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

Goal state before:
pp: case hn
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m1 m2 m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h1 : m1 ≤ m
h2 : m2 ≤ m
hp1 : IsPiSystem p1
hp2 : IsPiSystem p2
hpm1 : m1 = generateFrom p1
hpm2 : m2 = generateFrom p2
hyp : IndepSets p1 p2 κ μ
h✝ : IsMarkovKernel κ
t1 t2 : Set Ω
ht2 : t2 ∈ {s | MeasurableSet s}
f : ℕ → Set Ω
hf_disj : Pairwise (Function.onFun Disjoint f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), (κ a) (f i ∩ t2) = (κ a) (f i) * (κ a) t2
a : α
ha : ∀ (i : ℕ), (κ a) (f i ∩ t2) = (κ a) (f i) * (κ a) t2
⊢ Pairwise (Function.onFun Disjoint fun i ↦ t2 ∩ f i)

pp: case h
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m1 m2 m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h1 : m1 ≤ m
h2 : m2 ≤ m
hp1 : IsPiSystem p1
hp2 : IsPiSystem p2
hpm1 : m1 = generateFrom p1
hpm2 : m2 = generateFrom p2
hyp : IndepSets p1 p2 κ μ
h✝ : IsMarkovKernel κ
t1 t2 : Set Ω
ht2 : t2 ∈ {s | MeasurableSet s}
f : ℕ → Set Ω
hf_disj : Pairwise (Function.onFun Disjoint f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), (κ a) (f i ∩ t2) = (κ a) (f i) * (κ a) t2
a : α
ha : ∀ (i : ℕ), (κ a) (f i ∩ t2) = (κ a) (f i) * (κ a) t2
⊢ ∀ (i : ℕ), MeasurableSet (t2 ∩ f i)

Tactic:
· intro i j hij
      rw [Function.onFun, Set.inter_comm t2, Set.inter_comm t2]
      exact Disjoint.inter_left _ (Disjoint.inter_right _ (hf_disj hij))

Goal state after:
case h
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m1 m2 m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h1 : m1 ≤ m
h2 : m2 ≤ m
hp1 : IsPiSystem p1
hp2 : IsPiSystem p2
hpm1 : m1 = generateFrom p1
hpm2 : m2 = generateFrom p2
hyp : IndepSets p1 p2 κ μ
h✝ : IsMarkovKernel κ
t1 t2 : Set Ω
ht2 : t2 ∈ {s | MeasurableSet s}
f : ℕ → Set Ω
hf_disj : Pairwise (Function.onFun Disjoint f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), (κ a) (f i ∩ t2) = (κ a) (f i) * (κ a) t2
a : α
ha : ∀ (i : ℕ), (κ a) (f i ∩ t2) = (κ a) (f i) * (κ a) t2
⊢ ∀ (i : ℕ), MeasurableSet (t2 ∩ f i)
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m1 m2 m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h1 : m1 ≤ m
h2 : m2 ≤ m
hp1 : IsPiSystem p1
hp2 : IsPiSystem p2
hpm1 : m1 = generateFrom p1
hpm2 : m2 = generateFrom p2
hyp : IndepSets p1 p2 κ μ
⊢ Indep m1 m2 κ μ

Tactic:
rcases eq_zero_or_isMarkovKernel κ with rfl | h
  · simp
  intro t1 t2 ht1 ht2
  induction t1, ht1 using induction_on_inter hpm1 hp1 with
  | empty =>
    simp only [Set.empty_inter, measure_empty, zero_mul, Filter.eventually_true]
  | basic t ht =>
    refine IndepSets.indep_aux h2 hp2 hpm2 hyp ht (h1 _ ?_) ht2
    rw [hpm1]
    exact measurableSet_generateFrom ht
  | compl t ht iht =>
    filter_upwards [iht] with a ha
    have : tᶜ ∩ t2 = t2 \ (t ∩ t2) := by
      rw [Set.inter_comm t, Set.sdiff_self_inter, Set.sdiff_eq_compl_inter]
    rw [this, Set.inter_comm t t2,
      measure_sdiff Set.inter_subset_left ((h2 _ ht2).inter (h1 _ ht)).nullMeasurableSet
        (measure_ne_top (κ a) _),
      Set.inter_comm, ha, measure_compl (h1 _ ht) (measure_ne_top (κ a) t), measure_univ,
      mul_comm (1 - κ a t), ENNReal.mul_sub (fun _ _ ↦ measure_ne_top (κ a) _), mul_one, mul_comm]
  | iUnion f hf_disj hf_meas h =>
    rw [← ae_all_iff] at h
    filter_upwards [h] with a ha
    rw [Set.inter_comm, Set.inter_iUnion, measure_iUnion]
    · rw [measure_iUnion hf_disj (fun i ↦ h1 _ (hf_meas i))]
      rw [← ENNReal.tsum_mul_right]
      congr 1 with i
      rw [Set.inter_comm t2, ha i]
    · intro i j hij
      rw [Function.onFun, Set.inter_comm t2, Set.inter_comm t2]
      exact Disjoint.inter_left _ (Disjoint.inter_right _ (hf_disj hij))
    · exact fun i ↦ (h2 _ ht2).inter (h1 _ (hf_meas i))

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case refine_2.refine_3
α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
u : Set α
h : (μ ⊗ₘ κ.map fun ω ↦ (f ω, g ω)) (u ×ˢ s ×ˢ t) = (μ ⊗ₘ (κ.map f ×ₖ κ.map g)) (u ×ˢ s ×ˢ t)
hu : MeasurableSet u
hμu : μ u < ⊤
⊢ ∫⁻ (x : α) in u, (κ x) (f ⁻¹' s ∩ g ⁻¹' t) ∂μ = ∫⁻ (x : α) in u, (κ x) (f ⁻¹' s) * (κ x) (g ⁻¹' t) ∂μ

Tactic:
Measure.compProd_apply_prod hu (hs.prod ht),

Goal state after:
case refine_2.refine_3
α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
u : Set α
h : ∫⁻ (a : α) in u, ((κ.map fun ω ↦ (f ω, g ω)) a) (s ×ˢ t) ∂μ = (μ ⊗ₘ (κ.map f ×ₖ κ.map g)) (u ×ˢ s ×ˢ t)
hu : MeasurableSet u
hμu : μ u < ⊤
⊢ ∫⁻ (x : α) in u, (κ x) (f ⁻¹' s ∩ g ⁻¹' t) ∂μ = ∫⁻ (x : α) in u, (κ x) (f ⁻¹' s) * (κ x) (g ⁻¹' t) ∂μ
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: case inr
Ω : Type u_1
ι : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : SecondCountableTopology Ω
inst✝¹ : OpensMeasurableSpace Ω
S : Set (Set Ω)
hS : IsPiSystem S
μ : ι → ProbabilityMeasure Ω
ν : ProbabilityMeasure Ω
l : Filter ι
inst✝ : l.IsCountablyGenerated
hmeas : ∀ s ∈ S, MeasurableSet s
h : ∀ (u : Set Ω), IsOpen[inst✝³] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u
h' : ∀ s ∈ S, Tendsto (fun i ↦ (μ i) s) l (𝓝 (ν s))
hl : l.NeBot
⊢ ∀ (G : Set Ω), IsOpen[inst✝³] G → ν G ≤ liminf (fun i ↦ (μ i) G) l

Tactic:
intro G

Goal state after:
case inr
Ω : Type u_1
ι : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : SecondCountableTopology Ω
inst✝¹ : OpensMeasurableSpace Ω
S : Set (Set Ω)
hS : IsPiSystem S
μ : ι → ProbabilityMeasure Ω
ν : ProbabilityMeasure Ω
l : Filter ι
inst✝ : l.IsCountablyGenerated
hmeas : ∀ s ∈ S, MeasurableSet s
h : ∀ (u : Set Ω), IsOpen[inst✝³] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u
h' : ∀ s ∈ S, Tendsto (fun i ↦ (μ i) s) l (𝓝 (ν s))
hl : l.NeBot
G : Set Ω
⊢ IsOpen[inst✝³] G → ν G ≤ liminf (fun i ↦ (μ i) G) l
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: case hg
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y Z : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
hZ : MemLp Z 2 μ
⊢ Integrable (fun ω ↦ (Y ω - ∫ (x : Ω), Y x ∂μ) * (Z ω - ∫ (x : Ω), Z x ∂μ)) μ

Tactic:
· exact (hY.sub (memLp_const _)).integrable_mul (hZ.sub (memLp_const _))

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
c : ℝ
⊢ ∫ (ω : Ω), (c • X ω - ∫ (x : Ω), c • X x ∂μ) * (Y ω - ∫ (x : Ω), Y x ∂μ) ∂μ =
    c * ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) * (Y ω - ∫ (x : Ω), Y x ∂μ) ∂μ

Tactic:
smul_eq_mul,

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
c : ℝ
⊢ ∫ (ω : Ω), (c * X ω - ∫ (x : Ω), c * X x ∂μ) * (Y ω - ∫ (x : Ω), Y x ∂μ) ∂μ =
    c * ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) * (Y ω - ∫ (x : Ω), Y x ∂μ) ∂μ
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁸ : TopologicalSpace E
inst✝⁷ : ContinuousENorm E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
inst✝⁴ : NormedAddGroup F
inst✝³ : MeasurableSpace F
inst✝² : OpensMeasurableSpace F
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousENorm G
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
B : E → F → G
c : ℝ≥0
hc : c ≠ 0
hB : ∀ (x : E) (y : F), ↑c * ‖x‖ₑ * ‖y‖ₑ ≤ ‖B x y‖ₑ
h'XY : Integrable (fun ω ↦ B (X ω) (Y ω)) μ
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
h'Y : ¬Y =ᵐ[μ] 0
H : ∫⁻ (ω : Ω), ‖Y ω‖ₑ ∂μ = 0
⊢ False

Tactic:
have I : (fun ω => ‖Y ω‖ₑ : Ω → ℝ≥0∞) =ᵐ[μ] 0 := (lintegral_eq_zero_iff' hY.enorm).1 H

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁸ : TopologicalSpace E
inst✝⁷ : ContinuousENorm E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
inst✝⁴ : NormedAddGroup F
inst✝³ : MeasurableSpace F
inst✝² : OpensMeasurableSpace F
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousENorm G
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
B : E → F → G
c : ℝ≥0
hc : c ≠ 0
hB : ∀ (x : E) (y : F), ↑c * ‖x‖ₑ * ‖y‖ₑ ≤ ‖B x y‖ₑ
h'XY : Integrable (fun ω ↦ B (X ω) (Y ω)) μ
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
h'Y : ¬Y =ᵐ[μ] 0
H : ∫⁻ (ω : Ω), ‖Y ω‖ₑ ∂μ = 0
I : (fun ω ↦ ‖Y ω‖ₑ) =ᵐ[μ] 0
⊢ False
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
hm : AEStronglyMeasurable X μ
hX : ¬MemLp X 2 μ
⊢ ∫ (a : Ω), (X a - ∫ (x : Ω), X x ∂μ) ^ 2 ∂μ ≤ ∫ (x : Ω), (X ^ 2) x ∂μ

Tactic:
by_cases hint : Integrable X μ

Goal state after:
case pos
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
hm : AEStronglyMeasurable X μ
hX : ¬MemLp X 2 μ
hint : Integrable X μ
⊢ ∫ (a : Ω), (X a - ∫ (x : Ω), X x ∂μ) ^ 2 ∂μ ≤ ∫ (x : Ω), (X ^ 2) x ∂μ

case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
hm : AEStronglyMeasurable X μ
hX : ¬MemLp X 2 μ
hint : ¬Integrable X μ
⊢ ∫ (a : Ω), (X a - ∫ (x : Ω), X x ∂μ) ^ 2 ∂μ ≤ ∫ (x : Ω), (X ^ 2) x ∂μ
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
hX : MemLp X 2 μ
c : ℝ
hc : 0 < c
⊢ μ {ω | c ≤ |X ω - ∫ (x : Ω), X x ∂μ|} ≤ ENNReal.ofReal (Var[X; μ] / c ^ 2)

Tactic:
rw [ENNReal.ofReal_div_of_pos (sq_pos_of_ne_zero hc.ne.symm), hX.ofReal_variance_eq]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
hX : MemLp X 2 μ
c : ℝ
hc : 0 < c
⊢ μ {ω | c ≤ |X ω - ∫ (x : Ω), X x ∂μ|} ≤ eVar[X; μ] / ENNReal.ofReal (c ^ 2)
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case e_a
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_meas : ∀ (i : ι), Measurable (f i)
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[μ] ⇑η
hη : IsMarkovKernel η
πSβ : Set (Set ((i : ↥S) → β ↑i)) := univ.pi '' univ.pi fun i ↦ {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t ∈ πSβ, (fun a i ↦ f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i ↦ f (↑i) a) MeasurableSpace.pi = generateFrom πS
πTβ : Set (Set ((i : ↥T) → β ↑i)) := univ.pi '' univ.pi fun i ↦ {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t ∈ πTβ, (fun a i ↦ f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i ↦ f (↑i) a) MeasurableSpace.pi = generateFrom πT
s : Set ((i : ↥S) → β ↑i)
sets_s : (i : ↥S) → Set (β ↑i)
hs2 : univ.pi sets_s = s
t : Set ((i : ↥T) → β ↑i)
sets_t : (i : ↥T) → Set (β ↑i)
ht2 : univ.pi sets_t = t
hs1 : ∀ (i : ↥S), MeasurableSet (sets_s i)
ht1 : ∀ (i : ↥T), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i ↦ if hi : i ∈ S then sets_s ⟨i, hi⟩ else univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s ⟨i, hi⟩
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = univ
sets_t' : (i : ι) → Set (β i) := fun i ↦ if hi : i ∈ T then sets_t ⟨i, hi⟩ else univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = univ
h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i)
h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i ↦ f (↑i) ω) ⁻¹' univ.pi sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i ↦ f (↑i) ω) ⁻¹' univ.pi sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (η a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (η a) (f i ⁻¹' sets i)
h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)
h_meas_inter : ∀ i ∈ S ∪ T, MeasurableSet (sets_s' i ∩ sets_t' i)
a : α
h_indepS : (η a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i ∈ S, (η a) (f i ⁻¹' sets_s' i)
h_indepT : (η a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i ∈ T, (η a) (f i ⁻¹' sets_t' i)
h_indepST : (η a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i ∈ S ∪ T, (η a) (f i ⁻¹' (sets_s' i ∩ sets_t' i))
i : ι
hi : i ∈ T
⊢ (η a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) = (η a) (f i ⁻¹' sets_t' i)

Tactic:
rw [h_sets_s'_univ hi, Set.univ_inter]

Goal state after:
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f : ℕ → α → β
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε
ε : ℝ≥0
hε : 0 < ε
N : ℕ
hNε : ∀ n ≥ N, eLpNorm (f n) p μ ≤ ↑ε
F : Fin N → α → β := fun n ↦ f ↑n
hF : ∀ (n : Fin N), MemLp (F n) p μ
⊢ ∃ s, μ s ≠ ∞ ∧ ∀ (i : ℕ), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε

Tactic:
obtain ⟨s, hμs, hFε⟩ := unifTight_fin hp' hF hε

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f : ℕ → α → β
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε
ε : ℝ≥0
hε : 0 < ε
N : ℕ
hNε : ∀ n ≥ N, eLpNorm (f n) p μ ≤ ↑ε
F : Fin N → α → β := fun n ↦ f ↑n
hF : ∀ (n : Fin N), MemLp (F n) p μ
s : Set α
hμs : μ s ≠ ∞
hFε : ∀ (i : Fin N), eLpNorm (sᶜ.indicator (F i)) p μ ≤ ↑ε
⊢ ∃ s, μ s ≠ ∞ ∧ ∀ (i : ℕ), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε
Module: Mathlib.MeasureTheory.Integral.CircleTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
z w : ℂ
f : ℂ → E
⊢ ∫ (θ : ℝ) in 0..2 * π, circleTransform R z w f θ = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(z, R), (z - w)⁻¹ • f z

Tactic:
simp_rw [circleTransform, circleIntegral, deriv_circleMap, circleMap]
  simp

Goal state after:
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
U : Set E
hU : IsOpen U ∧ U ∩ ↑L = {0}
⊢ ↑L ⊓ {0} = ↑L ⊓ U

Tactic:
by simp [Set.inter_comm .. ▸ hU.2, zero_mem]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q : ℝ
hp : 0 < p
hq : -1 < q
⊢ ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * rexp (-x)) =
    ∫ (x : ℝ) in Ioi 0, 1 / p * rexp (-x) * x ^ (1 / p - 1 + q / p)

Tactic:
smul_eq_mul,

Goal state after:
p q : ℝ
hp : 0 < p
hq : -1 < q
⊢ ∫ (x : ℝ) in Ioi 0, 1 / p * x ^ (1 / p - 1) * ((x ^ (1 / p)) ^ q * rexp (-x)) =
    ∫ (x : ℝ) in Ioi 0, 1 / p * rexp (-x) * x ^ (1 / p - 1 + q / p)
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: case inl
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
a : ℝ
f : α →C_c ℝ
ha : 0 ≤ a
⊢ ↑(Λ (a • f).nnrealPart) - ↑(Λ (-(a • f)).nnrealPart) = a • (↑(Λ f.nnrealPart) - ↑(Λ (-f).nnrealPart))

Tactic:
smul_eq_mul,

Goal state after:
case inl
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
a : ℝ
f : α →C_c ℝ
ha : 0 ≤ a
⊢ ↑(Λ (a • f).nnrealPart) - ↑(Λ (-(a • f)).nnrealPart) = a * (↑(Λ f.nnrealPart) - ↑(Λ (-f).nnrealPart))
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ (∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-|x| ^ p)) * (2 * π) • 1 =
    2 * π * ∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-|x| ^ p)

Tactic:
smul_eq_mul,

Goal state after:
p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ (∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-|x| ^ p)) * (2 * π * 1) =
    2 * π * ∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-|x| ^ p)
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ (∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-b * |x| ^ p)) * (2 * π) • 1 =
    2 * π * ∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-b * |x| ^ p)

Tactic:
smul_eq_mul,

Goal state after:
p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ (∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-b * |x| ^ p)) * (2 * π * 1) =
    2 * π * ∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-b * |x| ^ p)
Module: Mathlib.MeasureTheory.Integral.MeanValue

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
⊢ ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤

Tactic:
simp [ρ]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.MeanValue

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
h : (∫ (x : α) in s, f x ∂μ.withDensity fun x ↦ ρ x) = ∫ (x : α) in s, (ρ x).toReal • f x ∂μ
⊢ ∫ (x : α) in s, (ρ x).toReal * f x ∂μ = ∫ (x : α) in s, f x ∂ν

Tactic:
simp [ν, h]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.MeanValue

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
h : (∫ (x : α) in s, 1 ∂μ.withDensity fun x ↦ ρ x) = ∫ (x : α) in s, (ρ x).toReal • 1 ∂μ
⊢ ∫ (x : α) in s, (ρ x).toReal ∂μ = ∫ (x : α) in s, 1 ∂ν

Tactic:
simp [ν, h]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b : E
ω₁ ω₂ : E → E →L[𝕜] F
γ : Path a b
h₁ : CurveIntegrable ω₁ γ
h₂ : CurveIntegrable ω₂ γ
⊢ curveIntegral (ω₁ - ω₂) γ = ∫ᶜ (x : E) in γ, ω₁ x - ∫ᶜ (x : E) in γ, ω₂ x

Tactic:
rw [sub_eq_add_neg, sub_eq_add_neg, curveIntegral_add h₁ h₂.neg, curveIntegral_neg]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C : ℝ
hab : a ≤ b
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hnorm : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ C
s : Set ℝ := toMeasurable volume {x | deriv f x ≠ 0}
hsm : MeasurableSet s
⊢ volume.real (s ∩ Ioo a b) • C = C * volume.real {x | x ∈ Ioo a b ∧ deriv f x ≠ 0}

Tactic:
smul_eq_mul,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C : ℝ
hab : a ≤ b
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hnorm : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ C
s : Set ℝ := toMeasurable volume {x | deriv f x ≠ 0}
hsm : MeasurableSet s
⊢ volume.real (s ∩ Ioo a b) * C = C * volume.real {x | x ∈ Ioo a b ∧ deriv f x ≠ 0}
Module: Mathlib.MeasureTheory.Integral.Regular

Goal state before:
pp: case a
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
k : Set X
hk : IsCompact k
μ : Measure X
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : μ.InnerRegularCompactLTTop
inst✝¹ : LocallyCompactSpace X
inst✝ : RegularSpace X
f : X → ℝ
⊢ Continuous f → HasCompactSupport f → EqOn f 1 k → 0 ≤ f → μ k ≤ ENNReal.ofReal (∫ (x : X), f x ∂μ)

Tactic:
f_cont

Goal state after:
case a
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
k : Set X
hk : IsCompact k
μ : Measure X
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : μ.InnerRegularCompactLTTop
inst✝¹ : LocallyCompactSpace X
inst✝ : RegularSpace X
f : X → ℝ
f_cont : Continuous f
⊢ HasCompactSupport f → EqOn f 1 k → 0 ≤ f → μ k ≤ ENNReal.ofReal (∫ (x : X), f x ∂μ)