ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Combinatorics/Schnirelmann.lean
A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
hk : 0 ∉ A
\u22a2 schnirelmannDensity A ≤ 1 - (↑0)⁻¹
-- Mathlib/Combinatorics/Schnirelmann.lean
A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
k : ℕ
hk : k ∉ A
hk' : k > 0
\u22a2 schnirelmannDensity A ≤ 1 - (↑k)⁻¹
-- Mathlib/Combinatorics/Schnirelmann.lean
A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
k : ℕ
hk : k ∉ A
hk' : k > 0
\u22a2 ↑(filter (fun x => x ∈ A) (Ioc 0 k)).card / ↑k ≤ 1 - (↑k)⁻¹
-- Mathlib/Combinatorics/Schnirelmann.lean
A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
k : ℕ
hk : k ∉ A
hk' : k > 0
\u22a2 ↑(filter (fun x => x ∈ A) (Ioc 0 k)).card / ↑k ≤ (↑k - 1) / ↑k
-- Mathlib/Combinatorics/Schnirelmann.lean
A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
k : ℕ
hk : k ∉ A
hk' : k > 0
\u22a2 ↑(filter (fun x => x ∈ A) (Ioc 0 k)).card ≤ ↑k - 1
-- Mathlib/Combinatorics/Schnirelmann.lean
A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
k : ℕ
hk : k ∉ A
hk' : k > 0
\u22a2 (filter (fun x => x ∈ A) (Ioc 0 k)).card ≤ k - 1
-- Mathlib/Combinatorics/Schnirelmann.lean
A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
k : ℕ
hk : k ∉ A
hk' : k > 0
\u22a2 filter (fun x => x ∈ A) (Ioc 0 k) ⊆ Ioo 0 k
-- Mathlib/Combinatorics/Schnirelmann.lean
A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
k : ℕ
hk : k ∉ A
hk' : k > 0
\u22a2 filter (fun x => x ∈ A) (Ioo 0 k) ⊆ Ioo 0 k
-- Mathlib/Algebra/Squarefree/Basic.lean
R : Type u_1
inst✝¹ : MonoidWithZero R
inst✝ : Nontrivial R
\u22a2 ¬Squarefree 0
-- Mathlib/Algebra/Squarefree/Basic.lean
R : Type u_1
inst✝¹ : MonoidWithZero R
inst✝ : Nontrivial R
\u22a2 ∃ x, ¬(x * x ∣ 0 → IsUnit x)
-- Mathlib/Analysis/Calculus/FDeriv/Equiv.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 : E
s t : Set E
L✝ L₁ L₂ : Filter E
iso : E ≃L[𝕜] F
L : G ≃L[𝕜] G'
f : E → F →L[𝕜] G
hs : UniqueDiffWithinAt 𝕜 s x
\u22a2 fderivWithin 𝕜 (⇑((ContinuousLinearEquiv.refl 𝕜 F).arrowCongr L) ∘ f) s x = (↑((ContinuousLinearEquiv.refl 𝕜 F).arrowCongr L)).comp (fderivWithin 𝕜 f s x)
-- Mathlib/CategoryTheory/Galois/Prorepresentability.lean
C : Type u₁
inst✝² : Category.{u₂, u₁} C
inst✝¹ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝ : FiberFunctor F
f : End F
A : PointedGaloisObject F
\u22a2 F.map ((AutGalois.π F A) ((endEquivAutGalois F) f)).hom A.pt = f.app A.obj A.pt
-- Mathlib/CategoryTheory/Galois/Prorepresentability.lean
C : Type u₁
inst✝² : Category.{u₂, u₁} C
inst✝¹ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝ : FiberFunctor F
f : End F
A : PointedGaloisObject F
\u22a2 F.map (↑(((sectionsFunctor (PointedGaloisObject F)).mapIso (autIsoFibers F)).symm.toEquiv ((endEquivSectionsFibers F) f)) A).hom A.pt = f.app A.obj A.pt
-- Mathlib/CategoryTheory/Galois/Prorepresentability.lean
C : Type u₁
inst✝² : Category.{u₂, u₁} C
inst✝¹ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝ : FiberFunctor F
f : End F
A : PointedGaloisObject F
\u22a2 F.map ((evaluationEquivOfIsGalois F A.obj A.pt).symm (↑((endEquivSectionsFibers F) f) A)).hom A.pt = f.app A.obj A.pt
-- Mathlib/CategoryTheory/Galois/Prorepresentability.lean
C : Type u₁
inst✝² : Category.{u₂, u₁} C
inst✝¹ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝ : FiberFunctor F
f : End F
A : PointedGaloisObject F
\u22a2 F.map ((evaluationEquivOfIsGalois F A.obj A.pt).symm (f.app A.obj A.pt)).hom A.pt = f.app A.obj A.pt
-- Mathlib/Data/Num/Lemmas.lean
α : Type u_1
inst✝ : AddMonoidWithOne α
n : PosNum
\u22a2 ↑n.succ = ↑n + 1
-- Mathlib/Analysis/Convex/Deriv.lean
D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x
\u22a2 ConvexOn ℝ D f
-- Mathlib/Analysis/Convex/Deriv.lean
D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x
this : EqOn (deriv f) f' (interior D)
\u22a2 ConvexOn ℝ D f
-- Mathlib/Analysis/Convex/Deriv.lean
D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x
this : EqOn (deriv f) f' (interior D)
\u22a2 DifferentiableOn ℝ (deriv f) (interior D)
-- Mathlib/Analysis/Convex/Deriv.lean
D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x
this : EqOn (deriv f) f' (interior D)
\u22a2 ∀ x ∈ interior D, 0 ≤ deriv^[2] f x
-- Mathlib/Analysis/Asymptotics/Asymptotics.lean
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
E' : Type u_6
F' : Type u_7
G' : Type u_8
E'' : Type u_9
F'' : Type u_10
G'' : Type u_11
E''' : Type u_12
R : Type u_13
R' : Type u_14
𝕜 : Type u_15
𝕜' : Type u_16
inst✝¹³ : Norm E
inst✝¹² : Norm F
inst✝¹¹ : Norm G
inst✝¹⁰ : SeminormedAddCommGroup E'
inst✝⁹ : SeminormedAddCommGroup F'
inst✝⁸ : SeminormedAddCommGroup G'
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedAddCommGroup F''
inst✝⁵ : NormedAddCommGroup G''
inst✝⁴ : SeminormedRing R
inst✝³ : SeminormedAddGroup E'''
inst✝² : SeminormedRing R'
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : NormedDivisionRing 𝕜'
c c' c₁ c₂ : ℝ
f : α → E
g : α → F
k : α → G
f' : α → E'
g' : α → F'
k' : α → G'
f'' : α → E''
g'' : α → F''
k'' : α → G''
l l' : Filter α
u v : α → ℝ
\u22a2 (f =O[l] fun x => ‖g' x‖) ↔ f =O[l] g'
-- Mathlib/Analysis/Asymptotics/Asymptotics.lean
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
E' : Type u_6
F' : Type u_7
G' : Type u_8
E'' : Type u_9
F'' : Type u_10
G'' : Type u_11
E''' : Type u_12
R : Type u_13
R' : Type u_14
𝕜 : Type u_15
𝕜' : Type u_16
inst✝¹³ : Norm E
inst✝¹² : Norm F
inst✝¹¹ : Norm G
inst✝¹⁰ : SeminormedAddCommGroup E'
inst✝⁹ : SeminormedAddCommGroup F'
inst✝⁸ : SeminormedAddCommGroup G'
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedAddCommGroup F''
inst✝⁵ : NormedAddCommGroup G''
inst✝⁴ : SeminormedRing R
inst✝³ : SeminormedAddGroup E'''
inst✝² : SeminormedRing R'
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : NormedDivisionRing 𝕜'
c c' c₁ c₂ : ℝ
f : α → E
g : α → F
k : α → G
f' : α → E'
g' : α → F'
k' : α → G'
f'' : α → E''
g'' : α → F''
k'' : α → G''
l l' : Filter α
u v : α → ℝ
\u22a2 (∃ c, IsBigOWith c l f fun x => ‖g' x‖) ↔ ∃ c, IsBigOWith c l f g'
-- Mathlib/AlgebraicTopology/SimplicialObject.lean
C : Type u
inst✝ : Category.{v, u} C
X : SimplicialObject C
n : ℕ
i : Fin (n + 2)
\u22a2 X.δ i.castSucc ≫ X.δ i = X.δ i.succ ≫ X.δ i
-- Mathlib/Topology/Compactness/Compact.lean
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
S : Set (Set X)
hf : S.Finite
hc : ∀ s ∈ S, IsCompact s
\u22a2 IsCompact (⋃₀ S)
-- Mathlib/Topology/Compactness/Compact.lean
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
S : Set (Set X)
hf : S.Finite
hc : ∀ s ∈ S, IsCompact s
\u22a2 IsCompact (⋃ i ∈ S, i)
-- Mathlib/MeasureTheory/Measure/Haar/Unique.lean
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : μ'.InnerRegularCompactLTTop
s : Set G
hs : μ s ≠ ⊤
h's : μ' s ≠ ⊤
c : ℝ≥0 := μ'.haarScalarFactor μ
\u22a2 μ' s = μ'.haarScalarFactor μ • μ s
-- Mathlib/MeasureTheory/Measure/Haar/Unique.lean
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : μ'.InnerRegularCompactLTTop
s : Set G
hs : μ s ≠ ⊤
h's : μ' s ≠ ⊤
c : ℝ≥0 := μ'.haarScalarFactor μ
B : ∀ (s : Set G), MeasurableSet s → μ s < ⊤ → μ' s < ⊤ → μ' s = (c • μ) s
\u22a2 μ' s = μ'.haarScalarFactor μ • μ s
-- Mathlib/MeasureTheory/Measure/Haar/Unique.lean
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : μ'.InnerRegularCompactLTTop
s : Set G
hs : μ s ≠ ⊤
h's : μ' s ≠ ⊤
c : ℝ≥0 := μ'.haarScalarFactor μ
B : ∀ (s : Set G), MeasurableSet s → μ s < ⊤ → μ' s < ⊤ → μ' s = (c • μ) s
t : Set G := toMeasurable μ' s ∩ toMeasurable μ s
\u22a2 μ' s = μ'.haarScalarFactor μ • μ s
-- Mathlib/MeasureTheory/Measure/Haar/Unique.lean
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : μ'.InnerRegularCompactLTTop
s : Set G
hs : μ s ≠ ⊤
h's : μ' s ≠ ⊤
c : ℝ≥0 := μ'.haarScalarFactor μ
B : ∀ (s : Set G), MeasurableSet s → μ s < ⊤ → μ' s < ⊤ → μ' s = (c • μ) s
t : Set G := toMeasurable μ' s ∩ toMeasurable μ s
st : s ⊆ t
\u22a2 μ' s = μ'.haarScalarFactor μ • μ s
-- Mathlib/MeasureTheory/Measure/Haar/Unique.lean
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : μ'.InnerRegularCompactLTTop
s : Set G
hs : μ s ≠ ⊤
h's : μ' s ≠ ⊤
c : ℝ≥0 := μ'.haarScalarFactor μ
B : ∀ (s : Set G), MeasurableSet s → μ s < ⊤ → μ' s < ⊤ → μ' s = (c • μ) s
t : Set G := toMeasurable μ' s ∩ toMeasurable μ s
st : s ⊆ t
mu'_t : μ' t = μ' s
mu_t : μ t = μ s
\u22a2 μ' s = μ'.haarScalarFactor μ • μ s
-- Mathlib/MeasureTheory/Measure/Haar/Unique.lean
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : μ'.InnerRegularCompactLTTop
s : Set G
hs : μ s ≠ ⊤
h's : μ' s ≠ ⊤
c : ℝ≥0 := μ'.haarScalarFactor μ
B : ∀ (s : Set G), MeasurableSet s → μ s < ⊤ → μ' s < ⊤ → μ' s = (c • μ) s
t : Set G := toMeasurable μ' s ∩ toMeasurable μ s
st : s ⊆ t
mu'_t : μ' t = μ' s
mu_t : μ t = μ s
\u22a2 μ' t = ↑(μ'.haarScalarFactor μ) * μ t
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean
R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
θ : ℂ
n : ℤ
\u22a2 eval (cos θ) (T ℂ n) = cos (↑n * θ)
-- Mathlib/CategoryTheory/Shift/Basic.lean
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddCommMonoid A
inst✝ : HasShift C A
X Y : C
f : X ⟶ Y
n : A
\u22a2 (shiftFunctorZero C A).inv.app ((shiftFunctor C n).obj X) = (shiftFunctor C n).map ((shiftFunctorZero C A).inv.app X) ≫ (shiftFunctorComm C n 0).inv.app X
-- Mathlib/CategoryTheory/Shift/Basic.lean
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddCommMonoid A
inst✝ : HasShift C A
X Y : C
f : X ⟶ Y
n : A
\u22a2 𝟙 ((𝟭 C).obj ((shiftFunctor C n).obj X)) = (shiftFunctor C n).map (𝟙 ((𝟭 C).obj X))
-- Mathlib/CategoryTheory/Shift/Basic.lean
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddCommMonoid A
inst✝ : HasShift C A
X Y : C
f : X ⟶ Y
n : A
\u22a2 𝟙 ((shiftFunctor C n).obj X) = (shiftFunctor C n).map (𝟙 X)
-- Mathlib/Order/Partition/Finpartition.lean
α : Type u_1
inst✝¹ : DecidableEq α
s t u : Finset α
P : Finpartition s
a : α
inst✝ : Fintype α
F : Finset (Finset α)
hs : s.Nonempty
\u22a2 (atomise s ∅).parts = {s}
-- Mathlib/Order/Partition/Finpartition.lean
α : Type u_1
inst✝¹ : DecidableEq α
s t u : Finset α
P : Finpartition s
a : α
inst✝ : Fintype α
F : Finset (Finset α)
hs : s.Nonempty
\u22a2 (ofErase {s} ⋯ ⋯).parts = {s}
-- Mathlib/RingTheory/MvPolynomial/Symmetric.lean
σ : Type u_1
R : Type u_2
τ : Type u_3
S : Type u_4
inst✝³ : CommSemiring R
inst✝² : CommSemiring S
inst✝¹ : Fintype σ
inst✝ : Fintype τ
n : ℕ
e : σ ≃ τ
\u22a2 (rename ⇑e) (esymm σ R n) = ∑ x ∈ powersetCard n univ, ∏ i ∈ x, X (e i)
-- Mathlib/RingTheory/MvPolynomial/Symmetric.lean
σ : Type u_1
R : Type u_2
τ : Type u_3
S : Type u_4
inst✝³ : CommSemiring R
inst✝² : CommSemiring S
inst✝¹ : Fintype σ
inst✝ : Fintype τ
n : ℕ
e : σ ≃ τ
\u22a2 ∑ x ∈ powersetCard n univ, ∏ i ∈ x, X (e i) = ∑ t ∈ powersetCard n (Finset.map e.toEmbedding univ), ∏ i ∈ t, X i
-- Mathlib/RingTheory/MvPolynomial/Symmetric.lean
σ : Type u_1
R : Type u_2
τ : Type u_3
S : Type u_4
inst✝³ : CommSemiring R
inst✝² : CommSemiring S
inst✝¹ : Fintype σ
inst✝ : Fintype τ
n : ℕ
e : σ ≃ τ
\u22a2 ∑ x ∈ powersetCard n univ, ∏ i ∈ x, X (e i) = ∑ x ∈ powersetCard n univ, ∏ i ∈ (mapEmbedding e.toEmbedding) x, X i
-- Mathlib/RingTheory/MvPolynomial/Symmetric.lean
σ : Type u_1
R : Type u_2
τ : Type u_3
S : Type u_4
inst✝³ : CommSemiring R
inst✝² : CommSemiring S
inst✝¹ : Fintype σ
inst✝ : Fintype τ
n : ℕ
e : σ ≃ τ
\u22a2 ∑ t ∈ powersetCard n (Finset.map e.toEmbedding univ), ∏ i ∈ t, X i = ∑ t ∈ powersetCard n univ, ∏ i ∈ t, X i
-- Mathlib/Data/Fin/Basic.lean
n m : ℕ
a b : Fin n
\u22a2 a.castSucc < b.succ ↔ a ≤ b
-- Mathlib/RingTheory/Ideal/Operations.lean
R : Type u
M : Type v
M'✝ : Type u_1
F : Type u_2
G : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid M'✝
inst✝² : Module R M'✝
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
s : Set M
x : M
\u22a2 x ∈ I • span R s ↔ x ∈ span R (⋃ a ∈ I, ⋃ b ∈ s, {a • b})
-- Mathlib/RingTheory/Ideal/Operations.lean
R : Type u
M : Type v
M'✝ : Type u_1
F : Type u_2
G : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid M'✝
inst✝² : Module R M'✝
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
s : Set M
x : M
\u22a2 x ∈ span R (⋃ s_1 ∈ ↑I, ⋃ t ∈ s, {s_1 • t}) ↔ x ∈ span R (⋃ a ∈ I, ⋃ b ∈ s, {a • b})
-- Mathlib/LinearAlgebra/Dual.lean
R : Type u_1
M : Type u_2
ι : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
e : ι → M
ε : ι → Dual R M
inst✝¹ : DecidableEq ι
h : DualBases e ε
inst✝ : _root_.Finite ι
i j : ι
\u22a2 (h.basis.dualBasis i) (h.basis j) = (ε i) (h.basis j)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
x : ℝ
\u22a2 sin (arctan x) = x / √(1 + x ^ 2)
-- Mathlib/Analysis/Calculus/LHopital.lean
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfb : Tendsto f (𝓝[<] b) (𝓝 0)
hgb : Tendsto g (𝓝[<] b) (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l
\u22a2 Tendsto (fun x => f x / g x) (𝓝[<] b) l
-- Mathlib/Analysis/Calculus/LHopital.lean
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfb : Tendsto f (𝓝[<] b) (𝓝 0)
hgb : Tendsto g (𝓝[<] b) (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l
hdnf : ∀ x ∈ -Ioo a b, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
\u22a2 Tendsto (fun x => f x / g x) (𝓝[<] b) l
-- Mathlib/Analysis/Calculus/LHopital.lean
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfb : Tendsto f (𝓝[<] b) (𝓝 0)
hgb : Tendsto g (𝓝[<] b) (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l
hdnf : ∀ x ∈ -Ioo a b, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ -Ioo a b, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
\u22a2 Tendsto (fun x => f x / g x) (𝓝[<] b) l
-- Mathlib/Analysis/Calculus/LHopital.lean
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfb : Tendsto f (𝓝[<] b) (𝓝 0)
hgb : Tendsto g (𝓝[<] b) (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l
hdnf : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ -Ioo a b, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
\u22a2 Tendsto (fun x => f x / g x) (𝓝[<] b) l
-- Mathlib/Analysis/Calculus/LHopital.lean
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfb : Tendsto f (𝓝[<] b) (𝓝 0)
hgb : Tendsto g (𝓝[<] b) (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l
hdnf : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
\u22a2 Tendsto (fun x => f x / g x) (𝓝[<] b) l
-- Mathlib/Analysis/Calculus/LHopital.lean
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfb : Tendsto f (𝓝[<] b) (𝓝 0)
hgb : Tendsto g (𝓝[<] b) (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l
hdnf : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
this : Tendsto (fun x => (f ∘ Neg.neg) x / (g ∘ Neg.neg) x) (𝓝[>] (-b)) l
\u22a2 Tendsto (fun x => f x / g x) (𝓝[<] b) l
-- Mathlib/Analysis/Calculus/LHopital.lean
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfb : Tendsto f (𝓝[<] b) (𝓝 0)
hgb : Tendsto g (𝓝[<] b) (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l
hdnf : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
this✝ : Tendsto (fun x => (f ∘ Neg.neg) x / (g ∘ Neg.neg) x) (𝓝[>] (-b)) l
this : Tendsto ((fun x => (f ∘ Neg.neg) x / (g ∘ Neg.neg) x) ∘ Neg.neg) (𝓝[<] b) l
\u22a2 Tendsto (fun x => f x / g x) (𝓝[<] b) l
-- Mathlib/Analysis/Calculus/LHopital.lean
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfb : Tendsto f (𝓝[<] b) (𝓝 0)
hgb : Tendsto g (𝓝[<] b) (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l
hdnf : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
this✝ : Tendsto (fun x => (f ∘ Neg.neg) x / (g ∘ Neg.neg) x) (𝓝[>] (-b)) l
this : Tendsto (fun x => (fun x => f (-x) / g (-x)) (-x)) (𝓝[<] b) l
\u22a2 Tendsto (fun x => f x / g x) (𝓝[<] b) l
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
\u22a2 μ.toFinite.withDensity μ.densityToFinite = μ
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
this : (μ.toFinite.withDensity fun a => ∑' (n : ℕ), (sFiniteSeq μ n).rnDeriv μ.toFinite a) = μ.toFinite.withDensity (∑' (n : ℕ), (sFiniteSeq μ n).rnDeriv μ.toFinite)
\u22a2 μ.toFinite.withDensity μ.densityToFinite = μ
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
this : (μ.toFinite.withDensity fun a => ∑' (n : ℕ), (sFiniteSeq μ n).rnDeriv μ.toFinite a) = μ.toFinite.withDensity (∑' (n : ℕ), (sFiniteSeq μ n).rnDeriv μ.toFinite)
\u22a2 (Measure.sum fun n => μ.toFinite.withDensity ((sFiniteSeq μ n).rnDeriv μ.toFinite)) = μ
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
this : (μ.toFinite.withDensity fun a => ∑' (n : ℕ), (sFiniteSeq μ n).rnDeriv μ.toFinite a) = μ.toFinite.withDensity (∑' (n : ℕ), (sFiniteSeq μ n).rnDeriv μ.toFinite)
\u22a2 (Measure.sum fun n => μ.toFinite.withDensity ((sFiniteSeq μ n).rnDeriv μ.toFinite)) = Measure.sum (sFiniteSeq μ)
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
this : (μ.toFinite.withDensity fun a => ∑' (n : ℕ), (sFiniteSeq μ n).rnDeriv μ.toFinite a) = μ.toFinite.withDensity (∑' (n : ℕ), (sFiniteSeq μ n).rnDeriv μ.toFinite)
n : ℕ
s✝ : Set α
a✝ : MeasurableSet s✝
\u22a2 (μ.toFinite.withDensity ((sFiniteSeq μ n).rnDeriv μ.toFinite)) s✝ = (sFiniteSeq μ n) s✝
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
this : (μ.toFinite.withDensity fun a => ∑' (n : ℕ), (sFiniteSeq μ n).rnDeriv μ.toFinite a) = μ.toFinite.withDensity (∑' (n : ℕ), (sFiniteSeq μ n).rnDeriv μ.toFinite)
n : ℕ
s✝ : Set α
a✝ : MeasurableSet s✝
\u22a2 sFiniteSeq μ n ≪ μ.toFinite
-- Mathlib/Order/SymmDiff.lean
ι : Type u_1
α : Type u_2
β : Type u_3
π : ι → Type u_4
inst✝ : GeneralizedBooleanAlgebra α
a b c d : α
\u22a2 Disjoint (a ∆ b) (a ⊓ b)
-- Mathlib/Order/SymmDiff.lean
ι : Type u_1
α : Type u_2
β : Type u_3
π : ι → Type u_4
inst✝ : GeneralizedBooleanAlgebra α
a b c d : α
\u22a2 Disjoint ((a ⊔ b) \ (a ⊓ b)) (a ⊓ b)
-- Mathlib/Topology/Inseparable.lean
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s✝ : Set X
f g : X → Y
s : Set X
\u22a2 StableUnderGeneralization s ↔ ∃ S, (∀ s ∈ S, IsOpen s) ∧ ⋂₀ S = s
-- Mathlib/Topology/Inseparable.lean
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s✝ : Set X
f g : X → Y
s : Set X
\u22a2 StableUnderGeneralization s → ∃ S, (∀ s ∈ S, IsOpen s) ∧ ⋂₀ S = s
-- Mathlib/Topology/Inseparable.lean
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s✝ : Set X
f g : X → Y
s : Set X
\u22a2 (∃ S, (∀ s ∈ S, IsClosed s) ∧ ⋃₀ S = sᶜ) → ∃ S, (∀ s ∈ S, IsOpen s) ∧ ⋂₀ S = s
-- .lake/packages/batteries/Batteries/Data/RBMap/WF.lean
α✝ : Type u_1
cmp : α✝ → α✝ → Ordering
t : RBNode α✝
v : α✝
h : Ordered cmp t
\u22a2 Ordered cmp (match t.isRed with | red => (ins cmp v t).setBlack | black => ins cmp v t)
-- Mathlib/Data/Stream/Init.lean
α : Type u
β : Type v
δ : Type w
a : α
s : Stream' α
b : α
x✝ : a ∈ s
n : ℕ
h : (fun b => a = b) (s.get n)
\u22a2 (fun b => a = b) ((b :: s).get n.succ)