mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Group.ModularCharacter

Goal state before:
pp: G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : LocallyCompactSpace G
inst✝² : MeasurableSpace G
inst✝¹ : BorelSpace G
μ : Measure G
inst✝ : μ.IsHaarMeasure
g : G
ν : Measure G := haar
f : G → ℝ
f_cont : Continuous f
f_comp : HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont }
f_nonneg : 0 ≤ { toFun := f, continuous_toFun := f_cont }
f_one : { toFun := f, continuous_toFun := f_cont } 1 ≠ 0
int_f_ne_zero : ∀ (μ₀ : Measure G) [μ₀.IsHaarMeasure], ∫ (x : G), f x ∂μ₀ ≠ 0
t : ∫ (x : G), f (x * g) ∂ν = ∫ (x : G), f (x * g) ∂ν.haarScalarFactor μ • μ
r : ↑(ν.haarScalarFactor μ) / ↑(ν.haarScalarFactor μ) = 1
⊢ (∫ (x : G), f (x * g) ∂μ) / ∫ (x : G), f x ∂μ = (∫ (x : G), f x ∂map (fun x ↦ x * g) μ) / ∫ (x : G), f x ∂μ

Tactic:
rw [integral_map (AEMeasurable.mul_const aemeasurable_id' _)
    (Continuous.aestronglyMeasurable f_cont)]

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

Goal state before:
pp: case refine_1
α : Type u_2
inst✝ : TopologicalSpace α
f₁ f₂ : α →C_c ℝ≥0
h : f₁ ≤ f₂
⊢ HasCompactSupport (f₂.toContinuousMap - f₁.toContinuousMap).toFun

pp: case refine_2
α : Type u_2
inst✝ : TopologicalSpace α
f₁ f₂ : α →C_c ℝ≥0
h : f₁ ≤ f₂
⊢ f₁ + { toContinuousMap := f₂.toContinuousMap - f₁.toContinuousMap, hasCompactSupport' := ?refine_1 } = f₂

Tactic:
· apply (f₁.hasCompactSupport'.union f₂.hasCompactSupport').of_isClosed_subset isClosed_closure
    rw [tsupport, tsupport, ← closure_union]
    apply closure_mono
    intro x hx
    contrapose hx
    simp only [ContinuousMap.toFun_eq_coe, coe_toContinuousMap, Set.mem_union, Function.mem_support,
      ne_eq, not_or, Decidable.not_not, ContinuousMap.coe_sub, Pi.sub_apply] at hx ⊢
    simp [hx.1, hx.2]

Goal state after:
case refine_2
α : Type u_2
inst✝ : TopologicalSpace α
f₁ f₂ : α →C_c ℝ≥0
h : f₁ ≤ f₂
⊢ f₁ + { toContinuousMap := f₂.toContinuousMap - f₁.toContinuousMap, hasCompactSupport' := ⋯ } = f₂
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral

Goal state before:
pp: case bound
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
a : E
r : ℝ
hr : 0 ≤ r
hdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x
hderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r
C : ℝ
hC₀ : C > 0
hC : ∀ᶠ (x : E) in 𝓝 a, ‖fderiv ℝ f x‖ ≤ C * ‖‖x - a‖ ^ r‖
ε : ℝ
hε₀ : ε > 0
hε : ∀ y ∈ ball a ε, DifferentiableAt ℝ f y ∧ ‖fderiv ℝ f y‖ ≤ C * ‖‖y - a‖ ^ r‖
y : E
hy : y ∈ ball a ε
hsub : closedBall a ‖y - a‖ ⊆ ball a ε
z : E
hz : z ∈ closedBall a ‖y - a‖
⊢ C * ‖z - a‖ ^ r ≤ C * ‖y - a‖ ^ r

Tactic:
mem_closedBall_iff_norm.mp hz

Goal state after:
case bound
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
a : E
r : ℝ
hr : 0 ≤ r
hdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x
hderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r
C : ℝ
hC₀ : C > 0
hC : ∀ᶠ (x : E) in 𝓝 a, ‖fderiv ℝ f x‖ ≤ C * ‖‖x - a‖ ^ r‖
ε : ℝ
hε₀ : ε > 0
hε : ∀ y ∈ ball a ε, DifferentiableAt ℝ f y ∧ ‖fderiv ℝ f y‖ ≤ C * ‖‖y - a‖ ^ r‖
y : E
hy : y ∈ ball a ε
hsub : closedBall a ‖y - a‖ ⊆ ball a ε
z : E
hz : z ∈ closedBall a ‖y - a‖
⊢ C * ‖y - a‖ ^ r ≤ C * ‖y - a‖ ^ r
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
f₁ f₂ : X →C_c ℝ≥0
h : f₁ ≤ f₂
g : X →C_c ℝ≥0
hg : f₁ + g = f₂
⊢ Λ f₁ ≤ Λ f₂

Tactic:
rw [← hg]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
f₁ f₂ : X →C_c ℝ≥0
h : f₁ ≤ f₂
g : X →C_c ℝ≥0
hg : f₁ + g = f₂
⊢ Λ f₁ ≤ Λ (f₁ + g)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ≥0
hf : ∀ (x : X), f x ≤ 1
K : Set X
hK : IsCompact K
h : tsupport ⇑f ⊆ K
ε : ℝ≥0
hε : 0 < ε
⊢ Λ f ≤ rieszContentAux Λ { carrier := K, isCompact' := hK } + ε

Tactic:
obtain ⟨g, hg⟩ := exists_lt_rieszContentAux_add_pos Λ ⟨K, hK⟩ hε

Goal state after:
X : Type u_1
inst✝⁴ : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ≥0
hf : ∀ (x : X), f x ≤ 1
K : Set X
hK : IsCompact K
h : tsupport ⇑f ⊆ K
ε : ℝ≥0
hε : 0 < ε
g : X →C_c ℝ≥0
hg :
  (∀ x ∈ { carrier := K, isCompact' := hK }, 1 ≤ g x) ∧ Λ g < rieszContentAux Λ { carrier := K, isCompact' := hK } + ε
⊢ Λ f ≤ rieszContentAux Λ { carrier := K, isCompact' := hK } + ε
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
⊢ (Disjoint on E) m n

Tactic:
apply Disjoint.preimage

Goal state after:
case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := ⋯
y : Fin N → ℝ := ⋯
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := ⋯
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
⊢ Disjoint {a | a ∈ ⇑f ⁻¹' Ioc (y m - ε) (y m) ∧ a ∈ tsupport ⇑f} {a | a ∈ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∧ a ∈ tsupport ⇑f}
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: α : Type u
s : Set (Set α)
hs : #↑s ≤ 𝔠
⊢ #↑{t | GenerateMeasurable s t} ≤ 𝔠

Tactic:
apply (cardinal_generateMeasurable_le s).trans
  rw [← continuum_power_aleph0]
  exact_mod_cast power_le_power_right (max_le hs (nat_lt_continuum 2).le)

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
mE : MeasurableSpace E
μ : Measure E
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
t : E
⊢ charFun μ t = ∫ (x : E), ↑(probChar ⟪x, t⟫) ∂μ

Tactic:
by
  simp [charFun_apply, probChar_apply]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
mE : MeasurableSpace E
μ : Measure E
inst✝ : OpensMeasurableSpace E
L : StrongDual ℝ E
this : ∫ (x : E), cexp (↑(L x) * I) ∂μ = ∫ (x : ℝ), cexp (↑x * I) ∂Measure.map (⇑L) μ
⊢ ∫ (v : E), cexp (↑(L v) * I) ∂μ = charFun (Measure.map (⇑L) μ) 1

Tactic:
rw [this, charFun_apply]

Goal state after:
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
mE : MeasurableSpace E
μ : Measure E
inst✝ : OpensMeasurableSpace E
L : StrongDual ℝ E
this : ∫ (x : E), cexp (↑(L x) * I) ∂μ = ∫ (x : ℝ), cexp (↑x * I) ∂Measure.map (⇑L) μ
⊢ ∫ (x : ℝ), cexp (↑x * I) ∂Measure.map (⇑L) μ = ∫ (x : ℝ), cexp (↑⟪x, 1⟫ * I) ∂Measure.map (⇑L) μ
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
mE : MeasurableSpace E
μ : Measure E
inst✝ : OpensMeasurableSpace E
L : StrongDual ℝ E
u : ℝ
this : ∫ (x : E), cexp (↑((u • L) x) * I) ∂μ = ∫ (x : ℝ), cexp (↑u * ↑x * I) ∂Measure.map (⇑L) μ
⊢ charFun (Measure.map (⇑L) μ) u = ∫ (v : E), cexp (↑((u • L) v) * I) ∂μ

Tactic:
rw [this, charFun_apply]

Goal state after:
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
mE : MeasurableSpace E
μ : Measure E
inst✝ : OpensMeasurableSpace E
L : StrongDual ℝ E
u : ℝ
this : ∫ (x : E), cexp (↑((u • L) x) * I) ∂μ = ∫ (x : ℝ), cexp (↑u * ↑x * I) ∂Measure.map (⇑L) μ
⊢ ∫ (x : ℝ), cexp (↑⟪x, u⟫ * I) ∂Measure.map (⇑L) μ = ∫ (x : ℝ), cexp (↑u * ↑x * I) ∂Measure.map (⇑L) μ
Module: Mathlib.MeasureTheory.Measure.DiracProba

Goal state before:
pp: X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
⊢ Tendsto diracProba L (𝓝 (diracProba x)) ↔ Tendsto id L (𝓝 x)

Tactic:
by
  constructor
  · contrapose
    exact not_tendsto_diracProba_of_not_tendsto L
  · intro h
    have aux := (@continuous_diracProba X _ _ _).continuousAt (x := x)
    simp only [ContinuousAt] at aux
    exact aux.comp h

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
inst✝⁶ : BorelSpace F
inst✝⁵ : SecondCountableTopology F
inst✝⁴ : CompleteSpace E
inst✝³ : CompleteSpace F
ξ : Measure (E × F)
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsFiniteMeasure ξ
h :
  ∀ (L : StrongDual ℝ (E × F)),
    charFunDual ξ L =
      charFunDual μ (L ∘SL ContinuousLinearMap.inl ℝ E F) * charFunDual ν (L ∘SL ContinuousLinearMap.inr ℝ E F)
t : StrongDual ℝ (E × F)
⊢ charFunDual ξ t = charFunDual (μ.prod ν) t

Tactic:
rw [h, charFunDual_prod]

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

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) :=
  {t | ∃ s, t = univ.pi s ∧ (∀ (i : ι), MeasurableSet (s i)) ∧ ∀ (i : ι), (μ i) (frontier (s i)) = 0}
s : (i : ι) → Set (α i)
smeas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (i : ι), (μ i) (frontier (s i)) = 0
s' : (i : ι) → Set (α i)
s'meas : ∀ (i : ι), MeasurableSet (s' i)
hs' : ∀ (i : ι), (μ i) (frontier (s' i)) = 0
⊢ univ.pi s ∩ univ.pi s' ∈ S

Tactic:
refine ⟨fun i ↦ s i ∩ s' i, pi_inter_distrib.symm, fun i ↦ (smeas i).inter (s'meas i),
      fun i ↦ ?_⟩

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) :=
  {t | ∃ s, t = univ.pi s ∧ (∀ (i : ι), MeasurableSet (s i)) ∧ ∀ (i : ι), (μ i) (frontier (s i)) = 0}
s : (i : ι) → Set (α i)
smeas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (i : ι), (μ i) (frontier (s i)) = 0
s' : (i : ι) → Set (α i)
s'meas : ∀ (i : ι), MeasurableSet (s' i)
hs' : ∀ (i : ι), (μ i) (frontier (s' i)) = 0
i : ι
⊢ (μ i) (frontier ((fun i ↦ s i ∩ s' i) i)) = 0
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁷ : TopologicalSpace X
inst✝⁶ : T2Space X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
μ ν : Measure X
inst✝³ : LocallyCompactSpace X
inst✝² : ν.OuterRegular
inst✝¹ : IsFiniteMeasureOnCompacts ν
inst✝ : IsFiniteMeasureOnCompacts μ
hμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂μ ≤ ∫ (x : X), f x ∂ν
K : Set X
hK : IsCompact K
ε : NNReal
hε : 0 < ε
hν : ν K < ∞
hνK : ν K ≠ ∞
hμK : μ K ≠ ∞
V : Set X
pV1 : V ⊇ K
pV2 : IsOpen[inst✝⁷] V
pV3 : ν V ≤ ν K + ↑ε
this : μ.real K ≤ ν.real K + ↑ε
⊢ μ K ≤ ν K + ↑ε

Tactic:
← ENNReal.toReal_le_toReal,

Goal state after:
X : Type u_1
inst✝⁷ : TopologicalSpace X
inst✝⁶ : T2Space X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
μ ν : Measure X
inst✝³ : LocallyCompactSpace X
inst✝² : ν.OuterRegular
inst✝¹ : IsFiniteMeasureOnCompacts ν
inst✝ : IsFiniteMeasureOnCompacts μ
hμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂μ ≤ ∫ (x : X), f x ∂ν
K : Set X
hK : IsCompact K
ε : NNReal
hε : 0 < ε
hν : ν K < ∞
hνK : ν K ≠ ∞
hμK : μ K ≠ ∞
V : Set X
pV1 : V ⊇ K
pV2 : IsOpen[inst✝⁷] V
pV3 : ν V ≤ ν K + ↑ε
this : μ.real K ≤ ν.real K + ↑ε
⊢ (μ K).toReal ≤ (ν K + ↑ε).toReal

case ha
X : Type u_1
inst✝⁷ : TopologicalSpace X
inst✝⁶ : T2Space X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
μ ν : Measure X
inst✝³ : LocallyCompactSpace X
inst✝² : ν.OuterRegular
inst✝¹ : IsFiniteMeasureOnCompacts ν
inst✝ : IsFiniteMeasureOnCompacts μ
hμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂μ ≤ ∫ (x : X), f x ∂ν
K : Set X
hK : IsCompact K
ε : NNReal
hε : 0 < ε
hν : ν K < ∞
hνK : ν K ≠ ∞
hμK : μ K ≠ ∞
V : Set X
pV1 : V ⊇ K
pV2 : IsOpen[inst✝⁷] V
pV3 : ν V ≤ ν K + ↑ε
this : μ.real K ≤ ν.real K + ↑ε
⊢ μ K ≠ ∞

case hb
X : Type u_1
inst✝⁷ : TopologicalSpace X
inst✝⁶ : T2Space X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
μ ν : Measure X
inst✝³ : LocallyCompactSpace X
inst✝² : ν.OuterRegular
inst✝¹ : IsFiniteMeasureOnCompacts ν
inst✝ : IsFiniteMeasureOnCompacts μ
hμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂μ ≤ ∫ (x : X), f x ∂ν
K : Set X
hK : IsCompact K
ε : NNReal
hε : 0 < ε
hν : ν K < ∞
hνK : ν K ≠ ∞
hμK : μ K ≠ ∞
V : Set X
pV1 : V ⊇ K
pV2 : IsOpen[inst✝⁷] V
pV3 : ν V ≤ ν K + ↑ε
this : μ.real K ≤ ν.real K + ↑ε
⊢ ν K + ↑ε ≠ ∞
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureProd

Goal state before:
pp: case refine_2
α : Type u_1
inst✝⁹ : MeasurableSpace α
β : Type u_2
inst✝⁸ : MeasurableSpace β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : SecondCountableTopology β
inst✝³ : PseudoMetrizableSpace α
inst✝² : PseudoMetrizableSpace β
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace β
μ : ProbabilityMeasure α × ProbabilityMeasure β
S : Set (Set (α × β)) :=
  {t | ∃ a b, MeasurableSet a ∧ μ.1 (frontier a) = 0 ∧ MeasurableSet b ∧ μ.2 (frontier b) = 0 ∧ t = a ×ˢ b}
a : Set α
b : Set β
ameas : MeasurableSet a
ha : μ.1 (frontier a) = 0
bmeas : MeasurableSet b
hb : ↑μ.2 (frontier b) = 0
a' : Set α
b' : Set β
a'meas : MeasurableSet a'
ha' : μ.1 (frontier a') = 0
b'meas : MeasurableSet b'
hb' : ↑μ.2 (frontier b') = 0
⊢ ↑μ.2 (frontier (b ∩ b')) = 0

Tactic:
exact null_frontier_inter hb hb'

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

Goal state before:
pp: case h'
α : Type u_1
inst✝⁹ : MeasurableSpace α
β : Type u_2
inst✝⁸ : MeasurableSpace β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : SecondCountableTopology β
inst✝³ : PseudoMetrizableSpace α
inst✝² : PseudoMetrizableSpace β
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace β
μ : ProbabilityMeasure α × ProbabilityMeasure β
S : Set (Set (α × β)) := ⋯
this : IsPiSystem S
⊢ ∀ s ∈ S, Tendsto (fun i ↦ (i.1.prod i.2) s) (𝓝 μ) (𝓝 (((fun μ ↦ μ.1.prod μ.2) μ) s))

Tactic:
rintro s ⟨a, b, ameas, ha, bmeas, hb, rfl⟩

Goal state after:
case h'
α : Type u_1
inst✝⁹ : MeasurableSpace α
β : Type u_2
inst✝⁸ : MeasurableSpace β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : SecondCountableTopology β
inst✝³ : PseudoMetrizableSpace α
inst✝² : PseudoMetrizableSpace β
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace β
μ : ProbabilityMeasure α × ProbabilityMeasure β
S : Set (Set (α × β)) :=
  {t | ∃ a b, MeasurableSet a ∧ μ.1 (frontier a) = 0 ∧ MeasurableSet b ∧ μ.2 (frontier b) = 0 ∧ t = a ×ˢ b}
this : IsPiSystem S
a : Set α
b : Set β
ameas : MeasurableSet a
ha : μ.1 (frontier a) = 0
bmeas : MeasurableSet b
hb : μ.2 (frontier b) = 0
⊢ Tendsto (fun i ↦ (i.1.prod i.2) (a ×ˢ b)) (𝓝 μ) (𝓝 (((fun μ ↦ μ.1.prod μ.2) μ) (a ×ˢ b)))
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹³ : Group A
inst✝¹² : Group B
inst✝¹¹ : Group C
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : TopologicalSpace B
inst✝⁸ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁷ : IsTopologicalGroup A
inst✝⁶ : IsTopologicalGroup B
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace A
inst✝³ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝² : NormedSpace ℝ E
inst✝¹ : IsTopologicalGroup C
inst✝ : LocallyCompactSpace B
f : CompactlySupportedContinuousMap B E
this : UniformSpace B := IsTopologicalGroup.rightUniformSpace B
b : B
ε : ℝ
hε : ε > 0
U₀ : Set B
hU₀ : IsCompact U₀
hb : U₀ ∈ nhds b
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
⊢ ∀ᶠ (x : B) in nhds b, dist (∫ (a : A), (H.pullback f x) a ∂μA) (∫ (a : A), (H.pullback f b) a ∂μA) < ε

Tactic:
let S : Set A := φ ⁻¹' (U₀⁻¹ * K)

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹³ : Group A
inst✝¹² : Group B
inst✝¹¹ : Group C
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : TopologicalSpace B
inst✝⁸ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁷ : IsTopologicalGroup A
inst✝⁶ : IsTopologicalGroup B
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace A
inst✝³ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝² : NormedSpace ℝ E
inst✝¹ : IsTopologicalGroup C
inst✝ : LocallyCompactSpace B
f : CompactlySupportedContinuousMap B E
this : UniformSpace B := IsTopologicalGroup.rightUniformSpace B
b : B
ε : ℝ
hε : ε > 0
U₀ : Set B
hU₀ : IsCompact U₀
hb : U₀ ∈ nhds b
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
⊢ ∀ᶠ (x : B) in nhds b, dist (∫ (a : A), (H.pullback f x) a ∂μA) (∫ (a : A), (H.pullback f b) a ∂μA) < ε
Module: Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar

Goal state before:
pp: G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
φ : G ≃ₜ* G
smul : haar = haar.haarScalarFactor μ • μ
⊢ (haar.haarScalarFactor μ • μ).haarScalarFactor (haar.haarScalarFactor μ • map (⇑φ) μ) =
    μ.haarScalarFactor (map (⇑φ) μ)

Tactic:
exact haarScalarFactor_smul_smul _ _ (haarScalarFactor_pos_of_isHaarMeasure haar μ).ne'

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: x : EuclideanSpace ℝ (Fin 2)
r : ℝ
⊢ volume (closedBall x r) = ENNReal.ofReal r ^ 2 * ENNReal.ofReal π

Tactic:
by
  rw [addHaar_closedBall_eq_addHaar_ball, volume_ball_fin_two x r]

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

Goal state before:
pp: case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
inst✝ : CompleteSpace F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
a : E
ha : a ∈ s
b : E
hb : b ∈ s
⊢ HasFDerivWithinAt (fun x ↦ curveIntegral ω (Path.segment a x)) (ω b) s b

Tactic:
exact hs.hasFDerivWithinAt_curveIntegral_segment_of_hasFDerivWithinAt_symmetric hω hdω ha hb

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

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖⟪y, x⟫_𝕜‖}) atTop (𝓝 0)
z : E
⊢ Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop) atTop (𝓝 0)

Tactic:
refine tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds (h z)
    (fun _ ↦ zero_le) fun r ↦ ?_

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖⟪y, x⟫_𝕜‖}) atTop (𝓝 0)
z : E
r : ℝ
⊢ limsup (fun n ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop ≤ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖⟪z, x⟫_𝕜‖}
Module: Mathlib.MeasureTheory.Measure.MeasuredSets

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s✝ : Set α
hs✝ : MeasurableSet s✝
ε✝ : ℝ≥0∞
hε : 0 < ε✝
s : Set α
hs : MeasurableSet s
h's : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ t ∈ C, μ (t ∆ s) < ε
ε : ℝ≥0∞
εpos : 0 < ε
t : Set α
tC : t ∈ C
ht : μ (t ∆ s) < ε / 2
⊢ ∃ t' ∈ C, μ t'ᶜ < ε / 2

Tactic:
obtain ⟨D, D_count, DC, hD, Dne⟩ :
          ∃ D : Set (Set α), D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0 ∧ D.Nonempty := by
        rcases h'C with ⟨D, D_count, DC, hD⟩
        refine ⟨D ∪ {∅}, D_count.union (by simp), ?_⟩
        simp only [union_subset_iff, DC, singleton_subset_iff, true_and, and_true, hC.empty_mem]
        simp only [union_singleton, sUnion_insert, empty_union, insert_nonempty, and_true, hD]
      obtain ⟨f, hf⟩ : ∃ f : ℕ → Set α, D = Set.range f := Set.Countable.exists_eq_range D_count Dne
      have fC n : Set.accumulate f n ∈ C := hC.accumulate_mem (fun n ↦ DC (by simp [hf])) n
      have : Tendsto (fun n ↦ μ (Set.accumulate f n)ᶜ) atTop (𝓝 0) := by
        have : ⋃₀ D = ⋃ n, Set.accumulate f n := by simp [hf, iUnion_accumulate]
        rw [show (⋃₀ D)ᶜ = ⋂ n, (Set.accumulate f n)ᶜ by simp [this, accumulate]] at hD
        rw [← hD]
        apply tendsto_measure_iInter_atTop (fun i ↦ ?_)
          (fun i j hij ↦ by simpa using monotone_accumulate hij) ⟨0, by simp⟩
        apply MeasurableSet.nullMeasurableSet
        rw [h]
        exact (measurableSet_generateFrom (fC i)).compl
      obtain ⟨n, hn⟩ : ∃ n, μ (accumulate f n)ᶜ < ε / 2 :=
        ((tendsto_order.1 this).2 _ (ENNReal.half_pos εpos.ne')).exists
      exact ⟨accumulate f n, fC n, hn⟩

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : MeasurableSpace X
μ : Measure X
x : X
⊢ x ∉ μ.support ↔ ∃ U ∈ 𝓝 x, μ U = 0

Tactic:
simp [mem_support_iff_forall]

Goal state after: