awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
i : ℕ
hi : p i → 0 < r i
pi : p i
nlinarith : r i ≤ 1 / 2 * r i → 2 / 2 = 1
⊢ 1 / 2 * r i < r i
have nlinarith₁ : r i ≤ 1 / 2 * r i → HMul.hMul (α := ℝ) 1 2 = 2
Tactic: nlinarith [hi pi]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
i : ℕ
hi : p i → 0 < r i
pi : p i
nlinarith : r i ≤ 1 / 2 * r i → 2 / 2 = 1
nlinarith₁ : r i ≤ 1 / 2 * r i → HMul.hMul (α := ℝ) 1 2 = 2
⊢ 1 / 2 * r i < r i
have nlinarith₂ : r i ≤ 1 / 2 * r i → HMul.hMul (α := ℝ) 2 1 = 2
Tactic: nlinarith [hi pi]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
i : ℕ
hi : p i → 0 < r i
pi : p i
nlinarith : r i ≤ 1 / 2 * r i → 2 / 2 = 1
nlinarith₁ : r i ≤ 1 / 2 * r i → HMul.hMul (α := ℝ) 1 2 = 2
nlinarith₂ : r i ≤ 1 / 2 * r i → HMul.hMul (α := ℝ) 2 1 = 2
⊢ 1 / 2 * r i < r i
have nlinarith₃ : r i ≤ 1 / 2 * r i → GT.gt (α := ℝ) 2 0
Tactic: nlinarith [hi pi]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
i : ℕ
hi : p i → 0 < r i
pi : p i
nlinarith : r i ≤ 1 / 2 * r i → 2 / 2 = 1
nlinarith₁ : r i ≤ 1 / 2 * r i → HMul.hMul (α := ℝ) 1 2 = 2
nlinarith₂ : r i ≤ 1 / 2 * r i → HMul.hMul (α := ℝ) 2 1 = 2
nlinarith₃ : r i ≤ 1 / 2 * r i → GT.gt (α := ℝ) 2 0
⊢ 1 / 2 * r i < r i
have nlinarith₄ : r i ≤ 1 / 2 * r i → 2 * r i - HMul.hMul (α := ℝ) 1 1 * (1 * r i) + (0 - r i) = 0
Tactic: nlinarith [hi pi]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (hypothesis)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
⊢ (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ thickening (M * r i) (s i)) atTop p)
    (blimsup (fun (i : ℕ) ↦ thickening (r i) (s i)) atTop p)
have h₁ : (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ cthickening (r i) (s i)) atTop p)
  (blimsup (fun (i : ℕ) ↦ thickening (r i) (s i)) atTop p)
Tactic: have h₁ := blimsup_cthickening_ae_eq_blimsup_thickening (s := s) μ hr hr'
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (hypothesis)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ :
  (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ cthickening (r i) (s i)) atTop p)
    (blimsup (fun (i : ℕ) ↦ thickening (r i) (s i)) atTop p)
⊢ (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ thickening (M * r i) (s i)) atTop p)
    (blimsup (fun (i : ℕ) ↦ thickening (r i) (s i)) atTop p)
have h₂ : (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ cthickening (M * r i) (s i)) atTop p)
  (blimsup (fun (i : ℕ) ↦ cthickening (r i) (s i)) atTop p)
Tactic: have h₂ := blimsup_cthickening_mul_ae_eq μ p s hM r hr
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ :
  (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ cthickening (r i) (s i)) atTop p)
    (blimsup (fun (i : ℕ) ↦ thickening (r i) (s i)) atTop p)
h₂ :
  (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ cthickening (M * r i) (s i)) atTop p)
    (blimsup (fun (i : ℕ) ↦ cthickening (r i) (s i)) atTop p)
⊢ Tendsto (β := ℝ) (fun (i : ℕ) ↦ M * r i) atTop (𝓝 0)
have h : 0 = M * 0
Tactic: convert! hr.const_mul M
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (hypothesis)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ :
  (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ cthickening (r i) (s i)) atTop p)
    (blimsup (fun (i : ℕ) ↦ thickening (r i) (s i)) atTop p)
h₂ :
  (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ cthickening (M * r i) (s i)) atTop p)
    (blimsup (fun (i : ℕ) ↦ cthickening (r i) (s i)) atTop p)
⊢ (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ thickening (M * r i) (s i)) atTop p)
    (blimsup (fun (i : ℕ) ↦ thickening (r i) (s i)) atTop p)
have hr₁ : Tendsto (β := ℝ) (fun (i : ℕ) ↦ M * r i) atTop (𝓝 0)
Tactic: replace hr : Tendsto (fun i => M * r i) atTop (𝓝 0) := by convert! hr.const_mul M; simp
Cleared hypotheses: hr
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (hypothesis)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i
h₁ :
  (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ cthickening (r i) (s i)) atTop p)
    (blimsup (fun (i : ℕ) ↦ thickening (r i) (s i)) atTop p)
h₂ :
  (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ cthickening (M * r i) (s i)) atTop p)
    (blimsup (fun (i : ℕ) ↦ cthickening (r i) (s i)) atTop p)
hr : Tendsto (β := ℝ) (fun (i : ℕ) ↦ M * r i) atTop (𝓝 0)
⊢ (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ thickening (M * r i) (s i)) atTop p)
    (blimsup (fun (i : ℕ) ↦ thickening (r i) (s i)) atTop p)
have hr'₁ : ∀ᶠ (i : ℕ) in atTop, p i → 0 < M * r i
Tactic: replace hr' : ∀ᶠ i in atTop, p i → 0 < M * r i := hr'.mono fun i hi hip ↦ mul_pos hM (hi hip)
Cleared hypotheses: hr'
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (hypothesis)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun (i : ℕ) ↦ p i ∧ 0 < r i
⊢ (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ thickening (M * r i) (s i)) atTop p)
    (blimsup (fun (i : ℕ) ↦ thickening (r i) (s i)) atTop p)
have hq : ∀ {u : ℕ → Set α} (hu : ∀ (i : ℕ), u i ≠ ∅ → 0 < r i), blimsup u atTop p = blimsup u atTop q
Tactic: have hq {u : ℕ → Set α} (hu : ∀ i, u i ≠ ∅ → 0 < r i) :
      blimsup u atTop p = blimsup u atTop q :=
    blimsup_congr' <| Eventually.of_forall fun i hi ↦ by simp [q, hu i hi]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
q : ℕ → Prop := fun (i : ℕ) ↦ p i ∧ 0 < r i
hq : ∀ {u : ℕ → Set α} (hu : ∀ (i : ℕ), u i ≠ ∅ → 0 < r i), blimsup u atTop p = blimsup u atTop q
⊢ (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ thickening (M * r i) (s i)) atTop p)
    (blimsup (fun (i : ℕ) ↦ thickening (r i) (s i)) atTop p)
have rw : (ae μ).EventuallyEq (β := Prop) (blimsup (fun (i : ℕ) ↦ thickening (M * r i) (s i)) atTop q)
  (blimsup (fun (i : ℕ) ↦ thickening (r i) (s i)) atTop q)
Tactic: rw [hq fun i hi ↦ (thickening_nonempty_iff.1 <| nonempty_iff_ne_empty.2 hi).1,
    hq fun i hi ↦ (mul_pos_iff_of_pos_left hM).1 <|
      (thickening_nonempty_iff.1 <| nonempty_iff_ne_empty.2 hi).1]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
⊢ Icc x y ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x
have rw : Metric.closedBall ((x + y) / 2) ((y - x) / 2) ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x
Tactic: rw [Icc_eq_closedBall]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
⊢ 0 < (y - x) / 2
have linarith : (y - x) / 2 ≤ 0 → 2 / 2 = 1
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : (y - x) / 2 ≤ 0 → 2 / 2 = 1
⊢ 0 < (y - x) / 2
have linarith₁ : (y - x) / 2 ≤ 0 → HMul.hMul (α := ℝ) 1 2 = 2
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : (y - x) / 2 ≤ 0 → 2 / 2 = 1
linarith₁ : (y - x) / 2 ≤ 0 → HMul.hMul (α := ℝ) 1 2 = 2
⊢ 0 < (y - x) / 2
have linarith₂ : (y - x) / 2 ≤ 0 → GT.gt (α := ℝ) 2 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : (y - x) / 2 ≤ 0 → 2 / 2 = 1
linarith₁ : (y - x) / 2 ≤ 0 → HMul.hMul (α := ℝ) 1 2 = 2
linarith₂ : (y - x) / 2 ≤ 0 → GT.gt (α := ℝ) 2 0
⊢ 0 < (y - x) / 2
have linarith₃ : (y - x) / 2 ≤ 0 → x - y + (1 * y - 1 * x - HMul.hMul (α := ℝ) 2 0) = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
⊢ Metric.closedBall ((x + y) / 2) ((y - x) / 2) ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x
have refine : dist x ((x + y) / 2) ≤ 1 * ((y - x) / 2)
Tactic: refine closedBall_mem_vitaliFamily_of_dist_le_mul _ ?_ (by linarith)
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
⊢ dist x ((x + y) / 2) ≤ 1 * ((y - x) / 2)
have rw : (x + y) / 2 - x ≤ 1 * ((y - x) / 2)
Tactic: rw [dist_comm, Real.dist_eq, abs_of_nonneg]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
rw : (x + y) / 2 - x ≤ 1 * ((y - x) / 2)
⊢ dist x ((x + y) / 2) ≤ 1 * ((y - x) / 2)
have rw₁ : 0 ≤ (x + y) / 2 - x
Tactic: rw [dist_comm, Real.dist_eq, abs_of_nonneg]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
⊢ (x + y) / 2 - x ≤ 1 * ((y - x) / 2)
have linarith : 1 * ((y - x) / 2) < (x + y) / 2 - x → 2 / 2 = 1
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : 1 * ((y - x) / 2) < (x + y) / 2 - x → 2 / 2 = 1
⊢ (x + y) / 2 - x ≤ 1 * ((y - x) / 2)
have linarith₁ : 1 * ((y - x) / 2) < (x + y) / 2 - x → HMul.hMul (α := ℝ) 1 2 = 2
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : 1 * ((y - x) / 2) < (x + y) / 2 - x → 2 / 2 = 1
linarith₁ linarith₂ : 1 * ((y - x) / 2) < (x + y) / 2 - x → HMul.hMul (α := ℝ) 1 2 = 2
linarith₃ : 1 * ((y - x) / 2) < (x + y) / 2 - x → 2 / 2 = 1
linarith₄ : 1 * ((y - x) / 2) < (x + y) / 2 - x → HMul.hMul (α := ℝ) 1 2 = 2
⊢ (x + y) / 2 - x ≤ 1 * ((y - x) / 2)
have linarith₅ : 1 * ((y - x) / 2) < (x + y) / 2 - x → GT.gt (α := ℝ) 2 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : 1 * ((y - x) / 2) < (x + y) / 2 - x → 2 / 2 = 1
linarith₁ linarith₂ : 1 * ((y - x) / 2) < (x + y) / 2 - x → HMul.hMul (α := ℝ) 1 2 = 2
linarith₃ : 1 * ((y - x) / 2) < (x + y) / 2 - x → 2 / 2 = 1
linarith₄ : 1 * ((y - x) / 2) < (x + y) / 2 - x → HMul.hMul (α := ℝ) 1 2 = 2
linarith₅ : 1 * ((y - x) / 2) < (x + y) / 2 - x → GT.gt (α := ℝ) 2 0
⊢ (x + y) / 2 - x ≤ 1 * ((y - x) / 2)
have linarith₆ : 1 * ((y - x) / 2) < (x + y) / 2 - x → HMul.hMul (α := ℝ) 1 1 * (1 * y - 1 * x) - (1 * x + 1 * y - 2 * x) = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
⊢ 0 ≤ (x + y) / 2 - x
have linarith : (x + y) / 2 - x < 0 → 2 / 2 = 1
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : (x + y) / 2 - x < 0 → 2 / 2 = 1
⊢ 0 ≤ (x + y) / 2 - x
have linarith₁ : (x + y) / 2 - x < 0 → HMul.hMul (α := ℝ) 1 2 = 2
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : (x + y) / 2 - x < 0 → 2 / 2 = 1
linarith₁ : (x + y) / 2 - x < 0 → HMul.hMul (α := ℝ) 1 2 = 2
⊢ 0 ≤ (x + y) / 2 - x
have linarith₂ : (x + y) / 2 - x < 0 → GT.gt (α := ℝ) 2 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : (x + y) / 2 - x < 0 → 2 / 2 = 1
linarith₁ : (x + y) / 2 - x < 0 → HMul.hMul (α := ℝ) 1 2 = 2
linarith₂ : (x + y) / 2 - x < 0 → GT.gt (α := ℝ) 2 0
⊢ 0 ≤ (x + y) / 2 - x
have linarith₃ : (x + y) / 2 - x < 0 → x - y + (1 * x + 1 * y - 2 * x - HMul.hMul (α := ℝ) 2 0) = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x : ℝ
⊢ Tendsto (fun (y : ℝ) ↦ Icc x y) (𝓝[>] x) (VitaliFamily.filterAt (μ := volume) (vitaliFamily volume 1) x)
have refine_1 : ∀ᶠ (i : ℝ) in 𝓝[>] x, Icc x i ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x
Tactic: refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x : ℝ
refine_1 : ∀ᶠ (i : ℝ) in 𝓝[>] x, Icc x i ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x
⊢ Tendsto (fun (y : ℝ) ↦ Icc x y) (𝓝[>] x) (VitaliFamily.filterAt (μ := volume) (vitaliFamily volume 1) x)
have refine_2 : ∀ ε > 0, ∀ᶠ (i : ℝ) in 𝓝[>] x, Icc x i ⊆ Metric.closedBall x ε
Tactic: refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x : ℝ
⊢ ∀ᶠ (i : ℝ) in 𝓝[>] x, Icc x i ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x
have h : ∀ (a : ℝ),
  Membership.mem (γ := Set ℝ)
    {x_1 : ℝ |
      x_1 ∈ Ioi x →
        Membership.mem (γ := Set ℝ)
          {x_2 : ℝ | (fun (i : ℝ) ↦ Icc x i ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x) x_2} x_1}
    a
Tactic: filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_right hy
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x : ℝ
h :
  ∀ (a : ℝ),
    Membership.mem (γ := Set ℝ)
      {x_1 : ℝ |
        x_1 ∈ Ioi x →
          Membership.mem (γ := Set ℝ)
            {x_2 : ℝ | (fun (i : ℝ) ↦ Icc x i ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x) x_2} x_1}
      a
⊢ ∀ᶠ (i : ℝ) in 𝓝[>] x, Icc x i ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x
have h₁ : ∀ a ∈ Ioi x, Icc x a ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x
Tactic: filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_right hy
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x ε : ℝ
εpos : ε > 0
⊢ x < x + ε
have linarith : x + ε ≤ x → 0 - ε + (x + ε - x) = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x ε : ℝ
εpos : ε > 0
⊢ ∀ᶠ (i : ℝ) in 𝓝[>] x, Icc x i ⊆ Metric.closedBall x ε
have h : ∀ (y : ℝ) (hy : y ∈ Icc x (x + ε)), Icc x y ⊆ Metric.closedBall x ε
Tactic: filter_upwards [Icc_mem_nhdsGT <| show x < x + ε by linarith] with y hy
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x ε : ℝ
εpos : ε > 0
y : ℝ
hy : y ∈ Icc x (x + ε)
⊢ Icc x y ⊆ Metric.closedBall x ε
have h : Icc x y ⊆ Icc (x - ε) (x + ε)
Tactic: rw [closedBall_eq_Icc]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x ε : ℝ
εpos : ε > 0
y : ℝ
hy : y ∈ Icc x (x + ε)
⊢ x - ε ≤ x
have linarith : x < x - ε → 0 - ε + (x - (x - ε)) = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
⊢ Icc x y ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) y
have rw : Metric.closedBall ((x + y) / 2) ((y - x) / 2) ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) y
Tactic: rw [Icc_eq_closedBall]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
⊢ Metric.closedBall ((x + y) / 2) ((y - x) / 2) ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) y
have refine : dist y ((x + y) / 2) ≤ 1 * ((y - x) / 2)
Tactic: refine closedBall_mem_vitaliFamily_of_dist_le_mul _ ?_ (by linarith)
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
⊢ dist y ((x + y) / 2) ≤ 1 * ((y - x) / 2)
have rw : y - (x + y) / 2 ≤ 1 * ((y - x) / 2)
Tactic: rw [Real.dist_eq, abs_of_nonneg]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
rw : y - (x + y) / 2 ≤ 1 * ((y - x) / 2)
⊢ dist y ((x + y) / 2) ≤ 1 * ((y - x) / 2)
have rw₁ : 0 ≤ y - (x + y) / 2
Tactic: rw [Real.dist_eq, abs_of_nonneg]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
⊢ y - (x + y) / 2 ≤ 1 * ((y - x) / 2)
have linarith : 1 * ((y - x) / 2) < y - (x + y) / 2 → 2 / 2 = 1
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : 1 * ((y - x) / 2) < y - (x + y) / 2 → 2 / 2 = 1
⊢ y - (x + y) / 2 ≤ 1 * ((y - x) / 2)
have linarith₁ : 1 * ((y - x) / 2) < y - (x + y) / 2 → HMul.hMul (α := ℝ) 1 2 = 2
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : 1 * ((y - x) / 2) < y - (x + y) / 2 → 2 / 2 = 1
linarith₁ linarith₂ : 1 * ((y - x) / 2) < y - (x + y) / 2 → HMul.hMul (α := ℝ) 1 2 = 2
linarith₃ : 1 * ((y - x) / 2) < y - (x + y) / 2 → 2 / 2 = 1
linarith₄ : 1 * ((y - x) / 2) < y - (x + y) / 2 → HMul.hMul (α := ℝ) 1 2 = 2
⊢ y - (x + y) / 2 ≤ 1 * ((y - x) / 2)
have linarith₅ : 1 * ((y - x) / 2) < y - (x + y) / 2 → GT.gt (α := ℝ) 2 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : 1 * ((y - x) / 2) < y - (x + y) / 2 → 2 / 2 = 1
linarith₁ linarith₂ : 1 * ((y - x) / 2) < y - (x + y) / 2 → HMul.hMul (α := ℝ) 1 2 = 2
linarith₃ : 1 * ((y - x) / 2) < y - (x + y) / 2 → 2 / 2 = 1
linarith₄ : 1 * ((y - x) / 2) < y - (x + y) / 2 → HMul.hMul (α := ℝ) 1 2 = 2
linarith₅ : 1 * ((y - x) / 2) < y - (x + y) / 2 → GT.gt (α := ℝ) 2 0
⊢ y - (x + y) / 2 ≤ 1 * ((y - x) / 2)
have linarith₆ : 1 * ((y - x) / 2) < y - (x + y) / 2 → HMul.hMul (α := ℝ) 1 1 * (1 * y - 1 * x) - (2 * y - (1 * x + 1 * y)) = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
⊢ 0 ≤ y - (x + y) / 2
have linarith : y - (x + y) / 2 < 0 → 2 / 2 = 1
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : y - (x + y) / 2 < 0 → 2 / 2 = 1
⊢ 0 ≤ y - (x + y) / 2
have linarith₁ : y - (x + y) / 2 < 0 → HMul.hMul (α := ℝ) 1 2 = 2
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : y - (x + y) / 2 < 0 → 2 / 2 = 1
linarith₁ : y - (x + y) / 2 < 0 → HMul.hMul (α := ℝ) 1 2 = 2
⊢ 0 ≤ y - (x + y) / 2
have linarith₂ : y - (x + y) / 2 < 0 → GT.gt (α := ℝ) 2 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x y : ℝ
hxy : x < y
linarith : y - (x + y) / 2 < 0 → 2 / 2 = 1
linarith₁ : y - (x + y) / 2 < 0 → HMul.hMul (α := ℝ) 1 2 = 2
linarith₂ : y - (x + y) / 2 < 0 → GT.gt (α := ℝ) 2 0
⊢ 0 ≤ y - (x + y) / 2
have linarith₃ : y - (x + y) / 2 < 0 → x - y + (2 * y - (1 * x + 1 * y) - HMul.hMul (α := ℝ) 2 0) = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x : ℝ
⊢ Tendsto (fun (y : ℝ) ↦ Icc y x) (𝓝[<] x) (VitaliFamily.filterAt (μ := volume) (vitaliFamily volume 1) x)
have refine_1 : ∀ᶠ (i : ℝ) in 𝓝[<] x, Icc i x ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x
Tactic: refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x : ℝ
refine_1 : ∀ᶠ (i : ℝ) in 𝓝[<] x, Icc i x ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x
⊢ Tendsto (fun (y : ℝ) ↦ Icc y x) (𝓝[<] x) (VitaliFamily.filterAt (μ := volume) (vitaliFamily volume 1) x)
have refine_2 : ∀ ε > 0, ∀ᶠ (i : ℝ) in 𝓝[<] x, Icc i x ⊆ Metric.closedBall x ε
Tactic: refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x : ℝ
⊢ ∀ᶠ (i : ℝ) in 𝓝[<] x, Icc i x ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x
have h : ∀ (a : ℝ),
  Membership.mem (γ := Set ℝ)
    {x_1 : ℝ |
      x_1 ∈ Iio x →
        Membership.mem (γ := Set ℝ)
          {x_2 : ℝ | (fun (i : ℝ) ↦ Icc i x ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x) x_2} x_1}
    a
Tactic: filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_left hy
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x : ℝ
h :
  ∀ (a : ℝ),
    Membership.mem (γ := Set ℝ)
      {x_1 : ℝ |
        x_1 ∈ Iio x →
          Membership.mem (γ := Set ℝ)
            {x_2 : ℝ | (fun (i : ℝ) ↦ Icc i x ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x) x_2} x_1}
      a
⊢ ∀ᶠ (i : ℝ) in 𝓝[<] x, Icc i x ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x
have h₁ : ∀ a ∈ Iio x, Icc a x ∈ VitaliFamily.setsAt (μ := volume) (vitaliFamily volume 1) x
Tactic: filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_left hy
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x ε : ℝ
εpos : ε > 0
⊢ x - ε < x
have linarith : x ≤ x - ε → 0 - ε + (x - (x - ε)) = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x ε : ℝ
εpos : ε > 0
⊢ ∀ᶠ (i : ℝ) in 𝓝[<] x, Icc i x ⊆ Metric.closedBall x ε
have h : ∀ (y : ℝ) (hy : y ∈ Icc (x - ε) x), Icc y x ⊆ Metric.closedBall x ε
Tactic: filter_upwards [Icc_mem_nhdsLT <| show x - ε < x by linarith] with y hy
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x ε : ℝ
εpos : ε > 0
y : ℝ
hy : y ∈ Icc (x - ε) x
⊢ Icc y x ⊆ Metric.closedBall x ε
have h : Icc y x ⊆ Icc (x - ε) (x + ε)
Tactic: rw [closedBall_eq_Icc]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.OneDim (goal)
Goal:
x ε : ℝ
εpos : ε > 0
y : ℝ
hy : y ∈ Icc (x - ε) x
⊢ x ≤ x + ε
have linarith : x + ε < x → 0 - ε + (x + ε - x) = 0
Tactic: linarith