ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
context: V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
v w : V
h : G.Adj v w
goal: (G \ fromEdgeSet {s(v, w)}).Reachable v w ↔ G.Adj v w ∧ (G \ fromEdgeSet {s(v, w)}).Reachable v w
context: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Module.Oriented ℝ V (Fin 2)
s : Sphere P
p₁ p₂ p₃ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃ : p₃ ∈ s
hp₁p₂ : p₁ ≠ p₂
hp₁p₃ : p₁ ≠ p₃
hp₂p₃ : p₂ ≠ p₃
goal: ((∡ p₁ p₂ p₃).tan⁻¹ / 2) • (o.rotation ↑(π / 2)) (p₃ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₃ = s.center
context: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Module.Oriented ℝ V (Fin 2)
s : Sphere P
p₁ p₂ p₃ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃ : p₃ ∈ s
hp₁p₂ : p₁ ≠ p₂
hp₁p₃ : p₁ ≠ p₃
hp₂p₃ : p₂ ≠ p₃
goal: (∡ p₁ p₂ p₃).tan⁻¹ = (∡ p₃ p₁ s.center).tan
context: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Module.Oriented ℝ V (Fin 2)
s : Sphere P
p₁ p₂ p₃ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃ : p₃ ∈ s
hp₁p₂ : p₁ ≠ p₂
hp₁p₃ : p₁ ≠ p₃
hp₂p₃ : p₂ ≠ p₃
goal: 2 • ∡ p₁ p₂ p₃ + 2 • ∡ p₃ p₁ s.center = ↑π
context: K : Type u
inst✝ : Field K
A : ValuationSubring K
a✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ
h : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝
this : A.idealOfLE ↑(A.primeSpectrumEquiv b✝) ?refine_2 ≤ A.idealOfLE ↑(A.primeSpectrumEquiv a✝) ?refine_1
goal: a✝ ≤ b✝
context: K : Type u
inst✝ : Field K
A : ValuationSubring K
a✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ
h : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝
goal: A ≤ ↑(A.primeSpectrumEquiv a✝)
context: K : Type u
inst✝ : Field K
A : ValuationSubring K
a✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ
h : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝
goal: A ≤ ↑(A.primeSpectrumEquiv b✝)
context: K : Type u
inst✝ : Field K
A : ValuationSubring K
a✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ
h : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝
goal: A ≤ ↑(A.primeSpectrumEquiv a✝)
context: K : Type u
inst✝ : Field K
A : ValuationSubring K
a✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ
h : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝
goal: A ≤ ↑(A.primeSpectrumEquiv b✝)
context: α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f g : α → M
a b : α
s t : Set α
h : a ≠ b
goal: a ∉ {b}
context: α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f g : α → M
a b : α
s t : Set α
h : a ≠ b
goal: {b}.Finite
context: a b : ℝ
n : ℕ
C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x
goal: IntervalIntegrable (fun x => -↑(n + 1) * sin x * cos x ^ n) MeasureTheory.volume a b
context: a b : ℝ
n : ℕ
C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x
goal: IntervalIntegrable cos MeasureTheory.volume a b
context: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
a✝ b✝ c d : ℝ
f g : ℝ → E
μ : Measure ℝ
a b : ℝ
ha : IntegrableOn f (Iic a) μ
hb : IntegrableOn f (Iic b) μ
hab : a ≤ b
goal: MeasurableSet (Ioc a b)
context: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
a✝ b✝ c d : ℝ
f g : ℝ → E
μ : Measure ℝ
a b : ℝ
ha : IntegrableOn f (Iic a) μ
hb : IntegrableOn f (Iic b) μ
hab : a ≤ b
goal: IntegrableOn f (Iic a) μ
context: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
a✝ b✝ c d : ℝ
f g : ℝ → E
μ : Measure ℝ
a b : ℝ
ha : IntegrableOn f (Iic a) μ
hb : IntegrableOn f (Iic b) μ
hab : a ≤ b
goal: IntegrableOn f (Ioc a b) μ
context: P : Game → Prop
α✝¹ β✝¹ : Type u_1
a✝³ : α✝¹ → PGame
a✝² : β✝¹ → PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
h : ∀ (k : (mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft k⟧
i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves
j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves
goal: ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧ = ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft (Sum.inr (i, j))⟧
context: P : Game → Prop
α✝¹ β✝¹ : Type u_1
a✝³ : α✝¹ → PGame
a✝² : β✝¹ → PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
h : (∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧) ∧ ∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧
i : β✝¹
j : β✝
goal: ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft (Sum.inr (i, j))⟧ = ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧
context: α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f g : α → M
a b : α
s t✝ t : Set α
h : (s ∩ mulSupport f).Finite
goal: (s ∩ t ∩ mulSupport f).Finite
context: α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f g : α → M
a b : α
s t✝ t : Set α
h : (s ∩ mulSupport f).Finite
goal: (s \ t ∩ mulSupport f).Finite
context: C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g
i j k : 𝒰.J
goal: ((fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) = (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)
context: C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g
i j k : 𝒰.J
goal: ((fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫ pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) = (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j) ≫ pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)
context: C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g
i j k : 𝒰.J
goal: (((fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫ pullback.fst (𝒰.map j ≫ f) g = ((pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫ pullback.fst (𝒰.map j ≫ f) g
context: C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g
i j k : 𝒰.J
goal: (((fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫ pullback.snd (𝒰.map j ≫ f) g = ((pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫ pullback.snd (𝒰.map j ≫ f) g
context: C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g
i j k : 𝒰.J
goal: ((fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫ pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) = (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j) ≫ pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
inst✝ : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
μ : Measure α
s : Set α
hs : ∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s
goal: MeasurableSet hs.choose
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
inst✝ : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
μ : Measure α
s : Set α
hs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s
h's : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)
goal: MeasurableSet h's.choose
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
inst✝ : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
μ : Measure α
s : Set α
hs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s
h's : ¬∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)
goal: MeasurableSet ⋯.choose
context: L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : IsDirected ι fun x x_1 => x ≤ x_1
inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)
inst✝ : Nonempty ι
n : ℕ
R : L.Relations n
i : ι
x : Fin n → G i
k : ι
left✝ : i ≤ k
right✝ : Classical.choose ⋯ ≤ k
goal: RelMap R (unify f (fun a => Structure.Sigma.mk f i (x a)) i ?intro.intro) = RelMap R x
context: L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : IsDirected ι fun x x_1 => x ≤ x_1
inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)
inst✝ : Nonempty ι
n : ℕ
R : L.Relations n
i : ι
x : Fin n → G i
k : ι
left✝ : i ≤ k
right✝ : Classical.choose ⋯ ≤ k
goal: i ∈ upperBounds (range (Sigma.fst ∘ fun a => Structure.Sigma.mk f i (x a)))
context: ι : Type u_1
inst✝¹ : Finite ι
I : Box ι
π : Prepartition I
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
goal: ∀ b ∈ π.boxes, MeasurableSet ↑b
context: ι : Type u_1
inst✝¹ : Finite ι
I : Box ι
π : Prepartition I
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
goal: ∀ a ∈ π.boxes, μ ↑a ≠ ⊤
context: G : Type u_1
M : Type u_2
N : Type u_3
α : Sort u_4
β : Sort u_5
ι : Sort u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f : α → M
p : M → Prop
hp₀ : p 1
hp₁ : ∀ (x y : M), p x → p y → p (x * y)
hp₂ : ∀ (i : α), p (f i)
h✝ : (mulSupport ((fun i => f i) ∘ PLift.down)).Finite
goal: p (∏ i ∈ h✝.toFinset, f i.down)
context: G : Type u_1
M : Type u_2
N : Type u_3
α : Sort u_4
β : Sort u_5
ι : Sort u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f : α → M
p : M → Prop
hp₀ : p 1
hp₁ : ∀ (x y : M), p x → p y → p (x * y)
hp₂ : ∀ (i : α), p (f i)
h✝ : ¬(mulSupport ((fun i => f i) ∘ PLift.down)).Finite
goal: p 1
context: ι : Type u_1
σ : Type u_2
S : Type u_3
R : Type u_4
inst✝⁹ : DecidableEq ι
inst✝⁸ : Semiring R
inst✝⁷ : SetLike σ R
inst✝⁶ : AddSubmonoidClass σ R
A : ι → σ
inst✝⁵ : CanonicallyOrderedAddCommMonoid ι
inst✝⁴ : SetLike.GradedMonoid A
inst✝³ : Sub ι
inst✝² : OrderedSub ι
inst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
r : ⨁ (i : ι), ↥(A i)
i : ι
r' : ↥(A i)
n : ι
inst✝ : Decidable (i ≤ n)
h : i ≤ n
goal: ↑((r * (of (fun i => ↥(A i)) i) r') n) = ↑(r (n - i)) * ↑r'
context: ι : Type u_1
σ : Type u_2
S : Type u_3
R : Type u_4
inst✝⁹ : DecidableEq ι
inst✝⁸ : Semiring R
inst✝⁷ : SetLike σ R
inst✝⁶ : AddSubmonoidClass σ R
A : ι → σ
inst✝⁵ : CanonicallyOrderedAddCommMonoid ι
inst✝⁴ : SetLike.GradedMonoid A
inst✝³ : Sub ι
inst✝² : OrderedSub ι
inst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
r : ⨁ (i : ι), ↥(A i)
i : ι
r' : ↥(A i)
n : ι
inst✝ : Decidable (i ≤ n)
h : ¬i ≤ n
goal: ↑((r * (of (fun i => ↥(A i)) i) r') n) = 0
context: a : ℝ
ha : 0 ≤ a
hS : ∀ x ∈ ∅, x ≤ a
goal: sSup ∅ ≤ a
context: S : Set ℝ
a : ℝ
hS : ∀ x ∈ S, x ≤ a
ha : 0 ≤ a
hS₂ : S.Nonempty
goal: sSup S ≤ a
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
inst✝ : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
μ : Measure α
s : Set α
hs : ∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s
goal: s ⊆ hs.choose
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
inst✝ : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
μ : Measure α
s : Set α
hs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s
h's : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)
goal: s ⊆ h's.choose
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
inst✝ : MeasurableSpace α
μ✝ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
μ : Measure α
s : Set α
hs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s
h's : ¬∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)
goal: s ⊆ ⋯.choose
context: C : Type u
inst✝² : Category.{v, u} C
I : MultispanIndex C
inst✝¹ : HasCoproduct I.left
inst✝ : HasCoproduct I.right
K₁ K₂ : Cofork I.fstSigmaMap I.sndSigmaMap
f : K₁ ⟶ K₂
a✝ : I.L
goal: (Multicofork.ofSigmaCofork I K₁).ι.app (WalkingMultispan.left a✝) ≫ f.hom = (Multicofork.ofSigmaCofork I K₂).ι.app (WalkingMultispan.left a✝)
context: C : Type u
inst✝² : Category.{v, u} C
I : MultispanIndex C
inst✝¹ : HasCoproduct I.left
inst✝ : HasCoproduct I.right
K₁ K₂ : Cofork I.fstSigmaMap I.sndSigmaMap
f : K₁ ⟶ K₂
a✝ : I.R
goal: (Multicofork.ofSigmaCofork I K₁).ι.app (WalkingMultispan.right a✝) ≫ f.hom = (Multicofork.ofSigmaCofork I K₂).ι.app (WalkingMultispan.right a✝)
context: u : ℕ → ℕ
f : ℕ → ℝ≥0∞
C : ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
h_nonneg : ∀ (n : ℕ), 0 ≤ f n
hu : Monotone u
h_succ_diff : SuccDiffBounded C u
n : ℕ
goal: ∑ a ∈ range n.succ, (↑(u (a + 1)) - ↑(u a)) * f (u a) ≤ (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑ x ∈ Ico (u 0 + 1) (u n + 1), f x
context: u : ℕ → ℕ
f : ℕ → ℝ≥0∞
C : ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
h_nonneg : ∀ (n : ℕ), 0 ≤ f n
hu : Monotone u
h_succ_diff : SuccDiffBounded C u
n : ℕ
goal: 0 ≤ ↑C