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

Goal state before:
pp: case mp
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsZeroOrMarkovKernel κ
hf : Measurable f
hg : Measurable g
h :
  ∀ (s : Set β) (t : Set β'),
    MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
s : Set β
t : Set β'
hs : MeasurableSet s
⊢ MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ

Tactic:
ht

Goal state after:
case mp
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsZeroOrMarkovKernel κ
hf : Measurable f
hg : Measurable g
h :
  ∀ (s : Set β) (t : Set β'),
    MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
s : Set β
t : Set β'
hs : MeasurableSet s
ht : MeasurableSet t
⊢ IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case mpr
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsZeroOrMarkovKernel κ
hf : Measurable f
hg : Measurable g
h : ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ
s : Set β
t : Set β'
hs : MeasurableSet s
⊢ MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)

Tactic:
ht

Goal state after:
case mpr
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsZeroOrMarkovKernel κ
hf : Measurable f
hg : Measurable g
h : ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ
s : Set β
t : Set β'
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case refine_1
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
h :
  ∀ (s : Set β) (t : Set γ),
    MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
u : Set α
s : Set β
t : Set γ
hu : MeasurableSet u
hs : MeasurableSet s
⊢ MeasurableSet t → (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)

Tactic:
ht

Goal state after:
case refine_1
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
h :
  ∀ (s : Set β) (t : Set γ),
    MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
u : Set α
s : Set β
t : Set γ
hu : MeasurableSet u
hs : MeasurableSet s
ht : MeasurableSet t
⊢ (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)
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case refine_2
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
h : (μ ⊗ₘ κ.map fun ω ↦ (f ω, g ω)) = μ ⊗ₘ (κ.map f ×ₖ κ.map g)
s : Set β
t : Set γ
hs : MeasurableSet s
⊢ MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)

Tactic:
ht

Goal state after:
case refine_2
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
h : (μ ⊗ₘ κ.map fun ω ↦ (f ω, g ω)) = μ ⊗ₘ (κ.map f ×ₖ κ.map g)
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: case pos
ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
g : ι → Set Ω := fun i' ↦ if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
i : ι
x✝ : i ∈ Finset.univ
hi : i ∈ s
⊢ (μ (Y i ⁻¹' t i))⁻¹ * μ (g i) = if i ∈ s then μ[f i | Y i ⁻¹' t i] else 1

Tactic:
simp only [hi, ↓reduceIte, g, cond_apply (hY i (ht i))]

Goal state after:
Module: Mathlib.Probability.Density

Goal state before:
pp: case refine_1
Ω : Type u_1
E : Type u_2
inst✝ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
X : Ω → E
hac : map X ℙ ≪ μ
hpdf : ¬pdf X ℙ μ =ᵐ[μ] 0
⊢ AEMeasurable X ℙ

Tactic:
exact aemeasurable_of_pdf_ne_zero X hpdf

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : SecondCountableTopology Ω
S : Set (Set Ω)
ν : ProbabilityMeasure Ω
h : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u
G : Set Ω
hG : IsOpen[inst✝¹] G
r : ℝ≥0
hr : r < ν G
⊢ ∃ T ⊆ S, T.Countable ∧ ⋃ t ∈ T, t = G

Tactic:
have : ∀ (x : G), ∃ s ∈ S, s ∈ 𝓝 (x : Ω) ∧ s ⊆ G := fun x ↦ h G hG x x.2
    choose! s hsS hs_nhds hsG using this
    rcases TopologicalSpace.isOpen_iUnion_countable (fun i ↦ interior (s i))
      (fun i ↦ isOpen_interior) with ⟨T₀, T₀_count, hT₀⟩
    refine ⟨s '' T₀, by grind, T₀_count.image s, ?_⟩
    refine Subset.antisymm (by simp; grind) ?_
    have : G ⊆ ⋃ i, interior (s i) := by
      intro y hy
      simpa using ⟨y, hy, mem_interior_iff_mem_nhds.2 (hs_nhds ⟨y, hy⟩)⟩
    apply this.trans
    rw [← hT₀, biUnion_image]
    exact iUnion₂_mono fun i j ↦ interior_subset

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
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ ∫ (ω : Ω),
      X ω * Y ω - X ω * ∫ (x : Ω), Y x ∂μ - ((∫ (x : Ω), X x ∂μ) * Y ω - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ) ∂μ =
    ∫ (x : Ω), (X * Y) x ∂μ - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ

Tactic:
integral_sub

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ ∫ (a : Ω), X a * Y a - X a * ∫ (x : Ω), Y x ∂μ ∂μ -
      ∫ (a : Ω), (∫ (x : Ω), X x ∂μ) * Y a - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ ∂μ =
    ∫ (x : Ω), (X * Y) x ∂μ - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ

case hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun ω ↦ X ω * Y ω - X ω * ∫ (x : Ω), Y x ∂μ) μ

case hg
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun ω ↦ (∫ (x : Ω), X x ∂μ) * Y ω - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ) μ
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ ∫ (a : Ω), X a * Y a - X a * ∫ (x : Ω), Y x ∂μ ∂μ -
      ∫ (a : Ω), (∫ (x : Ω), X x ∂μ) * Y a - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ ∂μ =
    ∫ (x : Ω), (X * Y) x ∂μ - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ

pp: case hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun ω ↦ X ω * Y ω - X ω * ∫ (x : Ω), Y x ∂μ) μ

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

Tactic:
integral_sub

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ ∫ (a : Ω), X a * Y a ∂μ - ∫ (a : Ω), X a * ∫ (x : Ω), Y x ∂μ ∂μ -
      ∫ (a : Ω), (∫ (x : Ω), X x ∂μ) * Y a - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ ∂μ =
    ∫ (x : Ω), (X * Y) x ∂μ - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ

case hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun a ↦ X a * Y a) μ

case hg
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun a ↦ X a * ∫ (x : Ω), Y x ∂μ) μ

case hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun ω ↦ X ω * Y ω - X ω * ∫ (x : Ω), Y x ∂μ) μ

case hg
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun ω ↦ (∫ (x : Ω), X x ∂μ) * Y ω - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ) μ
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ ∫ (a : Ω), X a * Y a ∂μ - ∫ (a : Ω), X a * ∫ (x : Ω), Y x ∂μ ∂μ -
      ∫ (a : Ω), (∫ (x : Ω), X x ∂μ) * Y a - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ ∂μ =
    ∫ (x : Ω), (X * Y) x ∂μ - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ

pp: case hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun a ↦ X a * Y a) μ

pp: case hg
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun a ↦ X a * ∫ (x : Ω), Y x ∂μ) μ

pp: case hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun ω ↦ X ω * Y ω - X ω * ∫ (x : Ω), Y x ∂μ) μ

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

Tactic:
integral_sub

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ ∫ (a : Ω), X a * Y a ∂μ - ∫ (a : Ω), X a * ∫ (x : Ω), Y x ∂μ ∂μ -
      (∫ (a : Ω), (∫ (x : Ω), X x ∂μ) * Y a ∂μ - ∫ (a : Ω), (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ ∂μ) =
    ∫ (x : Ω), (X * Y) x ∂μ - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ

case hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun a ↦ (∫ (x : Ω), X x ∂μ) * Y a) μ

case hg
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun a ↦ (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ) μ

case hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun a ↦ X a * Y a) μ

case hg
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun a ↦ X a * ∫ (x : Ω), Y x ∂μ) μ

case hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun ω ↦ X ω * Y ω - X ω * ∫ (x : Ω), Y x ∂μ) μ

case hg
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Integrable (fun ω ↦ (∫ (x : Ω), X x ∂μ) * Y ω - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ) μ
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case pos
α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) κ μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy✝ : ∀ᵐ (a : α) ∂μ, ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
val✝ : Fintype ι
g : ι → Set Ω := fun i' ↦ if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
hYt✝ : ∀ (i : ι), MeasurableSet (Y i ⁻¹' t i)
hg✝ : ∀ (i : ι), MeasurableSet (g i)
a : α
hy : ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
a✝ : IsProbabilityMeasure (κ a)
hYt : (κ a) (⋂ i, Y i ⁻¹' t i) = ∏ i, (κ a) (Y i ⁻¹' t i)
hg : (κ a) (⋂ i, g i) = ∏ i, (κ a) (g i)
i : ι
x✝ : i ∈ Finset.univ
hi : i ∈ s
⊢ ((κ a) (Y i ⁻¹' t i))⁻¹ * (κ a) (g i) = if i ∈ s then (κ a)[f i | Y i ⁻¹' t i] else 1

Tactic:
simp only [hi, ↓reduceIte, g, cond_apply (hY i (ht i))]

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

Goal state before:
pp: case neg.refine_1
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : ι → α → β
p : ℝ≥0∞
hf : UnifTight f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
hε_top : ¬ε = ∞
s✝ : Set α
hμs : μ s✝ ≠ ∞
hfs : ∀ (i : ι), eLpNorm (s✝ᶜ.indicator (f i)) p μ ≤ ↑ε.toNNReal
s t : Set α
hst : s ⊆ t
⊢ (∀ (i : ι), eLpNorm (t.indicator (f i)) p μ ≤ ε) → ∀ (i : ι), eLpNorm (s.indicator (f i)) p μ ≤ ε

Tactic:
ht

Goal state after:
case neg.refine_1
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : ι → α → β
p : ℝ≥0∞
hf : UnifTight f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
hε_top : ¬ε = ∞
s✝ : Set α
hμs : μ s✝ ≠ ∞
hfs : ∀ (i : ι), eLpNorm (s✝ᶜ.indicator (f i)) p μ ≤ ↑ε.toNNReal
s t : Set α
hst : s ⊆ t
ht : ∀ (i : ι), eLpNorm (t.indicator (f i)) p μ ≤ ε
⊢ ∀ (i : ι), eLpNorm (s.indicator (f i)) p μ ≤ ε
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : ι → α → β
p : ℝ≥0∞
hf : UnifTight f p μ
⊢ UnifTight (-f) p μ

Tactic:
UnifTight,

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : ι → α → β
p : ℝ≥0∞
hf : UnifTight f p μ
⊢ ∀ ⦃ε : ℝ≥0⦄, 0 < ε → ∃ s, μ s ≠ ∞ ∧ ∀ (i : ι), eLpNorm (sᶜ.indicator ((-f) i)) p μ ≤ ↑ε
Module: Mathlib.MeasureTheory.Group.IntegralConvolution

Goal state before:
pp: M : Type u_1
F : Type u_2
inst✝⁵ : Monoid M
mM : MeasurableSpace M
inst✝⁴ : MeasurableMul₂ M
inst✝³ : NormedAddCommGroup F
μ ν : Measure M
f : M → F
inst✝² : NormedSpace ℝ F
inst✝¹ : SFinite μ
inst✝ : SFinite ν
hf : Integrable f (μ ∗ₘ ν)
⊢ ∫ (x : M), f x ∂μ ∗ₘ ν = ∫ (x : M), ∫ (y : M), f (x * y) ∂ν ∂μ

Tactic:
unfold Measure.mconv

Goal state after:
M : Type u_1
F : Type u_2
inst✝⁵ : Monoid M
mM : MeasurableSpace M
inst✝⁴ : MeasurableMul₂ M
inst✝³ : NormedAddCommGroup F
μ ν : Measure M
f : M → F
inst✝² : NormedSpace ℝ F
inst✝¹ : SFinite μ
inst✝ : SFinite ν
hf : Integrable f (μ ∗ₘ ν)
⊢ ∫ (x : M), f x ∂Measure.map (fun x ↦ x.1 * x.2) (μ.prod ν) = ∫ (x : M), ∫ (y : M), f (x * y) ∂ν ∂μ
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
x : α
R : Type u_5
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring β
inst✝³ : IsTopologicalSemiring β
inst✝² : Module R β
inst✝¹ : ContinuousConstSMul R β
inst✝ : IsScalarTower R β β
r : R
f g : α →C_c β
⊢ (r • f) • g = r • f • g

Tactic:
ext
    simp only [smul_eq_mul, coe_mul, coe_smul, Pi.mul_apply, Pi.smul_apply]
    rw [← smul_eq_mul, ← smul_eq_mul, smul_assoc]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : AddMonoid β
inst✝¹ : StarAddMonoid β
inst✝ : ContinuousStar β
f : α →C_c β
⊢ HasCompactSupport fun x ↦ star (f x)

Tactic:
rw [HasCompactSupport, tsupport]

Goal state after:
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : AddMonoid β
inst✝¹ : StarAddMonoid β
inst✝ : ContinuousStar β
f : α →C_c β
⊢ IsCompact (closure (Function.support fun x ↦ star (f x)))
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
f : α →C_c ℝ≥0
⊢ (toRealPositiveLinear Λ) f.toReal = ↑(Λ f)

Tactic:
simp [toRealPositiveLinear_apply]

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✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace F
ω : E →L[𝕜] F
a b : E
this : NormedSpace ℝ F := NormedSpace.restrictScalars ℝ 𝕜 F
⊢ ∫ᶜ (x : E) in Path.segment a b, ω = ω (b - a)

Tactic:
simp [curveIntegral_segment]

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
hg1 : g =ᵐ[μ.restrict s] 0
hν0 : ∫ (x : α) in s, 1 ∂ν = 0
c : α
hc : c ∈ s
⊢ ∫ (x : α) in s, f x ∂ν = 0

Tactic:
have heq_zero : ∫ x in s, f x * g x ∂μ = 0 := by
          have heq_ae : (fun x ↦ f x * g x) =ᵐ[μ.restrict s] 0 := by
            apply hg1.mono
            intro x hx
            simp [hx]
          simpa using! integral_congr_ae heq_ae

Goal state after:
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
hg1 : g =ᵐ[μ.restrict s] 0
hν0 : ∫ (x : α) in s, 1 ∂ν = 0
c : α
hc : c ∈ s
heq_zero : ∫ (x : α) in s, f x * g x ∂μ = 0
⊢ ∫ (x : α) in s, f x ∂ν = 0
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
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ∫ (x : α) in s, 1 ∂ν = 0
c : α
hc : c ∈ s
⊢ 0 = f c * ∫ (x : α) in s, g x ∂μ

Tactic:
by simp [hν0, hg1]

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

Goal state before:
pp: case hderiv
E✝ : Type u_1
inst✝³ : NormedAddCommGroup E✝
inst✝² : NormedSpace ℝ E✝
f✝ : ℝ → E✝
a b : ℝ
B : ℝ → ℝ
hab : a ≤ b
hBi : IntervalIntegrable B volume a b
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t
hE : CompleteSpace E
hfB' : (fun x ↦ ‖deriv f x‖) ≤ᵐ[volume.restrict (uIoc a b)] B
t : ℝ
⊢ t ∈ Ioo a b → HasDerivWithinAt f (deriv f t) (Ioi t) t

Tactic:
ht

Goal state after:
case hderiv
E✝ : Type u_1
inst✝³ : NormedAddCommGroup E✝
inst✝² : NormedSpace ℝ E✝
f✝ : ℝ → E✝
a b : ℝ
B : ℝ → ℝ
hab : a ≤ b
hBi : IntervalIntegrable B volume a b
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t
hE : CompleteSpace E
hfB' : (fun x ↦ ‖deriv f x‖) ≤ᵐ[volume.restrict (uIoc a b)] B
t : ℝ
ht : t ∈ Ioo a b
⊢ HasDerivWithinAt f (deriv f t) (Ioi t) t
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
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
hνfin : ν s ≠ ⊤
hν0'' : ν s ≠ 0
hint : IntegrableOn f s ν
c : α
hc : c ∈ s
h_ave : f c = ⨍ (x : α) in s, f x ∂ν
⊢ f c * ∫ (x : α) in s, 1 ∂ν = f c * ∫ (x : α) in s, g x ∂μ

Tactic:
simp [hg1]

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

Goal state before:
pp: a b : ℝ
f g : ℝ → ℝ
μ : Measure ℝ
hf : ContinuousOn f [[a, b]]
hg : IntervalIntegrable g μ a b
hg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x
h : ¬a = b
this :
  ∀ {a b : ℝ},
    ContinuousOn f [[a, b]] →
      IntervalIntegrable g μ a b →
        (∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x) →
          ¬a = b → a < b → ∃ c ∈ [[a, b]], ∫ (x : ℝ) in a..b, f x * g x ∂μ = f c * ∫ (x : ℝ) in a..b, g x ∂μ
hab : b ≤ a
c : ℝ
c_in_uIcc : c ∈ [[b, a]]
that : ∫ (x : ℝ) in b..a, f x * g x ∂μ = f c * ∫ (x : ℝ) in b..a, g x ∂μ
⊢ ∫ (x : ℝ) in a..b, f x * g x ∂μ = f c * ∫ (x : ℝ) in a..b, g x ∂μ

Tactic:
simpa [integral_symm b a]

Goal state after: