-- Mathlib/MeasureTheory/Covering/DensityTheorem.lean
theorem closedBall_mem_vitaliFamily_of_dist_le_mul {K : ℝ} {x y : α} {r : ℝ} (h : dist x y ≤ K * r)
    (rpos : 0 < r) : closedBall y r ∈ (vitaliFamily μ K).setsAt x

Proof state:
case pos.h.inl
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : r ≤ R
⊢ ∃ r_1,
    closedBall y r ⊆ closedBall x r_1 ∧
      μ (closedBall x (3 * r_1)) ≤ ↑(scalingConstantOf μ ((4 * K + 3) ⊔ 3)) * μ (closedBall y r)

Tactic:
refine ⟨(K + 1) * r, ?_⟩

Result:
case pos.h.inl
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : r ≤ R
⊢ closedBall y r ⊆ closedBall x ((K + 1) * r) ∧
    μ (closedBall x (3 * ((K + 1) * r))) ≤ ↑(scalingConstantOf μ ((4 * K + 3) ⊔ 3)) * μ (closedBall y r)

Proof state:
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
⊢ (∃ r_1,
      closedBall y r ⊆ closedBall x r_1 ∧
        μ (closedBall x (3 * r_1)) ≤ ↑(scalingConstantOf μ ((4 * K + 3) ⊔ 3)) * μ (closedBall y r)) ∨
    ¬closedBall y r ⊆ closedBall x (scalingScaleOf μ ((4 * K + 3) ⊔ 3) / 4)

Tactic:
by_cases H : closedBall y r ⊆ closedBall x (R / 4)

Result:
case pos
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
H : closedBall y r ⊆ closedBall x (R / 4)
⊢ (∃ r_1,
      closedBall y r ⊆ closedBall x r_1 ∧
        μ (closedBall x (3 * r_1)) ≤ ↑(scalingConstantOf μ ((4 * K + 3) ⊔ 3)) * μ (closedBall y r)) ∨
    ¬closedBall y r ⊆ closedBall x (scalingScaleOf μ ((4 * K + 3) ⊔ 3) / 4)

case neg
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
H : ¬closedBall y r ⊆ closedBall x (R / 4)
⊢ (∃ r_1,
      closedBall y r ⊆ closedBall x r_1 ∧
        μ (closedBall x (3 * r_1)) ≤ ↑(scalingConstantOf μ ((4 * K + 3) ⊔ 3)) * μ (closedBall y r)) ∨
    ¬closedBall y r ⊆ closedBall x (scalingScaleOf μ ((4 * K + 3) ⊔ 3) / 4)

Proof state:
case pos
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
H : closedBall y r ⊆ closedBall x (R / 4)
⊢ (∃ r_1,
      closedBall y r ⊆ closedBall x r_1 ∧
        μ (closedBall x (3 * r_1)) ≤ ↑(scalingConstantOf μ ((4 * K + 3) ⊔ 3)) * μ (closedBall y r)) ∨
    ¬closedBall y r ⊆ closedBall x (scalingScaleOf μ ((4 * K + 3) ⊔ 3) / 4)

Tactic:
left

Result:
case pos.h
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
H : closedBall y r ⊆ closedBall x (R / 4)
⊢ ∃ r_1,
    closedBall y r ⊆ closedBall x r_1 ∧
      μ (closedBall x (3 * r_1)) ≤ ↑(scalingConstantOf μ ((4 * K + 3) ⊔ 3)) * μ (closedBall y r)

Proof state:
case pos.h.inl
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : r ≤ R
⊢ closedBall y r ⊆ closedBall x ((K + 1) * r) ∧
    μ (closedBall x (3 * ((K + 1) * r))) ≤ ↑(scalingConstantOf μ ((4 * K + 3) ⊔ 3)) * μ (closedBall y r)

Tactic:
constructor

Result:
case pos.h.inl.left
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : r ≤ R
⊢ closedBall y r ⊆ closedBall x ((K + 1) * r)

case pos.h.inl.right
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : r ≤ R
⊢ μ (closedBall x (3 * ((K + 1) * r))) ≤ ↑(scalingConstantOf μ ((4 * K + 3) ⊔ 3)) * μ (closedBall y r)

Proof state:
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
⊢ closedBall y r ∈ (vitaliFamily μ K).setsAt x

Tactic:
let R := scalingScaleOf μ (max (4 * K + 3) 3)

Result:
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
⊢ closedBall y r ∈ (vitaliFamily μ K).setsAt x

Proof state:
case pos.h.inr
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : R < r
⊢ ∃ r_1,
    closedBall y r ⊆ closedBall x r_1 ∧
      μ (closedBall x (3 * r_1)) ≤ ↑(scalingConstantOf μ ((4 * K + 3) ⊔ 3)) * μ (closedBall y r)

Tactic:
refine ⟨R / 4, H, ?_⟩

Result:
case pos.h.inr
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : R < r
⊢ μ (closedBall x (3 * (R / 4))) ≤ ↑(scalingConstantOf μ ((4 * K + 3) ⊔ 3)) * μ (closedBall y r)

Proof state:
case h
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : r ≤ R
⊢ 3 * ((K + 1) * r) + dist x y ≤ (4 * K + 3) * r

Tactic:
linarith

Result:
no goals