ ⊗ₘ traj κ b)
Module: Mathlib.Probability.Distributions.Exponential

Goal state before:
pp: r : ℝ
hr : 0 < r
x : ℝ
h : 0 ≤ x
⊢ AEStronglyMeasurable (fun x ↦ r * rexp (-(r * x))) (volume.restrict (Icc 0 x))

Tactic:
refine Integrable.aestronglyMeasurable (Integrable.const_mul ?_ _)

Goal state after:
r : ℝ
hr : 0 < r
x : ℝ
h : 0 ≤ x
⊢ Integrable (fun x ↦ rexp (-(r * x))) (volume.restrict (Icc 0 x))
Module: Mathlib.Probability.Distributions.Exponential

Goal state before:
pp: r : ℝ
hr : 0 < r
x : ℝ
⊢ ↑(cdf (expMeasure r)) x = if 0 ≤ x then 1 - rexp (-(r * x)) else 0

Tactic:
rw [cdf_expMeasure_eq_lintegral hr, lintegral_exponentialPDF_eq_antiDeriv hr x,
    ENNReal.toReal_ofReal_eq_iff]
  split_ifs with h
  · simp only [sub_nonneg, exp_le_one_iff, Left.neg_nonpos_iff]
    exact mul_nonneg hr.le h
  · exact le_rfl

Goal state after:
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

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

Tactic:
have i_f' : Integrable (fun x ↦ ∫ y, f y ∂(traj κ b) x)
      (((traj κ a) x₀).map (frestrictLe b)) := by
    rw [← map_apply _ (measurable_frestrictLe _), traj_map_frestrictLe _ _]
    rw [← traj_comp_partialTraj hab] at i_f
    exact i_f.integral_comp
  refine ae_eq_condExp_of_forall_setIntegral_eq (piLE.le _) i_f
    (fun s _ _ ↦ i_f'.comp_aemeasurable (measurable_frestrictLe b).aemeasurable |>.integrableOn)
    ?_ ?_ |>.symm <;> rw [piLE_eq_comap_frestrictLe]
  · rintro - ⟨t, mt, rfl⟩ -
    simp_rw [Function.comp_apply]
    rw [← setIntegral_map mt i_f'.1, ← map_apply, traj_map_frestrictLe,
      setIntegral_traj_partialTraj hab i_f mt]
    all_goals fun_prop
  · exact (i_f'.1.comp_ae_measurable' (measurable_frestrictLe b).aemeasurable)

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

Goal state before:
pp: t r : ℝ
ht : 0 < t
hr : 0 < r
leftSide : ∫⁻ (x : ℝ) in Iio t, paretoPDF t r x = 0
rightSide : ∫⁻ (x : ℝ) in Ici t, paretoPDF t r x = ∫⁻ (x : ℝ) in Ici t, ENNReal.ofReal (r * t ^ r * x ^ (-(r + 1)))
x : ℝ
hx : x ∈ Ici t
⊢ 0 x ≤ r * t ^ r * x ^ (-(r + 1))

Tactic:
by positivity [lt_of_lt_of_le ht hx]

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

Goal state before:
pp: Ω✝ : Type u_1
m✝ : MeasurableSpace Ω✝
X✝ : Ω✝ → ℝ
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
m₀ : MeasurableSpace Ω
hm : m ≤ m₀
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hXmeas✝ : AEMeasurable X μ
hX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1
hXmeas : Measurable X
⊢ μ[X | m] - μ[X | m] ^ 2 =ᵐ[μ] μ[X | m] * μ[1 - X | m]

Tactic:
rw [sq, ← one_sub_mul, mul_comm]

Goal state after:
Ω✝ : Type u_1
m✝ : MeasurableSpace Ω✝
X✝ : Ω✝ → ℝ
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
m₀ : MeasurableSpace Ω
hm : m ≤ m₀
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hXmeas✝ : AEMeasurable X μ
hX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1
hXmeas : Measurable X
⊢ μ[X | m] * (1 - μ[X | m]) =ᵐ[μ] μ[X | m] * μ[1 - X | m]
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: case refine_1
Ω : Type u_1
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
s1 s2 : Set (Set Ω)
hs1 : ∀ s ∈ s1, MeasurableSet s
hs2 : ∀ s ∈ s2, MeasurableSet s
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hs1_eq : ∀ s ∈ s1, (fun ω ↦ (((condExpKernel μ m') ω) s).toReal) =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m']
hs2_eq : ∀ s ∈ s2, (fun ω ↦ (((condExpKernel μ m') ω) s).toReal) =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m']
hs12_eq :
  ∀ s ∈ s1, ∀ t ∈ s2, (fun ω ↦ (((condExpKernel μ m') ω) (s ∩ t)).toReal) =ᵐ[μ] μ[(s ∩ t).indicator fun ω ↦ 1 | m']
s t : Set Ω
h : ∀ᵐ (a : Ω) ∂μ.trim hm', ((condExpKernel μ m') a) (s ∩ t) = ((condExpKernel μ m') a) s * ((condExpKernel μ m') a) t
hs : s ∈ s1
ht : t ∈ s2
⊢ μ[(s ∩ t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m'] * μ[t.indicator fun ω ↦ 1 | m']

pp: case refine_2
Ω : Type u_1
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
s1 s2 : Set (Set Ω)
hs1 : ∀ s ∈ s1, MeasurableSet s
hs2 : ∀ s ∈ s2, MeasurableSet s
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hs1_eq : ∀ s ∈ s1, (fun ω ↦ (((condExpKernel μ m') ω) s).toReal) =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m']
hs2_eq : ∀ s ∈ s2, (fun ω ↦ (((condExpKernel μ m') ω) s).toReal) =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m']
hs12_eq :
  ∀ s ∈ s1, ∀ t ∈ s2, (fun ω ↦ (((condExpKernel μ m') ω) (s ∩ t)).toReal) =ᵐ[μ] μ[(s ∩ t).indicator fun ω ↦ 1 | m']
s t : Set Ω
h : μ[(s ∩ t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m'] * μ[t.indicator fun ω ↦ 1 | m']
hs : s ∈ s1
ht : t ∈ s2
⊢ ∀ᵐ (a : Ω) ∂μ.trim hm', ((condExpKernel μ m') a) (s ∩ t) = ((condExpKernel μ m') a) s * ((condExpKernel μ m') a) t

Tactic:
· have h' := ae_eq_of_ae_eq_trim h
    filter_upwards [hs1_eq s hs, hs2_eq t ht, hs12_eq s hs t ht, h'] with ω hs_eq ht_eq hst_eq h'
    rw [← hst_eq, Pi.mul_apply, ← hs_eq, ← ht_eq, h', ENNReal.toReal_mul]

Goal state after:
case refine_2
Ω : Type u_1
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
s1 s2 : Set (Set Ω)
hs1 : ∀ s ∈ s1, MeasurableSet s
hs2 : ∀ s ∈ s2, MeasurableSet s
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hs1_eq : ∀ s ∈ s1, (fun ω ↦ (((condExpKernel μ m') ω) s).toReal) =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m']
hs2_eq : ∀ s ∈ s2, (fun ω ↦ (((condExpKernel μ m') ω) s).toReal) =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m']
hs12_eq :
  ∀ s ∈ s1, ∀ t ∈ s2, (fun ω ↦ (((condExpKernel μ m') ω) (s ∩ t)).toReal) =ᵐ[μ] μ[(s ∩ t).indicator fun ω ↦ 1 | m']
s t : Set Ω
h : μ[(s ∩ t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m'] * μ[t.indicator fun ω ↦ 1 | m']
hs : s ∈ s1
ht : t ∈ s2
⊢ ∀ᵐ (a : Ω) ∂μ.trim hm', ((condExpKernel μ m') a) (s ∩ t) = ((condExpKernel μ m') a) s * ((condExpKernel μ m') a) t
Module: Mathlib.Probability.Independence.BoundedContinuousFunction

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
G : Type u_6
H : Type u_7
inst✝⁸ : TopologicalSpace G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : HasOuterApproxClosed G
inst✝⁴ : TopologicalSpace H
inst✝³ : MeasurableSpace H
inst✝² : BorelSpace H
inst✝¹ : HasOuterApproxClosed H
Z : Ω → G
U : Ω → H
inst✝ : IsFiniteMeasure P
mZ : AEMeasurable Z P
mU : AEMeasurable U P
h :
  ∀ (f : G →ᵇ ℝ) (g : H →ᵇ ℝ), ∫ (x : Ω), (⇑f ∘ Z * ⇑g ∘ U) x ∂P = (∫ (x : Ω), (⇑f ∘ Z) x ∂P) * ∫ (x : Ω), (⇑g ∘ U) x ∂P
f : G →ᵇ ℝ
g : H →ᵇ ℝ
⊢ ∫ (x : Ω), f (Z x, U x).1 * g (Z x, U x).2 ∂P = (∫ (x : Ω), f (Z x) ∂P) * ∫ (x : Ω), g (U x) ∂P

Tactic:
exact h f g

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

Goal state before:
pp: case refine_2
Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
inst✝⁵ : (s : S) → TopologicalSpace (E s)
inst✝⁴ : (s : S) → MeasurableSpace (E s)
inst✝³ : ∀ (s : S), BorelSpace (E s)
inst✝² : ∀ (s : S), HasOuterApproxClosed (E s)
X : (s : S) → Ω → E s
inst✝¹ : Fintype S
inst✝ : IsProbabilityMeasure P
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, ∏ s, (f s) (X s ω) ∂P = P.real A * ∫ (ω : Ω), ∏ s, (f s) (X s ω) ∂P
f : ℝ →ᵇ ℝ
g : (t : S) → E t →ᵇ ℝ
h1 :
  ∀ (ω : Ω),
    f (A.indicator 1 ω) * ∏ s, (g s) (X s ω) =
      A.indicator (fun ω ↦ f 1 * ∏ s, (g s) (X s ω)) ω + f 0 * ∏ s, (g s) (X s ω) -
        A.indicator (fun ω ↦ f 0 * ∏ s, (g s) (X s ω)) ω
h2 : ∀ (ω : Ω), f (A.indicator 1 ω) = A.indicator (fun x ↦ f 1) ω + Aᶜ.indicator (fun x ↦ f 0) ω
c : ℝ
ω : Ω
⊢ ‖c‖ * ‖∏ s, (g s) (X s ω)‖ ≤ ‖c‖ * ∏ s, ‖g s‖

Tactic:
norm_prod

Goal state after:
case refine_2
Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
inst✝⁵ : (s : S) → TopologicalSpace (E s)
inst✝⁴ : (s : S) → MeasurableSpace (E s)
inst✝³ : ∀ (s : S), BorelSpace (E s)
inst✝² : ∀ (s : S), HasOuterApproxClosed (E s)
X : (s : S) → Ω → E s
inst✝¹ : Fintype S
inst✝ : IsProbabilityMeasure P
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, ∏ s, (f s) (X s ω) ∂P = P.real A * ∫ (ω : Ω), ∏ s, (f s) (X s ω) ∂P
f : ℝ →ᵇ ℝ
g : (t : S) → E t →ᵇ ℝ
h1 :
  ∀ (ω : Ω),
    f (A.indicator 1 ω) * ∏ s, (g s) (X s ω) =
      A.indicator (fun ω ↦ f 1 * ∏ s, (g s) (X s ω)) ω + f 0 * ∏ s, (g s) (X s ω) -
        A.indicator (fun ω ↦ f 0 * ∏ s, (g s) (X s ω)) ω
h2 : ∀ (ω : Ω), f (A.indicator 1 ω) = A.indicator (fun x ↦ f 1) ω + Aᶜ.indicator (fun x ↦ f 0) ω
c : ℝ
ω : Ω
⊢ ‖c‖ * ∏ b, ‖(g b) (X b ω)‖ ≤ ‖c‖ * ∏ s, ‖g s‖
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 ⋯)
this :
  ∀ᵐ (x : Ω) ∂μ,
    x ∈ {x | τ x ≤ σ x} → μ[stoppedValue f τ | ⋯.measurableSpace] x = μ[stoppedValue f τ | hσ.measurableSpace] x
x : Ω
hx : τ x ≤ σ x → μ[stoppedValue f τ | ⋯.measurableSpace] x = μ[stoppedValue f τ | hσ.measurableSpace] x
hx_mem : τ x < σ x
⊢ μ[stoppedValue f τ | ⋯.measurableSpace] x = μ[stoppedValue f τ | hσ.measurableSpace] x

Tactic:
exact hx hx_mem.le

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
t : ℝ
p : ℝ≥0
⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (⇑κ ∘ₘ ν)

Tactic:
by_cases hp0 : p = 0
  · simpa [hp0] using (h.integrable_exp_mul t).1
  constructor
  · exact (h.integrable_exp_mul t).1
  · rw [eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top (mod_cast hp0) (by simp)]
    simp only [ENNReal.coe_toReal]
    have h' := (h.integrable_exp_mul (p * t)).2
    rw [hasFiniteIntegral_def] at h'
    convert! h' using 3 with ω
    rw [enorm_eq_ofReal (by positivity), enorm_eq_ofReal (by positivity),
      ENNReal.ofReal_rpow_of_nonneg (by positivity), ← exp_mul, mul_comm, ← mul_assoc]
    positivity

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

Goal state before:

Tactic:
(κ ω') {ω | X ω ≠ 0}
  _ = (κ ω') {ω | X ω < 0 ∨ 0 < X ω} := by simp_rw [ne_iff_lt_or_gt]
  _ ≤ (κ ω') {ω | X ω < 0} + (κ ω') {ω | 0 < X ω} := measure_union_le _ _
  _ = 0 := by simp [h1, h2]

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case ha
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 := ⋯
ρ : Measure ℝ := ⋯
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
a✝ : k ∈ range K
⊢ 0 ≤ ∫ (x : ℝ) in ↑k..↑(k + 1), x ^ 2 ∂ρ

Tactic:
refine intervalIntegral.integral_nonneg_of_forall ?_ fun u => sq_nonneg _
        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
φ✝ φ : ρ.Equiv σ
ψ : σ.Equiv τ
g : G
⊢ ↑ψ.toLinearEquiv ∘ₗ ↑φ.toLinearEquiv ∘ₗ ρ g = τ g ∘ₗ ↑ψ.toLinearEquiv ∘ₗ ↑φ.toLinearEquiv

Tactic:
φ.1.2,

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
φ✝ φ : ρ.Equiv σ
ψ : σ.Equiv τ
g : G
⊢ ↑ψ.toLinearEquiv ∘ₗ σ g ∘ₗ (↑φ).toLinearMap = τ g ∘ₗ ↑ψ.toLinearEquiv ∘ₗ ↑φ.toLinearEquiv
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
⊢ mgf X μ t ≤ rexp ((↑‖b - a‖₊ / 2) ^ 2 * t ^ 2 / 2)

Tactic:
rw [← exp_cgf (hi t), exp_le_exp, h2]

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
⊢ iteratedDeriv 2 (cgf X μ) u * t ^ 2 / 2 ≤ (↑‖b - a‖₊ / 2) ^ 2 * t ^ 2 / 2
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: case inr.inl
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
inst✝ : IsProbabilityMeasure μ
a b : ℝ
hm : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hc : ∫ (x : Ω), X x ∂μ = 0
t : ℝ
ht : 0 = t
⊢ mgf X μ t ≤ rexp (↑((‖b - a‖₊ / 2) ^ 2) * t ^ 2 / 2)

pp: case inr.inr
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
inst✝ : IsProbabilityMeasure μ
a b : ℝ
hm : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hc : ∫ (x : Ω), X x ∂μ = 0
t : ℝ
ht : t < 0
⊢ mgf X μ t ≤ rexp (↑((‖b - a‖₊ / 2) ^ 2) * t ^ 2 / 2)

Tactic:
· simp [← ht]

Goal state after:
case inr.inr
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
inst✝ : IsProbabilityMeasure μ
a b : ℝ
hm : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hc : ∫ (x : Ω), X x ∂μ = 0
t : ℝ
ht : t < 0
⊢ mgf X μ t ≤ rexp (↑((‖b - a‖₊ / 2) ^ 2) * t ^ 2 / 2)
Module: Mathlib.RepresentationTheory.Irreducible

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
inst✝³ : Monoid G
inst✝² : Field k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
ρ : Representation k G V
⊢ ρ.IsIrreducible ↔ IsSimpleModule k[G] ρ.asModule

Tactic:
isSimpleModule_iff

Goal state after:
G : Type u_1
k : Type u_2
V : Type u_3
inst✝³ : Monoid G
inst✝² : Field k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
ρ : Representation k G V
⊢ ρ.IsIrreducible ↔ IsSimpleOrder (Submodule k[G] ρ.asModule)
Module: Mathlib.RepresentationTheory.Irreducible

Goal state before:
pp: G : Type u_1
k : Type u_2
inst✝³ : Monoid G
inst✝² : Field k
M : Type u_5
inst✝¹ : AddCommGroup M
inst✝ : Module k[G] M
⊢ IsSimpleModule k[G] M ↔ (ofModule M).IsIrreducible

Tactic:
isSimpleModule_iff

Goal state after:
G : Type u_1
k : Type u_2
inst✝³ : Monoid G
inst✝² : Field k
M : Type u_5
inst✝¹ : AddCommGroup M
inst✝ : Module k[G] M
⊢ IsSimpleOrder (Submodule k[G] M) ↔ (ofModule M).IsIrreducible
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✝⁷ : CommRing k
inst✝⁶ : Monoid G
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
inst✝³ : AddCommGroup W
inst✝² : Module k W
inst✝¹ : AddCommGroup X
inst✝ : Module k X
ρ : Representation k G V
τ : Representation k G W
υ : Representation k G X
f : V →ₗ[k] W
h : ∀ (x : G), f ∘ₗ ρ x = f
x : V
x✝ : x ∈ Set.range fun gv ↦ (ρ gv.1) gv.2 - gv.2
g : G
y : V
hy : (fun gv ↦ (ρ gv.1) gv.2 - gv.2) (g, y) = x
⊢ x ∈ ↑f.ker

Tactic:
simpa only [← hy, SetLike.mem_coe, LinearMap.mem_ker, map_sub, sub_eq_zero, LinearMap.coe_comp,
      Function.comp_apply] using LinearMap.ext_iff.1 (h g) y

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
⊢ cochainsMap (φ.comp ψ) ((resFunctor ↑ψ).map f ≫ g) = cochainsMap φ f ≫ cochainsMap ψ g

Tactic:
ext i v x

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

Goal state before:
pp: case a.a
k G : Type u
inst✝¹ : CommRing k
n✝ : ℕ
inst✝ : Group G
A : Rep k G
n : ℕ
| ((barComplex k G).linearYonedaObj k A).d n (n + 1) ≫
    𝟙 (((barComplex k G).linearYonedaObj k A).X (n + 1)) ≫ ((barComplex k G).linearYonedaObj k A).d (n + 1) (n + 1 + 1)

pp: case a.a
k G : Type u
inst✝¹ : CommRing k
n✝ : ℕ
inst✝ : Group G
A : Rep k G
n : ℕ
| (freeLiftLEquiv k G (Fin (n + 1 + 1) → G) A).toModuleIso.hom

pp: case a
k G : Type u
inst✝¹ : CommRing k
n✝ : ℕ
inst✝ : Group G
A : Rep k G
n : ℕ
| (freeLiftLEquiv k G (Fin n → G) A).toModuleIso.inv

Tactic:
Category.id_comp,

Goal state after:
case a.a
k G : Type u
inst✝¹ : CommRing k
n✝ : ℕ
inst✝ : Group G
A : Rep k G
n : ℕ
| ((barComplex k G).linearYonedaObj k A).d n (n + 1) ≫ ((barComplex k G).linearYonedaObj k A).d (n + 1) (n + 1 + 1)

case a.a
k G : Type u
inst✝¹ : CommRing k
n✝ : ℕ
inst✝ : Group G
A : Rep k G
n : ℕ
| (freeLiftLEquiv k G (Fin (n + 1 + 1) → G) A).toModuleIso.hom

case a
k G : Type u
inst✝¹ : CommRing k
n✝ : ℕ
inst✝ : Group G
A : Rep k G
n : ℕ
| (freeLiftLEquiv k G (Fin n → G) A).toModuleIso.inv
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
inst✝ : A.IsTrivial
x✝¹ : ↑(ModuleCat.of k ↑A)
x✝ : G
⊢ (ModuleCat.Hom.hom (d₀₁ A)) x✝¹ x✝ = (ModuleCat.Hom.hom 0) x✝¹ x✝

Tactic:
d₀₁_hom_apply,

Goal state after:
k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
inst✝ : A.IsTrivial
x✝¹ : ↑(ModuleCat.of k ↑A)
x✝ : G
⊢ (A.ρ x✝) x✝¹ - x✝¹ = (ModuleCat.Hom.hom 0) x✝¹ x✝
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.Hilbert90

Goal state before:
pp: case refine_2.e_a
K L : Type
inst✝¹⁶ : Field K
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra K L
inst✝¹³ : FiniteDimensional K L
inst✝¹² : IsGalois K L
inst✝¹¹ : IsCyclic Gal(L/K)
g : Gal(L/K)
A : Type u_1
B : Type u_2
inst✝¹⁰ : CommRing A
inst✝⁹ : CommRing B
inst✝⁸ : Algebra A B
inst✝⁷ : Algebra A L
inst✝⁶ : Algebra A K
inst✝⁵ : Algebra B L
inst✝⁴ : IsScalarTower A B L
inst✝³ : IsScalarTower A K L
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : IsIntegralClosure B A L
hg : ∀ (x : Gal(L/K)), x ∈ Subgroup.zpowers g
η : B
hη : (Algebra.norm K) ((algebraMap B L) η) = 1
this✝¹ : Module.IsTorsionFree A L
this✝ : IsLocalization (Algebra.algebraMapSubmonoid B A⁰) L
η' : Lˣ := Units.mk0 ((algebraMap B L) η) ⋯
ε : Lˣ
a : B
b : ↥(Algebra.algebraMapSubmonoid B A⁰)
h : IsLocalization.mk' L a b = ↑ε
t : A
ht : t ∈ ↑A⁰
ht' : (algebraMap A B) t = ↑b
this : t • IsLocalization.mk' L a b = (algebraMap B L) a
hε : (algebraMap B L) η * ↑g ((algebraMap B L) a) = (algebraMap B L) a
⊢ (algebraMap B L) (((galRestrict A K L B) g) a) = ↑g ((algebraMap B L) a)

Tactic:
exact algebraMap_galRestrictHom_apply A K L B g a

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
x y : G × G → ↑A
g : G × G × G
⊢ (A.ρ g.1) ((x + y) (g.2.1, g.2.2)) - (x + y) (g.1 * g.2.1, g.2.2) + (x + y) (g.1, g.2.1 * g.2.2) -
      (x + y) (g.1, g.2.1) =
    ((fun g ↦ (A.ρ g.1) (x (g.2.1, g.2.2)) - x (g.1 * g.2.1, g.2.2) + x (g.1, g.2.1 * g.2.2) - x (g.1, g.2.1)) + fun g ↦
        (A.ρ g.1) (y (g.2.1, g.2.2)) - y (g.1 * g.2.1, g.2.2) + y (g.1, g.2.1 * g.2.2) - y (g.1, g.2.1))
      g

Tactic:
dsimp
        rw [map_add, add_sub_add_comm (A.ρ _ _), add_sub_assoc, add_sub_add_comm, add_add_add_comm,
          add_sub_assoc, add_sub_assoc]

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

Goal state before:
pp: case a.a
k✝ G✝ : Type u
inst✝³ : CommRing k✝
inst✝² : Group G✝
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
n✝ n : ℕ
| (HomologicalComplex.coinvariantsTensorObj A (barComplex k G)).d (n + 1 + 1) (n + 1) ≫
    𝟙 ((HomologicalComplex.coinvariantsTensorObj A (barComplex k G)).X (n + 1)) ≫
      (HomologicalComplex.coinvariantsTensorObj A (barComplex k G)).d (n + 1) n

pp: case a.a
k✝ G✝ : Type u
inst✝³ : CommRing k✝
inst✝² : Group G✝
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
n✝ n : ℕ
| (A.coinvariantsTensorFreeLEquiv (Fin n → G)).toModuleIso.hom

pp: case a
k✝ G✝ : Type u
inst✝³ : CommRing k✝
inst✝² : Group G✝
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
n✝ n : ℕ
| (A.coinvariantsTensorFreeLEquiv (Fin (n + 1 + 1) → G)).toModuleIso.inv

Tactic:
Category.id_comp,

Goal state after:
case a.a
k✝ G✝ : Type u
inst✝³ : CommRing k✝
inst✝² : Group G✝
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
n✝ n : ℕ
| (HomologicalComplex.coinvariantsTensorObj A (barComplex k G)).d (n + 1 + 1) (n + 1) ≫
    (HomologicalComplex.coinvariantsTensorObj A (barComplex k G)).d (n + 1) n

case a.a
k✝ G✝ : Type u
inst✝³ : CommRing k✝
inst✝² : Group G✝
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
n✝ n : ℕ
| (A.coinvariantsTensorFreeLEquiv (Fin n → G)).toModuleIso.hom

case a
k✝ G✝ : Type u
inst✝³ : CommRing k✝
inst✝² : Group G✝
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
n✝ n : ℕ
| (A.coinvariantsTensorFreeLEquiv (Fin (n + 1 + 1) → G)).toModuleIso.inv
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
inst✝ : A.IsTrivial
⊢ cycles₁ A = ⊤

Tactic:
rw [cycles₁, d₁₀_eq_zero_of_isTrivial, ModuleCat.hom_zero, LinearMap.ker_zero]

Goal state after: