awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
⊢ (μ : (a : Set Ω) → ℝ≥0∞) B ≤
    HAdd.hAdd (α := ℝ≥0∞) ((κ : (a : Set Ω) → ℝ≥0∞) (thickening (s.toReal + r.toReal) B) : ℝ≥0∞) (s + r)
have calc : HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal B) : ℝ≥0∞) r ≤
  HAdd.hAdd (α := ℝ≥0∞) ((κ : (a : Set Ω) → ℝ≥0∞) (thickening s.toReal (thickening r.toReal B)) : ℝ≥0∞) s + r
Tactic: calc μ B ≤ ν (thickening r.toReal B) + r :=
      left_measure_le_of_levyProkhorovEDist_lt lt_r B_mble
    _ ≤ κ (thickening s.toReal (thickening r.toReal B)) + s + r := by
      grw [left_measure_le_of_levyProkhorovEDist_lt lt_s isOpen_thickening.measurableSet]
    _ = κ (thickening s.toReal (thickening r.toReal B)) + (s + r) := add_assoc _ _ _
    _ ≤ κ (thickening (s.toReal + r.toReal) B) + (s + r) := by grw [thickening_thickening_subset]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
calc :
  HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal B) : ℝ≥0∞) r ≤
    HAdd.hAdd (α := ℝ≥0∞) ((κ : (a : Set Ω) → ℝ≥0∞) (thickening s.toReal (thickening r.toReal B)) : ℝ≥0∞) s + r
⊢ (μ : (a : Set Ω) → ℝ≥0∞) B ≤
    HAdd.hAdd (α := ℝ≥0∞) ((κ : (a : Set Ω) → ℝ≥0∞) (thickening (s.toReal + r.toReal) B) : ℝ≥0∞) (s + r)
have calc₁ : HAdd.hAdd (α := ℝ≥0∞) ((κ : (a : Set Ω) → ℝ≥0∞) (thickening s.toReal (thickening r.toReal B)) : ℝ≥0∞) (s + r) ≤
  HAdd.hAdd (α := ℝ≥0∞) ((κ : (a : Set Ω) → ℝ≥0∞) (thickening (s.toReal + r.toReal) B) : ℝ≥0∞) (s + r)
Tactic: calc μ B ≤ ν (thickening r.toReal B) + r :=
      left_measure_le_of_levyProkhorovEDist_lt lt_r B_mble
    _ ≤ κ (thickening s.toReal (thickening r.toReal B)) + s + r := by
      grw [left_measure_le_of_levyProkhorovEDist_lt lt_s isOpen_thickening.measurableSet]
    _ = κ (thickening s.toReal (thickening r.toReal B)) + (s + r) := add_assoc _ _ _
    _ ≤ κ (thickening (s.toReal + r.toReal) B) + (s + r) := by grw [thickening_thickening_subset]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
⊢ HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set Ω) → ℝ≥0∞) (thickening s.toReal B) : ℝ≥0∞) s ≤
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) r + s
have grw : HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) r + s ≤
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) r + s
Tactic: grw [right_measure_le_of_levyProkhorovEDist_lt lt_r isOpen_thickening.measurableSet]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
⊢ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) r + s =
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) (s + r)
have rw : HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) (r + s) =
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) (s + r)
Tactic: rw [add_assoc, add_comm r]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
rw :
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) (r + s) =
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) (s + r)
⊢ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) r + s =
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) (s + r)
have rw₁ : HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) (s + r) =
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) (s + r)
Tactic: rw [add_assoc, add_comm r]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
⊢ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) (s + r) ≤
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (r.toReal + s.toReal) B) : ℝ≥0∞) (s + r)
have grw : HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (r.toReal + s.toReal) B) : ℝ≥0∞) (s + r) ≤
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (r.toReal + s.toReal) B) : ℝ≥0∞) (s + r)
Tactic: grw [thickening_thickening_subset]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
⊢ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (r.toReal + s.toReal) B) : ℝ≥0∞) (s + r) =
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (s.toReal + r.toReal) B) : ℝ≥0∞) (s + r)
have rw : HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (s.toReal + r.toReal) B) : ℝ≥0∞) (s + r) =
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (s.toReal + r.toReal) B) : ℝ≥0∞) (s + r)
Tactic: rw [add_comm r.toReal]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
⊢ (κ : (a : Set Ω) → ℝ≥0∞) B ≤
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (s.toReal + r.toReal) B) : ℝ≥0∞) (s + r)
have calc : HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set Ω) → ℝ≥0∞) (thickening s.toReal B) : ℝ≥0∞) s ≤
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) r + s
Tactic: calc κ B ≤ ν (thickening s.toReal B) + s :=
      right_measure_le_of_levyProkhorovEDist_lt lt_s B_mble
    _ ≤ μ (thickening r.toReal (thickening s.toReal B)) + r + s := by
      grw [right_measure_le_of_levyProkhorovEDist_lt lt_r isOpen_thickening.measurableSet]
    _ = μ (thickening r.toReal (thickening s.toReal B)) + (s + r) := by rw [add_assoc, add_comm r]
    _ ≤ μ (thickening (r.toReal + s.toReal) B) + (s + r) := by grw [thickening_thickening_subset]
    _ = μ (thickening (s.toReal + r.toReal) B) + (s + r) := by rw [add_comm r.toReal]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
calc :
  HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set Ω) → ℝ≥0∞) (thickening s.toReal B) : ℝ≥0∞) s ≤
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) r + s
⊢ (κ : (a : Set Ω) → ℝ≥0∞) B ≤
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (s.toReal + r.toReal) B) : ℝ≥0∞) (s + r)
have calc₁ : HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) r + s =
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) (s + r)
Tactic: calc κ B ≤ ν (thickening s.toReal B) + s :=
      right_measure_le_of_levyProkhorovEDist_lt lt_s B_mble
    _ ≤ μ (thickening r.toReal (thickening s.toReal B)) + r + s := by
      grw [right_measure_le_of_levyProkhorovEDist_lt lt_r isOpen_thickening.measurableSet]
    _ = μ (thickening r.toReal (thickening s.toReal B)) + (s + r) := by rw [add_assoc, add_comm r]
    _ ≤ μ (thickening (r.toReal + s.toReal) B) + (s + r) := by grw [thickening_thickening_subset]
    _ = μ (thickening (s.toReal + r.toReal) B) + (s + r) := by rw [add_comm r.toReal]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
calc :
  HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set Ω) → ℝ≥0∞) (thickening s.toReal B) : ℝ≥0∞) s ≤
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) r + s
calc₁ :
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) r + s =
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) (s + r)
⊢ (κ : (a : Set Ω) → ℝ≥0∞) B ≤
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (s.toReal + r.toReal) B) : ℝ≥0∞) (s + r)
have calc₂ : HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) (s + r) ≤
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (r.toReal + s.toReal) B) : ℝ≥0∞) (s + r)
Tactic: calc κ B ≤ ν (thickening s.toReal B) + s :=
      right_measure_le_of_levyProkhorovEDist_lt lt_s B_mble
    _ ≤ μ (thickening r.toReal (thickening s.toReal B)) + r + s := by
      grw [right_measure_le_of_levyProkhorovEDist_lt lt_r isOpen_thickening.measurableSet]
    _ = μ (thickening r.toReal (thickening s.toReal B)) + (s + r) := by rw [add_assoc, add_comm r]
    _ ≤ μ (thickening (r.toReal + s.toReal) B) + (s + r) := by grw [thickening_thickening_subset]
    _ = μ (thickening (s.toReal + r.toReal) B) + (s + r) := by rw [add_comm r.toReal]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
hs_add_r' : s.toReal + r.toReal = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal
calc :
  HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set Ω) → ℝ≥0∞) (thickening s.toReal B) : ℝ≥0∞) s ≤
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) r + s
calc₁ :
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) r + s =
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) (s + r)
calc₂ :
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening r.toReal (thickening s.toReal B)) : ℝ≥0∞) (s + r) ≤
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (r.toReal + s.toReal) B) : ℝ≥0∞) (s + r)
⊢ (κ : (a : Set Ω) → ℝ≥0∞) B ≤
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (s.toReal + r.toReal) B) : ℝ≥0∞) (s + r)
have calc₃ : HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (r.toReal + s.toReal) B) : ℝ≥0∞) (s + r) =
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) (thickening (s.toReal + r.toReal) B) : ℝ≥0∞) (s + r)
Tactic: calc κ B ≤ ν (thickening s.toReal B) + s :=
      right_measure_le_of_levyProkhorovEDist_lt lt_s B_mble
    _ ≤ μ (thickening r.toReal (thickening s.toReal B)) + r + s := by
      grw [right_measure_le_of_levyProkhorovEDist_lt lt_r isOpen_thickening.measurableSet]
    _ = μ (thickening r.toReal (thickening s.toReal B)) + (s + r) := by rw [add_assoc, add_comm r]
    _ ≤ μ (thickening (r.toReal + s.toReal) B) + (s + r) := by grw [thickening_thickening_subset]
    _ = μ (thickening (s.toReal + r.toReal) B) + (s + r) := by rw [add_comm r.toReal]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (hypothesis)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
inst_3 : IsFiniteMeasure μ
inst_4 : IsFiniteMeasure ν
inst_5 : IsFiniteMeasure κ
⊢ levyProkhorovDist μ κ ≤ levyProkhorovDist μ ν + levyProkhorovDist ν κ
have dμν_finite : levyProkhorovEDist μ ν ≠ ∞
Tactic: have dμν_finite := (levyProkhorovEDist_lt_top μ ν).ne
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (hypothesis)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
inst_3 : IsFiniteMeasure μ
inst_4 : IsFiniteMeasure ν
inst_5 : IsFiniteMeasure κ
dμν_finite : levyProkhorovEDist μ ν ≠ ∞
⊢ levyProkhorovDist μ κ ≤ levyProkhorovDist μ ν + levyProkhorovDist ν κ
have dνκ_finite : levyProkhorovEDist ν κ ≠ ∞
Tactic: have dνκ_finite := (levyProkhorovEDist_lt_top ν κ).ne
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
inst_3 : IsFiniteMeasure μ
inst_4 : IsFiniteMeasure ν
inst_5 : IsFiniteMeasure κ
dμν_finite : levyProkhorovEDist μ ν ≠ ∞
dνκ_finite : levyProkhorovEDist ν κ ≠ ∞
⊢ levyProkhorovDist μ κ ≤ levyProkhorovDist μ ν + levyProkhorovDist ν κ
have h : levyProkhorovDist μ ν + levyProkhorovDist ν κ = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ).toReal
Tactic: convert! ENNReal.toReal_mono ?_ <| levyProkhorovEDist_triangle μ ν κ
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
inst_3 : IsFiniteMeasure μ
inst_4 : IsFiniteMeasure ν
inst_5 : IsFiniteMeasure κ
dμν_finite : levyProkhorovEDist μ ν ≠ ∞
dνκ_finite : levyProkhorovEDist ν κ ≠ ∞
h : levyProkhorovDist μ ν + levyProkhorovDist ν κ = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ).toReal
⊢ levyProkhorovDist μ κ ≤ levyProkhorovDist μ ν + levyProkhorovDist ν κ
have convert! : levyProkhorovEDist μ ν + levyProkhorovEDist ν κ ≠ ∞
Tactic: convert! ENNReal.toReal_mono ?_ <| levyProkhorovEDist_triangle μ ν κ
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν : Measure Ω
hLP : levyProkhorovEDist μ ν = 0
s : Set Ω
s_mble : MeasurableSet s
h_finite : ∃ δ > 0, Ne (α := ℝ≥0∞) ((ν : (a : Set Ω) → ℝ≥0∞) (thickening δ s) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ Tendsto ENNReal.toReal (𝓝[>] 0) (𝓝[>] 0)
have h1 : Tendsto ENNReal.toReal (𝓝[>] 0) (𝓝 0)
Tactic: apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within (s := Ioi 0) ENNReal.toReal
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν : Measure Ω
hLP : levyProkhorovEDist μ ν = 0
s : Set Ω
s_mble : MeasurableSet s
h_finite : ∃ δ > 0, Ne (α := ℝ≥0∞) ((ν : (a : Set Ω) → ℝ≥0∞) (thickening δ s) : ℝ≥0∞) (∞ : ℝ≥0∞)
h1 : Tendsto ENNReal.toReal (𝓝[>] 0) (𝓝 0)
⊢ Tendsto ENNReal.toReal (𝓝[>] 0) (𝓝[>] 0)
have h2 : ∀ᶠ (x : ℝ≥0∞) in 𝓝[>] 0, Membership.mem (γ := Set ℝ) (Ioi 0) x.toReal
Tactic: apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within (s := Ioi 0) ENNReal.toReal
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (goal)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν : Measure Ω
hLP : levyProkhorovEDist μ ν = 0
s : Set Ω
s_mble : MeasurableSet s
h_finite : ∃ δ > 0, Ne (α := ℝ≥0∞) ((ν : (a : Set Ω) → ℝ≥0∞) (thickening δ s) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ ∀ᶠ (x : ℝ≥0∞) in 𝓝[>] 0, Membership.mem (γ := Set ℝ) (Ioi 0) x.toReal
have h : ∀ (x : ℝ≥0∞) (hx : Membership.mem (γ := Set ℝ≥0∞) (Ioo 0 1) x), Membership.mem (γ := Set ℝ) (Ioi 0) x.toReal
Tactic: filter_upwards [Ioo_mem_nhdsGT zero_lt_one] with x hx
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (hypothesis)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν : Measure Ω
hLP : levyProkhorovEDist μ ν = 0
s : Set Ω
s_mble : MeasurableSet s
h_finite : ∃ δ > 0, Ne (α := ℝ≥0∞) ((ν : (a : Set Ω) → ℝ≥0∞) (thickening δ s) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ Tendsto (β := ℝ≥0∞) (fun (ε : ℝ≥0∞) ↦ ((ν : (a : Set Ω) → ℝ≥0∞) (thickening ε.toReal s) : ℝ≥0∞)) (𝓝[>] 0)
    (𝓝 ((ν : (a : Set Ω) → ℝ≥0∞) (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s)) : Filter ℝ≥0∞)
have aux : Tendsto ENNReal.toReal (𝓝[>] 0) (𝓝[>] 0)
Tactic: have aux : Tendsto ENNReal.toReal (𝓝[>] 0) (𝓝[>] 0) := by
      apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within (s := Ioi 0) ENNReal.toReal
      · exact tendsto_nhdsWithin_of_tendsto_nhds (continuousAt_toReal zero_ne_top).tendsto
      · filter_upwards [Ioo_mem_nhdsGT zero_lt_one] with x hx
        exact toReal_pos hx.1.ne' <| ne_top_of_lt hx.2
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.LevyProkhorovMetric (hypothesis)
Goal:
Ω : Type u_1
inst : MeasurableSpace Ω
inst_1 : PseudoEMetricSpace Ω
inst_2 : OpensMeasurableSpace Ω
μ ν : Measure Ω
hLP : levyProkhorovEDist μ ν = 0
s : Set Ω
s_mble : MeasurableSet s
h_finite : ∃ δ > 0, Ne (α := ℝ≥0∞) ((ν : (a : Set Ω) → ℝ≥0∞) (thickening δ s) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Set Ω) → ℝ≥0∞) s : ℝ≥0∞)
    ((ν : (a : Set Ω) → ℝ≥0∞) (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s) : ℝ≥0∞)
have key : Tendsto (β := ℝ≥0∞) (fun (ε : ℝ≥0∞) ↦ ((ν : (a : Set Ω) → ℝ≥0∞) (thickening ε.toReal s) : ℝ≥0∞)) (𝓝[>] 0)
  (𝓝 ((ν : (a : Set Ω) → ℝ≥0∞) (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s)) : Filter ℝ≥0∞)
Tactic: have key : Tendsto (fun ε ↦ ν (thickening ε.toReal s)) (𝓝[>] (0 : ℝ≥0∞)) (𝓝 (ν (closure s))) := by
    have aux : Tendsto ENNReal.toReal (𝓝[>] 0) (𝓝[>] 0) := by
      apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within (s := Ioi 0) ENNReal.toReal
      · exact tendsto_nhdsWithin_of_tendsto_nhds (continuousAt_toReal zero_ne_top).tendsto
      · filter_upwards [Ioo_mem_nhdsGT zero_lt_one] with x hx
        exact toReal_pos hx.1.ne' <| ne_top_of_lt hx.2
    exact (tendsto_measure_thickening h_finite).comp aux