mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ ∫ (x : ℝ) in Ioc 0 ‖f‖, ν.real (thickening ε {a | x ≤ f a}) + ε ≤
    (∫ (t : ℝ) in Ioc 0 ‖f‖, ν.real (thickening ε {a | t ≤ f a})) + ε * ‖f‖

Tactic:
rw [integral_add]
    · simp [(mul_comm _ ε).le]
    · exact intble₂
    · exact integrable_const ε

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)) :=
  {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 ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u

Tactic:
x

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✝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
⊢ x ∈ u → ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case pos
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
VltTop : ν V < ∞
f : X →C_c ℝ
pf1 : EqOn (⇑f) 1 K
pf2 : tsupport ⇑f ⊆ V
pf3 : ∀ (x : X), f x ∈ Icc 0 1
x : X
hx : x ∈ tsupport ⇑f
⊢ f x ≤ V.indicator 1 x

pp: case neg
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
VltTop : ν V < ∞
f : X →C_c ℝ
pf1 : EqOn (⇑f) 1 K
pf2 : tsupport ⇑f ⊆ V
pf3 : ∀ (x : X), f x ∈ Icc 0 1
x : X
hx : x ∉ tsupport ⇑f
⊢ f x ≤ V.indicator 1 x

Tactic:
· simp [(pf2 hx), (pf3 x).2]

Goal state after:
case neg
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
VltTop : ν V < ∞
f : X →C_c ℝ
pf1 : EqOn (⇑f) 1 K
pf2 : tsupport ⇑f ⊆ V
pf3 : ∀ (x : X), f x ∈ Icc 0 1
x : X
hx : x ∉ tsupport ⇑f
⊢ f x ≤ V.indicator 1 x
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toMeasure
Ps : ℕ → ProbabilityMeasure Ω := toMeasure ∘ μs
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
εs : ℕ → ℝ
left✝ : StrictAnti εs
εs_lim : Tendsto εs atTop (𝓝 0)
ε_of_room : Tendsto (fun x ↦ dist (μs x) ν + εs x) atTop (𝓝 0)
ε_of_room' : Tendsto (fun n ↦ dist (μs n) ν + εs n) atTop (𝓝[>] 0)
key :
  Tendsto (fun x ↦ ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real (thickening (dist (μs x) ν + εs x) {a | t ≤ f a})) atTop
    (𝓝 (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real {a | t ≤ f a}))
aux : ∀ (z : ℝ), Iio (z + δ / 2) ∈ 𝓝 z
n : ℕ
εs_pos : 0 < εs n
hn :
  ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real (thickening (dist (μs n) ν + εs n) {a | t ≤ f a}) <
    (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real {a | t ≤ f a}) + δ / 2
hn' : dist (μs n) ν + εs n < ‖f‖⁻¹ * δ / 2
bound :
  ∫ (ω : Ω), f ω ∂↑(Ps n) ≤
    (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real (thickening (dist (μs n) ν + εs n) {a | t ≤ f a})) + (dist (μs n) ν + εs n) * ‖f‖
⊢ δ / 2 + ‖f‖ * (dist (μs n) ν + εs n) ≤ δ / 2 + ‖f‖ * (‖f‖⁻¹ * δ / 2)

Tactic:
by gcongr

Goal state after:
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)
⊢ (∫ (y : ℝ) in Set.Ioi 0, y ^ (↑(finrank ℝ E) - ↑(Nat.succ 0)) * Real.exp (-y ^ p)) /
        Real.Gamma (↑(finrank ℝ E) / p + 1) *
      ↑(finrank ℝ E) =
    1

Tactic:
Nat.cast_one

Goal state after:
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)
⊢ (∫ (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
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
n : ℕ
⊢ ∃ ν ∈ {ρ | ρ.mass ≤ C ∧ ρ ((partialSups K) n)ᶜ = 0}, Filter.map (fun ρ ↦ ρ.restrict (disjointed K n)) ↑f ≤ 𝓝 ν

Tactic:
← Ultrafilter.coe_map

Goal state after:
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
n : ℕ
⊢ ∃ ν ∈ {ρ | ρ.mass ≤ C ∧ ρ ((partialSups K) n)ᶜ = 0}, ↑(Ultrafilter.map (fun ρ ↦ ρ.restrict (disjointed K n)) f) ≤ 𝓝 ν
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
⊢ ∃ x ∈ {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n}, ↑f ≤ 𝓝 x

Tactic:
let μ : FiniteMeasure E := ⟨Measure.sum (fun n ↦ (ν n : Measure E)), ⟨B.trans_lt (by simp)⟩⟩

Goal state after:
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
⊢ ∃ x ∈ {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n}, ↑f ≤ 𝓝 x
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case inr
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
h₂ : ∀ (x : ι → ℂ), 0 ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p)
hr : 0 < r
⊢ volume {x | (∑ i, ‖x i‖ ^ p) ^ (1 / p) < r} =
    ENNReal.ofReal r ^ (2 * card ι) *
      ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Tactic:
← Complex.volume_sum_rpow_lt_one _ hp,

Goal state after:
case inr
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
h₂ : ∀ (x : ι → ℂ), 0 ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p)
hr : 0 < r
⊢ volume {x | (∑ i, ‖x i‖ ^ p) ^ (1 / p) < r} = ENNReal.ofReal r ^ (2 * card ι) * volume {x | ∑ i, ‖x i‖ ^ p < 1}
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nontrivial E
x : E
r : ℝ
this✝ : Nonempty (Fin (finrank ℝ E))
this :
  volume (ball ((stdOrthonormalBasis ℝ E).repr x) r) =
    ENNReal.ofReal r ^ finrank ℝ E * ENNReal.ofReal (√π ^ finrank ℝ E / Gamma (↑(finrank ℝ E) / 2 + 1))
⊢ volume (⇑(stdOrthonormalBasis ℝ E).repr.symm ⁻¹' ball x r) =
    ENNReal.ofReal r ^ finrank ℝ E * ENNReal.ofReal (√π ^ finrank ℝ E / Gamma (↑(finrank ℝ E) / 2 + 1))

Tactic:
convert! this

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nontrivial E
x : E
r : ℝ
this✝ : Nonempty (Fin (finrank ℝ E))
this :
  volume (ball ((stdOrthonormalBasis ℝ E).repr x) r) =
    ENNReal.ofReal r ^ finrank ℝ E * ENNReal.ofReal (√π ^ finrank ℝ E / Gamma (↑(finrank ℝ E) / 2 + 1))
⊢ ⇑(stdOrthonormalBasis ℝ E).repr.symm ⁻¹' ball x r = ball ((stdOrthonormalBasis ℝ E).repr x) r
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ volume (ball x r) =
    ENNReal.ofReal r ^ finrank ℝ E * ENNReal.ofReal (√π ^ finrank ℝ E / Gamma (↑(finrank ℝ E) / 2 + 1))

Tactic:
InnerProductSpace.volume_ball _

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ ENNReal.ofReal r ^ finrank ℝ E * ENNReal.ofReal (√π ^ finrank ℝ E / Gamma (↑(finrank ℝ E) / 2 + 1)) =
    ENNReal.ofReal r ^ finrank ℝ E * ENNReal.ofReal (√π ^ finrank ℝ E / Gamma (↑(finrank ℝ E) / 2 + 1))
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : BorelSpace E
inst✝¹ : ProperSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖x‖}) atTop) atTop (𝓝 0)
n : ℕ
u v : ℝ
huv : u ≤ v
⊢ (μ n) {x | v < ‖x‖} ≤ (μ n) {x | u < ‖x‖}

Tactic:
by gcongr

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
⊢ ∀ a_1 ∈ s, HasFDerivWithinAt (fun x ↦ curveIntegral ω (Path.segment a x)) (ω a_1) s a_1

Tactic:
intro b hb

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

Goal state before:
pp: case hdω
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : 𝕜 → E
s : Set 𝕜
hs : Convex ℝ s
hf : DifferentiableOn 𝕜 f s
this : NormedSpace ℝ E := ⋯
⊢ ∀ a ∈ s,
    ∀ x ∈ tangentConeAt ℝ s a,
      ∀ y ∈ tangentConeAt ℝ s a,
        ((ContinuousLinearMap.restrictScalars ℝ
                (ContinuousLinearMap.toSpanSingleton 𝕜
                  (((ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1) (derivWithin f s a))))
              x)
            y =
          ((ContinuousLinearMap.restrictScalars ℝ
                (ContinuousLinearMap.toSpanSingleton 𝕜
                  (((ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1) (derivWithin f s a))))
              y)
            x

Tactic:
rintro a ha x - y -

Goal state after:
case hdω
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : 𝕜 → E
s : Set 𝕜
hs : Convex ℝ s
hf : DifferentiableOn 𝕜 f s
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
a : 𝕜
ha : a ∈ s
x y : 𝕜
⊢ ((ContinuousLinearMap.restrictScalars ℝ
          (ContinuousLinearMap.toSpanSingleton 𝕜 (((ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1) (derivWithin f s a))))
        x)
      y =
    ((ContinuousLinearMap.restrictScalars ℝ
          (ContinuousLinearMap.toSpanSingleton 𝕜 (((ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1) (derivWithin f s a))))
        y)
      x
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: case hbc.refine_1
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ 0 ≤ᵐ[volume.restrict (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹))] fun x ↦ ‖1 - charFun (μ n) (x • z)‖

Tactic:
exact ae_of_all _ fun _ ↦ by positivity

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 ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
z : E
n : ℕ
u v : ℝ
huv : u ≤ v
⊢ (μ n) {x | v < ‖⟪z, x⟫_𝕜‖} ≤ (μ n) {x | u < ‖⟪z, x⟫_𝕜‖}

Tactic:
by gcongr

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

Goal state before:
pp: case e'_2
ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁸ : (i : ι) → TopologicalSpace (X i)
inst✝⁷ : ∀ (i : ι), BorelSpace (X i)
inst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁵ : (j : κ) → TopologicalSpace (Y j)
inst✝⁴ : ∀ (j : κ), BorelSpace (Y j)
inst✝³ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝² : Fintype ι
inst✝¹ : Fintype κ
μ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝ : IsFiniteMeasure μ
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ≥0) (g : (j : κ) → Y j →ᵇ ℝ≥0),
    ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂μ =
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂ν
hμν : μ univ = ν univ
this✝ : IsFiniteMeasure ν
π : Set (Set (((i : ι) → X i) × ((j : κ) → Y j))) :=
  image2 (fun s t ↦ s ×ˢ t) (univ.pi '' univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s})
    (univ.pi '' univ.pi fun x ↦ {t | IsClosed[inst✝⁵ x] t})
hπ1 : IsPiSystem π
hπ2 : Prod.instMeasurableSpace = generateFrom π
s : (i : ι) → Set (X i)
t : (i : κ) → Set (Y i)
hs : ∀ (i : ι), IsClosed[inst✝⁸ i] (s i)
ht : ∀ (i : κ), IsClosed[inst✝⁵ i] (t i)
hp1 : ∀ (x : (i : ι) → X i), ∏ i, (s i).indicator (fun x ↦ 1) (x i) = (univ.pi s).indicator 1 x
hp2 : ∀ (y : (j : κ) → Y j), ∏ j, (t j).indicator (fun x ↦ 1) (y j) = (univ.pi t).indicator 1 y
this :
  ∀ (p : ((i : ι) → X i) × ((j : κ) → Y j)),
    Tendsto (ofNNReal ∘ fun x ↦ (∏ c, (⋯.apprSeq x) (p.1 c)) * ∏ c, (⋯.apprSeq x) (p.2 c)) atTop
      (𝓝 ↑((univ.pi s ×ˢ univ.pi t).indicator 1 p))
h1 :
  Tendsto
    (fun n ↦
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑((∏ i, (⋯.apprSeq n) (p.1 i)) * ∏ j, (⋯.apprSeq n) (p.2 j)) ∂μ)
    atTop (𝓝 (∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑((univ.pi s ×ˢ univ.pi t).indicator 1 p) ∂μ))
h2 :
  Tendsto
    (fun n ↦
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑((∏ i, (⋯.apprSeq n) (p.1 i)) * ∏ j, (⋯.apprSeq n) (p.2 j)) ∂μ)
    atTop (𝓝 (∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑((univ.pi s ×ˢ univ.pi t).indicator 1 p) ∂ν))
⊢ μ ((fun s t ↦ s ×ˢ t) (univ.pi s) (univ.pi t)) =
    ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑((univ.pi s ×ˢ univ.pi t).indicator 1 p) ∂μ

Tactic:
simp [(MeasurableSet.univ_pi (fun i ↦ (hs i).measurableSet)).prod
      (.univ_pi (fun j ↦ (ht j).measurableSet))]

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

Goal state before:
pp: case e'_3
ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁸ : (i : ι) → TopologicalSpace (X i)
inst✝⁷ : ∀ (i : ι), BorelSpace (X i)
inst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁵ : (j : κ) → TopologicalSpace (Y j)
inst✝⁴ : ∀ (j : κ), BorelSpace (Y j)
inst✝³ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝² : Fintype ι
inst✝¹ : Fintype κ
μ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝ : IsFiniteMeasure μ
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ≥0) (g : (j : κ) → Y j →ᵇ ℝ≥0),
    ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂μ =
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂ν
hμν : μ univ = ν univ
this✝ : IsFiniteMeasure ν
π : Set (Set (((i : ι) → X i) × ((j : κ) → Y j))) :=
  image2 (fun s t ↦ s ×ˢ t) (univ.pi '' univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s})
    (univ.pi '' univ.pi fun x ↦ {t | IsClosed[inst✝⁵ x] t})
hπ1 : IsPiSystem π
hπ2 : Prod.instMeasurableSpace = generateFrom π
s : (i : ι) → Set (X i)
t : (i : κ) → Set (Y i)
hs : ∀ (i : ι), IsClosed[inst✝⁸ i] (s i)
ht : ∀ (i : κ), IsClosed[inst✝⁵ i] (t i)
hp1 : ∀ (x : (i : ι) → X i), ∏ i, (s i).indicator (fun x ↦ 1) (x i) = (univ.pi s).indicator 1 x
hp2 : ∀ (y : (j : κ) → Y j), ∏ j, (t j).indicator (fun x ↦ 1) (y j) = (univ.pi t).indicator 1 y
this :
  ∀ (p : ((i : ι) → X i) × ((j : κ) → Y j)),
    Tendsto (ofNNReal ∘ fun x ↦ (∏ c, (⋯.apprSeq x) (p.1 c)) * ∏ c, (⋯.apprSeq x) (p.2 c)) atTop
      (𝓝 ↑((univ.pi s ×ˢ univ.pi t).indicator 1 p))
h1 :
  Tendsto
    (fun n ↦
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑((∏ i, (⋯.apprSeq n) (p.1 i)) * ∏ j, (⋯.apprSeq n) (p.2 j)) ∂μ)
    atTop (𝓝 (∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑((univ.pi s ×ˢ univ.pi t).indicator 1 p) ∂μ))
h2 :
  Tendsto
    (fun n ↦
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑((∏ i, (⋯.apprSeq n) (p.1 i)) * ∏ j, (⋯.apprSeq n) (p.2 j)) ∂μ)
    atTop (𝓝 (∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑((univ.pi s ×ˢ univ.pi t).indicator 1 p) ∂ν))
⊢ ν ((fun s t ↦ s ×ˢ t) (univ.pi s) (univ.pi t)) =
    ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑((univ.pi s ×ˢ univ.pi t).indicator 1 p) ∂ν

Tactic:
simp [(MeasurableSet.univ_pi (fun i ↦ (hs i).measurableSet)).prod
      (.univ_pi (fun j ↦ (ht j).measurableSet))]

Goal state after:
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' : Set α
t'C : t' ∈ C
ht' : μ t'ᶜ < ε / 2
⊢ μ (t ∆ s) + μ t'ᶜ < ε / 2 + ε / 2

Tactic:
by gcongr

Goal state after:
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✝
f : ℕ → Set α
f_disj : Pairwise (Function.onFun Disjoint f)
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf : ∀ (i : ℕ) (ε : ℝ≥0∞), 0 < ε → ∃ t ∈ C, μ (t ∆ f i) < ε
ε : ℝ≥0∞
εpos : 0 < ε
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε / 2
t : ℕ → Set α
tC : ∀ (i : ℕ), t i ∈ C
ht : ∀ (i : ℕ), μ (t i ∆ f i) < δ i
this : Tendsto (fun n ↦ μ (⋃ i ∈ Ici n, f i)) atTop (𝓝 0)
n : ℕ
hn : μ (⋃ i ∈ Ici n, f i) < ε / 2
⊢ ∑' (i : ℕ), δ i + μ (⋃ i ∈ Ici n, f i) < ε / 2 + ε / 2

Tactic:
by gcongr

Goal state after: