ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν : μ ≪ ν
hνν' : ν ⟂ₘ ν'
t : Set α := hνν'.nullSet
ht : MeasurableSet t
this : (ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) = ((ν + ν').restrict tᶜ).withDensity (μ.rnDeriv (ν + ν'))
⊢ (ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) = (ν.restrict tᶜ).withDensity (μ.rnDeriv ν)
α : Type u_1
n : ℕ
s t : Multiset α
l : List α
⊢ s ∈ powersetCard n ⟦l⟧ ↔ s ≤ ⟦l⟧ ∧ card s = n
X : Type u
Y : Type v
ι : Sort w
α : Type u_1
β : Type u_2
x : X
s s₁ s₂ t : Set X
p p₁ p₂ : X → Prop
inst✝ : TopologicalSpace X
⊢ x ∉ closure s ↔ 𝓝[s] x = ⊥
α : Type u_1
β : Type u_2
s t : Set α
⊢ s.encard = 0 ↔ s = ∅
a✝ b✝ c p q : ℚ
a : ℤ
ha : 0 ≤ a
b : ℕ
⊢ 0 ≤ mkRat a b