ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/GroupTheory/PGroup.lean
p : ℕ
G : Type u_1
inst✝³ : Group G
G₂ : Type u_2
inst✝² : Group G₂
p₁ p₂ : ℕ
hp₁ : Fact (Nat.Prime p₁)
hp₂ : Fact (Nat.Prime p₂)
hne : p₁ ≠ p₂
H₁ : Subgroup G
H₂ : Subgroup G₂
inst✝¹ : Finite ↥H₁
inst✝ : Finite ↥H₂
hH₁ : IsPGroup p₁ ↥H₁
hH₂ : IsPGroup p₂ ↥H₂
\u22a2 (Nat.card ↥H₁).Coprime (Nat.card ↥H₂)
-- Mathlib/GroupTheory/PGroup.lean
p : ℕ
G : Type u_1
inst✝³ : Group G
G₂ : Type u_2
inst✝² : Group G₂
p₁ p₂ : ℕ
hp₁ : Fact (Nat.Prime p₁)
hp₂ : Fact (Nat.Prime p₂)
hne : p₁ ≠ p₂
H₁ : Subgroup G
H₂ : Subgroup G₂
inst✝¹ : Finite ↥H₁
inst✝ : Finite ↥H₂
hH₁ : IsPGroup p₁ ↥H₁
hH₂ : IsPGroup p₂ ↥H₂
n₁ : ℕ
heq₁ : Nat.card ↥H₁ = p₁ ^ n₁
\u22a2 (Nat.card ↥H₁).Coprime (Nat.card ↥H₂)
-- Mathlib/GroupTheory/PGroup.lean
p : ℕ
G : Type u_1
inst✝³ : Group G
G₂ : Type u_2
inst✝² : Group G₂
p₁ p₂ : ℕ
hp₁ : Fact (Nat.Prime p₁)
hp₂ : Fact (Nat.Prime p₂)
hne : p₁ ≠ p₂
H₁ : Subgroup G
H₂ : Subgroup G₂
inst✝¹ : Finite ↥H₁
inst✝ : Finite ↥H₂
hH₁ : IsPGroup p₁ ↥H₁
hH₂ : IsPGroup p₂ ↥H₂
n₁ : ℕ
\u22a2 (p₁ ^ n₁).Coprime (Nat.card ↥H₂)
-- Mathlib/GroupTheory/PGroup.lean
p : ℕ
G : Type u_1
inst✝³ : Group G
G₂ : Type u_2
inst✝² : Group G₂
p₁ p₂ : ℕ
hp₁ : Fact (Nat.Prime p₁)
hp₂ : Fact (Nat.Prime p₂)
hne : p₁ ≠ p₂
H₁ : Subgroup G
H₂ : Subgroup G₂
inst✝¹ : Finite ↥H₁
inst✝ : Finite ↥H₂
hH₁ : IsPGroup p₁ ↥H₁
hH₂ : IsPGroup p₂ ↥H₂
n₁ n₂ : ℕ
heq₂ : Nat.card ↥H₂ = p₂ ^ n₂
\u22a2 (p₁ ^ n₁).Coprime (Nat.card ↥H₂)
-- Mathlib/GroupTheory/PGroup.lean
p : ℕ
G : Type u_1
inst✝³ : Group G
G₂ : Type u_2
inst✝² : Group G₂
p₁ p₂ : ℕ
hp₁ : Fact (Nat.Prime p₁)
hp₂ : Fact (Nat.Prime p₂)
hne : p₁ ≠ p₂
H₁ : Subgroup G
H₂ : Subgroup G₂
inst✝¹ : Finite ↥H₁
inst✝ : Finite ↥H₂
hH₁ : IsPGroup p₁ ↥H₁
hH₂ : IsPGroup p₂ ↥H₂
n₁ n₂ : ℕ
\u22a2 (p₁ ^ n₁).Coprime (p₂ ^ n₂)
-- Mathlib/Order/Disjoint.lean
α : Type u_1
inst✝¹ : DistribLattice α
inst✝ : BoundedOrder α
a b x y z x' y' : α
h : IsCompl x y
h' : IsCompl x' y'
\u22a2 (x ⊔ x') ⊓ (y ⊓ y') = ⊥
-- Mathlib/Order/Disjoint.lean
α : Type u_1
inst✝¹ : DistribLattice α
inst✝ : BoundedOrder α
a b x y z x' y' : α
h : IsCompl x y
h' : IsCompl x' y'
\u22a2 x ⊔ x' ⊔ y ⊓ y' = ⊤
-- Mathlib/MeasureTheory/OuterMeasure/AE.lean
α : Type u_1
β : Type u_2
F : Type u_3
inst✝¹ : FunLike F (Set α) ℝ≥0∞
inst✝ : OuterMeasureClass F α
μ : F
s t : Set α
\u22a2 s \ t =ᶠ[ae μ] s ↔ μ (s ∩ t) = 0
-- Mathlib/Data/Finset/Grade.lean
α : Type u_1
s t : Finset α
a : α
ha : a ∉ s
\u22a2 s ⋖ cons a s ha
-- Mathlib/Analysis/Convolution.lean
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
P : Type uP
inst✝¹⁸ : NormedAddCommGroup E
inst✝¹⁷ : NormedAddCommGroup E'
inst✝¹⁶ : NormedAddCommGroup E''
inst✝¹⁵ : NormedAddCommGroup F
f f' : G → E
g g' : G → E'
x x' : G
y y' : E
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : NormedSpace 𝕜 E
inst✝¹² : NormedSpace 𝕜 E'
inst✝¹¹ : NormedSpace 𝕜 E''
inst✝¹⁰ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁹ : MeasurableSpace G
μ ν : Measure G
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : AddCommGroup G
inst✝⁶ : μ.IsAddLeftInvariant
inst✝⁵ : μ.IsNegInvariant
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalAddGroup G
inst✝² : BorelSpace G
inst✝¹ : FirstCountableTopology G
inst✝ : SecondCountableTopologyEither G E
hbf : BddAbove (range fun x => ‖f x‖)
hf : Continuous f
hg : Integrable g μ
\u22a2 Continuous (f ⋆[L, μ] g)
-- Mathlib/Analysis/Convolution.lean
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
P : Type uP
inst✝¹⁸ : NormedAddCommGroup E
inst✝¹⁷ : NormedAddCommGroup E'
inst✝¹⁶ : NormedAddCommGroup E''
inst✝¹⁵ : NormedAddCommGroup F
f f' : G → E
g g' : G → E'
x x' : G
y y' : E
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : NormedSpace 𝕜 E
inst✝¹² : NormedSpace 𝕜 E'
inst✝¹¹ : NormedSpace 𝕜 E''
inst✝¹⁰ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁹ : MeasurableSpace G
μ ν : Measure G
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : AddCommGroup G
inst✝⁶ : μ.IsAddLeftInvariant
inst✝⁵ : μ.IsNegInvariant
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalAddGroup G
inst✝² : BorelSpace G
inst✝¹ : FirstCountableTopology G
inst✝ : SecondCountableTopologyEither G E
hbf : BddAbove (range fun x => ‖f x‖)
hf : Continuous f
hg : Integrable g μ
\u22a2 Continuous (g ⋆[L.flip, μ] f)
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
f : α ≃ₜ β
\u22a2 (map (⇑f) μ).Regular ↔ μ.Regular
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
f : α ≃ₜ β
h : (map (⇑f) μ).Regular
\u22a2 μ.Regular
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
f : α ≃ₜ β
h : (map (⇑f) μ).Regular
\u22a2 μ = map (⇑f.symm) (map (⇑f) μ)
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
f : α ≃ₜ β
h : (map (⇑f) μ).Regular
\u22a2 μ = map (⇑f.symm ∘ ⇑f) μ
-- Mathlib/CategoryTheory/Monoidal/Preadditive.lean
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
X : C
f : J → C
j : J
\u22a2 (leftDistributor X f).inv ≫ X ◁ biproduct.π f j = biproduct.π (fun j => X ⊗ f j) j
-- Mathlib/Topology/MetricSpace/Polish.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : Countable ι
t : ι → TopologicalSpace α
ht₀ : ∃ t₀, T2Space α ∧ ∀ (i : ι), t i ≤ t₀
u : ι → UniformSpace α
hcomp : ∀ (i : ι), CompleteSpace α
hcount : ∀ (i : ι), (𝓤 α).IsCountablyGenerated
hut : ∀ (i : ι), UniformSpace.toTopologicalSpace = t i
\u22a2 ∃ u, CompleteSpace α ∧ (𝓤 α).IsCountablyGenerated ∧ UniformSpace.toTopologicalSpace = ⨅ i, t i
-- Mathlib/Topology/MetricSpace/Polish.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : Countable ι
u : ι → UniformSpace α
hcomp : ∀ (i : ι), CompleteSpace α
hcount : ∀ (i : ι), (𝓤 α).IsCountablyGenerated
ht₀ : ∃ t₀, T2Space α ∧ ∀ (i : ι), (fun i => UniformSpace.toTopologicalSpace) i ≤ t₀
hut : ∀ (i : ι), UniformSpace.toTopologicalSpace = (fun i => UniformSpace.toTopologicalSpace) i
\u22a2 ∃ u_1, CompleteSpace α ∧ (𝓤 α).IsCountablyGenerated ∧ UniformSpace.toTopologicalSpace = ⨅ i, (fun i => UniformSpace.toTopologicalSpace) i
-- Mathlib/Topology/MetricSpace/Polish.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : Countable ι
u : ι → UniformSpace α
hcomp : ∀ (i : ι), CompleteSpace α
hcount : ∀ (i : ι), (𝓤 α).IsCountablyGenerated
ht₀ : ∃ t₀, T2Space α ∧ ∀ (i : ι), (fun i => UniformSpace.toTopologicalSpace) i ≤ t₀
hut : ∀ (i : ι), UniformSpace.toTopologicalSpace = (fun i => UniformSpace.toTopologicalSpace) i
\u22a2 (𝓤 α).IsCountablyGenerated
-- Mathlib/Topology/MetricSpace/Polish.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : Countable ι
u : ι → UniformSpace α
hcomp : ∀ (i : ι), CompleteSpace α
hcount : ∀ (i : ι), (𝓤 α).IsCountablyGenerated
ht₀ : ∃ t₀, T2Space α ∧ ∀ (i : ι), (fun i => UniformSpace.toTopologicalSpace) i ≤ t₀
hut : ∀ (i : ι), UniformSpace.toTopologicalSpace = (fun i => UniformSpace.toTopologicalSpace) i
\u22a2 (⨅ i, 𝓤 α).IsCountablyGenerated
-- Mathlib/CategoryTheory/GradedObject/Monoidal.lean
I : Type u
inst✝²⁵ : AddMonoid I
C : Type u_1
inst✝²⁴ : Category.{u_2, u_1} C
inst✝²³ : MonoidalCategory C
X₁ X₂ X₃ X₄ : GradedObject I C
inst✝²² : X₁.HasTensor X₂
inst✝²¹ : X₂.HasTensor X₃
inst✝²⁰ : X₃.HasTensor X₄
inst✝¹⁹ : (tensorObj X₁ X₂).HasTensor X₃
inst✝¹⁸ : X₁.HasTensor (tensorObj X₂ X₃)
inst✝¹⁷ : (tensorObj X₂ X₃).HasTensor X₄
inst✝¹⁶ : X₂.HasTensor (tensorObj X₃ X₄)
inst✝¹⁵ : (tensorObj (tensorObj X₁ X₂) X₃).HasTensor X₄
inst✝¹⁴ : (tensorObj X₁ (tensorObj X₂ X₃)).HasTensor X₄
inst✝¹³ : X₁.HasTensor (tensorObj (tensorObj X₂ X₃) X₄)
inst✝¹² : X₁.HasTensor (tensorObj X₂ (tensorObj X₃ X₄))
inst✝¹¹ : (tensorObj X₁ X₂).HasTensor (tensorObj X₃ X₄)
inst✝¹⁰ : X₁.HasGoodTensor₁₂Tensor X₂ X₃
inst✝⁹ : X₁.HasGoodTensorTensor₂₃ X₂ X₃
inst✝⁸ : X₁.HasGoodTensor₁₂Tensor (tensorObj X₂ X₃) X₄
inst✝⁷ : X₁.HasGoodTensorTensor₂₃ (tensorObj X₂ X₃) X₄
inst✝⁶ : X₂.HasGoodTensor₁₂Tensor X₃ X₄
inst✝⁵ : X₂.HasGoodTensorTensor₂₃ X₃ X₄
inst✝⁴ : (tensorObj X₁ X₂).HasGoodTensor₁₂Tensor X₃ X₄
inst✝³ : (tensorObj X₁ X₂).HasGoodTensorTensor₂₃ X₃ X₄
inst✝² : X₁.HasGoodTensor₁₂Tensor X₂ (tensorObj X₃ X₄)
inst✝¹ : X₁.HasGoodTensorTensor₂₃ X₂ (tensorObj X₃ X₄)
inst✝ : X₁.HasTensor₄ObjExt X₂ X₃ X₄
j i₁ i₂ i₃ i₄ : I
h : i₁ + i₂ + i₃ + i₄ = j
\u22a2 ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ tensorHom (𝟙 X₁) (associator X₂ X₃ X₄).inv j ≫ (associator X₁ (tensorObj X₂ X₃) X₄).inv j ≫ tensorHom (associator X₁ X₂ X₃).inv (𝟙 X₄) j = ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ (associator X₁ X₂ (tensorObj X₃ X₄)).inv j ≫ (associator (tensorObj X₁ X₂) X₃ X₄).inv j
-- Mathlib/CategoryTheory/GradedObject/Monoidal.lean
I : Type u
inst✝²⁵ : AddMonoid I
C : Type u_1
inst✝²⁴ : Category.{u_2, u_1} C
inst✝²³ : MonoidalCategory C
X₁ X₂ X₃ X₄ : GradedObject I C
inst✝²² : X₁.HasTensor X₂
inst✝²¹ : X₂.HasTensor X₃
inst✝²⁰ : X₃.HasTensor X₄
inst✝¹⁹ : (tensorObj X₁ X₂).HasTensor X₃
inst✝¹⁸ : X₁.HasTensor (tensorObj X₂ X₃)
inst✝¹⁷ : (tensorObj X₂ X₃).HasTensor X₄
inst✝¹⁶ : X₂.HasTensor (tensorObj X₃ X₄)
inst✝¹⁵ : (tensorObj (tensorObj X₁ X₂) X₃).HasTensor X₄
inst✝¹⁴ : (tensorObj X₁ (tensorObj X₂ X₃)).HasTensor X₄
inst✝¹³ : X₁.HasTensor (tensorObj (tensorObj X₂ X₃) X₄)
inst✝¹² : X₁.HasTensor (tensorObj X₂ (tensorObj X₃ X₄))
inst✝¹¹ : (tensorObj X₁ X₂).HasTensor (tensorObj X₃ X₄)
inst✝¹⁰ : X₁.HasGoodTensor₁₂Tensor X₂ X₃
inst✝⁹ : X₁.HasGoodTensorTensor₂₃ X₂ X₃
inst✝⁸ : X₁.HasGoodTensor₁₂Tensor (tensorObj X₂ X₃) X₄
inst✝⁷ : X₁.HasGoodTensorTensor₂₃ (tensorObj X₂ X₃) X₄
inst✝⁶ : X₂.HasGoodTensor₁₂Tensor X₃ X₄
inst✝⁵ : X₂.HasGoodTensorTensor₂₃ X₃ X₄
inst✝⁴ : (tensorObj X₁ X₂).HasGoodTensor₁₂Tensor X₃ X₄
inst✝³ : (tensorObj X₁ X₂).HasGoodTensorTensor₂₃ X₃ X₄
inst✝² : X₁.HasGoodTensor₁₂Tensor X₂ (tensorObj X₃ X₄)
inst✝¹ : X₁.HasGoodTensorTensor₂₃ X₂ (tensorObj X₃ X₄)
inst✝ : X₁.HasTensor₄ObjExt X₂ X₃ X₄
j i₁ i₂ i₃ i₄ : I
h : i₁ + i₂ + i₃ + i₄ = j
\u22a2 (α_ (X₁ i₁) (X₂ i₂) (X₃ i₃ ⊗ X₄ i₄)).inv ≫ (α_ (X₁ i₁ ⊗ X₂ i₂) (X₃ i₃) (X₄ i₄)).inv ≫ ιTensorObj₃' X₁ X₂ X₃ i₁ i₂ i₃ (i₁ + i₂ + i₃) ⋯ ▷ X₄ i₄ ≫ ιTensorObj (tensorObj (tensorObj X₁ X₂) X₃) X₄ (i₁ + i₂ + i₃) i₄ j ⋯ = ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ (associator X₁ X₂ (tensorObj X₃ X₄)).inv j ≫ (associator (tensorObj X₁ X₂) X₃ X₄).inv j
-- Mathlib/RingTheory/Nilpotent/Defs.lean
R : Type u_1
S : Type u_2
x y : R
inst✝³ : MonoidWithZero R
inst✝² : MonoidWithZero S
r : R
F : Type u_3
inst✝¹ : FunLike F R S
inst✝ : MonoidWithZeroHomClass F R S
f : F
hf : Injective ⇑f
x✝ : IsNilpotent (f r)
k : ℕ
hk : f r ^ k = 0
\u22a2 f (r ^ k) = 0
-- Mathlib/Data/Set/Lattice.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : (i : ι) → κ i → Set α
\u22a2 (⋂ i, ⋂ j, s i j)ᶜ = ⋃ i, ⋃ j, (s i j)ᶜ
-- Mathlib/CategoryTheory/Shift/Basic.lean
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a : A
X : C
\u22a2 (eqToHom ⋯).app X ≫ (shiftFunctorAdd C 0 a).hom.app X = 𝟙 (((shiftMonoidalFunctor C A).obj { as := a }).obj X) ≫ ((shiftMonoidalFunctor C A).obj { as := a }).map ((shiftMonoidalFunctor C A).ε.app X)
-- Mathlib/CategoryTheory/Shift/Basic.lean
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a : A
X : C
\u22a2 eqToHom ⋯ ≫ (shiftFunctorAdd C 0 a).hom.app X = eqToHom ⋯ ≫ ((shiftMonoidalFunctor C A).μIso (𝟙_ (Discrete A)) { as := a }).inv.app X
-- Mathlib/SetTheory/Ordinal/FixedPoint.lean
f✝ f : Ordinal.{u} → Ordinal.{u}
\u22a2 (fun a => sup fun n => f^[n] a) = nfp f
-- Mathlib/SetTheory/Ordinal/FixedPoint.lean
f✝ f : Ordinal.{u} → Ordinal.{u}
a : Ordinal.{u}
\u22a2 (sup fun n => f^[n] a) ≤ nfp f a
-- Mathlib/SetTheory/Ordinal/FixedPoint.lean
f✝ f : Ordinal.{u} → Ordinal.{u}
a : Ordinal.{u}
l : List Unit
\u22a2 List.foldr (fun x => f) a l ≤ sup fun n => f^[n] a
-- Mathlib/Algebra/Order/Ring/WithTop.lean
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b✝ : WithBot α
b : α
hb : b ≠ 0
\u22a2 ⊥ * ↑b = Option.bind ⊥ fun a => Option.some (a * b)
-- Mathlib/Algebra/Order/Ring/WithTop.lean
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b✝ : WithBot α
b : α
hb : b ≠ 0
\u22a2 ⊥ = Option.bind ⊥ fun a => Option.some (a * b)
-- Mathlib/Data/Rat/Defs.lean
q a b c : ℚ
x : ℤ
hx : x ≠ 0
n d : ℤ
hd : ¬d = 0
\u22a2 n /. x * (x /. d) = n /. d
-- Mathlib/Order/Bounds/Basic.lean
α : Type u
β : Type v
γ : Type w
ι : Sort x
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : Preorder γ
f : α → β → γ
s : Set α
t : Set β
a✝ : α
b✝ : β
h₀ : ∀ (b : β), Antitone (swap f b)
h₁ : ∀ (a : α), Monotone (f a)
a : α
ha : a ∈ lowerBounds s
b : β
hb : b ∈ upperBounds t
\u22a2 BddAbove (image2 f s t)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace.lean
C : Type u_1
inst✝ : Category.{?u.79019, u_1} C
X : PresheafedSpace C
\u22a2 X.presheaf = (Presheaf.pushforward C (X.ofRestrict ⋯).base).obj (X.restrict ⋯).presheaf
-- Mathlib/Geometry/RingedSpace/PresheafedSpace.lean
C : Type u_1
inst✝ : Category.{?u.79019, u_1} C
X : PresheafedSpace C
\u22a2 X.presheaf = (Presheaf.pushforward C ((Opens.inclusionTopIso ↑X).inv ≫ (X.ofRestrict ⋯).base)).obj X.presheaf
-- Mathlib/Geometry/RingedSpace/PresheafedSpace.lean
C : Type u_1
inst✝ : Category.{?u.79019, u_1} C
X : PresheafedSpace C
\u22a2 X.presheaf = (Presheaf.pushforward C (𝟙 ↑X)).obj X.presheaf
-- Mathlib/Geometry/RingedSpace/PresheafedSpace.lean
C : Type u_1
inst✝ : Category.{?u.79019, u_1} C
X : PresheafedSpace C
\u22a2 X.presheaf = (𝟭 (Presheaf C ↑X)).obj X.presheaf
-- Mathlib/Geometry/RingedSpace/PresheafedSpace.lean
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X : PresheafedSpace C
U✝ : Opens ↑↑X
\u22a2 X.presheaf.map (⋯.adjunction.counit.app U✝).op = (eqToHom ⋯).app (op U✝)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace.lean
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X : PresheafedSpace C
U✝ : Opens ↑↑X
\u22a2 op U✝ = op (⋯.functor.obj ((Opens.map ⊤.inclusion).obj U✝))
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n✝ : ℤ
γ✝ γ₁ γ₂ : Cochain K L n✝
n' a : ℤ
γ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'
n : ℤ
hn : n' + a = n
p q : ℤ
hpq : p + n = q
p' : ℤ
hp' : p + n' = p'
\u22a2 q = p' + a
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n✝ : ℤ
γ✝ γ₁ γ₂ : Cochain K L n✝
n' a : ℤ
γ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'
n : ℤ
hn : n' + a = n
p q : ℤ
hpq : p + n = q
\u22a2 (mk fun p q hpq => γ.v p (p + n') ⋯ ≫ (L.shiftFunctorObjXIso a (p + n') q ⋯).hom).v p q hpq = γ.v p (p + n') ⋯ ≫ (L.shiftFunctorObjXIso a (p + n') q ⋯).hom
-- Mathlib/Data/Nat/PartENat.lean
x y z : PartENat
hz : z ≠ ⊤
\u22a2 z + x < z + y ↔ x < y
-- Mathlib/Combinatorics/Derangements/Exponential.lean
s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1) ^ k / ↑k.factorial
\u22a2 Tendsto (fun n => ↑(numDerangements n) / ↑n.factorial) atTop (𝓝 (Real.exp (-1)))
-- Mathlib/Combinatorics/Derangements/Exponential.lean
s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1) ^ k / ↑k.factorial
\u22a2 ∀ (n : ℕ), ↑(numDerangements n) / ↑n.factorial = s (n + 1)
-- Mathlib/Combinatorics/Derangements/Exponential.lean
s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1) ^ k / ↑k.factorial
n : ℕ
\u22a2 ↑(numDerangements n) / ↑n.factorial = s (n + 1)
-- Mathlib/Combinatorics/Derangements/Exponential.lean
s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1) ^ k / ↑k.factorial
n : ℕ
\u22a2 ↑(∑ k ∈ Finset.range (n + 1), (-1) ^ k * ↑((k + 1).ascFactorial (n - k))) / ↑n.factorial = s (n + 1)
-- Mathlib/Combinatorics/Derangements/Exponential.lean
s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1) ^ k / ↑k.factorial
n : ℕ
\u22a2 (∑ x ∈ Finset.range (n + 1), (-1) ^ x * ↑((x + 1).ascFactorial (n - x))) / ↑n.factorial = s (n + 1)
-- Mathlib/Combinatorics/Derangements/Exponential.lean
s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1) ^ k / ↑k.factorial
n : ℕ
\u22a2 ∑ i ∈ Finset.range (n + 1), (-1) ^ i * ↑((i + 1).ascFactorial (n - i)) / ↑n.factorial = s (n + 1)
-- Mathlib/Combinatorics/Derangements/Exponential.lean
s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1) ^ k / ↑k.factorial
n : ℕ
\u22a2 ∀ x ∈ Finset.range (n + 1), (-1) ^ x * ↑((x + 1).ascFactorial (n - x)) / ↑n.factorial = (-1) ^ x / ↑x.factorial
-- Mathlib/Combinatorics/Derangements/Exponential.lean
s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1) ^ k / ↑k.factorial
n k : ℕ
hk : k ∈ Finset.range (n + 1)
\u22a2 (-1) ^ k * ↑((k + 1).ascFactorial (n - k)) / ↑n.factorial = (-1) ^ k / ↑k.factorial
-- Mathlib/Combinatorics/Derangements/Exponential.lean
s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1) ^ k / ↑k.factorial
n k : ℕ
hk : k ∈ Finset.range (n + 1)
h_le : k ≤ n
\u22a2 (-1) ^ k * ↑((k + 1).ascFactorial (n - k)) / ↑n.factorial = (-1) ^ k / ↑k.factorial
-- Mathlib/Combinatorics/Derangements/Exponential.lean
s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1) ^ k / ↑k.factorial
n k : ℕ
hk : k ∈ Finset.range (n + 1)
h_le : k ≤ n
\u22a2 (-1) ^ k * ↑(n.factorial / k.factorial) / ↑n.factorial = (-1) ^ k / ↑k.factorial
-- Mathlib/Combinatorics/Derangements/Exponential.lean
s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1) ^ k / ↑k.factorial
n k : ℕ
hk : k ∈ Finset.range (n + 1)
h_le : k ≤ n
\u22a2 (-1) ^ k * (↑n.factorial / ↑k.factorial) / ↑n.factorial = (-1) ^ k / ↑k.factorial
-- Mathlib/Combinatorics/Derangements/Exponential.lean
s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1) ^ k / ↑k.factorial
n k : ℕ
hk : k ∈ Finset.range (n + 1)
h_le : k ≤ n
\u22a2 (-1) ^ k * ↑n.factorial * ↑k.factorial = (-1) ^ k * (↑k.factorial * ↑n.factorial)
-- Mathlib/Topology/SeparatedMap.lean
X : Type u_1
Y : Sort u_2
A : Type ?u.10594
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
\u22a2 IsSeparatedMap f ↔ ClosedEmbedding (toPullbackDiag f)
-- Mathlib/Topology/SeparatedMap.lean
X : Type u_1
Y : Sort u_2
A : Type ?u.10594
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
\u22a2 IsClosed (Set.range (toPullbackDiag f)) ↔ ClosedEmbedding (toPullbackDiag f)
-- Mathlib/RingTheory/HahnSeries/Basic.lean
Γ : Type u_1
R : Type u_2
inst✝³ : Zero R
inst✝² : LinearOrder Γ
inst✝¹ : LocallyFiniteOrder Γ
inst✝ : Nonempty Γ
\u22a2 BddBelow (Function.support 0)
-- Mathlib/FieldTheory/Perfect.lean
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
p n : ℕ
inst✝¹ : ExpChar R p
f : R[X]
inst✝ : PerfectRing R p
\u22a2 Multiset.map (⇑(frobenius R p)) ((expand R p) f).roots = p • f.roots
-- Mathlib/Data/Finsupp/Defs.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝¹ : AddZeroClass M
inst✝ : DecidableEq α
g₁ g₂ : α →₀ M
h : Disjoint g₁.support g₂.support
a : α
ha✝ : a ∈ g₁.support ∪ g₂.support
ha : a ∈ g₁.support
\u22a2 a ∈ (g₁ + g₂).support
-- Mathlib/Data/Finsupp/Defs.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝¹ : AddZeroClass M
inst✝ : DecidableEq α
g₁ g₂ : α →₀ M
h : Disjoint g₁.support g₂.support
a : α
ha✝ : a ∈ g₁.support ∪ g₂.support
ha : a ∈ g₁.support
this : a ∉ g₂.support
\u22a2 a ∈ (g₁ + g₂).support
-- Mathlib/Data/Finsupp/Defs.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝¹ : AddZeroClass M
inst✝ : DecidableEq α
g₁ g₂ : α →₀ M
h : Disjoint g₁.support g₂.support
a : α
ha✝ : a ∈ g₁.support ∪ g₂.support
ha : g₁ a ≠ 0
this : g₂ a = 0
\u22a2 (g₁ + g₂) a ≠ 0
-- Mathlib/Data/Finsupp/Defs.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝¹ : AddZeroClass M
inst✝ : DecidableEq α
g₁ g₂ : α →₀ M
h : Disjoint g₁.support g₂.support
a : α
ha✝ : a ∈ g₁.support ∪ g₂.support
ha : a ∈ g₂.support
\u22a2 a ∈ (g₁ + g₂).support
-- Mathlib/Data/Finsupp/Defs.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝¹ : AddZeroClass M
inst✝ : DecidableEq α
g₁ g₂ : α →₀ M
h : Disjoint g₁.support g₂.support
a : α
ha✝ : a ∈ g₁.support ∪ g₂.support
ha : a ∈ g₂.support
this : a ∉ g₁.support
\u22a2 a ∈ (g₁ + g₂).support
-- Mathlib/Data/Finsupp/Defs.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝¹ : AddZeroClass M
inst✝ : DecidableEq α
g₁ g₂ : α →₀ M
h : Disjoint g₁.support g₂.support
a : α
ha✝ : a ∈ g₁.support ∪ g₂.support
ha : g₂ a ≠ 0
this : g₁ a = 0
\u22a2 (g₁ + g₂) a ≠ 0
-- Mathlib/Algebra/Polynomial/Degree/Definitions.lean
R : Type u
S : Type v
a b c d : R
n✝¹ m : ℕ
inst✝ : Semiring R
p q r : R[X]
h : p.degree ≤ 1
n✝ n : ℕ
\u22a2 p.coeff Nat.zero.succ = (C (p.coeff 1) * X + C (p.coeff 0)).coeff Nat.zero.succ
-- Mathlib/Algebra/Polynomial/Degree/Definitions.lean
R : Type u
S : Type v
a b c d : R
n✝¹ m✝ : ℕ
inst✝ : Semiring R
p q r : R[X]
h : p.degree ≤ 1
n✝ n m : ℕ
\u22a2 p.coeff m.succ.succ = (C (p.coeff 1) * X + C (p.coeff 0)).coeff m.succ.succ
-- Mathlib/Algebra/Polynomial/Degree/Definitions.lean
R : Type u
S : Type v
a b c d : R
n✝¹ m✝ : ℕ
inst✝ : Semiring R
p q r : R[X]
h : p.degree ≤ 1
n✝ n m : ℕ
this : p.degree < ↑m.succ.succ
\u22a2 p.coeff m.succ.succ = (C (p.coeff 1) * X + C (p.coeff 0)).coeff m.succ.succ
-- Mathlib/MeasureTheory/Integral/CircleIntegral.lean
E : Type u_1
inst✝ : NormedAddCommGroup E
c : ℂ
R : ℝ
hR : 0 ≤ R
θ : ℝ
\u22a2 circleMap c R θ ∈ sphere c R
-- Mathlib/Analysis/SpecialFunctions/Arsinh.lean
x✝ y x : ℝ
\u22a2 rexp (arsinh x) = x + √(1 + x ^ 2)
-- Mathlib/Analysis/SpecialFunctions/Arsinh.lean
x✝ y x : ℝ
\u22a2 0 < x + √(1 + x ^ 2)
-- Mathlib/Analysis/SpecialFunctions/Arsinh.lean
x✝ y x : ℝ
\u22a2 -x < √(1 + x ^ 2)
-- Mathlib/Analysis/SpecialFunctions/Arsinh.lean
x✝ y x : ℝ
\u22a2 (-x) ^ 2 < 1 + x ^ 2
-- Mathlib/Algebra/GroupWithZero/Basic.lean
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝ : MonoidWithZero M₀
m n : ℕ
hn : n ≠ 0
ha : 0 ^ n ≠ 0
\u22a2 False
-- Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
s : ℂ
X : ℝ
hs : 0 < s.re
hX : 0 ≤ X
\u22a2 IntervalIntegrable (fun x => ↑(rexp (-x)) * ↑x ^ (s - 1)) volume 0 X
-- Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
s : ℂ
X : ℝ
hs : 0 < s.re
hX : 0 ≤ X
\u22a2 IntegrableOn (fun x => ↑(rexp (-x)) * ↑x ^ (s - 1)) (Ioc 0 X) volume
-- Mathlib/Data/Complex/Exponential.lean
x y : ℂ
\u22a2 cosh (-x) = cosh x
-- Mathlib/Analysis/Normed/Algebra/Spectrum.lean
𝕜 : Type u_1
A✝ : Type u_2
A : Type u_3
inst✝³ : NonUnitalRing A
inst✝² : Module ℂ A
inst✝¹ : IsScalarTower ℂ A A
inst✝ : SMulCommClass ℂ A A
a : A
\u22a2 QuasispectrumRestricts a ⇑Complex.reCLM ↔ ∀ x ∈ σₙ ℂ a, x = ↑x.re
-- Mathlib/GroupTheory/MonoidLocalization/Basic.lean
M : Type u_1
inst✝² : CommMonoid M
S : Submonoid M
N : Type u_2
inst✝¹ : CommMonoid N
P : Type u_3
inst✝ : CommMonoid P
f : S.LocalizationMap N
z : N
\u22a2 f.toMap (f.sec z).1 = f.toMap ↑(f.sec z).2 * z