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

Goal state before:
pp: case inr
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a b c✝ : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
c d : ℕ
hac : a ≤ c
had : a ≤ d
hcd : c ≤ d
hcb : c ≤ b
⊢ lmarginalPartialTraj κ b c f = lmarginalPartialTraj κ b d f

Tactic:
· rw [hf.lmarginalPartialTraj_of_le c mf (hac.trans hcb),
      hf.lmarginalPartialTraj_of_le d mf (hac.trans hcb)]

Goal state after:
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
i : ι
s : Set (X i)
hs : MeasurableSet s
⊢ (map (Function.eval i) (infinitePi μ)) s = (μ i) s

Tactic:
have : @Function.eval ι X i =
        (@Function.eval ({i} : Finset ι) (fun j ↦ X j) ⟨i, by simp⟩) ∘
        (Finset.restrict {i}) := by ext; simp

Goal state after:
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
i : ι
s : Set (X i)
hs : MeasurableSet s
this : Function.eval i = Function.eval ⟨i, ⋯⟩ ∘ {i}.restrict
⊢ (map (Function.eval i) (infinitePi μ)) s = (μ i) s
Module: Mathlib.Probability.Decision.Risk.Basic

Goal state before:
pp: case inr
Θ : Type u_1
𝓧 : Type u_2
𝓨 : Type u_4
mΘ : MeasurableSpace Θ
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
ℓ : Θ → 𝓨 → ℝ≥0∞
P : Kernel Θ 𝓧
π : Measure Θ
inst✝¹ : Subsingleton 𝓧
inst✝ : Nonempty 𝓨
hX : Nonempty 𝓧
x : 𝓧
⊢ ⨅ κ, ⨅ (_ : IsMarkovKernel κ), avgRisk ℓ P κ π = ⨅ μ, ⨅ (_ : IsProbabilityMeasure μ), avgRisk ℓ P (Kernel.const 𝓧 μ) π

Tactic:
iInf_subtype',

Goal state after:
case inr
Θ : Type u_1
𝓧 : Type u_2
𝓨 : Type u_4
mΘ : MeasurableSpace Θ
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
ℓ : Θ → 𝓨 → ℝ≥0∞
P : Kernel Θ 𝓧
π : Measure Θ
inst✝¹ : Subsingleton 𝓧
inst✝ : Nonempty 𝓨
hX : Nonempty 𝓧
x : 𝓧
⊢ ⨅ x, avgRisk ℓ P (↑x) π = ⨅ μ, ⨅ (_ : IsProbabilityMeasure μ), avgRisk ℓ P (Kernel.const 𝓧 μ) π
Module: Mathlib.Probability.Decision.Risk.Basic

Goal state before:
pp: Θ : Type u_1
𝓧 : Type u_2
𝓧' : Type u_3
𝓨 : Type u_4
mΘ : MeasurableSpace Θ
m𝓧 : MeasurableSpace 𝓧
m𝓧' : MeasurableSpace 𝓧'
m𝓨 : MeasurableSpace 𝓨
ℓ : Θ → 𝓨 → ℝ≥0∞
P : Kernel Θ 𝓧
inst✝¹ : IsSFiniteKernel P
π : Measure Θ
η : Kernel (Θ × 𝓧) 𝓧'
inst✝ : IsMarkovKernel η
⊢ P = (P ⊗ₖ η).fst

Tactic:
Kernel.fst_compProd

Goal state after:
Θ : Type u_1
𝓧 : Type u_2
𝓧' : Type u_3
𝓨 : Type u_4
mΘ : MeasurableSpace Θ
m𝓧 : MeasurableSpace 𝓧
m𝓧' : MeasurableSpace 𝓧'
m𝓨 : MeasurableSpace 𝓨
ℓ : Θ → 𝓨 → ℝ≥0∞
P : Kernel Θ 𝓧
inst✝¹ : IsSFiniteKernel P
π : Measure Θ
η : Kernel (Θ × 𝓧) 𝓧'
inst✝ : IsMarkovKernel η
⊢ P = P
Module: Mathlib.Probability.Distributions.Bernoulli

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : MeasurableSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : MeasurableSingletonClass X
inst✝ : MeasurableSingletonClass Y
x y : X
f : X → Y
p : ↑I
⊢ map f Ber(x, y, p) = Ber(f x, f y, p)

Tactic:
by
  have hf (x : X) : AEMeasurable f (dirac x) := by fun_prop
  simp only [bernoulliMeasure_def]
  rw [AEMeasurable.map_add₀ (by fun_prop) (by fun_prop)]
  simp

Goal state after:
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
u : (i : ↥(Iic a)) → X ↑i
f : ((i : ↥(Iic b)) → X ↑i) → ((n : ℕ) → X n) → E
hf : Integrable (uncurry f) ((partialTraj κ a b) u ⊗ₘ traj κ b)
A : Set ((i : ↥(Iic b)) → X ↑i)
hA : MeasurableSet A
⊢ ∫ (x : (i : ↥(Iic b)) → X ↑i),
      ∫ (y : (n : ℕ) → X n), A.indicator (fun x ↦ f x y) x ∂(traj κ b) x ∂(partialTraj κ a b) u =
    ∫ (y : (n : ℕ) → X n) in frestrictLe b ⁻¹' A, f (frestrictLe b y) y ∂(traj κ a) u

Tactic:
integral_traj_partialTraj' hab

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
u : (i : ↥(Iic a)) → X ↑i
f : ((i : ↥(Iic b)) → X ↑i) → ((n : ℕ) → X n) → E
hf : Integrable (uncurry f) ((partialTraj κ a b) u ⊗ₘ traj κ b)
A : Set ((i : ↥(Iic b)) → X ↑i)
hA : MeasurableSet A
⊢ ∫ (x : (n : ℕ) → X n), A.indicator (fun x_1 ↦ f x_1 x) (frestrictLe b x) ∂(traj κ a) u =
    ∫ (y : (n : ℕ) → X n) in frestrictLe b ⁻¹' A, f (frestrictLe b y) y ∂(traj κ a) u

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
u : (i : ↥(Iic a)) → X ↑i
f : ((i : ↥(Iic b)) → X ↑i) → ((n : ℕ) → X n) → E
hf : Integrable (uncurry f) ((partialTraj κ a b) u ⊗ₘ traj κ b)
A : Set ((i : ↥(Iic b)) → X ↑i)
hA : MeasurableSet A
⊢ Integrable (uncurry fun x y ↦ A.indicator (fun x ↦ f x y) x) ((partialTraj κ a b) u ⊗ₘ traj κ b)
Module: Mathlib.Probability.Distributions.Poisson.PoissonLimitThm

Goal state before:
pp: p : ℕ → ℝ
r : ℝ
hr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)
⊢ (fun n ↦ ↑n * p n * (1 / ↑n)) =ᶠ[atTop] p

Tactic:
filter_upwards [eventually_ge_atTop 1] with n hn

Goal state after:
p : ℕ → ℝ
r : ℝ
hr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)
n : ℕ
hn : 1 ≤ n
⊢ ↑n * p n * (1 / ↑n) = p n
Module: Mathlib.Probability.Distributions.TwoValued

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsZeroOrProbabilityMeasure μ
hXmeas : AEMeasurable X μ
hX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1
⊢ Var[X; μ] = μ.real {ω | X ω = 0} * μ.real {ω | X ω = 1}

Tactic:
by
  obtain rfl | hμ := eq_zero_or_isProbabilityMeasure μ
  · simp
  simpa [Pi.mul_def, integral_of_ae_eq_zero_or_one, integral_one_sub_of_ae_eq_zero_or_one, mul_comm,
    *] using condVar_of_ae_eq_zero_or_one bot_le hXmeas hX

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

Goal state before:
pp: E : Type u_1
inst✝ : MeasurableSpace E
μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → E
s : Set E
hns : μ s ≠ 0
hnt : μ s ≠ ∞
hu : Measure.map X ℙ = (μ s)⁻¹ • μ.restrict s
⊢ AEMeasurable X ℙ

Tactic:
by_contra h

Goal state after:
E : Type u_1
inst✝ : MeasurableSpace E
μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → E
s : Set E
hns : μ s ≠ 0
hnt : μ s ≠ ∞
hu : Measure.map X ℙ = (μ s)⁻¹ • μ.restrict s
h : ¬AEMeasurable X ℙ
⊢ False
Module: Mathlib.Probability.Distributions.Uniform

Goal state before:
pp: α : Type u_1
inst✝² : Fintype α
inst✝¹ : Nonempty α
s : Set α
inst✝ : Fintype ↑s
⊢ (uniformOfFintype α).toOuterMeasure s = ↑(Fintype.card ↑s) / ↑(Fintype.card α)

Tactic:
classical
  rw [uniformOfFintype, toOuterMeasure_uniformOfFinset_apply, Fintype.card_subtype,
    Finset.card_univ]

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

Goal state before:
pp: Ω : Type u_1
S : Type u_2
T : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
F : T → Type u_5
inst✝¹⁰ : (s : S) → TopologicalSpace (E s)
inst✝⁹ : (s : S) → MeasurableSpace (E s)
inst✝⁸ : ∀ (s : S), BorelSpace (E s)
inst✝⁷ : ∀ (s : S), HasOuterApproxClosed (E s)
inst✝⁶ : (t : T) → TopologicalSpace (F t)
inst✝⁵ : (t : T) → MeasurableSpace (F t)
inst✝⁴ : ∀ (t : T), BorelSpace (F t)
inst✝³ : ∀ (t : T), HasOuterApproxClosed (F t)
X : (s : S) → Ω → E s
Y : (t : T) → Ω → F t
inst✝² : Fintype S
inst✝¹ : Fintype T
inst✝ : IsFiniteMeasure P
mX : ∀ (s : S), AEMeasurable (X s) P
mY : ∀ (t : T), AEMeasurable (Y t) P
h :
  ∀ (f : (s : S) → E s →ᵇ ℝ) (g : (t : T) → F t →ᵇ ℝ),
    ∫ (x : Ω), ((∏ s, ⇑(f s) ∘ X s) * ∏ t, ⇑(g t) ∘ Y t) x ∂P =
      (∫ (x : Ω), (∏ s, ⇑(f s) ∘ X s) x ∂P) * ∫ (x : Ω), (∏ t, ⇑(g t) ∘ Y t) x ∂P
⊢ (fun ω s ↦ X s ω) ⟂ᵢ[P] fun ω t ↦ Y t ω

Tactic:
indepFun_iff_map_prod_eq_prod_map_map (aemeasurable_pi_lambda _ mX)
    (aemeasurable_pi_lambda _ mY)

Goal state after:
Ω : Type u_1
S : Type u_2
T : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
F : T → Type u_5
inst✝¹⁰ : (s : S) → TopologicalSpace (E s)
inst✝⁹ : (s : S) → MeasurableSpace (E s)
inst✝⁸ : ∀ (s : S), BorelSpace (E s)
inst✝⁷ : ∀ (s : S), HasOuterApproxClosed (E s)
inst✝⁶ : (t : T) → TopologicalSpace (F t)
inst✝⁵ : (t : T) → MeasurableSpace (F t)
inst✝⁴ : ∀ (t : T), BorelSpace (F t)
inst✝³ : ∀ (t : T), HasOuterApproxClosed (F t)
X : (s : S) → Ω → E s
Y : (t : T) → Ω → F t
inst✝² : Fintype S
inst✝¹ : Fintype T
inst✝ : IsFiniteMeasure P
mX : ∀ (s : S), AEMeasurable (X s) P
mY : ∀ (t : T), AEMeasurable (Y t) P
h :
  ∀ (f : (s : S) → E s →ᵇ ℝ) (g : (t : T) → F t →ᵇ ℝ),
    ∫ (x : Ω), ((∏ s, ⇑(f s) ∘ X s) * ∏ t, ⇑(g t) ∘ Y t) x ∂P =
      (∫ (x : Ω), (∏ s, ⇑(f s) ∘ X s) x ∂P) * ∫ (x : Ω), (∏ t, ⇑(g t) ∘ Y t) x ∂P
⊢ map (fun ω ↦ (fun a ↦ X a ω, fun a ↦ Y a ω)) P = (map (fun c a ↦ X a c) P).prod (map (fun c a ↦ Y a c) P)
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
π : ι → Set (Set Ω)
hπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
h_eq' :
  ∀ (s : Finset ι) (f : ι → Set Ω),
    (∀ i ∈ s, f i ∈ π i) →
      ∀ i ∈ s, (fun ω ↦ (((condExpKernel μ m') ω) (f i)).toReal) =ᵐ[μ] μ[(f i).indicator fun ω ↦ 1 | m']
h_eq :
  ∀ (s : Finset ι) (f : ι → Set Ω),
    (∀ i ∈ s, f i ∈ π i) →
      ∀ᵐ (ω : Ω) ∂μ, ∀ i ∈ s, (((condExpKernel μ m') ω) (f i)).toReal = μ[(f i).indicator fun ω ↦ 1 | m'] ω
⊢ (∀ (s : Finset ι) {f : ι → Set Ω},
      (∀ i ∈ s, f i ∈ π i) →
        ∀ᵐ (a : Ω) ∂μ.trim hm', ((condExpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condExpKernel μ m') a) (f i)) ↔
    ∀ (s : Finset ι) {f : ι → Set Ω},
      (∀ i ∈ s, f i ∈ π i) → μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ i ∈ s, μ[(f i).indicator fun ω ↦ 1 | m']

Tactic:
have h_inter_eq : ∀ (s : Finset ι) (f : ι → Set Ω) (_H : ∀ i, i ∈ s → f i ∈ π i),
      (fun ω ↦ ENNReal.toReal (condExpKernel μ m' ω (⋂ i ∈ s, f i)))
        =ᵐ[μ] μ⟦⋂ i ∈ s, f i | m'⟧ := by
    refine fun s f H ↦ condExpKernel_ae_eq_condExp hm' ?_
    exact MeasurableSet.biInter (Finset.countable_toSet _) (fun i hi ↦ hπ i _ (H i hi))

Goal state after:
Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
π : ι → Set (Set Ω)
hπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
h_eq' :
  ∀ (s : Finset ι) (f : ι → Set Ω),
    (∀ i ∈ s, f i ∈ π i) →
      ∀ i ∈ s, (fun ω ↦ (((condExpKernel μ m') ω) (f i)).toReal) =ᵐ[μ] μ[(f i).indicator fun ω ↦ 1 | m']
h_eq :
  ∀ (s : Finset ι) (f : ι → Set Ω),
    (∀ i ∈ s, f i ∈ π i) →
      ∀ᵐ (ω : Ω) ∂μ, ∀ i ∈ s, (((condExpKernel μ m') ω) (f i)).toReal = μ[(f i).indicator fun ω ↦ 1 | m'] ω
h_inter_eq :
  ∀ (s : Finset ι) (f : ι → Set Ω),
    (∀ i ∈ s, f i ∈ π i) →
      (fun ω ↦ (((condExpKernel μ m') ω) (⋂ i ∈ s, f i)).toReal) =ᵐ[μ] μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m']
⊢ (∀ (s : Finset ι) {f : ι → Set Ω},
      (∀ i ∈ s, f i ∈ π i) →
        ∀ᵐ (a : Ω) ∂μ.trim hm', ((condExpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condExpKernel μ m') a) (f i)) ↔
    ∀ (s : Finset ι) {f : ι → Set Ω},
      (∀ i ∈ s, f i ∈ π i) → μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ i ∈ s, μ[(f i).indicator fun ω ↦ 1 | m']
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}
measurable_f : Measurable (uncurry f)
a : X
x : ↑I
Iic_to_Icc : Iic x = Icc 0 x
κ_in_I : (κ a).real (Icc 0 x) ∈ I
⊢ volume {x_1 | f a x_1 ≤ x} = (κ a) (Icc 0 x)

Tactic:
← ofReal_measureReal (measure_ne_top (κ a) _),

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}
measurable_f : Measurable (uncurry f)
a : X
x : ↑I
Iic_to_Icc : Iic x = Icc 0 x
κ_in_I : (κ a).real (Icc 0 x) ∈ I
⊢ volume {x_1 | f a x_1 ≤ x} = ENNReal.ofReal ((κ a).real (Icc 0 x))
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: case refine_1
Ω : Type u_1
β : Type u_3
β' : Type u_4
mΩ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
f : Ω → β
g : Ω → β'
γ : Type u_5
mγ : MeasurableSpace γ
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝³ : StandardBorelSpace β
inst✝² : Nonempty β
inst✝¹ : StandardBorelSpace β'
inst✝ : Nonempty β'
hf : Measurable f
hg : Measurable g
k : Ω → γ
hk : Measurable k
hk_meas : ∀ {s : Set γ}, MeasurableSet s → MeasurableSet (k ⁻¹' s)
h_left :
  ∀ {s : Set γ} {t : Set β} {u : Set β'},
    MeasurableSet s →
      MeasurableSet t →
        MeasurableSet u →
          (Measure.map (fun ω ↦ (k ω, f ω, g ω)) μ) (s ×ˢ t ×ˢ u) =
            (Measure.map (fun ω ↦ (ω, f ω, g ω)) μ) ((k ⁻¹' s) ×ˢ t ×ˢ u)
h_right :
  ∀ {s : Set γ} {t : Set β} {u : Set β'},
    MeasurableSet s →
      MeasurableSet t →
        MeasurableSet u →
          (⇑(Kernel.id ×ₖ (condDistrib f k μ ×ₖ condDistrib g k μ)) ∘ₘ Measure.map k μ) (s ×ˢ t ×ˢ u) =
            (⇑(Kernel.id ×ₖ
                    ((condExpKernel μ (MeasurableSpace.comap k mγ)).map f ×ₖ
                      (condExpKernel μ (MeasurableSpace.comap k mγ)).map g)) ∘ₘ
                μ.trim ⋯)
              ((k ⁻¹' s) ×ˢ t ×ˢ u)
h :
  ∀ {s : Set Ω} {t : Set β} {u : Set β'},
    MeasurableSet s →
      MeasurableSet t →
        MeasurableSet u →
          (Measure.map (fun ω ↦ (ω, f ω, g ω)) μ) (s ×ˢ t ×ˢ u) =
            (⇑(Kernel.id ×ₖ
                    ((condExpKernel μ (MeasurableSpace.comap k inferInstance)).map f ×ₖ
                      (condExpKernel μ (MeasurableSpace.comap k inferInstance)).map g)) ∘ₘ
                μ.trim ⋯)
              (s ×ˢ t ×ˢ u)
s : Set γ
t : Set β
u : Set β'
hs : MeasurableSet s
ht : MeasurableSet t
hu : MeasurableSet u
⊢ (Measure.map (fun ω ↦ (k ω, f ω, g ω)) μ) (s ×ˢ t ×ˢ u) =
    (⇑(Kernel.id ×ₖ (condDistrib f k μ ×ₖ condDistrib g k μ)) ∘ₘ Measure.map k μ) (s ×ˢ t ×ˢ u)

Tactic:
convert! h (hk_meas hs) ht hu
    · exact h_left hs ht hu
    · exact h_right hs ht hu

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
t : ℝ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
g : Ω → E
s : Set Ω
ht : Integrable (fun ω ↦ rexp (t * X ω)) μ
⊢ (∫ (x : Ω) in s, g x ∂μ.tilted fun x ↦ t * X x) = ∫ (x : Ω) in s, rexp (t * X x - cgf X μ t) • g x ∂μ

Tactic:
by
  rcases eq_zero_or_neZero μ with rfl | hμ
  · simp
  · simp_rw [setIntegral_tilted_mul_eq_mgf, exp_sub, exp_cgf ht]

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
β : ι → Type u_5
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
h_indep : iCondIndepFun m' hm' f μ
hf : ∀ (i : ι), Measurable (f i)
i j k l : ι
hik : i ≠ k
hil : i ≠ l
hjk : j ≠ k
hjl : j ≠ l
⊢ CondIndepFun m' hm' (fun a ↦ (f i a, f j a)) (fun a ↦ (f k a, f l a)) μ

Tactic:
classical
  let g (i j : ι) (v : Π x : ({i, j} : Finset ι), β x) : β i × β j :=
    ⟨v ⟨i, mem_insert_self _ _⟩, v ⟨j, mem_insert_of_mem <| mem_singleton_self _⟩⟩
  have hg (i j : ι) : Measurable (g i j) := by fun_prop
  exact (h_indep.indepFun_finset {i, j} {k, l} (by aesop) hf).comp (hg i j) (hg k l)

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

Goal state before:

Tactic:
(κ ω').real {ω | ε ≤ X ω}
  _ ≤ exp (-t * ε) * mgf X (κ ω') t := measure_ge_le_exp_mul_mgf ε ht (h2 t)
  _ ≤ exp (-t * ε + c * t ^ 2 / 2) := by
    rw [exp_add]
    gcongr
    exact h1 t

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✝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
⊢ ∀ᵐ (ω : Ω) ∂P, Tendsto (fun x ↦ ⨅ j, ⨅ (_ : j ≥ x), τ j ω) atTop (𝓝 ⊤)

Tactic:
filter_upwards [hτ.tendsto_top] with ω hω

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 : Tendsto (fun x ↦ τ x ω) atTop (𝓝 ⊤)
⊢ Tendsto (fun x ↦ ⨅ j, ⨅ (_ : j ≥ x), τ j ω) atTop (𝓝 ⊤)
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
⊢ (∫ (x : Ω), (fun ω ↦ rexp (t * X ω) ^ ↑p) x ∂κ ω') ^ (1 / ↑p) *
      (∫ (x : Ω), (fun ω ↦ rexp (t * Y ω) ^ ↑q) x ∂κ ω') ^ (1 / ↑q) ≤
    rexp (↑cX * (t * ↑p) ^ 2 / 2) ^ (1 / ↑p) * rexp (↑cY * (t * ↑q) ^ 2 / 2) ^ (1 / ↑q)

Tactic:
simp_rw [← exp_mul _ p, ← exp_mul _ q, mul_right_comm t _ p, mul_right_comm t _ q]
        gcongr
        · exact hmX (t * p)
        · exact hmY (t * q)

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
⊢ (∫ (x : Ω), (fun ω ↦ rexp (t * X ω) ^ ↑p) x ∂κ ω') ^ (1 / ↑p) *
      (∫ (x : Ω), (fun ω ↦ rexp (t * Y ω) ^ ↑q) x ∂κ ω') ^ (1 / ↑q) ≤
    rexp (↑cX * (t * ↑p) ^ 2 / 2) ^ (1 / ↑p) * rexp (↑cY * (t * ↑q) ^ 2 / 2) ^ (1 / ↑q)

Tactic:
by
        simp_rw [← exp_mul _ p, ← exp_mul _ q, mul_right_comm t _ p, mul_right_comm t _ q]
        gcongr
        · exact hmX (t * p)
        · exact hmY (t * q)

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 i : ℕ
x✝ : i ∈ range K
⊢ Tendsto (fun N ↦ ℙ {ω | X ω ∈ Set.Ioc ↑i ↑N}) atTop (𝓝 (ℙ {ω | X ω ∈ Set.Ioi ↑i}))

Tactic:
have : {ω | X ω ∈ Set.Ioi (i : ℝ)} = ⋃ N : ℕ, {ω | X ω ∈ Set.Ioc (i : ℝ) N} := by
      apply Set.Subset.antisymm _ _
      · intro ω hω
        obtain ⟨N, hN⟩ : ∃ N : ℕ, X ω ≤ N := exists_nat_ge (X ω)
        exact Set.mem_iUnion.2 ⟨N, hω, hN⟩
      · simp +contextual only [Set.mem_Ioc, Set.mem_Ioi,
          Set.iUnion_subset_iff, Set.ofPred_subset_ofPred, imp_true_iff]

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 i : ℕ
x✝ : i ∈ range K
this : {ω | X ω ∈ Set.Ioi ↑i} = ⋃ N, {ω | X ω ∈ Set.Ioc ↑i ↑N}
⊢ Tendsto (fun N ↦ ℙ {ω | X ω ∈ Set.Ioc ↑i ↑N}) atTop (𝓝 (ℙ {ω | X ω ∈ Set.Ioi ↑i}))
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: k : Type u
inst✝³ : Field k
G : Type v
inst✝² : Group G
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Nat.card G)
V : FDRep k G
⊢ Invertible ↑(Fintype.card G)

Tactic:
rwa [Fintype.card_eq_nat_card]

Goal state after:
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
h : H
x : H → B
hx : x ∈ coindV φ ρ
g : G
h₁ : H
⊢ (LinearMap.funLeft k B fun x ↦ x * h) x (φ g * h₁) = (ρ g) ((LinearMap.funLeft k B fun x ↦ x * h) x h₁)

Tactic:
simpa [mul_assoc] using hx g (h₁ * h)

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

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹⁸ : Monoid G
inst✝¹⁷ : Ring R
inst✝¹⁶ : AddCommGroup V
inst✝¹⁵ : TopologicalSpace V
inst✝¹⁴ : IsTopologicalAddGroup V
inst✝¹³ : Module R V
inst✝¹² : AddCommGroup W
inst✝¹¹ : TopologicalSpace W
inst✝¹⁰ : IsTopologicalAddGroup W
inst✝⁹ : Module R W
inst✝⁸ : AddCommGroup U
inst✝⁷ : Module R U
inst✝⁶ : TopologicalSpace U
inst✝⁵ : IsTopologicalAddGroup U
π₁ : ContRepresentation R G V
π₂ : ContRepresentation R G W
π₃ : ContRepresentation R G U
S : Type u_6
inst✝⁴ : Monoid S
inst✝³ : DistribMulAction S W
inst✝² : SMulCommClass R S W
inst✝¹ : ContinuousConstSMul S W
inst✝ : LinearMap.CompatibleSMul W W S R
s : S
f : π₁ →ⁱL π₂
g : G
⊢ (s • f.toContinuousLinearMap) ∘SL π₁ g = π₂ g ∘SL (s • f.toContinuousLinearMap)

Tactic:
by
    rw [ContinuousLinearMap.smul_comp, f.2, ContinuousLinearMap.comp_smul]

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

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Ring k
inst✝ : Monoid G
A✝ B✝ C A B : Rep k G
⊢ ofHom (Representation.IntertwiningMap.inl k A.ρ B.ρ) ≫ ofHom (Representation.IntertwiningMap.fst k A.ρ B.ρ) = 𝟙 A

Tactic:
ext1; simp

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

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Ring k
inst✝ : Monoid G
A✝ B✝ C A B : Rep k G
⊢ ofHom (Representation.IntertwiningMap.inr k A.ρ B.ρ) ≫ ofHom (Representation.IntertwiningMap.snd k A.ρ B.ρ) = 𝟙 B

Tactic:
ext1; simp

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

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : CommRing k
inst✝ : Monoid G
x✝² x✝¹ x✝ : Rep k G
⊢ x✝² ◁ 0 = 0

Tactic:
ext1; simp

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

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : CommRing k
inst✝ : Monoid G
x✝² x✝¹ x✝ : Rep k G
⊢ 0 ▷ x✝² = 0

Tactic:
ext1; simp

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
f : res (↑φ) X ⟶ Y
i j : ℕ
hij : i + 1 = j
⊢ invariantsResMap (↑φ) (resolutionMap φ f (i + 1)) ≫ Y.homogeneousCochains.d i j =
    X.homogeneousCochains.d i j ≫ invariantsResMap (↑φ) (resolutionMap φ f (j + 1))

Tactic:
subst hij
    rw [homogeneousCochains.d_eq, homogeneousCochains.d_eq, ← invariantsResMap_comp,
      resolutionMap_comp_d, invariantsResMap_map_comp]

Goal state after: