awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x : α | (↑q : ℝ≥0∞) < v.limRatioMeas hρ x}
t : Set α :=
  {x : α |
    Tendsto
      (fun (a : Set α) ↦
        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
      (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
x : α
hx : x ∈ s ∩ t
I :
  ∀ᶠ (a : Set α) in v.filterAt x,
    (↑q : ℝ≥0∞) <
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
apply :
  ∀ (a : Set α)
    (ha :
      (↑q : ℝ≥0∞) <
        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)),
    LE.le (α := ℝ≥0∞) ((q • μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
⊢ ∃ᶠ (a : Set α) in v.filterAt x,
    LE.le (α := ℝ≥0∞) ((q • μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)

Tactic:
apply I.frequently.mono fun a ha => ?_

Premises used: Filter.Frequently.mono, Filter.Eventually.frequently
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x : α | (↑q : ℝ≥0∞) < v.limRatioMeas hρ x}
t : Set α :=
  {x : α |
    Tendsto
      (fun (a : Set α) ↦
        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
      (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
x : α
hx : x ∈ s ∩ t
I :
  ∀ᶠ (a : Set α) in v.filterAt x,
    (↑q : ℝ≥0∞) <
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
a : Set α
ha :
  (↑q : ℝ≥0∞) <
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
rw : HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ≤ (ρ : (a : Set α) → ℝ≥0∞) a
⊢ LE.le (α := ℝ≥0∞) ((q • μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)

Tactic:
rw [coe_nnreal_smul_apply]

Premises used: MeasureTheory.Measure.coe_nnreal_smul_apply
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x : α | (↑q : ℝ≥0∞) < v.limRatioMeas hρ x}
t : Set α :=
  {x : α |
    Tendsto
      (fun (a : Set α) ↦
        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
      (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
x : α
hx : x ∈ s ∩ t
I :
  ∀ᶠ (a : Set α) in v.filterAt x,
    (↑q : ℝ≥0∞) <
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
a : Set α
ha :
  (↑q : ℝ≥0∞) <
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
⊢ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ≤ (ρ : (a : Set α) → ℝ≥0∞) a

Tactic:
ENNReal.mul_le_of_le_div ha.le

Premises used: LT.lt.le, ENNReal.mul_le_of_le_div
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x : α | (↑q : ℝ≥0∞) < v.limRatioMeas hρ x}
t : Set α :=
  {x : α |
    Tendsto
      (fun (a : Set α) ↦
        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
      (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
A : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
x : α
hx : x ∈ s ∩ t
I :
  ∀ᶠ (a : Set α) in v.filterAt x,
    (↑q : ℝ≥0∞) <
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
a : Set α
ha :
  (↑q : ℝ≥0∞) <
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
⊢ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ≤ (ρ : (a : Set α) → ℝ≥0∞) a

Tactic:
exact ENNReal.mul_le_of_le_div ha.le

Premises used: LT.lt.le, ENNReal.mul_le_of_le_div
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {x : α | v.limRatioMeas hρ x = ∞} : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
measure_null_of_locally_null _ fun x _ => ?_

Premises used: MeasureTheory.measure_null_of_locally_null
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
refine :
  ∀ (x : α),
    Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x →
      ∃ u ∈ 𝓝[{x : α | v.limRatioMeas hρ x = ∞}] x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {x : α | v.limRatioMeas hρ x = ∞} : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
refine measure_null_of_locally_null _ fun x _ => ?_

Premises used: MeasureTheory.measure_null_of_locally_null
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
⊢ ∃ (o : Set α),
    x ∈ o ∧
      IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o ∧
        LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
Measure.exists_isOpen_measure_lt_top ρ x

Premises used: MeasureTheory.Measure.exists_isOpen_measure_lt_top
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
obtain :
  ∀ (o : Set α) (xo : x ∈ o) (o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o)
    (μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)),
    ∃ u ∈ 𝓝[{x : α | v.limRatioMeas hρ x = ∞}] x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ ∃ u ∈ 𝓝[{x : α | v.limRatioMeas hρ x = ∞}] x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ ρ o < ∞ :=
    Measure.exists_isOpen_measure_lt_top ρ x

Premises used: MeasureTheory.Measure.exists_isOpen_measure_lt_top, Set
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
⊢ ∃ u ∈ 𝓝[{x : α | v.limRatioMeas hρ x = ∞}] x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
⟨s, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), le_antisymm ?_ bot_le⟩

Premises used: inter_mem_nhdsWithin, le_antisymm, And.intro, IsOpen.mem_nhds, Exists.intro, bot_le
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
refine : LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ ∃ u ∈ 𝓝[{x : α | v.limRatioMeas hρ x = ∞}] x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
refine ⟨s, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), le_antisymm ?_ bot_le⟩
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
⊢ Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
have ρs : ρ s ≠ ∞ := measure_ne_top_of_subset inter_subset_right μo.ne
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
ρs : Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
q : ℝ≥0
hq : 1 ≤ q
rw : HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ≤ (ρ : (a : Set α) → ℝ≥0∞) s
rw₁ : (↑q : ℝ≥0∞) ≠ 0 ∨ Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ (μ : (a : Set α) → ℝ≥0∞) s ≤ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)⁻¹ ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)

Tactic:
rw [mul_comm, ← div_eq_mul_inv, ENNReal.le_div_iff_mul_le _ (Or.inr ρs), mul_comm]

Premises used: ENNReal.le_div_iff_mul_le, div_eq_mul_inv, mul_comm, Or.inr
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
ρs : Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
q : ℝ≥0
hq : 1 ≤ q
apply : s ⊆ {x : α | (↑q : ℝ≥0∞) < v.limRatioMeas hρ x}
⊢ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ≤ (ρ : (a : Set α) → ℝ≥0∞) s

Tactic:
apply v.mul_measure_le_of_subset_lt_limRatioMeas hρ

Premises used: VitaliFamily.mul_measure_le_of_subset_lt_limRatioMeas
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
ρs : Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
q : ℝ≥0
hq : 1 ≤ q
y : α
hy : y ∈ s
⊢ v.limRatioMeas hρ y = ∞

Tactic:
have : v.limRatioMeas hρ y = ∞ := hy.1
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
ρs : Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
q : ℝ≥0
hq : 1 ≤ q
y : α
hy : y ∈ s
this : v.limRatioMeas hρ y = ∞
⊢ Membership.mem (γ := Set α) {x : α | (↑q : ℝ≥0∞) < v.limRatioMeas hρ x} y

Tactic:
simp only [this, ENNReal.coe_lt_top, mem_setOf_eq]

Premises used: Set.mem_setOf_eq, ENNReal.coe_lt_top
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
ρs : Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
q : ℝ≥0
hq : 1 ≤ q
⊢ (↑q : ℝ≥0∞) ≠ 0 ∨ Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
simp only [(zero_lt_one.trans_le hq).ne', true_or, ENNReal.coe_eq_zero, Ne,
        not_false_iff]

Premises used: true_or, Ne, ENNReal.coe_eq_zero, LT.lt.ne', zero_lt_one, LT.lt.trans_le, not_false_iff
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
ρs : Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ ∀ (q : ℝ≥0),
    1 ≤ q → (μ : (a : Set α) → ℝ≥0∞) s ≤ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)⁻¹ ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)

Tactic:
have A : ∀ q : ℝ≥0, 1 ≤ q → μ s ≤ (q : ℝ≥0∞)⁻¹ * ρ s := by
    intro q hq
    rw [mul_comm, ← div_eq_mul_inv, ENNReal.le_div_iff_mul_le _ (Or.inr ρs), mul_comm]
    · apply v.mul_measure_le_of_subset_lt_limRatioMeas hρ
      intro y hy
      have : v.limRatioMeas hρ y = ∞ := hy.1
      simp only [this, ENNReal.coe_lt_top, mem_setOf_eq]
    · simp only [(zero_lt_one.trans_le hq).ne', true_or, ENNReal.coe_eq_zero, Ne,
        not_false_iff]
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
ρs : Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
A :
  ∀ (q : ℝ≥0),
    1 ≤ q → (μ : (a : Set α) → ℝ≥0∞) s ≤ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)⁻¹ ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
apply : Tendsto (β := ℝ≥0∞) (fun (x : ℝ≥0) ↦ (↑x : ℝ≥0∞)⁻¹) atTop (𝓝 ∞⁻¹)
⊢ Tendsto (fun (q : ℝ≥0) ↦ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)⁻¹ ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)) atTop
    (𝓝 (HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞⁻¹ : ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)))

Tactic:
apply ENNReal.Tendsto.mul_const _ (Or.inr ρs)

Premises used: ENNReal.Tendsto.mul_const, Or.inr
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
ρs : Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
A :
  ∀ (q : ℝ≥0),
    1 ≤ q → (μ : (a : Set α) → ℝ≥0∞) s ≤ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)⁻¹ ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
⊢ Tendsto (β := ℝ≥0∞) (fun (x : ℝ≥0) ↦ (↑x : ℝ≥0∞)⁻¹) atTop (𝓝 ∞⁻¹)

Tactic:
tendsto_inv_iff.2 (ENNReal.tendsto_coe_nhds_top.2 tendsto_id)

Premises used: Iff.mpr, tendsto_inv_iff, Filter.tendsto_id, ENNReal.tendsto_coe_nhds_top
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
ρs : Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
A :
  ∀ (q : ℝ≥0),
    1 ≤ q → (μ : (a : Set α) → ℝ≥0∞) s ≤ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)⁻¹ ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
⊢ Tendsto (β := ℝ≥0∞) (fun (x : ℝ≥0) ↦ (↑x : ℝ≥0∞)⁻¹) atTop (𝓝 ∞⁻¹)

Tactic:
exact tendsto_inv_iff.2 (ENNReal.tendsto_coe_nhds_top.2 tendsto_id)

Premises used: Iff.mpr, tendsto_inv_iff, Filter.tendsto_id, ENNReal.tendsto_coe_nhds_top
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
ρs : Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
A :
  ∀ (q : ℝ≥0),
    1 ≤ q → (μ : (a : Set α) → ℝ≥0∞) s ≤ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)⁻¹ ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
⊢ Tendsto (fun (q : ℝ≥0) ↦ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)⁻¹ ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)) atTop
    (𝓝 (HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞⁻¹ : ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)))

Tactic:
have B : Tendsto (fun q : ℝ≥0 => (q : ℝ≥0∞)⁻¹ * ρ s) atTop (𝓝 (∞⁻¹ * ρ s)) := by
    apply ENNReal.Tendsto.mul_const _ (Or.inr ρs)
    exact tendsto_inv_iff.2 (ENNReal.tendsto_coe_nhds_top.2 tendsto_id)
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
ρs : Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
A :
  ∀ (q : ℝ≥0),
    1 ≤ q → (μ : (a : Set α) → ℝ≥0∞) s ≤ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)⁻¹ ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
B :
  Tendsto (fun (q : ℝ≥0) ↦ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)⁻¹ ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)) atTop
    (𝓝 (HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) (∞⁻¹ : ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)))
⊢ Tendsto (β := ℝ≥0∞) (fun (q : ℝ≥0) ↦ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)⁻¹ ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)) atTop
    (𝓝 0)

Tactic:
simp only [zero_mul, ENNReal.inv_top] at B

Premises used: MulZeroClass.zero_mul, ENNReal.inv_top
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
h : Membership.mem (γ := Set α) {x : α | v.limRatioMeas hρ x = ∞} x
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
ρs : Ne (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
A :
  ∀ (q : ℝ≥0),
    1 ≤ q → (μ : (a : Set α) → ℝ≥0∞) s ≤ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)⁻¹ ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
B :
  Tendsto (β := ℝ≥0∞) (fun (q : ℝ≥0) ↦ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)⁻¹ ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)) atTop
    (𝓝 0)
apply :
  ∀ᶠ (c : ℝ≥0) in atTop,
    (μ : (a : Set α) → ℝ≥0∞) s ≤ HMul.hMul (β := ℝ≥0∞) (↑c : ℝ≥0∞)⁻¹ ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
apply ge_of_tendsto B

Premises used: ge_of_tendsto