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

Goal state before:
pp: E : Type u_1
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : μ F * 2 ^ finrank ℝ E < μ s
h_vol : μ F < μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : v ∈ 2⁻¹ • s
w : E
hw : w ∈ 2⁻¹ • s
hvw : (fun x ↦ y +ᵥ x) w = (fun x_1 ↦ x +ᵥ x_1) v
⊢ ↑(x - y) ∈ s

Tactic:
rw [Set.mem_inv_smul_set_iff₀ (two_ne_zero' ℝ)] at hv hw

Goal state after:
E : Type u_1
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : μ F * 2 ^ finrank ℝ E < μ s
h_vol : μ F < μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : 2 • v ∈ s
w : E
hw : 2 • w ∈ s
hvw : (fun x ↦ y +ᵥ x) w = (fun x_1 ↦ x +ᵥ x_1) v
⊢ ↑(x - y) ∈ s
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
this : (↑K ∩ (↑L \ {0})).Nonempty
⊢ ∃ x, x ≠ 0 ∧ ↑x ∈ s

Tactic:
obtain ⟨x, hx⟩ := this
      exact ⟨⟨x, by simp_all⟩, by aesop⟩

Goal state after:
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) = (RingHom.id ℝ) a • (↑(Λ f.nnrealPart) - ↑(Λ (-f).nnrealPart))

pp: case inr
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 : a ≤ 0
⊢ ↑(Λ (a • f).nnrealPart) - ↑(Λ (-(a • f)).nnrealPart) = (RingHom.id ℝ) a • (↑(Λ f.nnrealPart) - ↑(Λ (-f).nnrealPart))

Tactic:
· rw [RingHom.id_apply, smul_eq_mul, ← (smul_neg a f), nnrealPart_smul_pos f ha,
            nnrealPart_smul_pos (-f) ha]
          simp [sup_of_le_left ha, mul_sub]

Goal state after:
case inr
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 : a ≤ 0
⊢ ↑(Λ (a • f).nnrealPart) - ↑(Λ (-(a • f)).nnrealPart) = (RingHom.id ℝ) 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 : ℂ), ‖x‖ ^ q * rexp (-‖x‖ ^ p) = ∫ (x : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (-|x.1| ^ p))

Tactic:
← Complex.integral_comp_polarCoord_symm,

Goal state after:
p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ ∫ (p_1 : ℝ × ℝ) in polarCoord.target,
      p_1.1 • (‖↑Complex.polarCoord.symm p_1‖ ^ q * rexp (-‖↑Complex.polarCoord.symm p_1‖ ^ p)) =
    ∫ (x : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (-|x.1| ^ p))
Module: Mathlib.MeasureTheory.Integral.Gamma

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

Tactic:
← Complex.integral_comp_polarCoord_symm,

Goal state after:
p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ ∫ (p_1 : ℝ × ℝ) in polarCoord.target,
      p_1.1 • (‖↑Complex.polarCoord.symm p_1‖ ^ q * rexp (-b * ‖↑Complex.polarCoord.symm p_1‖ ^ p)) =
    ∫ (x : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (-b * |x.1| ^ p))
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
a b : E
ω : E → E →L[𝕜] F
inst✝ : NormedSpace ℝ E
C : ℝ
h : ∀ z ∈ [a -[ℝ] b], ‖ω z‖ ≤ C
this : NormedSpace ℝ F := NormedSpace.restrictScalars ℝ 𝕜 F
t : ℝ
ht : t ∈ Ι 0 1
⊢ ‖(ω ((lineMap a b) t)) (b - a)‖ ≤ C * ‖b - a‖

Tactic:
rw [segment_eq_image_lineMap] at h

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
a b : E
ω : E → E →L[𝕜] F
inst✝ : NormedSpace ℝ E
C : ℝ
h : ∀ z ∈ ⇑(lineMap a b) '' Icc 0 1, ‖ω z‖ ≤ C
this : NormedSpace ℝ F := NormedSpace.restrictScalars ℝ 𝕜 F
t : ℝ
ht : t ∈ Ι 0 1
⊢ ‖(ω ((lineMap a b) t)) (b - a)‖ ≤ C * ‖b - a‖
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‖
⊢ ‖fderiv ℝ f z‖ ≤ C * ‖y - a‖ ^ r

Tactic:
(hε z <| hsub hz).2,

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 * ‖‖z - a‖ ^ r‖ ≤ C * ‖y - a‖ ^ r
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.NNReal

Goal state before:
pp: X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : T2Space X
inst✝⁴ : LocallyCompactSpace X
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
μ ν : Measure X
inst✝¹ : μ.Regular
inst✝ : ν.Regular
hμν : ∀ (f : X →C_c ℝ≥0), ∫ (x : X), ↑(f x) ∂μ = ∫ (x : X), ↑(f x) ∂ν
⊢ μ = ν

Tactic:
by
  apply Measure.ext_of_integral_eq_on_compactlySupported
  intro f
  repeat rw [integral_eq_integral_pos_part_sub_integral_neg_part f.integrable]
  erw [hμν f.nnrealPart, hμν (-f).nnrealPart]
  rfl

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case hg
a' b' ε : ℝ
hε : 0 < ε
⊢ Tendsto (fun x ↦ b' + a' / x) atTop (𝓝 (b' + 0))

Tactic:
· exact Tendsto.add tendsto_const_nhds (Tendsto.div_atTop tendsto_const_nhds tendsto_id)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
⊢ range ⇑f ⊆ Ioo a (a + ↑N * ε')

Tactic:
by
    dsimp [ε']
    field_simp
    simp [hab.2]

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

Goal state before:
pp: ι : Type u_4
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
E : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
mE : (i : ι) → MeasurableSpace (E i)
μ : (i : ι) → Measure (E i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
L : StrongDual ℝ ((i : ι) → E i)
⊢ ∫ (v : (i : ι) → E i), cexp (↑(L v) * I) ∂Measure.pi μ =
    ∏ i, ∫ (v : E i), cexp (↑((L ∘SL ContinuousLinearMap.single ℝ E i) v) * I) ∂μ i

Tactic:
← L.sum_comp_single,

Goal state after:
ι : Type u_4
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
E : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
mE : (i : ι) → MeasurableSpace (E i)
μ : (i : ι) → Measure (E i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
L : StrongDual ℝ ((i : ι) → E i)
⊢ ∫ (v : (i : ι) → E i), cexp (↑(∑ i, (L ∘SL ContinuousLinearMap.single ℝ E i) (v i)) * I) ∂Measure.pi μ =
    ∏ i, ∫ (v : E i), cexp (↑((L ∘SL ContinuousLinearMap.single ℝ E i) v) * I) ∂μ i
Module: Mathlib.MeasureTheory.Measure.DiracProba

Goal state before:
pp: case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhds : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhds : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ∀ x ∉ U, ↑(f x) ∉ Ioi 0

Tactic:
intro x x_notin_U

Goal state after:
case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x✝ : X
L : Filter X
h : ¬Tendsto id L (𝓝 x✝)
U : Set X
U_nhds : U ∈ 𝓝 x✝
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhds : interior U ∈ 𝓝 x✝
f : X →ᵇ ℝ≥0
fx_eq_one : f x✝ = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
x : X
x_notin_U : x ∉ U
⊢ ↑(f x) ∉ Ioi 0
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
h : ∀ (n : Fin N), (y n - ε') * μ.real (E n) ≤ ∫ (x : X) in E n, f x ∂μ
⊢ ∫ (x : X) in ⋃ n, E n, f x ∂μ = ∫ (x : X) in tsupport ⇑f, f x ∂μ

Tactic:
simp_rw [hE.1]

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

Goal state before:
pp: Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
key : (fun t ↦ μ.real {a | t ≤ f a}) ≤ fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε
intble₁ : IntegrableOn (fun t ↦ μ.real {a | t ≤ f a}) (Ioc 0 ‖f‖) volume
intble₂ : IntegrableOn (fun t ↦ ν.real (thickening ε {a | t ≤ f a})) (Ioc 0 ‖f‖) volume
⊢ IntegrableOn (fun t ↦ μ.real {a | t ≤ f a}) (Ioc 0 ‖f‖) volume

pp: Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
key : (fun t ↦ μ.real {a | t ≤ f a}) ≤ fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε
intble₁ : IntegrableOn (fun t ↦ μ.real {a | t ≤ f a}) (Ioc 0 ‖f‖) volume
intble₂ : IntegrableOn (fun t ↦ ν.real (thickening ε {a | t ≤ f a})) (Ioc 0 ‖f‖) volume
⊢ IntegrableOn (fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε) (Ioc 0 ‖f‖) volume

Tactic:
· exact intble₁

Goal state after:
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
key : (fun t ↦ μ.real {a | t ≤ f a}) ≤ fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε
intble₁ : IntegrableOn (fun t ↦ μ.real {a | t ≤ f a}) (Ioc 0 ‖f‖) volume
intble₂ : IntegrableOn (fun t ↦ ν.real (thickening ε {a | t ≤ f a})) (Ioc 0 ‖f‖) volume
⊢ IntegrableOn (fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε) (Ioc 0 ‖f‖) volume
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)) :=
  {t | ∃ a b, MeasurableSet a ∧ μ.1 (frontier a) = 0 ∧ MeasurableSet b ∧ μ.2 (frontier b) = 0 ∧ t = a ×ˢ b}
this✝¹ : IsPiSystem S
this✝ : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
u : Set (α × β)
u_open : IsOpen[instTopologicalSpaceProd] u
x : α × β
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ℝ
hr : r ∈ Ioo 0 ε
r' : ℝ
hr' : r' ∈ Ioo 0 ε
μr : ↑μ.1 (frontier (ball x.1 r)) = 0
μr' : ↑μ.2 (frontier (ball x.2 r')) = 0
⊢ ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u

Tactic:
refine ⟨ball x.1 r ×ˢ ball x.2 r', ⟨ball x.1 r, ball x.2 r', measurableSet_ball,
      by simp [coeFn_def, μr], measurableSet_ball, by simp [coeFn_def, μr'], rfl⟩, ?_, ?_⟩

Goal state after:
case h.refine_1
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✝¹ : IsPiSystem S
this✝ : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
u : Set (α × β)
u_open : IsOpen[instTopologicalSpaceProd] u
x : α × β
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ℝ
hr : r ∈ Ioo 0 ε
r' : ℝ
hr' : r' ∈ Ioo 0 ε
μr : ↑μ.1 (frontier (ball x.1 r)) = 0
μr' : ↑μ.2 (frontier (ball x.2 r')) = 0
⊢ ball x.1 r ×ˢ ball x.2 r' ∈ 𝓝 x

case h.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}
this✝¹ : IsPiSystem S
this✝ : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
u : Set (α × β)
u_open : IsOpen[instTopologicalSpaceProd] u
x : α × β
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ℝ
hr : r ∈ Ioo 0 ε
r' : ℝ
hr' : r' ∈ Ioo 0 ε
μr : ↑μ.1 (frontier (ball x.1 r)) = 0
μr' : ↑μ.2 (frontier (ball x.2 r')) = 0
⊢ ball x.1 r ×ˢ ball x.2 r' ⊆ u
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
⊢ mulEquivHaarChar φ • μ = comap (⇑φ) μ

Tactic:
by
  let e := φ.toHomeomorph.toMeasurableEquiv
  rw [show ⇑φ = ⇑e from rfl, ← e.map_symm, show ⇑e.symm = ⇑φ.symm from rfl]
  have : (map (φ.symm) μ).Regular := Regular.map φ.symm.toHomeomorph
  rw [← mulEquivHaarChar_smul_map (map φ.symm μ) φ, map_map]
  · simp
  · fun_prop
  · fun_prop

Goal state after:
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
f : G → ℝ
φ : G ≃ₜ* G
⊢ mulEquivHaarChar φ • ∫ (a : G), f a ∂map (⇑φ) μ = ∫ (a : G), f a ∂μ

Tactic:
nth_rw 2 [← mulEquivHaarChar_smul_map μ φ]

Goal state after:
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
f : G → ℝ
φ : G ≃ₜ* G
⊢ mulEquivHaarChar φ • ∫ (a : G), f a ∂map (⇑φ) μ = ∫ (a : G), f a ∂mulEquivHaarChar φ • map (⇑φ) μ
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
X : Set G
φ : G ≃ₜ* G
⊢ mulEquivHaarChar φ • μ (⇑φ ⁻¹' X) = μ X

Tactic:
nth_rw 2 [← mulEquivHaarChar_smul_map μ φ]

Goal state after:
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
X : Set G
φ : G ≃ₜ* G
⊢ mulEquivHaarChar φ • μ (⇑φ ⁻¹' X) = (mulEquivHaarChar φ • map (⇑φ) μ) X
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case inr
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Nontrivial E
this : 0 < ↑(finrank ℝ E)
⊢ μ (Metric.ball 0 1) = ENNReal.ofReal ((∫ (x : E), Real.exp (-‖x‖ ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))

Tactic:
have : ((∫ y in Set.Ioi (0 : ℝ), y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) /
        Real.Gamma ((finrank ℝ E) / p + 1)) * (finrank ℝ E) = 1 := by
      simp_rw [← Real.rpow_natCast _ (finrank ℝ E - 1), smul_eq_mul, Nat.cast_sub finrank_pos,
        Nat.cast_one]
      rw [integral_rpow_mul_exp_neg_rpow hp (by linarith), sub_add_cancel,
        Real.Gamma_add_one (ne_of_gt (by positivity))]
      field

Goal state after:
case inr
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Nontrivial E
this✝ : 0 < ↑(finrank ℝ E)
this :
  (∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) / Real.Gamma (↑(finrank ℝ E) / p + 1) *
      ↑(finrank ℝ E) =
    1
⊢ μ (Metric.ball 0 1) = ENNReal.ofReal ((∫ (x : E), Real.exp (-‖x‖ ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Nontrivial E
this✝ : 0 < ↑(finrank ℝ E)
this :
  (∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) / Real.Gamma (↑(finrank ℝ E) / p + 1) *
      ↑(finrank ℝ E) =
    1
⊢ μ (Metric.ball 0 1) ≠ ∞

Tactic:
exact ne_of_lt measure_ball_lt_top

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

Goal state before:
pp: case e'_3
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
L : StrongDual ℝ E
r : ℝ
hr : 0 < r
this : IsProbabilityMeasure (Measure.map (⇑L) μ)
⊢ μ.real {x | r < |L x|} = (Measure.map (⇑L) μ).real {x | r < |x|}

Tactic:
rw [map_measureReal_apply (by fun_prop)]

Goal state after:
case e'_3
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
L : StrongDual ℝ E
r : ℝ
hr : 0 < r
this : IsProbabilityMeasure (Measure.map (⇑L) μ)
⊢ μ.real {x | r < |L x|} = μ.real (⇑L ⁻¹' {x | r < |x|})

case e'_3
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
L : StrongDual ℝ E
r : ℝ
hr : 0 < r
this : IsProbabilityMeasure (Measure.map (⇑L) μ)
⊢ MeasurableSet {x | r < |x|}
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case e'_3
E : Type u_1
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
mE : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
a : E
r : ℝ
hr : 0 < r
this : IsProbabilityMeasure (Measure.map (fun x ↦ ⟪a, x⟫) μ)
⊢ μ.real {x | r < |⟪a, x⟫|} = (Measure.map (fun x ↦ ⟪a, x⟫) μ).real {x | r < |x|}

Tactic:
rw [map_measureReal_apply (by fun_prop)]

Goal state after:
case e'_3
E : Type u_1
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
mE : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
a : E
r : ℝ
hr : 0 < r
this : IsProbabilityMeasure (Measure.map (fun x ↦ ⟪a, x⟫) μ)
⊢ μ.real {x | r < |⟪a, x⟫|} = μ.real ((fun x ↦ ⟪a, x⟫) ⁻¹' {x | r < |x|})

case e'_3
E : Type u_1
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
mE : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
a : E
r : ℝ
hr : 0 < r
this : IsProbabilityMeasure (Measure.map (fun x ↦ ⟪a, x⟫) μ)
⊢ MeasurableSet {x | r < |x|}