ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/CategoryTheory/Limits/HasLimits.lean
J : Type u₁
inst✝² : Category.{v₁, u₁} J
K : Type u₂
inst✝¹ : Category.{v₂, u₂} K
C : Type u
inst✝ : Category.{v, u} C
F✝ : J ⥤ C
L : (J ⥤ C) ⥤ C
adj : const J ⊣ L
F : J ⥤ C
s : Cone F
j : J
eq : L.map s.π ≫ (adj.counit.app F).app j = (adj.counit.app ((const J).obj s.pt)).app j ≫ s.π.app j
eq' : adj.unit.app s.pt ≫ (adj.counit.app ((const J).obj s.pt)).app j = 𝟙 s.pt
\u22a2 (adj.homEquiv s.pt F) s.π ≫ (adj.counit.app F).app j = s.π.app j
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
J : Type w₁
f : J → C
inst✝¹ : HasBiproduct f
inst✝ : PreservesBiproduct f F
W : C
g : (j : J) → f j ⟶ W
j : J
\u22a2 ι (F.obj ∘ f) j ≫ (F.mapBiproduct f).inv ≫ F.map (desc g) = ι (F.obj ∘ f) j ≫ desc fun j => F.map (g j)
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
J : Type w₁
f : J → C
inst✝¹ : HasBiproduct f
inst✝ : PreservesBiproduct f F
W : C
g : (j : J) → f j ⟶ W
j : J
\u22a2 ι (fun x => F.obj (f x)) j ≫ (F.mapBiproduct f).inv ≫ F.map (desc g) = ι (fun x => F.obj (f x)) j ≫ desc fun j => F.map (g j)
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
J : Type w₁
f : J → C
inst✝¹ : HasBiproduct f
inst✝ : PreservesBiproduct f F
W : C
g : (j : J) → f j ⟶ W
j : J
this : HasBiproduct fun j => F.obj (f j)
\u22a2 ι (fun x => F.obj (f x)) j ≫ (F.mapBiproduct f).inv ≫ F.map (desc g) = ι (fun x => F.obj (f x)) j ≫ desc fun j => F.map (g j)
-- Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean
C : Type u_2
inst✝⁵ : Category.{u_1, u_2} C
inst✝⁴ : HasZeroMorphisms C
S₁ S₂ S₃ S₄ : ShortComplex C
inst✝³ : S₁.HasHomology
inst✝² : S₂.HasHomology
inst✝¹ : S₃.HasHomology
inst✝ : S₄.HasHomology
φ : S₁ ⟶ S₂
γ : HomologyMapData φ S₁.homologyData S₂.homologyData
\u22a2 QuasiIso (opMap φ) ↔ QuasiIso φ
-- Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean
C : Type u_2
inst✝⁵ : Category.{u_1, u_2} C
inst✝⁴ : HasZeroMorphisms C
S₁ S₂ S₃ S₄ : ShortComplex C
inst✝³ : S₁.HasHomology
inst✝² : S₂.HasHomology
inst✝¹ : S₃.HasHomology
inst✝ : S₄.HasHomology
φ : S₁ ⟶ S₂
γ : HomologyMapData φ S₁.homologyData S₂.homologyData
\u22a2 IsIso γ.op.right.φH ↔ IsIso γ.left.φH
-- Mathlib/Data/Complex/Exponential.lean
x y : ℝ
\u22a2 sinh x + cosh x = rexp x
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
x y : ℝ≥0∞
hx : x ≠ ⊤
hy : y ≠ ⊤
z : ℝ
\u22a2 (x * y) ^ z = x ^ z * y ^ z
-- Mathlib/RingTheory/FractionalIdeal/Operations.lean
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁸ : CommRing P
inst✝⁷ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝⁶ : CommRing P'
inst✝⁵ : Algebra R P'
loc' : IsLocalization S P'
P''✝ : Type u_4
inst✝⁴ : CommRing P''✝
inst✝³ : Algebra R P''✝
loc'' : IsLocalization S P''✝
I✝ J : FractionalIdeal S P
g : P →ₐ[R] P'
P'' : Type u_5
inst✝² : CommRing P''
inst✝¹ : Algebra R P''
inst✝ : IsLocalization S P''
I : FractionalIdeal S P
x✝ : P''
\u22a2 x✝ ∈ (canonicalEquiv S P' P'') ((canonicalEquiv S P P') I) ↔ x✝ ∈ (canonicalEquiv S P P'') I
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
f : ℝ → E
c ca cb : E
l l' la la' lb lb' : Filter ℝ
lt : Filter ι
a b z : ℝ
u v ua ub va vb : ι → ℝ
inst✝² : FTCFilter a la la'
inst✝¹ : FTCFilter b lb lb'
hf : IntervalIntegrable f volume a b
s t : Set ℝ
inst✝ : FTCFilter a (𝓝[s] a) (𝓝[t] a)
hmeas : StronglyMeasurableAtFilter f (𝓝[t] a) volume
ha : Tendsto f (𝓝[t] a ⊓ ae volume) (𝓝 c)
\u22a2 HasDerivWithinAt (fun u => ∫ (x : ℝ) in u..b, f x) (-c) s a
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
f : ℝ → E
c ca cb : E
l l' la la' lb lb' : Filter ℝ
lt : Filter ι
a b z : ℝ
u v ua ub va vb : ι → ℝ
inst✝² : FTCFilter a la la'
inst✝¹ : FTCFilter b lb lb'
hf : IntervalIntegrable f volume a b
s t : Set ℝ
inst✝ : FTCFilter a (𝓝[s] a) (𝓝[t] a)
hmeas : StronglyMeasurableAtFilter f (𝓝[t] a) volume
ha : Tendsto f (𝓝[t] a ⊓ ae volume) (𝓝 c)
\u22a2 HasDerivWithinAt (fun u => -∫ (x : ℝ) in b..u, f x) (-c) s a
-- Mathlib/Data/ZMod/Units.lean
n✝ m n : ℕ
\u22a2 unitsMap ⋯ = MonoidHom.id (ZMod n)ˣ
-- Mathlib/Analysis/Convex/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
β : Type u_4
ι : Type u_5
inst✝² : OrderedSemiring 𝕜
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i : ι
\u22a2 (fun x => if i = x then 1 else 0) ∈ stdSimplex 𝕜 ι
-- Mathlib/Topology/FiberBundle/Basic.lean
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝³ : TopologicalSpace X
E : B → Type u_5
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.pretrivializationAtlas
\u22a2 ContinuousOn (↑e.symm) e.target
-- Mathlib/Topology/FiberBundle/Basic.lean
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝³ : TopologicalSpace X
E : B → Type u_5
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.pretrivializationAtlas
z : B × F
H : z ∈ e.target
U : Set (TotalSpace F E)
h : U ∈ 𝓝 (↑e.symm z)
\u22a2 coinduced (fun x => ↑e.symm ↑x) inferInstance ≤ a.totalSpaceTopology
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o₁ o₂ : Ordinal.{u}
h : o₁ ≤ o₂
hl : isGood (π C fun x => ord I x < o₁) l
hl' : eval (π C fun x => ord I x < o₂) l ∈ Submodule.span ℤ (eval (π C fun x => ord I x < o₂) '' {m | m < l})
\u22a2 eval (π C fun x => ord I x < o₁) l ∈ Submodule.span ℤ (eval (π C fun x => ord I x < o₁) '' {m | m < l})
-- Mathlib/NumberTheory/Padics/PadicVal.lean
p n : ℕ
hp : 1 ≤ padicValNat p n
\u22a2 p ∣ n
-- Mathlib/NumberTheory/Padics/PadicVal.lean
p n : ℕ
hp : 1 ≤ padicValNat p n
h : ¬p ∣ n
\u22a2 False
-- Mathlib/NumberTheory/Padics/PadicVal.lean
p n : ℕ
hp : 1 ≤ 0
h : ¬p ∣ n
\u22a2 False
-- Mathlib/Data/Finset/Card.lean
α : Type u_1
β : Type u_2
R : Type u_3
s t u : Finset α
f : α → β
n : ℕ
\u22a2 1 < s.card ↔ s.Nontrivial
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
R : CommRingCat
\u22a2 identityToΓSpec.app (Spec.toLocallyRingedSpace.obj (op R)) ≫ Spec.toLocallyRingedSpace.map (SpecΓIdentity.inv.app R).op = 𝟙 ((𝟭 LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
R : CommRingCat
\u22a2 ((𝟭 LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R))).toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap (SpecΓIdentity.inv.app R).op.unop).val.base = (𝟙 ((𝟭 LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))).val.base
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
R : CommRingCat
\u22a2 ∀ (r : ↑(unop (op ((𝟭 CommRingCat).obj R)))), (SpecΓIdentity.inv.app R).op.unop ≫ ((𝟭 LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R))).presheaf.map (homOfLE ⋯).op = toOpen (↑(unop (op ((𝟭 CommRingCat).obj R)))) (basicOpen r) ≫ (𝟙 ((𝟭 LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))).val.c.app (op (basicOpen r))
-- Mathlib/Algebra/Polynomial/Reverse.lean
R✝ : Type u_1
inst✝² : Semiring R✝
f : R✝[X]
R : Type u_2
inst✝¹ : Ring R
inst✝ : NoZeroDivisors R
p q : R[X]
\u22a2 (p * q).trailingCoeff = p.trailingCoeff * q.trailingCoeff
-- Mathlib/MeasureTheory/Integral/CircleIntegral.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → ℂ
a : E
c : ℂ
R : ℝ
\u22a2 (∮ (z : ℂ) in C(c, R), f z • a) = (∮ (z : ℂ) in C(c, R), f z) • a
-- Mathlib/Algebra/BigOperators/Ring/Multiset.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝² : CommMonoidWithZero α
s✝ : Multiset α
inst✝¹ : NoZeroDivisors α
inst✝ : Nontrivial α
s : Multiset α
l : List α
\u22a2 prod ⟦l⟧ = 0 ↔ 0 ∈ ⟦l⟧
-- Mathlib/Algebra/BigOperators/Ring/Multiset.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝² : CommMonoidWithZero α
s✝ : Multiset α
inst✝¹ : NoZeroDivisors α
inst✝ : Nontrivial α
s : Multiset α
l : List α
\u22a2 l.prod = 0 ↔ 0 ∈ ↑l
-- Mathlib/RingTheory/AdjoinRoot.lean
R : Type u
S : Type v
K : Type w
inst✝ : CommRing R
f p : R[X]
x : R
\u22a2 (aeval (root f)) (C x) = (mk f) (C x)
-- Mathlib/RingTheory/AdjoinRoot.lean
R : Type u
S : Type v
K : Type w
inst✝ : CommRing R
f p : R[X]
x : R
\u22a2 (algebraMap R (AdjoinRoot f)) x = (mk f) (C x)
-- Mathlib/RingTheory/AdjoinRoot.lean
R : Type u
S : Type v
K : Type w
inst✝ : CommRing R
f p✝ p q : R[X]
ihp : (aeval (root f)) p = (mk f) p
ihq : (aeval (root f)) q = (mk f) q
\u22a2 (aeval (root f)) (p + q) = (mk f) (p + q)
-- Mathlib/RingTheory/AdjoinRoot.lean
R : Type u
S : Type v
K : Type w
inst✝ : CommRing R
f p : R[X]
n : ℕ
x : R
x✝ : (aeval (root f)) (C x * X ^ n) = (mk f) (C x * X ^ n)
\u22a2 (aeval (root f)) (C x * X ^ (n + 1)) = (mk f) (C x * X ^ (n + 1))
-- Mathlib/RingTheory/AdjoinRoot.lean
R : Type u
S : Type v
K : Type w
inst✝ : CommRing R
f p : R[X]
n : ℕ
x : R
x✝ : (aeval (root f)) (C x * X ^ n) = (mk f) (C x * X ^ n)
\u22a2 (algebraMap R (AdjoinRoot f)) x * root f ^ (n + 1) = (of f) x * root f ^ (n + 1)
-- Mathlib/NumberTheory/Bernoulli.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
\u22a2 bernoulli' n = 1 - ∑ k ∈ range n, ↑(n.choose k) / (↑n - ↑k + 1) * bernoulli' k
-- Mathlib/RingTheory/Noetherian.lean
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
\u22a2 IsNilpotent (nilradical R)
-- Mathlib/RingTheory/Noetherian.lean
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
n : ℕ
hn : ⊥.radical ^ n ≤ ⊥
\u22a2 IsNilpotent (nilradical R)
-- Mathlib/CategoryTheory/Limits/Shapes/Products.lean
β : Type w
α : Type w₂
γ : Type w₃
C : Type u
inst✝³ : Category.{v, u} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
f : β → C
inst✝¹ : HasProduct f
inst✝ : HasProduct fun b => G.obj (f b)
P : C
g : (j : β) → P ⟶ f j
j : β
\u22a2 (G.map (Pi.lift g) ≫ piComparison G f) ≫ Pi.π (fun b => G.obj (f b)) j = (Pi.lift fun j => G.map (g j)) ≫ Pi.π (fun b => G.obj (f b)) j
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
s : Multiset α
hs : 0 ∉ s
\u22a2 normalizedFactors s.prod = (Multiset.map normalizedFactors s).sum
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
s : Multiset α
hs : 0 ∉ s
h✝ : Subsingleton α
\u22a2 normalizedFactors s.prod = (Multiset.map normalizedFactors s).sum
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
s : Multiset α
hs : 0 ∉ s
h✝ : Nontrivial α
\u22a2 normalizedFactors s.prod = (Multiset.map normalizedFactors s).sum
-- Mathlib/Combinatorics/Enumerative/Partition.lean
n : ℕ
c : Composition n
\u22a2 (↑c.blocks).sum = n
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
M : Type u_1
inst✝¹² : AddCommGroup M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module S M
inst✝⁹ : IsScalarTower R S M
A : Type u_2
B : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : SMulCommClass S A B
h : Function.Surjective ⇑(algebraMap A B)
\u22a2 Submodule.map (Finsupp.mapRange.linearMap (Algebra.linearMap A B) ∘ₗ Finsupp.lmapDomain A A ⇑(algebraMap A B)) (kerTotal R A) ⊔ Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap S B) x) 1) = Submodule.restrictScalars A (kerTotal S B)
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
M : Type u_1
inst✝¹² : AddCommGroup M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module S M
inst✝⁹ : IsScalarTower R S M
A : Type u_2
B : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : SMulCommClass S A B
h : Function.Surjective ⇑(algebraMap A B)
\u22a2 Submodule.span A (⇑(Finsupp.mapRange.linearMap (Algebra.linearMap A B) ∘ₗ Finsupp.lmapDomain A A ⇑(algebraMap A B)) '' (((Set.range fun x => Finsupp.single x.1 1 + Finsupp.single x.2 1 - Finsupp.single (x.1 + x.2) 1) ∪ Set.range fun x => Finsupp.single x.2 x.1 + Finsupp.single x.1 x.2 - Finsupp.single (x.1 * x.2) 1) ∪ Set.range fun x => Finsupp.single ((algebraMap R A) x) 1)) ⊔ Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap S B) x) 1) = Submodule.span A (((Set.range fun x => Finsupp.single x.1 1 + Finsupp.single x.2 1 - Finsupp.single (x.1 + x.2) 1) ∪ Set.range fun x => Finsupp.single x.2 x.1 + Finsupp.single x.1 x.2 - Finsupp.single (x.1 * x.2) 1) ∪ Set.range fun x => Finsupp.single ((algebraMap S B) x) 1)
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
M : Type u_1
inst✝¹² : AddCommGroup M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module S M
inst✝⁹ : IsScalarTower R S M
A : Type u_2
B : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : SMulCommClass S A B
h : Function.Surjective ⇑(algebraMap A B)
\u22a2 Submodule.span A (Set.range fun x => (Finsupp.mapRange.linearMap (Algebra.linearMap A B) ∘ₗ Finsupp.lmapDomain A A ⇑(algebraMap A B)) (Finsupp.single x.1 1) + (Finsupp.mapRange.linearMap (Algebra.linearMap A B) ∘ₗ Finsupp.lmapDomain A A ⇑(algebraMap A B)) (Finsupp.single x.2 1) - (Finsupp.mapRange.linearMap (Algebra.linearMap A B) ∘ₗ Finsupp.lmapDomain A A ⇑(algebraMap A B)) (Finsupp.single (x.1 + x.2) 1)) ⊔ Submodule.span A (Set.range fun x => (Finsupp.mapRange.linearMap (Algebra.linearMap A B) ∘ₗ Finsupp.lmapDomain A A ⇑(algebraMap A B)) (Finsupp.single x.2 x.1) + (Finsupp.mapRange.linearMap (Algebra.linearMap A B) ∘ₗ Finsupp.lmapDomain A A ⇑(algebraMap A B)) (Finsupp.single x.1 x.2) - (Finsupp.mapRange.linearMap (Algebra.linearMap A B) ∘ₗ Finsupp.lmapDomain A A ⇑(algebraMap A B)) (Finsupp.single (x.1 * x.2) 1)) ⊔ Submodule.span A (Set.range fun x => (Finsupp.mapRange.linearMap (Algebra.linearMap A B) ∘ₗ Finsupp.lmapDomain A A ⇑(algebraMap A B)) (Finsupp.single ((algebraMap R A) x) 1)) ⊔ Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap S B) x) 1) = Submodule.span A (Set.range fun x => Finsupp.single x.1 1 + Finsupp.single x.2 1 - Finsupp.single (x.1 + x.2) 1) ⊔ Submodule.span A (Set.range fun x => Finsupp.single x.2 x.1 + Finsupp.single x.1 x.2 - Finsupp.single (x.1 * x.2) 1) ⊔ Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap S B) x) 1)
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
M : Type u_1
inst✝¹² : AddCommGroup M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module S M
inst✝⁹ : IsScalarTower R S M
A : Type u_2
B : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : SMulCommClass S A B
h : Function.Surjective ⇑(algebraMap A B)
\u22a2 Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap A B) x.1) 1 + Finsupp.single ((algebraMap A B) x.2) 1 - Finsupp.single ((algebraMap A B) x.1 + (algebraMap A B) x.2) 1) ⊔ Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap A B) x.2) ((algebraMap A B) x.1) + Finsupp.single ((algebraMap A B) x.1) ((algebraMap A B) x.2) - Finsupp.single ((algebraMap A B) x.1 * (algebraMap A B) x.2) 1) ⊔ Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap A B) ((algebraMap R A) x)) 1) ⊔ Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap S B) x) 1) = Submodule.span A (Set.range fun x => Finsupp.single x.1 1 + Finsupp.single x.2 1 - Finsupp.single (x.1 + x.2) 1) ⊔ Submodule.span A (Set.range fun x => Finsupp.single x.2 x.1 + Finsupp.single x.1 x.2 - Finsupp.single (x.1 * x.2) 1) ⊔ Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap S B) x) 1)
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
M : Type u_1
inst✝¹² : AddCommGroup M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module S M
inst✝⁹ : IsScalarTower R S M
A : Type u_2
B : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : SMulCommClass S A B
h : Function.Surjective ⇑(algebraMap A B)
\u22a2 Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap A B) x.1) 1 + Finsupp.single ((algebraMap A B) x.2) 1 - Finsupp.single ((algebraMap A B) x.1 + (algebraMap A B) x.2) 1) ⊔ (Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap A B) x.2) ((algebraMap A B) x.1) + Finsupp.single ((algebraMap A B) x.1) ((algebraMap A B) x.2) - Finsupp.single ((algebraMap A B) x.1 * (algebraMap A B) x.2) 1) ⊔ (Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap A B) ((algebraMap R A) x)) 1) ⊔ Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap S B) x) 1))) = Submodule.span A (Set.range ((fun x => Finsupp.single x.1 1 + Finsupp.single x.2 1 - Finsupp.single (x.1 + x.2) 1) ∘ Prod.map ⇑(algebraMap A B) ⇑(algebraMap A B))) ⊔ (Submodule.span A (Set.range ((fun x => Finsupp.single x.2 x.1 + Finsupp.single x.1 x.2 - Finsupp.single (x.1 * x.2) 1) ∘ Prod.map ⇑(algebraMap A B) ⇑(algebraMap A B))) ⊔ Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap S B) x) 1))
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
M : Type u_1
inst✝¹² : AddCommGroup M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module S M
inst✝⁹ : IsScalarTower R S M
A : Type u_2
B : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : SMulCommClass S A B
h : Function.Surjective ⇑(algebraMap A B)
\u22a2 Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap A B) ((algebraMap R A) x)) 1) ⊔ Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap S B) x) 1) = Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap S B) x) 1)
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
M : Type u_1
inst✝¹² : AddCommGroup M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module S M
inst✝⁹ : IsScalarTower R S M
A : Type u_2
B : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : SMulCommClass S A B
h : Function.Surjective ⇑(algebraMap A B)
\u22a2 Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap R B) x) 1) ⊔ Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap S B) x) 1) = Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap S B) x) 1)
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
M : Type u_1
inst✝¹² : AddCommGroup M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module S M
inst✝⁹ : IsScalarTower R S M
A : Type u_2
B : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : SMulCommClass S A B
h : Function.Surjective ⇑(algebraMap A B)
\u22a2 Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap R B) x) 1) ≤ Submodule.span A (Set.range fun x => Finsupp.single ((algebraMap S B) x) 1)
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
M : Type u_1
inst✝¹² : AddCommGroup M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module S M
inst✝⁹ : IsScalarTower R S M
A : Type u_2
B : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : SMulCommClass S A B
h : Function.Surjective ⇑(algebraMap A B)
\u22a2 (Set.range fun x => Finsupp.single ((algebraMap R B) x) 1) ⊆ Set.range fun x => Finsupp.single ((algebraMap S B) x) 1
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
M : Type u_1
inst✝¹² : AddCommGroup M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module S M
inst✝⁹ : IsScalarTower R S M
A : Type u_2
B : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : SMulCommClass S A B
h : Function.Surjective ⇑(algebraMap A B)
\u22a2 (Set.range fun x => Finsupp.single ((algebraMap S B) ((algebraMap R S) x)) 1) ⊆ Set.range fun x => Finsupp.single ((algebraMap S B) x) 1
-- Mathlib/Data/Nat/Cast/Commute.lean
α : Type u_1
β : Type u_2
inst✝ : NonAssocSemiring α
n : ℕ
x : α
\u22a2 Commute (↑n) x
-- Mathlib/Data/Nat/Choose/Factorization.lean
p n k : ℕ
n_big : 2 < n
p_le_n : p ≤ n
big : 2 * n < 3 * p
\u22a2 n.centralBinom.factorization p = 0
-- Mathlib/Data/Nat/Choose/Factorization.lean
p n k : ℕ
n_big : 2 < n
p_le_n : p ≤ n
big : 2 * n < 3 * p
\u22a2 p ≠ 2
-- Mathlib/Data/Nat/Choose/Factorization.lean
p n k : ℕ
n_big : 2 < n
p_le_n : p ≤ n
big : 2 * n < 3 * p
\u22a2 n ≤ 2 * n - n
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
t : Measure α := μ.withDensity (ρ.rnDeriv μ)
\u22a2 ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x))
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
t : Measure α := μ.withDensity (ρ.rnDeriv μ)
eq_add : ρ = ρ.singularPart μ + t
\u22a2 ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x))
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
t : Measure α := μ.withDensity (ρ.rnDeriv μ)
eq_add : ρ = ρ.singularPart μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (ρ.singularPart μ) a / μ a) (v.filterAt x) (𝓝 0)
\u22a2 ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x))