ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' : ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[>] 0) → (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) → Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ μ (B j) ≤ ↑C * μ (b j)
hB : μ (B j) = ⊤

⊢ μ (W ∩ B j) / μ (B j) ≤ ↑(1 - C⁻¹)
-- Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' : ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[>] 0) → (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) → Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ μ (B j) ≤ ↑C * μ (b j)
hB : μ (B j) ≠ ⊤

⊢ μ (W ∩ B j) / μ (B j) ≤ ↑(1 - C⁻¹)
-- Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' : ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[>] 0) → (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) → Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ μ (B j) ≤ ↑C * μ (b j)
hB : μ (B j) ≠ ⊤

⊢ μ (W ∩ B j) ≤ ↑(1 - C⁻¹) * μ (B j)
-- Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' : ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[>] 0) → (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) → Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ μ (B j) ≤ ↑C * μ (b j)
hB : μ (B j) ≠ ⊤

⊢ μ (W ∩ B j) ≤ μ (B j) - ↑C⁻¹ * μ (B j)
-- Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' : ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[>] 0) → (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) → Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ μ (B j) ≤ ↑C * μ (b j)
hB : ↑C⁻¹ * μ (B j) ≠ ⊤

⊢ μ (W ∩ B j) ≤ μ (B j) - ↑C⁻¹ * μ (B j)
-- Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' : ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[>] 0) → (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) → Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)
j : ℕ
hB : ↑C⁻¹ * μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : μ (B j) ≤ ↑C * μ (b j)

⊢ μ (W ∩ B j) ≤ μ (B j) - ↑C⁻¹ * μ (B j)
-- Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' : ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[>] 0) → (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) → Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)
j : ℕ
hB : ↑C⁻¹ * μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑C⁻¹ * μ (B j) ≤ μ (b j)

⊢ μ (W ∩ B j) ≤ μ (B j) - ↑C⁻¹ * μ (B j)
-- Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' : ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[>] 0) → (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) → Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)
j : ℕ
hB : ↑C⁻¹ * μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑C⁻¹ * μ (B j) ≤ μ (b j)
hj₃ : ↑C⁻¹ * μ (B j) + μ (W ∩ B j) ≤ μ (B j)

⊢ μ (W ∩ B j) ≤ μ (B j) - ↑C⁻¹ * μ (B j)
-- Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' : ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[>] 0) → (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) → Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)
j : ℕ
hB : ↑C⁻¹ * μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑C⁻¹ * μ (B j) ≤ μ (b j)
hj₃ : ↑C⁻¹ * μ (B j) + μ (W ∩ B j) - ↑C⁻¹ * μ (B j) ≤ μ (B j) - ↑C⁻¹ * μ (B j)

⊢ μ (W ∩ B j) ≤ μ (B j) - ↑C⁻¹ * μ (B j)
-- Mathlib/Order/Filter/AtTopBot.lean
ι✝ : Type u_1
ι' : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
ι : Type u_6
l : Filter ι
p : ι → Prop
inst✝ : l.IsCountablyGenerated
h : ∃ᶠ (n : ι) in l, p n

⊢ ∃ ns, Tendsto ns atTop l ∧ ∀ (n : ℕ), p (ns n)
-- Mathlib/Order/Filter/AtTopBot.lean
ι✝ : Type u_1
ι' : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
ι : Type u_6
l : Filter ι
p : ι → Prop
inst✝ : l.IsCountablyGenerated
h : ∃ x, Tendsto x atTop l ∧ ∃ᶠ (n : ℕ) in atTop, p (x n)

⊢ ∃ ns, Tendsto ns atTop l ∧ ∀ (n : ℕ), p (ns n)
-- Mathlib/Order/Filter/AtTopBot.lean
ι✝ : Type u_1
ι' : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
ι : Type u_6
l : Filter ι
p : ι → Prop
inst✝ : l.IsCountablyGenerated
x : ℕ → ι
hx_tendsto : Tendsto x atTop l
hx_freq : ∃ᶠ (n : ℕ) in atTop, p (x n)

⊢ ∃ ns, Tendsto ns atTop l ∧ ∀ (n : ℕ), p (ns n)
-- Mathlib/Order/Filter/AtTopBot.lean
ι✝ : Type u_1
ι' : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
ι : Type u_6
l : Filter ι
p : ι → Prop
inst✝ : l.IsCountablyGenerated
x : ℕ → ι
hx_tendsto : Tendsto x atTop l
hx_freq : ∃ᶠ (n : ℕ) in atTop, p (x n)
n_to_n : ℕ → ℕ
h_tendsto : Tendsto (fun n => x (n_to_n n)) atTop l
h_freq : ∀ (n : ℕ), p (x (n_to_n n))

⊢ ∃ ns, Tendsto ns atTop l ∧ ∀ (n : ℕ), p (ns n)
-- Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
u v : V
p : G.Walk u v

⊢ (induce {v_1 | v_1 ∈ p.support} G).Connected
-- Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
u v : V
p : G.Walk u v

⊢ (induce p.toSubgraph.verts G).Connected
-- Mathlib/Analysis/Normed/Order/UpperLower.lean
α : Type u_1
ι : Type u_2
inst✝ : NormedOrderedGroup α
s : Set α
hs : IsLowerSet s
ε : ℝ

⊢ IsLowerSet (cthickening ε s)
-- Mathlib/Analysis/Normed/Order/UpperLower.lean
α : Type u_1
ι : Type u_2
inst✝ : NormedOrderedGroup α
s : Set α
hs : IsLowerSet s
ε : ℝ

⊢ IsLowerSet (⋂ ε_1, ⋂ (_ : max 0 ε < ε_1), thickening ε_1 s)
-- Mathlib/Algebra/Order/Module/Defs.lean
α : Type u_1
β : Type u_2
a a₁✝ a₂✝ : α
b✝ b₁ b₂ : β
γ : Type u_3
inst✝⁸ : Zero α
inst✝⁷ : Preorder α
inst✝⁶ : Zero β
inst✝⁵ : Preorder β
inst✝⁴ : Zero γ
inst✝³ : Preorder γ
inst✝² : SMul α β
inst✝¹ : SMul α γ
f : β → γ
inst✝ : SMulPosReflectLT α γ
hf : ∀ {b₁ b₂ : β}, f b₁ ≤ f b₂ ↔ b₁ ≤ b₂
smul : ∀ (a : α) (b : β), f (a • b) = a • f b
zero : f 0 = 0
b : β
hb : 0 ≤ b
a₁ a₂ : α
h : a₁ • b < a₂ • b

⊢ a₁ < a₂
-- Mathlib/Algebra/Order/Module/Defs.lean
α : Type u_1
β : Type u_2
a a₁✝ a₂✝ : α
b✝ b₁ b₂ : β
γ : Type u_3
inst✝⁸ : Zero α
inst✝⁷ : Preorder α
inst✝⁶ : Zero β
inst✝⁵ : Preorder β
inst✝⁴ : Zero γ
inst✝³ : Preorder γ
inst✝² : SMul α β
inst✝¹ : SMul α γ
f : β → γ
inst✝ : SMulPosReflectLT α γ
b : β
a₁ a₂ : α
hf : ∀ {b₁ b₂ : β}, True
smul : α → β → True
zero : True
hb : 0 ≤ f b
h : a₁ • f b < a₂ • f b

⊢ a₁ < a₂
-- Mathlib/Order/Heyting/Boundary.lean
α : Type u_1
inst✝ : CoheytingAlgebra α
a✝ b a : α

⊢ ∂ ∂ a = ∂ a
-- Mathlib/Topology/Instances/NNReal.lean
α : Type u_1
f : α → ℝ
y : ℝ
hf₀ : ∀ (n : α), 0 ≤ f n
hy : HasSum f y

⊢ HasSum (fun x => (f x).toNNReal) y.toNNReal
-- Mathlib/Topology/Instances/NNReal.lean
α : Type u_1
f : α → ℝ
hf₀ : ∀ (n : α), 0 ≤ f n
y : ℝ≥0
hy : HasSum f ↑y

⊢ HasSum (fun x => (f x).toNNReal) (↑y).toNNReal
-- Mathlib/Topology/Instances/NNReal.lean
α : Type u_1
y : ℝ≥0
f : α → ℝ≥0
hy : HasSum (fun i => ↑(f i)) ↑y

⊢ HasSum (fun x => ((fun i => ↑(f i)) x).toNNReal) (↑y).toNNReal
-- Mathlib/RingTheory/Localization/Basic.lean
R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommSemiring P
inst✝³ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
T : Submonoid P
Q : Type u_4
inst✝² : CommSemiring Q
hy : M ≤ Submonoid.comap g T
inst✝¹ : Algebra P Q
inst✝ : IsLocalization T Q
h : R ≃+* P
H : Submonoid.map h.toMonoidHom M = T

⊢ Submonoid.map h.symm.toMonoidHom T = M
-- Mathlib/RingTheory/Localization/Basic.lean
R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommSemiring P
inst✝³ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
T : Submonoid P
Q : Type u_4
inst✝² : CommSemiring Q
hy : M ≤ Submonoid.comap g T
inst✝¹ : Algebra P Q
inst✝ : IsLocalization T Q
h : R ≃+* P
H : Submonoid.map h.toMonoidHom M = T

⊢ Submonoid.map (h.symm.toMonoidHom.comp h.toMonoidHom) M = Submonoid.map (MonoidHom.id R) M
-- Mathlib/RingTheory/Localization/Basic.lean
R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommSemiring P
inst✝³ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
T : Submonoid P
Q : Type u_4
inst✝² : CommSemiring Q
hy : M ≤ Submonoid.comap g T
inst✝¹ : Algebra P Q
inst✝ : IsLocalization T Q
h : R ≃+* P
H : Submonoid.map h.toMonoidHom M = T
x✝ : R

⊢ (h.symm.toMonoidHom.comp h.toMonoidHom) x✝ = (MonoidHom.id R) x✝
-- Mathlib/RingTheory/Localization/Basic.lean
R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommSemiring P
inst✝³ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
T : Submonoid P
Q : Type u_4
inst✝² : CommSemiring Q
hy : M ≤ Submonoid.comap g T
inst✝¹ : Algebra P Q
inst✝ : IsLocalization T Q
h : R ≃+* P
H : Submonoid.map h.toMonoidHom M = T
H' : Submonoid.map h.symm.toMonoidHom T = M
x : S

⊢ (map S ↑h.symm ⋯) ((map Q ↑h ⋯) x) = x
-- Mathlib/RingTheory/Localization/Basic.lean
R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommSemiring P
inst✝³ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
T : Submonoid P
Q : Type u_4
inst✝² : CommSemiring Q
hy : M ≤ Submonoid.comap g T
inst✝¹ : Algebra P Q
inst✝ : IsLocalization T Q
h : R ≃+* P
H : Submonoid.map h.toMonoidHom M = T
H' : Submonoid.map h.symm.toMonoidHom T = M
x : S

⊢ ∀ (x : R), (RingHom.id S) ((algebraMap R S) x) = (algebraMap R S) (((↑h.symm).comp ↑h) x)
-- Mathlib/RingTheory/Localization/Basic.lean
R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommSemiring P
inst✝³ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
T : Submonoid P
Q : Type u_4
inst✝² : CommSemiring Q
hy : M ≤ Submonoid.comap g T
inst✝¹ : Algebra P Q
inst✝ : IsLocalization T Q
h : R ≃+* P
H : Submonoid.map h.toMonoidHom M = T
H' : Submonoid.map h.symm.toMonoidHom T = M
x : Q

⊢ (map Q ↑h ⋯) ((map S ↑h.symm ⋯) x) = x
-- Mathlib/RingTheory/Localization/Basic.lean
R : Type u_1
inst✝⁷ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommSemiring S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommSemiring P
inst✝³ : IsLocalization M S
g : R →+* P
hg : ∀ (y : ↥M), IsUnit (g ↑y)
T : Submonoid P
Q : Type u_4
inst✝² : CommSemiring Q
hy : M ≤ Submonoid.comap g T
inst✝¹ : Algebra P Q
inst✝ : IsLocalization T Q
h : R ≃+* P
H : Submonoid.map h.toMonoidHom M = T
H' : Submonoid.map h.symm.toMonoidHom T = M
x : Q

⊢ ∀ (x : P), (RingHom.id Q) ((algebraMap P Q) x) = (algebraMap P Q) (((↑h).comp ↑h.symm) x)
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
as : List α

⊢ 𝓝 as = traverse 𝓝 as
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
as : List α

⊢ pure ≤ traverse 𝓝
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
as : List α

⊢ ∀ (a : List α), ∀ s ∈ traverse 𝓝 a, ∀ᶠ (y : List α) in traverse 𝓝 a, s ∈ traverse 𝓝 y
-- Mathlib/RingTheory/Localization/Integral.lean
R✝ : Type u_1
inst✝¹⁷ : CommRing R✝
M✝ : Submonoid R✝
S✝ : Type u_2
inst✝¹⁶ : CommRing S✝
inst✝¹⁵ : Algebra R✝ S✝
P : Type u_3
inst✝¹⁴ : CommRing P
Rₘ✝ : Type u_4
Sₘ✝ : Type u_5
inst✝¹³ : CommRing Rₘ✝
inst✝¹² : CommRing Sₘ✝
inst✝¹¹ : Algebra R✝ Rₘ✝
inst✝¹⁰ : IsLocalization M✝ Rₘ✝
inst✝⁹ : Algebra S✝ Sₘ✝
inst✝⁸ : IsLocalization (Algebra.algebraMapSubmonoid S✝ M✝) Sₘ✝
R : Type u_6
S : Type u_7
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
x : S
p : R[X]
hf : eval₂ f x p = 0
M : Submonoid R
hM : p.leadingCoeff ∈ M
Rₘ : Type u_8
Sₘ : Type u_9
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
inst✝³ : Algebra R Rₘ
inst✝² : IsLocalization M Rₘ
inst✝¹ : Algebra S Sₘ
inst✝ : IsLocalization (Submonoid.map f M) Sₘ
triv : ¬1 = 0

⊢ (IsLocalization.map Sₘ f ⋯).IsIntegralElem ((algebraMap S Sₘ) x)
-- Mathlib/RingTheory/Localization/Integral.lean
R✝ : Type u_1
inst✝¹⁷ : CommRing R✝
M✝ : Submonoid R✝
S✝ : Type u_2
inst✝¹⁶ : CommRing S✝
inst✝¹⁵ : Algebra R✝ S✝
P : Type u_3
inst✝¹⁴ : CommRing P
Rₘ✝ : Type u_4
Sₘ✝ : Type u_5
inst✝¹³ : CommRing Rₘ✝
inst✝¹² : CommRing Sₘ✝
inst✝¹¹ : Algebra R✝ Rₘ✝
inst✝¹⁰ : IsLocalization M✝ Rₘ✝
inst✝⁹ : Algebra S✝ Sₘ✝
inst✝⁸ : IsLocalization (Algebra.algebraMapSubmonoid S✝ M✝) Sₘ✝
R : Type u_6
S : Type u_7
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
x : S
p : R[X]
hf : eval₂ f x p = 0
M : Submonoid R
hM : p.leadingCoeff ∈ M
Rₘ : Type u_8
Sₘ : Type u_9
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
inst✝³ : Algebra R Rₘ
inst✝² : IsLocalization M Rₘ
inst✝¹ : Algebra S Sₘ
inst✝ : IsLocalization (Submonoid.map f M) Sₘ
triv : ¬1 = 0
this : Nontrivial Rₘ

⊢ (IsLocalization.map Sₘ f ⋯).IsIntegralElem ((algebraMap S Sₘ) x)