awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
n : ℕ
this : γ - (1 / 2) ^ n < γ
⊢ ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
have rcases : ∀ (s : Set α) (hs : Membership.mem (γ := Set (Set α)) {s : Set α | MeasurableSet s} s) (hlt : γ - (1 / 2) ^ n < d s),
  ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
Tactic: rcases exists_lt_of_lt_csSup c_nonempty this with ⟨r, ⟨s, hs, rfl⟩, hlt⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (hypothesis)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
⊢ ∃ (s : Set α),
    MeasurableSet s ∧
      (∀ (t : Set α),
          MeasurableSet t →
            t ⊆ s → LE.le (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)) ∧
        ∀ (t : Set α),
          MeasurableSet t →
            t ⊆ sᶜ → LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
have this : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
Tactic: have (n : ℕ) : ∃ s : Set α, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s := by
    have : γ - (1 / 2) ^ n < γ := sub_lt_self γ (pow_pos (half_pos zero_lt_one) n)
    rcases exists_lt_of_lt_csSup c_nonempty this with ⟨r, ⟨s, hs, rfl⟩, hlt⟩
    exact ⟨s, hs, hlt⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
⊢ ∃ (s : Set α),
    MeasurableSet s ∧
      (∀ (t : Set α),
          MeasurableSet t →
            t ⊆ s → LE.le (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)) ∧
        ∀ (t : Set α),
          MeasurableSet t →
            t ⊆ sᶜ → LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
have rcases : ∀ (e : (x : ℕ) → Set α) (he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)),
  ∃ (s : Set α),
    MeasurableSet s ∧
      (∀ (t : Set α),
          MeasurableSet t →
            t ⊆ s → LE.le (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)) ∧
        ∀ (t : Set α),
          MeasurableSet t →
            t ⊆ sᶜ → LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
Tactic: rcases Classical.axiom_of_choice this with ⟨e, he⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : (x : ℕ) → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
⊢ ∃ (s : Set α),
    MeasurableSet s ∧
      (∀ (t : Set α),
          MeasurableSet t →
            t ⊆ s → LE.le (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)) ∧
        ∀ (t : Set α),
          MeasurableSet t →
            t ⊆ sᶜ → LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
have change : ∀ (e : ℕ → Set α) (he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)),
  ∃ (s : Set α),
    MeasurableSet s ∧
      (∀ (t : Set α),
          MeasurableSet t →
            t ⊆ s → LE.le (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)) ∧
        ∀ (t : Set α),
          MeasurableSet t →
            t ⊆ sᶜ → LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
Tactic: change ℕ → Set α at e
Cleared hypotheses: e, he
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (hypothesis)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
⊢ ∃ (s : Set α),
    MeasurableSet s ∧
      (∀ (t : Set α),
          MeasurableSet t →
            t ⊆ s → LE.le (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)) ∧
        ∀ (t : Set α),
          MeasurableSet t →
            t ⊆ sᶜ → LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
have he₁ : ∀ (n : ℕ), MeasurableSet (e n)
Tactic: have he₁ : ∀ n, MeasurableSet (e n) := fun n => (he n).1
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (hypothesis)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
⊢ ∃ (s : Set α),
    MeasurableSet s ∧
      (∀ (t : Set α),
          MeasurableSet t →
            t ⊆ s → LE.le (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)) ∧
        ∀ (t : Set α),
          MeasurableSet t →
            t ⊆ sᶜ → LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
have he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
Tactic: have he₂ : ∀ n, γ - (1 / 2) ^ n < d (e n) := fun n => (he n).2
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun (n m : ℕ) ↦ (Finset.Ico n (m + 1)).inf e
n m : ℕ
⊢ MeasurableSet (f n m)
have simp : MeasurableSet.{u_1} (α := α) (⨅ a ∈ Finset.Ico n (m + 1), e a)
Tactic: simp only [f, Finset.inf_eq_iInf]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (hypothesis)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun (n m : ℕ) ↦ (Finset.Ico n (m + 1)).inf e
⊢ ∃ (s : Set α),
    MeasurableSet s ∧
      (∀ (t : Set α),
          MeasurableSet t →
            t ⊆ s → LE.le (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)) ∧
        ∀ (t : Set α),
          MeasurableSet t →
            t ⊆ sᶜ → LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
have hf : ∀ (n m : ℕ), MeasurableSet (f n m)
Tactic: have hf n m : MeasurableSet (f n m) := by
    simp only [f, Finset.inf_eq_iInf]
    exact MeasurableSet.biInter (to_countable _) fun i _ => he₁ _
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d_1 : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c_1 : Set ℝ := d_1 '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c_1
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d_1 s = d_1 (s \ t) + d_1 (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d_1 (s n)) atTop (𝓝 (d_1 (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d_1 (s n)) atTop (𝓝 (d_1 (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c_1
c_nonempty : c_1.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d_1 s ≤ γ
this : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d_1 s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d_1 (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d_1 (e n)
f : ℕ → ℕ → Set α := fun (n m : ℕ) ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
a b c d : ℕ
hab : a ≤ b
hcd : c ≤ d
⊢ f a d ⊆ f b c
have simp_rw : Subset.{u_1} (α := Set α) (⨅ a_1 ∈ Finset.Ico a (d + 1), e a_1) (⨅ a ∈ Finset.Ico b (c + 1), e a)
Tactic: simp_rw [f, Finset.inf_eq_iInf]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (hypothesis)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun (n m : ℕ) ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
⊢ ∃ (s : Set α),
    MeasurableSet s ∧
      (∀ (t : Set α),
          MeasurableSet t →
            t ⊆ s → LE.le (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)) ∧
        ∀ (t : Set α),
          MeasurableSet t →
            t ⊆ sᶜ → LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
have f_subset_f : ∀ {a b c d : ℕ} (hab : a ≤ b) (hcd : c ≤ d), f a d ⊆ f b c
Tactic: have f_subset_f {a b c d} (hab : a ≤ b) (hcd : c ≤ d) : f a d ⊆ f b c := by
    simp_rw [f, Finset.inf_eq_iInf]
    exact biInter_subset_biInter_left (Finset.Ico_subset_Ico hab <| Nat.succ_le_succ hcd)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (hypothesis)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun (n m : ℕ) ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ} (hab : a ≤ b) (hcd : c ≤ d), f a d ⊆ f b c
n m : ℕ
hnm : n ≤ m
⊢ f n (m + 1) = f n m ∩ e (m + 1)
have this₁ : n ≤ m + 1
Tactic: have : n ≤ m + 1 := le_of_lt (Nat.succ_le_succ hnm)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this_1 : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun (n m : ℕ) ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ} (hab : a ≤ b) (hcd : c ≤ d), f a d ⊆ f b c
n m : ℕ
hnm : n ≤ m
this : n ≤ m + 1
⊢ f n (m + 1) = f n m ∩ e (m + 1)
have simp_rw : e (m + 1) ⊓ (Finset.Ico n (m + 1)).inf e = e (m + 1) ∩ (Finset.Ico n (m + 1)).inf e
Tactic: simp_rw [f, ← Finset.insert_Ico_right_eq_Ico_add_one this, Finset.inf_insert,
      Set.inter_comm]