l3lab/ntp-mathlib huggingface dataset:
/-- As an intermediate step to show that `μ.withDensity (v.limRatioMeas hρ) = ρ`, we show here
that `ρ ≤ t μ.withDensity (v.limRatioMeas hρ)` for any `t > 1`. -/
theorem le_mul_withDensity {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
    ρ s ≤ t * μ.withDensity (v.limRatioMeas hρ) s := by
  /- We cut `s` into the sets where `v.limRatioMeas hρ = 0`, where `v.limRatioMeas hρ = ∞`, and
    where `v.limRatioMeas hρ ∈ [t^n, t^(n+1))` for `n : ℤ`. The first and second have measure `0`.
    For the latter, since `v.limRatioMeas hρ` fluctuates by at most `t` on this slice, we can use
    `measure_le_mul_of_subset_limRatioMeas_lt` and `mul_measure_le_of_subset_lt_limRatioMeas` to
    show that the two measures are comparable up to `t`. -/
  have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'
  have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
  let ν := μ.withDensity (v.limRatioMeas hρ)
  let f := v.limRatioMeas hρ
  have f_meas : Measurable f := v.limRatioMeas_measurable hρ
  have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by
    

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
⊢ ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})

Next tactic:
refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))
/-- As an intermediate step to show that `μ.withDensity (v.limRatioMeas hρ) = ρ`, we show here
that `ρ ≤ t μ.withDensity (v.limRatioMeas hρ)` for any `t > 1`. -/
theorem le_mul_withDensity {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
    ρ s ≤ t * μ.withDensity (v.limRatioMeas hρ) s := by
  /- We cut `s` into the sets where `v.limRatioMeas hρ = 0`, where `v.limRatioMeas hρ = ∞`, and
    where `v.limRatioMeas hρ ∈ [t^n, t^(n+1))` for `n : ℤ`. The first and second have measure `0`.
    For the latter, since `v.limRatioMeas hρ` fluctuates by at most `t` on this slice, we can use
    `measure_le_mul_of_subset_limRatioMeas_lt` and `mul_measure_le_of_subset_lt_limRatioMeas` to
    show that the two measures are comparable up to `t`. -/
  have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'
  have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
  let ν := μ.withDensity (v.limRatioMeas hρ)
  let f := v.limRatioMeas hρ
  have f_meas : Measurable f := v.limRatioMeas_measurable hρ
  have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by
    refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))
    

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
⊢ ↑↑ρ (f ⁻¹' {0}) = 0

Next tactic:
exact v.measure_limRatioMeas_zero hρ
/-- As an intermediate step to show that `μ.withDensity (v.limRatioMeas hρ) = ρ`, we show here
that `ρ ≤ t μ.withDensity (v.limRatioMeas hρ)` for any `t > 1`. -/
theorem le_mul_withDensity {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
    ρ s ≤ t * μ.withDensity (v.limRatioMeas hρ) s := by
  /- We cut `s` into the sets where `v.limRatioMeas hρ = 0`, where `v.limRatioMeas hρ = ∞`, and
    where `v.limRatioMeas hρ ∈ [t^n, t^(n+1))` for `n : ℤ`. The first and second have measure `0`.
    For the latter, since `v.limRatioMeas hρ` fluctuates by at most `t` on this slice, we can use
    `measure_le_mul_of_subset_limRatioMeas_lt` and `mul_measure_le_of_subset_lt_limRatioMeas` to
    show that the two measures are comparable up to `t`. -/
  have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'
  have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
  let ν := μ.withDensity (v.limRatioMeas hρ)
  let f := v.limRatioMeas hρ
  have f_meas : Measurable f := v.limRatioMeas_measurable hρ
  have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by
    refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))
    exact v.measure_limRatioMeas_zero hρ
  

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s

Next tactic:
have B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞}) := by
    apply le_trans (le_of_eq _) (zero_le _)
    apply hρ
    rw [← nonpos_iff_eq_zero]
    exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le
/-- As an intermediate step to show that `μ.withDensity (v.limRatioMeas hρ) = ρ`, we show here
that `ρ ≤ t μ.withDensity (v.limRatioMeas hρ)` for any `t > 1`. -/
theorem le_mul_withDensity {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
    ρ s ≤ t * μ.withDensity (v.limRatioMeas hρ) s := by
  /- We cut `s` into the sets where `v.limRatioMeas hρ = 0`, where `v.limRatioMeas hρ = ∞`, and
    where `v.limRatioMeas hρ ∈ [t^n, t^(n+1))` for `n : ℤ`. The first and second have measure `0`.
    For the latter, since `v.limRatioMeas hρ` fluctuates by at most `t` on this slice, we can use
    `measure_le_mul_of_subset_limRatioMeas_lt` and `mul_measure_le_of_subset_lt_limRatioMeas` to
    show that the two measures are comparable up to `t`. -/
  have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'
  have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
  let ν := μ.withDensity (v.limRatioMeas hρ)
  let f := v.limRatioMeas hρ
  have f_meas : Measurable f := v.limRatioMeas_measurable hρ
  have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by
    refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))
    exact v.measure_limRatioMeas_zero hρ
  have B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞}) := by
    

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
⊢ ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})

Next tactic:
apply le_trans (le_of_eq _) (zero_le _)
/-- As an intermediate step to show that `μ.withDensity (v.limRatioMeas hρ) = ρ`, we show here
that `ρ ≤ t μ.withDensity (v.limRatioMeas hρ)` for any `t > 1`. -/
theorem le_mul_withDensity {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
    ρ s ≤ t * μ.withDensity (v.limRatioMeas hρ) s := by
  /- We cut `s` into the sets where `v.limRatioMeas hρ = 0`, where `v.limRatioMeas hρ = ∞`, and
    where `v.limRatioMeas hρ ∈ [t^n, t^(n+1))` for `n : ℤ`. The first and second have measure `0`.
    For the latter, since `v.limRatioMeas hρ` fluctuates by at most `t` on this slice, we can use
    `measure_le_mul_of_subset_limRatioMeas_lt` and `mul_measure_le_of_subset_lt_limRatioMeas` to
    show that the two measures are comparable up to `t`. -/
  have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'
  have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
  let ν := μ.withDensity (v.limRatioMeas hρ)
  let f := v.limRatioMeas hρ
  have f_meas : Measurable f := v.limRatioMeas_measurable hρ
  have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by
    refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))
    exact v.measure_limRatioMeas_zero hρ
  have B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞}) := by
    apply le_trans (le_of_eq _) (zero_le _)
    

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
⊢ ↑↑ρ (s ∩ f ⁻¹' {⊤}) = 0

Next tactic:
apply hρ
/-- As an intermediate step to show that `μ.withDensity (v.limRatioMeas hρ) = ρ`, we show here
that `ρ ≤ t μ.withDensity (v.limRatioMeas hρ)` for any `t > 1`. -/
theorem le_mul_withDensity {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
    ρ s ≤ t * μ.withDensity (v.limRatioMeas hρ) s := by
  /- We cut `s` into the sets where `v.limRatioMeas hρ = 0`, where `v.limRatioMeas hρ = ∞`, and
    where `v.limRatioMeas hρ ∈ [t^n, t^(n+1))` for `n : ℤ`. The first and second have measure `0`.
    For the latter, since `v.limRatioMeas hρ` fluctuates by at most `t` on this slice, we can use
    `measure_le_mul_of_subset_limRatioMeas_lt` and `mul_measure_le_of_subset_lt_limRatioMeas` to
    show that the two measures are comparable up to `t`. -/
  have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'
  have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
  let ν := μ.withDensity (v.limRatioMeas hρ)
  let f := v.limRatioMeas hρ
  have f_meas : Measurable f := v.limRatioMeas_measurable hρ
  have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by
    refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))
    exact v.measure_limRatioMeas_zero hρ
  have B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞}) := by
    apply le_trans (le_of_eq _) (zero_le _)
    apply hρ
    

Proof state:
case a
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) = 0

Next tactic:
rw [← nonpos_iff_eq_zero]
/-- As an intermediate step to show that `μ.withDensity (v.limRatioMeas hρ) = ρ`, we show here
that `ρ ≤ t μ.withDensity (v.limRatioMeas hρ)` for any `t > 1`. -/
theorem le_mul_withDensity {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
    ρ s ≤ t * μ.withDensity (v.limRatioMeas hρ) s := by
  /- We cut `s` into the sets where `v.limRatioMeas hρ = 0`, where `v.limRatioMeas hρ = ∞`, and
    where `v.limRatioMeas hρ ∈ [t^n, t^(n+1))` for `n : ℤ`. The first and second have measure `0`.
    For the latter, since `v.limRatioMeas hρ` fluctuates by at most `t` on this slice, we can use
    `measure_le_mul_of_subset_limRatioMeas_lt` and `mul_measure_le_of_subset_lt_limRatioMeas` to
    show that the two measures are comparable up to `t`. -/
  have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'
  have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
  let ν := μ.withDensity (v.limRatioMeas hρ)
  let f := v.limRatioMeas hρ
  have f_meas : Measurable f := v.limRatioMeas_measurable hρ
  have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by
    refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))
    exact v.measure_limRatioMeas_zero hρ
  have B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞}) := by
    apply le_trans (le_of_eq _) (zero_le _)
    apply hρ
    rw [← nonpos_iff_eq_zero]
    

Proof state:
case a
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) ≤ 0

Next tactic:
exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le
/-- As an intermediate step to show that `μ.withDensity (v.limRatioMeas hρ) = ρ`, we show here
that `ρ ≤ t μ.withDensity (v.limRatioMeas hρ)` for any `t > 1`. -/
theorem le_mul_withDensity {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
    ρ s ≤ t * μ.withDensity (v.limRatioMeas hρ) s := by
  /- We cut `s` into the sets where `v.limRatioMeas hρ = 0`, where `v.limRatioMeas hρ = ∞`, and
    where `v.limRatioMeas hρ ∈ [t^n, t^(n+1))` for `n : ℤ`. The first and second have measure `0`.
    For the latter, since `v.limRatioMeas hρ` fluctuates by at most `t` on this slice, we can use
    `measure_le_mul_of_subset_limRatioMeas_lt` and `mul_measure_le_of_subset_lt_limRatioMeas` to
    show that the two measures are comparable up to `t`. -/
  have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'
  have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
  let ν := μ.withDensity (v.limRatioMeas hρ)
  let f := v.limRatioMeas hρ
  have f_meas : Measurable f := v.limRatioMeas_measurable hρ
  have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by
    refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))
    exact v.measure_limRatioMeas_zero hρ
  have B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞}) := by
    apply le_trans (le_of_eq _) (zero_le _)
    apply hρ
    rw [← nonpos_iff_eq_zero]
    exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le
  have C :
    ∀ n : ℤ,
      ρ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) ≤
        (t • ν) (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) := by
    

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
⊢ ∀ (n : ℤ), ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))

Next tactic:
intro n
/-- As an intermediate step to show that `μ.withDensity (v.limRatioMeas hρ) = ρ`, we show here
that `ρ ≤ t μ.withDensity (v.limRatioMeas hρ)` for any `t > 1`. -/
theorem le_mul_withDensity {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
    ρ s ≤ t * μ.withDensity (v.limRatioMeas hρ) s := by
  /- We cut `s` into the sets where `v.limRatioMeas hρ = 0`, where `v.limRatioMeas hρ = ∞`, and
    where `v.limRatioMeas hρ ∈ [t^n, t^(n+1))` for `n : ℤ`. The first and second have measure `0`.
    For the latter, since `v.limRatioMeas hρ` fluctuates by at most `t` on this slice, we can use
    `measure_le_mul_of_subset_limRatioMeas_lt` and `mul_measure_le_of_subset_lt_limRatioMeas` to
    show that the two measures are comparable up to `t`. -/
  have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'
  have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
  let ν := μ.withDensity (v.limRatioMeas hρ)
  let f := v.limRatioMeas hρ
  have f_meas : Measurable f := v.limRatioMeas_measurable hρ
  have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by
    refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))
    exact v.measure_limRatioMeas_zero hρ
  have B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞}) := by
    apply le_trans (le_of_eq _) (zero_le _)
    apply hρ
    rw [← nonpos_iff_eq_zero]
    exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le
  have C :
    ∀ n : ℤ,
      ρ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) ≤
        (t • ν) (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) := by
    intro n
    

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
n : ℤ
⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))

Next tactic:
let I := Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))
/-- As an intermediate step to show that `μ.withDensity (v.limRatioMeas hρ) = ρ`, we show here
that `ρ ≤ t μ.withDensity (v.limRatioMeas hρ)` for any `t > 1`. -/
theorem le_mul_withDensity {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
    ρ s ≤ t * μ.withDensity (v.limRatioMeas hρ) s := by
  /- We cut `s` into the sets where `v.limRatioMeas hρ = 0`, where `v.limRatioMeas hρ = ∞`, and
    where `v.limRatioMeas hρ ∈ [t^n, t^(n+1))` for `n : ℤ`. The first and second have measure `0`.
    For the latter, since `v.limRatioMeas hρ` fluctuates by at most `t` on this slice, we can use
    `measure_le_mul_of_subset_limRatioMeas_lt` and `mul_measure_le_of_subset_lt_limRatioMeas` to
    show that the two measures are comparable up to `t`. -/
  have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'
  have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
  let ν := μ.withDensity (v.limRatioMeas hρ)
  let f := v.limRatioMeas hρ
  have f_meas : Measurable f := v.limRatioMeas_measurable hρ
  have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by
    refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))
    exact v.measure_limRatioMeas_zero hρ
  have B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞}) := by
    apply le_trans (le_of_eq _) (zero_le _)
    apply hρ
    rw [← nonpos_iff_eq_zero]
    exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le
  have C :
    ∀ n : ℤ,
      ρ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) ≤
        (t • ν) (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) := by
    intro n
    let I := Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))
    

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))

Next tactic:
have M : MeasurableSet (s ∩ f ⁻¹' I) := hs.inter (f_meas measurableSet_Ico)
/-- As an intermediate step to show that `μ.withDensity (v.limRatioMeas hρ) = ρ`, we show here
that `ρ ≤ t μ.withDensity (v.limRatioMeas hρ)` for any `t > 1`. -/
theorem le_mul_withDensity {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
    ρ s ≤ t * μ.withDensity (v.limRatioMeas hρ) s := by
  /- We cut `s` into the sets where `v.limRatioMeas hρ = 0`, where `v.limRatioMeas hρ = ∞`, and
    where `v.limRatioMeas hρ ∈ [t^n, t^(n+1))` for `n : ℤ`. The first and second have measure `0`.
    For the latter, since `v.limRatioMeas hρ` fluctuates by at most `t` on this slice, we can use
    `measure_le_mul_of_subset_limRatioMeas_lt` and `mul_measure_le_of_subset_lt_limRatioMeas` to
    show that the two measures are comparable up to `t`. -/
  have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'
  have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
  let ν := μ.withDensity (v.limRatioMeas hρ)
  let f := v.limRatioMeas hρ
  have f_meas : Measurable f := v.limRatioMeas_measurable hρ
  have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by
    refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))
    exact v.measure_limRatioMeas_zero hρ
  have B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞}) := by
    apply le_trans (le_of_eq _) (zero_le _)
    apply hρ
    rw [← nonpos_iff_eq_zero]
    exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le
  have C :
    ∀ n : ℤ,
      ρ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) ≤
        (t • ν) (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) := by
    intro n
    let I := Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))
    have M : MeasurableSet (s ∩ f ⁻¹' I) := hs.inter (f_meas measurableSet_Ico)
    

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))

Next tactic:
simp only [M, withDensity_apply, coe_nnreal_smul_apply]