ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/GroupTheory/PresentedGroup.lean
α : Type u_1
G : Type u_2
inst✝ : Group G
f : α → G
rels : Set (FreeGroup α)
h : ∀ r ∈ rels, (FreeGroup.lift f) r = 1
g : PresentedGroup rels →* G
hg : ∀ (x : α), g (PresentedGroup.of x) = f x
x : PresentedGroup rels
\u22a2 g x = (toGroup h) x
-- Mathlib/GroupTheory/PresentedGroup.lean
α : Type u_1
G : Type u_2
inst✝ : Group G
f : α → G
rels : Set (FreeGroup α)
h : ∀ r ∈ rels, (FreeGroup.lift f) r = 1
g : PresentedGroup rels →* G
hg : ∀ (x : α), g (PresentedGroup.of x) = f x
x : PresentedGroup rels
\u22a2 ∀ (z : FreeGroup α), g ↑z = (toGroup h) ↑z
-- Mathlib/Algebra/Module/LocalizedModule.lean
R : Type u
inst✝⁵ : CommSemiring R
S : Submonoid R
M : Type v
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
T : Type u_1
inst✝² : CommSemiring T
inst✝¹ : Algebra R T
inst✝ : IsLocalization S T
x : LocalizedModule S M
m : M
s : ↥S
\u22a2 mk m s + 0 = mk m s
-- Mathlib/Algebra/Module/LocalizedModule.lean
R : Type u
inst✝⁵ : CommSemiring R
S : Submonoid R
M : Type v
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
T : Type u_1
inst✝² : CommSemiring T
inst✝¹ : Algebra R T
inst✝ : IsLocalization S T
x : LocalizedModule S M
m : M
s : ↥S
\u22a2 ∃ u, u • s • s • m = u • (s * s) • m
-- Mathlib/CategoryTheory/Galois/Decomposition.lean
C : Type u₁
inst✝² : Category.{u₂, u₁} C
inst✝¹ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝ : FiberFunctor F
X : C
x : ↑(F.obj X)
\u22a2 ∃ A f a, IsGalois A ∧ F.map f a = x
-- Mathlib/CategoryTheory/Galois/Decomposition.lean
C : Type u₁
inst✝² : Category.{u₂, u₁} C
inst✝¹ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝ : FiberFunctor F
X : C
x : ↑(F.obj X)
A : C
a : ↑(F.obj A)
h1 : IsGalois A
h2 : Function.Bijective fun f => F.map f a
\u22a2 ∃ A f a, IsGalois A ∧ F.map f a = x
-- Mathlib/CategoryTheory/Galois/Decomposition.lean
C : Type u₁
inst✝² : Category.{u₂, u₁} C
inst✝¹ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝ : FiberFunctor F
X : C
x : ↑(F.obj X)
A : C
a : ↑(F.obj A)
h1 : IsGalois A
h2 : Function.Bijective fun f => F.map f a
f : A ⟶ X
hf : (fun f => F.map f a) f = x
\u22a2 ∃ A f a, IsGalois A ∧ F.map f a = x
-- Mathlib/Probability/Kernel/Composition.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
η : Kernel (α × β) γ
h : IsSFiniteKernel η
\u22a2 κ ⊗ₖ η = Kernel.sum fun n => κ.seq n ⊗ₖ η
-- Mathlib/Probability/Kernel/Composition.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
η : Kernel (α × β) γ
h : IsSFiniteKernel η
\u22a2 (Kernel.sum fun n => Kernel.sum fun m => κ.seq n ⊗ₖ η.seq m) = Kernel.sum fun n => κ.seq n ⊗ₖ η
-- Mathlib/Probability/Kernel/Composition.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
s✝ : Set (β × γ)
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
η : Kernel (α × β) γ
h : IsSFiniteKernel η
n : ℕ
a : α
s : Set (β × γ)
hs : MeasurableSet s
\u22a2 ((Kernel.sum fun m => κ.seq n ⊗ₖ η.seq m) a) s = ((κ.seq n ⊗ₖ η) a) s
-- Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean
α : Type u_1
inst✝ : MeasurableSpace α
f g : α → α
s : Set α
hs : MeasurableSet s
\u22a2 f ∘ g ⁻¹' s = s
-- Mathlib/Topology/Separation.lean
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
s₁ s₂ t₁ t₂ : Set X
s_sub : s₁ ⊆ s₂
t_sub : t₁ ⊆ t₂
u : ℕ → Set X
u_cov : s₂ ⊆ ⋃ n, u n
u_props : ∀ (n : ℕ), IsOpen (u n) ∧ Disjoint (closure (u n)) t₂
\u22a2 HasSeparatingCover s₁ t₁
-- Mathlib/Tactic/NormNum/Ineq.lean
u : Level
α : Type u_1
inst✝ : LinearOrderedRing α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h : decide (num✝¹.mul (Int.ofNat db) ≤ num✝.mul (Int.ofNat da)) = true
\u22a2 ↑num✝¹ * ⅟↑da ≤ ↑num✝ * ⅟↑db
-- Mathlib/Tactic/NormNum/Ineq.lean
u : Level
α : Type u_1
inst✝ : LinearOrderedRing α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝ : decide (num✝¹.mul (Int.ofNat db) ≤ num✝.mul (Int.ofNat da)) = true
h : ↑(num✝¹.mul (Int.ofNat db)) ≤ ↑(num✝.mul (Int.ofNat da))
\u22a2 ↑num✝¹ * ⅟↑da ≤ ↑num✝ * ⅟↑db
-- Mathlib/Tactic/NormNum/Ineq.lean
u : Level
α : Type u_1
inst✝ : LinearOrderedRing α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝ : decide (num✝¹.mul (Int.ofNat db) ≤ num✝.mul (Int.ofNat da)) = true
h : ↑(num✝¹.mul (Int.ofNat db)) ≤ ↑(num✝.mul (Int.ofNat da))
ha : 0 ≤ ⅟↑da
\u22a2 ↑num✝¹ * ⅟↑da ≤ ↑num✝ * ⅟↑db
-- Mathlib/Tactic/NormNum/Ineq.lean
u : Level
α : Type u_1
inst✝ : LinearOrderedRing α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝ : decide (num✝¹.mul (Int.ofNat db) ≤ num✝.mul (Int.ofNat da)) = true
h : ↑(num✝¹.mul (Int.ofNat db)) ≤ ↑(num✝.mul (Int.ofNat da))
ha : 0 ≤ ⅟↑da
hb : 0 ≤ ⅟↑db
\u22a2 ↑num✝¹ * ⅟↑da ≤ ↑num✝ * ⅟↑db
-- Mathlib/Tactic/NormNum/Ineq.lean
u : Level
α : Type u_1
inst✝ : LinearOrderedRing α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝¹ : decide (num✝¹.mul (Int.ofNat db) ≤ num✝.mul (Int.ofNat da)) = true
h✝ : ↑(num✝¹.mul (Int.ofNat db)) ≤ ↑(num✝.mul (Int.ofNat da))
ha : 0 ≤ ⅟↑da
hb : 0 ≤ ⅟↑db
h : ⅟↑db * (↑(num✝¹.mul (Int.ofNat db)) * ⅟↑da) ≤ ⅟↑db * (↑(num✝.mul (Int.ofNat da)) * ⅟↑da)
\u22a2 ↑num✝¹ * ⅟↑da ≤ ↑num✝ * ⅟↑db
-- Mathlib/Tactic/NormNum/Ineq.lean
u : Level
α : Type u_1
inst✝ : LinearOrderedRing α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝¹ : decide (num✝¹.mul (Int.ofNat db) ≤ num✝.mul (Int.ofNat da)) = true
h✝ : ↑(num✝¹.mul (Int.ofNat db)) ≤ ↑(num✝.mul (Int.ofNat da))
ha : 0 ≤ ⅟↑da
hb : 0 ≤ ⅟↑db
h : ↑(num✝¹.mul (Int.ofNat db)) * ⅟↑db * ⅟↑da ≤ ⅟↑db * (↑(num✝.mul (Int.ofNat da)) * ⅟↑da)
\u22a2 ↑num✝¹ * ⅟↑da ≤ ↑num✝ * ⅟↑db
-- Mathlib/Tactic/NormNum/Ineq.lean
u : Level
α : Type u_1
inst✝ : LinearOrderedRing α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝¹ : decide (num✝¹.mul (Int.ofNat db) ≤ num✝.mul (Int.ofNat da)) = true
h✝ : ↑(num✝¹.mul (Int.ofNat db)) ≤ ↑(num✝.mul (Int.ofNat da))
ha : 0 ≤ ⅟↑da
hb : 0 ≤ ⅟↑db
h : ↑num✝¹ * ⅟↑da ≤ ⅟↑db * ↑num✝
\u22a2 ↑num✝¹ * ⅟↑da ≤ ↑num✝ * ⅟↑db
-- Mathlib/Data/Finset/NAry.lean
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
ζ : Type u_11
ζ' : Type u_12
ν : Type u_13
inst✝⁸ : DecidableEq α'
inst✝⁷ : DecidableEq β'
inst✝⁶ : DecidableEq γ
inst✝⁵ : DecidableEq γ'
inst✝⁴ : DecidableEq δ
inst✝³ : DecidableEq δ'
inst✝² : DecidableEq ε
inst✝¹ : DecidableEq ε'
f f' : α → β → γ
g g' : α → β → γ → δ
s s' : Finset α
t t' : Finset β
u u' : Finset γ
a a' : α
b b' : β
c : γ
inst✝ : DecidableEq α
\u22a2 image2 f (↑s ∩ ↑s') ↑t ⊆ image2 f ↑s ↑t ∩ image2 f ↑s' ↑t
-- Mathlib/Analysis/Calculus/FDeriv/Linear.lean
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x✝¹ : E
s t : Set E
L L₁ L₂ : Filter E
x✝ : E
x : E × E
\u22a2 0 = e x.1 - e x.2 - e (x.1 - x.2)
-- Mathlib/Computability/PartrecCode.lean
k : ℕ
c : Code
n x : ℕ
h : x ∈ evaln (k + 1) c n
\u22a2 x ∈ c.eval n
-- Mathlib/Computability/PartrecCode.lean
k n x : ℕ
left✝ : n ≤ k
h : 0 = x
\u22a2 x ∈ pure 0 n
-- Mathlib/Computability/PartrecCode.lean
k n x : ℕ
left✝ : n ≤ k
h : n + 1 = x
\u22a2 x = n + 1
-- Mathlib/Computability/PartrecCode.lean
k n x : ℕ
left✝ : n ≤ k
h : (unpair n).1 = x
\u22a2 x = (unpair n).1
-- Mathlib/Computability/PartrecCode.lean
k n x : ℕ
left✝ : n ≤ k
h : (unpair n).2 = x
\u22a2 x = (unpair n).2
-- Mathlib/Computability/PartrecCode.lean
k : ℕ
cf cg : Code
hf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n
hg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ cg.eval n
n x : ℕ
left✝ : n ≤ k
h : ∃ a, evaln (k + 1) cf n = some a ∧ ∃ a_1, evaln (k + 1) cg n = some a_1 ∧ Nat.pair a a_1 = x
\u22a2 ∃ a ∈ cf.eval n, ∃ a_1 ∈ cg.eval n, Nat.pair a a_1 = x
-- Mathlib/Computability/PartrecCode.lean
k : ℕ
cf cg : Code
hf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n
hg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ cg.eval n
n x : ℕ
left✝ : n ≤ k
h : ∃ a, evaln (k + 1) cg n = some a ∧ evaln (k + 1) cf a = some x
\u22a2 ∃ a ∈ cg.eval n, x ∈ cf.eval a
-- Mathlib/Computability/PartrecCode.lean
k : ℕ
cf cg : Code
hf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n
hg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ cg.eval n
n x : ℕ
left✝ : n ≤ k
h : Nat.rec (evaln (k + 1) cf (unpair n).1) (fun n_1 n_ih => (evaln k (cf.prec cg) (Nat.pair (unpair n).1 n_1)).bind fun i => evaln (k + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i))) (unpair n).2 = some x
\u22a2 x ∈ Nat.rec (cf.eval (unpair n).1) (fun y IH => IH.bind fun i => cg.eval (Nat.pair (unpair n).1 (Nat.pair y i))) (unpair n).2
-- Mathlib/Computability/PartrecCode.lean
k : ℕ
cf : Code
hf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n
n x : ℕ
left✝ : n ≤ k
h : ∃ a, evaln (k + 1) cf n = some a ∧ (if a = 0 then some (unpair n).2 else evaln k cf.rfind' (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x
\u22a2 ∃ a, (0 ∈ cf.eval (Nat.pair (unpair n).1 (a + (unpair n).2)) ∧ ∀ {m : ℕ}, m < a → ∃ a ∈ cf.eval (Nat.pair (unpair n).1 (m + (unpair n).2)), ¬a = 0) ∧ a + (unpair n).2 = x
-- Mathlib/Algebra/Homology/HomologySequenceLemmas.lean
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₁ i)
h₂ : IsIso (homologyMap φ.τ₂ i)
h₃ : ∀ (j : ι), c.Rel i j → IsIso (homologyMap φ.τ₁ j)
h₄ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)
\u22a2 IsIso (homologyMap φ.τ₃ i)
-- Mathlib/Algebra/Homology/HomologySequenceLemmas.lean
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₁ i)
h₂ : IsIso (homologyMap φ.τ₂ i)
h₃ : ∀ (j : ι), c.Rel i j → IsIso (homologyMap φ.τ₁ j)
h₄ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)
this : Mono (homologyMap φ.τ₃ i)
\u22a2 IsIso (homologyMap φ.τ₃ i)
-- Mathlib/Algebra/Homology/HomologySequenceLemmas.lean
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₁ i)
h₂ : IsIso (homologyMap φ.τ₂ i)
h₃ : ∀ (j : ι), c.Rel i j → IsIso (homologyMap φ.τ₁ j)
h₄ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)
this✝ : Mono (homologyMap φ.τ₃ i)
this : Epi (homologyMap φ.τ₃ i)
\u22a2 IsIso (homologyMap φ.τ₃ i)
-- Mathlib/Algebra/Order/Group/PosPart.lean
α : Type u_1
β : Type u_2
inst✝² : LinearOrder α
inst✝¹ : Group α
a : α
inst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
\u22a2 a⁻ᵐ = if 1 ≤ a⁻¹ then a⁻¹ else 1
-- Mathlib/Algebra/Order/Group/PosPart.lean
α : Type u_1
β : Type u_2
inst✝² : LinearOrder α
inst✝¹ : Group α
a : α
inst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
\u22a2 a⁻¹ ⊔ 1 = (fun x x_1 => x ⊔ x_1) 1 a⁻¹
-- Mathlib/Algebra/Periodic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
f g : α → β
c c₁ c₂ x✝ : α
inst✝ : AddGroup α
h1 : Periodic f c₁
h2 : Periodic f c₂
x : α
\u22a2 f (x + (c₁ - c₂)) = f x
-- Mathlib/CategoryTheory/Localization/HomEquiv.lean
C : Type u_1
C₁ : Type u_2
C₂ : Type u_3
C₃ : Type u_4
D₁ : Type u_5
D₂ : Type u_6
D₃ : Type u_7
inst✝⁹ : Category.{?u.7894, u_1} C
inst✝⁸ : Category.{u_10, u_2} C₁
inst✝⁷ : Category.{u_9, u_3} C₂
inst✝⁶ : Category.{?u.7906, u_4} C₃
inst✝⁵ : Category.{u_11, u_5} D₁
inst✝⁴ : Category.{u_8, u_6} D₂
inst✝³ : Category.{?u.7918, u_7} D₃
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
W₃ : MorphismProperty C₃
Φ : LocalizerMorphism W₁ W₂
Ψ : LocalizerMorphism W₂ W₃
L₁ : C₁ ⥤ D₁
inst✝² : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
L₃ : C₃ ⥤ D₃
inst✝ : L₃.IsLocalization W₃
X Y Z : C₁
\u22a2 Φ.homMap L₁ L₂ (𝟙 (L₁.obj X)) = 𝟙 (L₂.obj (Φ.functor.obj X))
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
\u22a2 Continuous toProbabilityMeasure
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
\u22a2 SeqContinuous toProbabilityMeasure
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toProbabilityMeasure
Ps : ℕ → ProbabilityMeasure Ω := fun n => (μs n).toProbabilityMeasure
\u22a2 Tendsto (toProbabilityMeasure ∘ μs) atTop (𝓝 P)
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toProbabilityMeasure
Ps : ℕ → ProbabilityMeasure Ω := fun n => (μs n).toProbabilityMeasure
\u22a2 ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (ω : Ω), f ω ∂↑((toProbabilityMeasure ∘ μs) i)) atTop (𝓝 (∫ (ω : Ω), f ω ∂↑P))
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toProbabilityMeasure
Ps : ℕ → ProbabilityMeasure Ω := fun n => (μs n).toProbabilityMeasure
f : Ω →ᵇ ℝ
\u22a2 ∀ (f : Ω →ᵇ ℝ), 0 ≤ f → limsup (fun i => ∫ (x : Ω), f x ∂↑((toProbabilityMeasure ∘ μs) i)) atTop ≤ ∫ (x : Ω), f x ∂↑P
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toProbabilityMeasure
Ps : ℕ → ProbabilityMeasure Ω := fun n => (μs n).toProbabilityMeasure
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
\u22a2 limsup (fun i => ∫ (x : Ω), f x ∂↑((toProbabilityMeasure ∘ μs) i)) atTop ≤ ∫ (x : Ω), f x ∂↑P
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toProbabilityMeasure
Ps : ℕ → ProbabilityMeasure Ω := fun n => (μs n).toProbabilityMeasure
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
\u22a2 limsup (fun i => ∫ (x : Ω), f x ∂↑((toProbabilityMeasure ∘ μs) i)) atTop ≤ ∫ (x : Ω), f x ∂↑P
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toProbabilityMeasure
Ps : ℕ → ProbabilityMeasure Ω := fun n => (μs n).toProbabilityMeasure
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
\u22a2 ∀ (ε : ℝ), 0 < ε → limsup (fun i => ∫ (x : Ω), f x ∂↑((toProbabilityMeasure ∘ μs) i)) atTop ≤ ∫ (x : Ω), f x ∂↑P + ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toProbabilityMeasure
Ps : ℕ → ProbabilityMeasure Ω := fun n => (μs n).toProbabilityMeasure
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
\u22a2 limsup (fun i => ∫ (x : Ω), f x ∂↑((toProbabilityMeasure ∘ μs) i)) atTop ≤ ∫ (x : Ω), f x ∂↑P + δ
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toProbabilityMeasure
Ps : ℕ → ProbabilityMeasure Ω := fun n => (μs n).toProbabilityMeasure
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
\u22a2 ∀ᶠ (n : ℕ) in atTop, ∫ (x : Ω), f x ∂↑((toProbabilityMeasure ∘ μs) n) ≤ ∫ (x : Ω), f x ∂↑P + δ
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toProbabilityMeasure
Ps : ℕ → ProbabilityMeasure Ω := fun n => (μs n).toProbabilityMeasure
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
\u22a2 IsCoboundedUnder (fun x x_1 => x ≤ x_1) atTop fun i => ∫ (x : Ω), f x ∂↑((toProbabilityMeasure ∘ μs) i)
-- Mathlib/FieldTheory/PurelyInseparable.lean
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
q : ℕ
hF : ExpChar F q
x : E
\u22a2 IsPurelyInseparable F ↥F⟮x⟯ ↔ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
-- Mathlib/RingTheory/DedekindDomain/Factorization.lean
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
n : R
hn : n ≠ 0
d : ↥R⁰
\u22a2 ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ (↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n})).factors) - ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑d})).factors)) = spanSingleton R⁰ (mk' K n d)
-- Mathlib/RingTheory/DedekindDomain/Factorization.lean
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
n : R
hn : n ≠ 0
d : ↥R⁰
hd_ne_zero : (algebraMap R K) ↑d ≠ 0
\u22a2 ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ (↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n})).factors) - ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑d})).factors)) = spanSingleton R⁰ (mk' K n d)
-- Mathlib/RingTheory/DedekindDomain/Factorization.lean
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
n : R
hn : n ≠ 0
d : ↥R⁰
hd_ne_zero : (algebraMap R K) ↑d ≠ 0
h0 : spanSingleton R⁰ (mk' K n d) ≠ 0
\u22a2 ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ (↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n})).factors) - ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑d})).factors)) = spanSingleton R⁰ (mk' K n d)
-- Mathlib/RingTheory/DedekindDomain/Factorization.lean
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
n : R
hn : n ≠ 0
d : ↥R⁰
hd_ne_zero : (algebraMap R K) ↑d ≠ 0
h0 : spanSingleton R⁰ (mk' K n d) ≠ 0
hI : spanSingleton R⁰ (mk' K n d) = spanSingleton R⁰ ((algebraMap R K) ↑d)⁻¹ * ↑(Ideal.span {n})
\u22a2 ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ (↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n})).factors) - ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑d})).factors)) = spanSingleton R⁰ (mk' K n d)
-- Mathlib/MeasureTheory/Measure/Hausdorff.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝³ : EMetricSpace X
inst✝² : EMetricSpace Y
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
μ : Measure X
ε : ℝ≥0∞
h₀ : 0 < ε
h : ∀ (s : Set X), diam s ≤ ε → μ s ≤ m (diam s)
\u22a2 μ ≤ mkMetric m
-- Mathlib/MeasureTheory/Measure/Hausdorff.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝³ : EMetricSpace X
inst✝² : EMetricSpace Y
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
μ : Measure X
ε : ℝ≥0∞
h₀ : 0 < ε
h : ∀ (s : Set X), diam s ≤ ε → μ s ≤ m (diam s)
\u22a2 μ.toOuterMeasure ≤ OuterMeasure.mkMetric m
-- Mathlib/LinearAlgebra/TensorProduct/Basic.lean
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
g✝ : P →ₗ[R] Q
f✝ : N →ₗ[R] P
f : M →ₗ[R] P
g : N →ₗ[R] Q
\u22a2 rTensor Q f ∘ₗ lTensor M g = map f g
-- Mathlib/Algebra/Homology/DerivedCategory/Ext.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y Z T : C
f : X ⟶ Y
g : Y ⟶ Z
n : ℕ
α : Ext Z T n
\u22a2 (mk₀ f).comp ((mk₀ g).comp α ⋯) ⋯ = (mk₀ (f ≫ g)).comp α ⋯
-- Mathlib/Algebra/Homology/DerivedCategory/Ext.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y Z T : C
f : X ⟶ Y
g : Y ⟶ Z
n : ℕ
α : Ext Z T n
\u22a2 0 + 0 + n = n
-- Mathlib/NumberTheory/ArithmeticFunction.lean
R : Type u_1
inst✝ : AddMonoidWithOne R
n : ℕ
\u22a2 ↑1 n = 1 n
-- Mathlib/Data/ENNReal/Operations.lean
a✝ b c d : ℝ≥0∞
r p q : ℝ≥0
α : Type u_1
a : ℝ≥0∞
\u22a2 a * a < ⊤ ↔ a < ⊤
-- Mathlib/Data/ENNReal/Operations.lean
a✝ b c d : ℝ≥0∞
r p q : ℝ≥0
α : Type u_1
a : ℝ≥0∞
\u22a2 a = 0 → a < ⊤
-- Mathlib/Data/ENNReal/Operations.lean
a b c d : ℝ≥0∞
r p q : ℝ≥0
α : Type u_1
\u22a2 0 < ⊤
-- Mathlib/LinearAlgebra/SymplecticGroup.lean
l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A ∈ symplecticGroup l R
\u22a2 Aᵀ ∈ symplecticGroup l R
-- Mathlib/LinearAlgebra/SymplecticGroup.lean
l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
\u22a2 Aᵀ * J l R * Aᵀᵀ = J l R
-- Mathlib/LinearAlgebra/SymplecticGroup.lean
l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
\u22a2 Aᵀ * J l R * A = J l R
-- Mathlib/LinearAlgebra/SymplecticGroup.lean
l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
\u22a2 Aᵀ * J l R * A = J l R
-- Mathlib/LinearAlgebra/SymplecticGroup.lean
l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
huAT : IsUnit Aᵀ.det
\u22a2 Aᵀ * J l R * A = J l R
-- Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean
s : ℝ
h : IntegrableOn (fun x => x ^ s) (Ioi 0) volume
\u22a2 False
-- Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean
s : ℝ
h : IntegrableOn (fun x => x ^ s) (Ioi 0) volume
hs : s ≤ -1
\u22a2 False
-- Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean
s : ℝ
h : IntegrableOn (fun x => x ^ s) (Ioi 0) volume
hs : -1 < s
\u22a2 False