ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n)))
bdd_c : BddAbove c

⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t
-- Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty

⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t
-- Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ

⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t
-- Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ 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, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s

⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t
-- Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ 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, 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, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t
-- Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ 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, 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, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t
-- Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ 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, 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)

⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t
-- Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ 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, 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 => (Finset.Ico n (m + 1)).inf e

⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t
-- Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ 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, 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 => (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)

⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t
-- Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ 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, 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 => (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c

⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t
-- Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ 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, 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 => (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)

⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t
-- Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ 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, 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 => (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ m, ⋂ n, f m n
γ_le_d_s : γ ≤ d s

⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t
-- Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ 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, 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 => (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ m, ⋂ n, f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s

⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t
-- Mathlib/Algebra/Polynomial/FieldDivision.lean
R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝ : Field R
p q : R[X]
h : p.degree = 1

⊢ p.IsRoot (-(p.coeff 0 / p.coeff 1))
-- Mathlib/Algebra/Polynomial/FieldDivision.lean
R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝ : Field R
p q : R[X]
h : p.degree = 1
this : p.coeff 1 ≠ 0

⊢ p.IsRoot (-(p.coeff 0 / p.coeff 1))
-- Mathlib/Algebra/Polynomial/FieldDivision.lean
R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝ : Field R
p q : R[X]
h : p.degree = 1
this : p.coeff 1 ≠ 0

⊢ (C (p.coeff 1) * X + C (p.coeff 0)).IsRoot (-(p.coeff 0 / p.coeff 1))
-- Mathlib/Logic/Equiv/Set.lean
α✝ : Sort u
β : Sort v
γ : Sort w
α : Type u_1
s t : Set α
h : s ⊆ t
inst✝ : DecidablePred fun x => x ∈ s
x : ↑t
hx : ↑x ∉ s

⊢ (Set.sumDiffSubset h).symm x = Sum.inr ⟨↑x, ⋯⟩
-- Mathlib/Logic/Equiv/Set.lean
α✝ : Sort u
β : Sort v
γ : Sort w
α : Type u_1
s t : Set α
h : s ⊆ t
inst✝ : DecidablePred fun x => x ∈ s
x : ↑t
hx : ↑x ∉ s

⊢ (Set.sumDiffSubset h) ((Set.sumDiffSubset h).symm x) = (Set.sumDiffSubset h) (Sum.inr ⟨↑x, ⋯⟩)
-- Mathlib/Logic/Equiv/Set.lean
α✝ : Sort u
β : Sort v
γ : Sort w
α : Type u_1
s t : Set α
h : s ⊆ t
inst✝ : DecidablePred fun x => x ∈ s
x : ↑t
hx : ↑x ∉ s

⊢ x = inclusion ⋯ ⟨↑x, ⋯⟩
-- Mathlib/Order/RelSeries.lean
α : Type u_1
r : Rel α α
β : Type u_2
s✝ : Rel β β
s : RelSeries r
x : α
irrefl : Irreflexive r

⊢ s.length ≠ 0 ↔ {x | x ∈ s}.Nontrivial
-- Mathlib/Order/RelSeries.lean
α : Type u_1
r : Rel α α
β : Type u_2
s✝ : Rel β β
s : RelSeries r
x : α
irrefl : Irreflexive r
h : s.length ≠ 0
rid : s.toFun 0 = s.toFun 1

⊢ r (s.toFun 0) (s.toFun 0)
-- Mathlib/Order/RelSeries.lean
α : Type u_1
r : Rel α α
β : Type u_2
s✝ : Rel β β
s : RelSeries r
x : α
irrefl : Irreflexive r
h : s.length ≠ 0
rid : s.toFun 0 = s.toFun 1

⊢ r (s.toFun 0) (s.toFun 1)
-- Mathlib/Order/RelSeries.lean
α : Type u_1
r : Rel α α
β : Type u_2
s✝ : Rel β β
s : RelSeries r
x : α
irrefl : Irreflexive r
h : s.length ≠ 0
rid : s.toFun 0 = s.toFun 1

⊢ 1 = ⟨0, ⋯⟩.succ
-- Mathlib/Order/RelSeries.lean
α : Type u_1
r : Rel α α
β : Type u_2
s✝ : Rel β β
s : RelSeries r
x : α
irrefl : Irreflexive r
h : s.length ≠ 0
rid : s.toFun 0 = s.toFun 1

⊢ ↑1 = ↑⟨0, ⋯⟩.succ
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f✝ : V → E
f : ℝ → E
hf : Integrable f volume
h'f : Differentiable ℝ f
hf' : Integrable (deriv f) volume
x : ℝ

⊢ 𝓕 (deriv f) x = (2 * ↑π * I * ↑x) • 𝓕 f x
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f✝ : V → E
f : ℝ → E
hf : Integrable f volume
h'f : Differentiable ℝ f
hf' : Integrable (deriv f) volume
x : ℝ
I : Integrable (fun x => fderiv ℝ f x) volume

⊢ 𝓕 (deriv f) x = (2 * ↑π * Complex.I * ↑x) • 𝓕 f x
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f✝ : V → E
f : ℝ → E
hf : Integrable f volume
h'f : Differentiable ℝ f
hf' : Integrable (deriv f) volume
x : ℝ
I : Integrable (fun x => fderiv ℝ f x) volume
this : 𝓕 (deriv f) x = (𝓕 (fderiv ℝ f) x) 1

⊢ 𝓕 (deriv f) x = (2 * ↑π * Complex.I * ↑x) • 𝓕 f x
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f✝ : V → E
f : ℝ → E
hf : Integrable f volume
h'f : Differentiable ℝ f
hf' : Integrable (deriv f) volume
x : ℝ
I : Integrable (fun x => fderiv ℝ f x) volume
this : 𝓕 (deriv f) x = (𝓕 (fderiv ℝ f) x) 1

⊢ (fourierSMulRight (-innerSL ℝ) (𝓕 f) x) 1 = (2 * ↑π * Complex.I * ↑x) • 𝓕 f x
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
ι : Type u
f : ι → Ordinal.{max u v}
a : Ordinal.{max v u}

⊢ lsub f ≤ a ↔ ∀ (i : ι), f i < a
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
ι : Type u
f : ι → Ordinal.{max u v}
a : Ordinal.{max v u}
a✝ : ι

⊢ f a✝ < a ↔ (succ ∘ f) a✝ ≤ a
-- Mathlib/Algebra/Homology/TotalComplex.lean
C : Type u_1
inst✝⁴ : Category.{u_5, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K L M : HomologicalComplex₂ C c₁ c₂
φ : K ⟶ L
e : K ≅ L
ψ : L ⟶ M
c₁₂ : ComplexShape I₁₂
inst✝² : DecidableEq I₁₂
inst✝¹ : TotalComplexShape c₁ c₂ c₁₂
inst✝ : K.HasTotal c₁₂
i₁₂ i₁₂' : I₁₂
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂

⊢ K.ιTotal c₁₂ i₁ i₂ i₁₂ h ≫ K.D₁ c₁₂ i₁₂ i₁₂' = K.d₁ c₁₂ i₁ i₂ i₁₂'
-- Mathlib/Order/Filter/Bases.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
l l' : Filter α
p : ι → Prop
s : ι → Set α
t : Set α
i : ι
p' : ι' → Prop
s' : ι' → Set α
i' : ι'
l₁ l₂ : Filter α

⊢ (diagonal α)ᶜ ∈ l₁ ×ˢ l₂ ↔ Disjoint l₁ l₂
-- Mathlib/Order/Atoms.lean
ι : Sort u_1
α : Type u_2
β : Type u_3
inst✝⁴ : PartialOrder α
inst✝³ : PartialOrder β
inst✝² : OrderBot α
inst✝¹ : IsAtomic α
inst✝ : OrderBot β
l : α → β
u : β → α
gi : GaloisInsertion l u
hbot : u ⊥ = ⊥
h_atom : ∀ (a : α), IsAtom a → u (l a) = a
b : β

⊢ IsAtom (u b) ↔ IsAtom b
-- Mathlib/MeasureTheory/Measure/AEMeasurable.lean
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f g : α → β
μ✝ ν : Measure α
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i => toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}

⊢ AEMeasurable f (sum μ)
-- Mathlib/MeasureTheory/Measure/AEMeasurable.lean
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f g : α → β
μ✝ ν : Measure α
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i => toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) (s i) = 0

⊢ AEMeasurable f (sum μ)
-- Mathlib/MeasureTheory/Measure/AEMeasurable.lean
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f g : α → β
μ✝ ν : Measure α
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i => toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)

⊢ AEMeasurable f (sum μ)
-- Mathlib/MeasureTheory/Measure/AEMeasurable.lean
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f g : α → β
μ✝ ν : Measure α
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i => toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)
hs : ∀ (i : ι), ∀ x ∉ s i, f x = mk f ⋯ x

⊢ AEMeasurable f (sum μ)
-- Mathlib/MeasureTheory/Measure/AEMeasurable.lean
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f g✝ : α → β
μ✝ ν : Measure α
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i => toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)
hs : ∀ (i : ι), ∀ x ∉ s i, f x = mk f ⋯ x
g : α → β := (⋂ i, s i).piecewise (const α default) f

⊢ AEMeasurable f (sum μ)