mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
Λ : (E →C_c ℝ) → ℝ
h₀Λ : ∀ (g : E →C_c ℝ), Λ g ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖)
hΛ : ∀ (g : E →C_c ℝ), Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 (Λ g))
g g' : E →C_c ℝ
x✝ : FiniteMeasure E
⊢ ∫ (x : E), g x + g' x ∂↑x✝ = ∫ (x : E), g x ∂↑x✝ + ∫ (x : E), g' x ∂↑x✝

Tactic:
integral_add

Goal state after:
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
Λ : (E →C_c ℝ) → ℝ
h₀Λ : ∀ (g : E →C_c ℝ), Λ g ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖)
hΛ : ∀ (g : E →C_c ℝ), Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 (Λ g))
g g' : E →C_c ℝ
x✝ : FiniteMeasure E
⊢ ∫ (a : E), g a ∂↑x✝ + ∫ (a : E), g' a ∂↑x✝ = ∫ (x : E), g x ∂↑x✝ + ∫ (x : E), g' x ∂↑x✝

case hf
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
Λ : (E →C_c ℝ) → ℝ
h₀Λ : ∀ (g : E →C_c ℝ), Λ g ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖)
hΛ : ∀ (g : E →C_c ℝ), Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 (Λ g))
g g' : E →C_c ℝ
x✝ : FiniteMeasure E
⊢ Integrable ⇑g ↑x✝

case hg
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
Λ : (E →C_c ℝ) → ℝ
h₀Λ : ∀ (g : E →C_c ℝ), Λ g ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖)
hΛ : ∀ (g : E →C_c ℝ), Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 (Λ g))
g g' : E →C_c ℝ
x✝ : FiniteMeasure E
⊢ Integrable ⇑g' ↑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 → ℝ), 0 ≤ ∑ i, |x i| ^ p
h₂ : ∀ (x : ι → ℝ), 0 ≤ (∑ i, |x i| ^ p) ^ (1 / p)
hr : 0 < r
⊢ volume {x | (∑ i, |x i| ^ p) ^ (1 / p) < r} = ENNReal.ofReal (r ^ card ι) * volume {x | ∑ i, |x i| ^ p < 1}

Tactic:
← finrank_pi ℝ

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 → ℝ), 0 ≤ ∑ i, |x i| ^ p
h₂ : ∀ (x : ι → ℝ), 0 ≤ (∑ i, |x i| ^ p) ^ (1 / p)
hr : 0 < r
⊢ volume {x | (∑ i, |x i| ^ p) ^ (1 / p) < r} = ENNReal.ofReal (r ^ finrank ℝ (ι → ℝ)) * volume {x | ∑ i, |x i| ^ p < 1}
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℝ), 0 ≤ ∑ i, |x i| ^ p
h₂ : ∀ (x : ι → ℝ), 0 ≤ (∑ i, |x i| ^ p) ^ (1 / p)
hr : 0 < r
⊢ {x | (∑ i, |x i| ^ p) ^ (1 / p) < r} = {a | ∑ x, |r⁻¹| ^ p * |a x| ^ p < 1}

Tactic:
abs_inv,

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

Goal state before:
pp: case e'_2
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
⊢ {x | (∑ i, ‖x i‖ ^ p) ^ (1 / p) < r} = {a | ∑ x, |r⁻¹| ^ p * ‖a x‖ ^ p < 1}

Tactic:
abs_inv,

Goal state after:
case e'_2
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
⊢ {x | (∑ i, ‖x i‖ ^ p) ^ (1 / p) < r} = {a | ∑ x, |r|⁻¹ ^ p * ‖a x‖ ^ p < 1}
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: ι : Type u_1
inst✝¹ : Nonempty ι
inst✝ : Fintype ι
x : EuclideanSpace ℝ ι
r : ℝ
⊢ volume (Metric.closedBall x r) = ENNReal.ofReal r ^ card ι * ENNReal.ofReal (√π ^ card ι / Gamma (↑(card ι) / 2 + 1))

Tactic:
rw [addHaar_closedBall_eq_addHaar_ball, EuclideanSpace.volume_ball]

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

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

Tactic:
have : Nontrivial E := Module.nontrivial_of_finrank_pos (R := ℝ) (hk ▸ (2 * k).succ_pos)
  rw [volume_ball, hk, pow_succ (√π), pow_mul, sq_sqrt pi_nonneg, mul_div_assoc, mul_div_assoc]
  congr 3
  suffices √π / (↑(2 * k + 1)‼ * √π / 2 ^ (k + 1)) = 2 ^ (k + 1) / ↑(2 * k + 1)‼ by
    simpa [add_div, add_right_comm, -one_div, Gamma_nat_add_one_add_half]
  field

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

Goal state before:
pp: case inl
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
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
inst✝ : CompleteSpace F
hs : Convex ℝ ∅
hω : ∀ x ∈ ∅, HasFDerivWithinAt ω (dω x) ∅ x
hdω : ∀ a ∈ ∅, ∀ x ∈ tangentConeAt ℝ ∅ a, ∀ y ∈ tangentConeAt ℝ ∅ a, ((dω a) x) y = ((dω a) y) x
⊢ ∃ f, ∀ a ∈ ∅, HasFDerivWithinAt f (ω a) ∅ a

pp: case inr
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
⊢ ∃ f, ∀ a ∈ s, HasFDerivWithinAt f (ω a) s a

Tactic:
· simp

Goal state after:
case inr
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
⊢ ∃ f, ∀ a ∈ s, HasFDerivWithinAt f (ω a) s a
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
h_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {Measure.map (fun x ↦ ⟪z, x⟫_𝕜) (μ n)}, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
r : ℝ
⊢ (μ n) ((fun x ↦ ⟪z, x⟫_𝕜) ⁻¹' {x | r < ‖x‖}) = (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}

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
h_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {Measure.map (fun x ↦ ⟪z, x⟫_𝕜) (μ n)}, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
r : ℝ
⊢ MeasurableSet {x | r < ‖x‖}

Tactic:
· simp

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 ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
z : E
n : ℕ
h_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {Measure.map (fun x ↦ ⟪z, x⟫_𝕜) (μ n)}, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
r : ℝ
⊢ MeasurableSet {x | r < ‖x‖}
Module: Mathlib.MeasureTheory.Measure.MeasuredSets

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
⊢ ∀ (x y : MeasuredSets μ), μ (↑x ∆ ↑y) = μ (↑y ∆ ↑x)

Tactic:
by grind

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' ∈ 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]

Goal state after:
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
D : Set (Set α)
D_count : D.Countable
DC : D ⊆ C
hD : μ (⋃₀ D)ᶜ = 0
Dne : D.Nonempty
⊢ ∃ t' ∈ C, μ t'ᶜ < ε / 2
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)
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), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
n : ℕ
h : NormalSpace E
this : (↑μ).InnerRegular
g : E → ℝ
g_cont : Continuous[inst✝², _] g
gK : EqOn g 0 (K n)
g_nonneg : 0 ≤ g
g_le : g ≤ 1
g' : E →ᵇ ℝ := { toFun := g, continuous_toFun := g_cont, map_bounded' := ⋯ }
⊢ Tendsto (fun ρ ↦ ∫ (x : E), g x ∂↑ρ) (↑f) (𝓝 (∫ (x : E), g x ∂↑μ))

Tactic:
exact tendsto_iff_forall_integral_tendsto.1 L g'

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

Goal state before:
pp: case inl
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : Set X
hs : MeasurableSet s
h : preVariationFun f s ≠ ∞
hε : 0 < ∞
⊢ ∃ P, preVariationFun f s ≤ ∑ p ∈ P.parts, f ↑p + ∞

pp: case inr
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : Set X
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
h : preVariationFun f s ≠ ∞
h'ε : ε < ∞
⊢ ∃ P, preVariationFun f s ≤ ∑ p ∈ P.parts, f ↑p + ε

Tactic:
· simp

Goal state after:
case inr
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : Set X
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
h : preVariationFun f s ≠ ∞
h'ε : ε < ∞
⊢ ∃ P, preVariationFun f s ≤ ∑ p ∈ P.parts, f ↑p + ε
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

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
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
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
h_limsup_le :
  ∀ (r : ℝ),
    0 < r →
      limsup (fun n ↦ (μ n).real {x | r < |⟪z, x⟫|}) atTop ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z)‖
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
r : ℝ
hrδ : 4 * δ⁻¹ ≤ r
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
x : ℝ
hx : x ∈ Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)
⊢ |x| ≤ 2 * r⁻¹

Tactic:
by grind

Goal state after:
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)
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), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
n : ℕ
h : Monotone K
this : Tendsto (fun m ↦ ∑ i ∈ Finset.range (m + 1), ↑(ν i) (K n)ᶜ) atTop (𝓝 (∑' (i : ℕ), ↑(ν i) (K n)ᶜ))
m : ℕ
hm : n ≤ m
⊢ Finset.Ioc n m ⊆ Finset.range (m + 1)

Tactic:
by grind

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

Goal state before:
pp: case inl
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : PolishSpace E
inst✝ : BorelSpace E
ι : Type u_4
μ : ι → ProbabilityMeasure E
h_tight : IsTightMeasureSet {x | ∃ n, ↑(μ n) = x}
μ₀ : ProbabilityMeasure E
A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)
hA : (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).SeparatesPoints
this : TopologicalSpace.UpgradedIsCompletelyMetrizableSpace E := TopologicalSpace.upgradeIsCompletelyMetrizable E
hμ : ∀ g ∈ A, Tendsto (fun n ↦ ∫ (x : E), g x ∂↑(μ n)) ⊥ (𝓝 (∫ (x : E), g x ∂↑μ₀))
⊢ Tendsto μ ⊥ (𝓝 μ₀)

pp: case inr
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : PolishSpace E
inst✝ : BorelSpace E
ι : Type u_4
𝓕 : Filter ι
μ : ι → ProbabilityMeasure E
h_tight : IsTightMeasureSet {x | ∃ n, ↑(μ n) = x}
μ₀ : ProbabilityMeasure E
A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)
hA : (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).SeparatesPoints
hμ : ∀ g ∈ A, Tendsto (fun n ↦ ∫ (x : E), g x ∂↑(μ n)) 𝓕 (𝓝 (∫ (x : E), g x ∂↑μ₀))
this : TopologicalSpace.UpgradedIsCompletelyMetrizableSpace E := TopologicalSpace.upgradeIsCompletelyMetrizable E
h✝ : 𝓕.NeBot
⊢ Tendsto μ 𝓕 (𝓝 μ₀)

Tactic:
· simp

Goal state after:
case inr
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : PolishSpace E
inst✝ : BorelSpace E
ι : Type u_4
𝓕 : Filter ι
μ : ι → ProbabilityMeasure E
h_tight : IsTightMeasureSet {x | ∃ n, ↑(μ n) = x}
μ₀ : ProbabilityMeasure E
A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)
hA : (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).SeparatesPoints
hμ : ∀ g ∈ A, Tendsto (fun n ↦ ∫ (x : E), g x ∂↑(μ n)) 𝓕 (𝓝 (∫ (x : E), g x ∂↑μ₀))
this : TopologicalSpace.UpgradedIsCompletelyMetrizableSpace E := TopologicalSpace.upgradeIsCompletelyMetrizable E
h✝ : 𝓕.NeBot
⊢ Tendsto μ 𝓕 (𝓝 μ₀)
Module: Mathlib.MeasureTheory.Measure.ResolventTransform

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
x : 𝕜
hx : x ∈ μ.support
w : A
hw : w ∈ s
⊢ ‖((algebraMap 𝕜 A) x - w)⁻¹‖ ^ 2 ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)⁻¹ ^ 2

Tactic:
norm_inv

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
x : 𝕜
hx : x ∈ μ.support
w : A
hw : w ∈ s
⊢ ‖(algebraMap 𝕜 A) x - w‖⁻¹ ^ 2 ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)⁻¹ ^ 2
Module: Mathlib.MeasureTheory.Measure.ResolventTransform

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
x : 𝕜
hx : x ∈ μ.support
w : A
hw : w ∈ s
⊢ infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2 ≤
    infDist a (⇑(algebraMap 𝕜 A) '' μ.support) - infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2

Tactic:
by grind

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hg : Measurable g
hgf : g ≤ᵐ[μ] f
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, (f - g) a ∂μ = ∫⁻ (a : α) in s, f a - g a ∂μ

pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hg : Measurable g
hgf : g ≤ᵐ[μ] f
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, g a ∂μ ≠ ∞

Tactic:
· simp

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hg : Measurable g
hgf : g ≤ᵐ[μ] f
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, g a ∂μ ≠ ∞
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
μ₀ : ProbabilityMeasure E
μ : ℕ → ProbabilityMeasure E
⊢ Tendsto μ atTop (𝓝 μ₀) ↔ ∀ (t : E), Tendsto (fun n ↦ charFun (↑(μ n)) t) atTop (𝓝 (charFun (↑μ₀) t))

Tactic:
refine ⟨fun h t ↦ ?_, tendsto_of_tendsto_charFun⟩
  rw [ProbabilityMeasure.tendsto_iff_forall_integral_rclike_tendsto ℂ] at h
  simp_rw [charFun_eq_integral_innerProbChar]
  exact h (innerProbChar t)

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

Goal state before:
pp: case refine_1
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hf : Measurable f
hg : Measurable g
t : Set α := {x | f x ≤ g x}
ht : MeasurableSet t
h_zero : (μ.withDensity (f - g)).restrict t = 0
this : IsFiniteMeasure ((μ.restrict tᶜ).withDensity g)
⊢ g ≤ᵐ[μ.restrict tᶜ] f

Tactic:
refine ae_restrict_of_forall_mem ht.compl fun x hx ↦ ?_

Goal state after:
case refine_1
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hf : Measurable f
hg : Measurable g
t : Set α := {x | f x ≤ g x}
ht : MeasurableSet t
h_zero : (μ.withDensity (f - g)).restrict t = 0
this : IsFiniteMeasure ((μ.restrict tᶜ).withDensity g)
x : α
hx : x ∈ tᶜ
⊢ g x ≤ f x
Module: Mathlib.MeasureTheory.Measure.WithDensityFinite

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
h✝ : IsFiniteMeasure μ
⊢ ae μ = ae μ

pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
h✝ : ¬IsFiniteMeasure μ
⊢ ae ⋯.choose = ae μ

Tactic:
· simp

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
h✝ : ¬IsFiniteMeasure μ
⊢ ae ⋯.choose = ae μ
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case h
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
S : Set (ProbabilityMeasure E)
hS : IsTightMeasureSet {x | ∃ μ ∈ S, ↑μ = x}
u : ℕ → ℝ≥0
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
K : ℕ → Set E
K_comp : ∀ (n : ℕ), IsCompact (K n)
hK : ∀ (n : ℕ), ∀ μ ∈ S, μ (K n)ᶜ ≤ u n
K' : ℕ → Set E := ⋯
h'K : IsCompact {μ | ∀ (n : ℕ), μ (K' n)ᶜ ≤ u n}
μ : ProbabilityMeasure E
hμ : μ ∈ S
n : ℕ
⊢ K n ⊆ K' n

Tactic:
simp only [mem_Iic, K']

Goal state after:
case h
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
S : Set (ProbabilityMeasure E)
hS : IsTightMeasureSet {x | ∃ μ ∈ S, ↑μ = x}
u : ℕ → ℝ≥0
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
K : ℕ → Set E
K_comp : ∀ (n : ℕ), IsCompact (K n)
hK : ∀ (n : ℕ), ∀ μ ∈ S, μ (K n)ᶜ ≤ u n
K' : ℕ → Set E := fun n ↦ ⋃ i ∈ Iic n, K i
h'K : IsCompact {μ | ∀ (n : ℕ), μ (K' n)ᶜ ≤ u n}
μ : ProbabilityMeasure E
hμ : μ ∈ S
n : ℕ
⊢ K n ⊆ ⋃ i, ⋃ (_ : i ≤ n), K i
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
S : Set (ProbabilityMeasure E)
hS : IsTightMeasureSet {x | ∃ μ ∈ S, ↑μ = x}
u : ℕ → ℝ≥0
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
K : ℕ → Set E
K_comp : ∀ (n : ℕ), IsCompact (K n)
hK : ∀ (n : ℕ), ∀ μ ∈ S, μ (K n)ᶜ ≤ u n
K' : ℕ → Set E := fun n ↦ ⋃ i ∈ Iic n, K i
h'K : IsCompact {μ | ∀ (n : ℕ), μ (K' n)ᶜ ≤ u n}
μ : ProbabilityMeasure E
hμ : μ ∈ S
n : ℕ
⊢ μ (K n)ᶜ ≤ u n

Tactic:
by grind

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsZeroOneMeasure μ
inst✝¹ : StandardBorelSpace α
inst✝ : NeZero μ
this : IsProbabilityMeasure μ
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y
B : ℕ → Set α := fun n ↦ if h : μ (A n) = 1 then A n else (A n)ᶜ
mBn : MeasurableSet (⋂ n, B n)
hBn : μ (⋂ n, B n) = 1
x₀ : α
hx₀ : ⋂ n, B n = {x₀}
s : Set α
hs : MeasurableSet s
h : x₀ ∈ s
⊢ {x₀} ⊆ s

Tactic:
by grind

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsZeroOneMeasure μ
inst✝¹ : StandardBorelSpace α
inst✝ : NeZero μ
this : IsProbabilityMeasure μ
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y
B : ℕ → Set α := fun n ↦ if h : μ (A n) = 1 then A n else (A n)ᶜ
mBn : MeasurableSet (⋂ n, B n)
hBn : μ (⋂ n, B n) = 1
x₀ : α
hx₀ : ⋂ n, B n = {x₀}
s : Set α
hs : MeasurableSet s
h : x₀ ∉ s
⊢ s ⊆ {x₀}ᶜ

Tactic:
by grind

Goal state after: