ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/CategoryTheory/Limits/VanKampen.lean
J : Type v'
inst✝⁷ : Category.{u', v'} J
C : Type u
inst✝⁶ : Category.{v, u} C
K : Type u_1
inst✝⁵ : Category.{?u.162150, u_1} K
D : Type u_2
inst✝⁴ : Category.{u_3, u_2} D
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝³ : Gr.Full
inst✝² : Gr.Faithful
F : J ⥤ D
c : Cocone (F ⋙ Gr)
H : IsUniversalColimit c
inst✝¹ : ∀ (X : D) (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)
inst✝ : (X : D) → (f : X ⟶ Gl.obj c.pt) → PreservesLimit (cospan (Gr.map f) (adj.unit.app c.pt)) Gl
this✝⁴ : PreservesLimitsOfSize.{0, 0, u_3, v, u_2, u} Gr
this✝³ : PreservesColimitsOfSize.{u', v', v, u_3, u, u_2} Gl
F' : J ⥤ D
c' : Cocone F'
α : F' ⟶ (F ⋙ Gr) ⋙ Gl
f : c'.pt ⟶ (Gl.mapCocone c).pt
h : α ≫ (Gl.mapCocone c).ι = c'.ι ≫ (Functor.const J).map f
hα : NatTrans.Equifibered α
hc' : ∀ (j : J), IsPullback (c'.ι.app j) (α.app j) f ((Gl.mapCocone c).ι.app j)
this✝² : HasPullback (Gl.map (Gr.map f)) (Gl.map (adj.unit.app c.pt))
α' : F' ⟶ F := α ≫ (F.associator Gr Gl).hom ≫ whiskerLeft F adj.counit ≫ F.rightUnitor.hom
hα' : NatTrans.Equifibered α'
hadj : ∀ (X : C), Gl.map (adj.unit.app X) = inv (adj.counit.app (Gl.toPrefunctor.1 ((𝟭 C).obj X)))
this✝¹ : ∀ (X : C), IsIso (Gl.map (adj.unit.app X))
hα'' : ∀ (j : J), Gl.map (Gr.map (α'.app j)) = adj.counit.app (F'.toPrefunctor.1 j) ≫ α.app j
hc'' : ∀ (j : J), α.app j ≫ Gl.map (c.ι.app j) = c'.ι.app j ≫ f
β : F' ⋙ Gr ⋙ Gl ≅ F' := isoWhiskerLeft F' (asIso adj.counit) ≪≫ F'.rightUnitor
c'' : Cocone (F' ⋙ Gr) := { pt := pullback (Gr.map f) (adj.unit.app c.pt), ι := { app := fun j => pullback.lift (Gr.map (c'.ι.app j)) (Gr.map (α'.app j) ≫ c.ι.app j) ⋯, naturality := ⋯ } }
cf : (Cocones.precompose β.hom).obj c' ⟶ Gl.mapCocone c'' := { hom := pullback.lift (inv (adj.counit.app c'.pt)) f ⋯ ≫ (PreservesPullback.iso Gl (Gr.map f) (adj.unit.app c.pt)).inv, w := ⋯ }
this✝ : cf.hom ≫ (PreservesPullback.iso Gl (Gr.map f) (adj.unit.app c.pt)).hom ≫ pullback.fst (Gl.map (Gr.map f)) (Gl.map (adj.unit.app c.pt)) ≫ adj.counit.app c'.pt = 𝟙 ((Cocones.precompose β.hom).obj c').pt
this : IsIso cf
\u22a2 ∀ (j : J), IsPullback (c''.ι.app j) ((whiskerRight α' Gr).app j) (pullback.snd (Gr.map f) (adj.unit.app c.pt)) (c.ι.app j)
-- Mathlib/Data/Nat/Digits.lean
n : ℕ
\u22a2 11 ∣ n ↔ 11 ∣ (List.map (fun n => ↑n) (digits 10 n)).alternatingSum
-- Mathlib/Data/Nat/Digits.lean
n : ℕ
t : 11 ∣ n ↔ ↑11 ∣ ofDigits (-1) (digits 10 n)
\u22a2 11 ∣ n ↔ 11 ∣ (List.map (fun n => ↑n) (digits 10 n)).alternatingSum
-- Mathlib/Data/Nat/Digits.lean
n : ℕ
t : 11 ∣ n ↔ ↑11 ∣ (List.map (fun n => ↑n) (digits 10 n)).alternatingSum
\u22a2 11 ∣ n ↔ 11 ∣ (List.map (fun n => ↑n) (digits 10 n)).alternatingSum
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
\u22a2 FromSpec.toFun f_deg hm ((toSpec 𝒜 f) x) = x
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
\u22a2 ∀ (i : ℕ), ∀ x_1 ∈ 𝒜 i, x_1 ∈ (↑(FromSpec.toFun f_deg hm ((toSpec 𝒜 f) x))).asHomogeneousIdeal ↔ x_1 ∈ (↑x).asHomogeneousIdeal
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
i : ℕ
z : A
hzi : z ∈ 𝒜 i
\u22a2 z ∈ (↑(FromSpec.toFun f_deg hm ((toSpec 𝒜 f) x))).asHomogeneousIdeal ↔ z ∈ (↑x).asHomogeneousIdeal
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
i : ℕ
z : A
hzi : z ∈ 𝒜 i
\u22a2 HomogeneousLocalization.mk { deg := m * i, num := ⟨z ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈ ((toSpec 𝒜 f) x).asIdeal ↔ z ∈ (↑x).asHomogeneousIdeal
-- Mathlib/NumberTheory/MulChar/Basic.lean
R : Type u_1
inst✝³ : CommMonoid R
inst✝² : Fintype R
R' : Type u_2
inst✝¹ : CommRing R'
inst✝ : DecidableEq R
\u22a2 ∑ a : R, 1 a = ↑(Fintype.card Rˣ)
-- Mathlib/NumberTheory/MulChar/Basic.lean
R : Type u_1
inst✝³ : CommMonoid R
inst✝² : Fintype R
R' : Type u_2
inst✝¹ : CommRing R'
inst✝ : DecidableEq R
a : R
x✝ : a ∈ Finset.univ
\u22a2 1 a = if IsUnit a then 1 else 0
-- Mathlib/NumberTheory/MulChar/Basic.lean
R : Type u_1
inst✝³ : CommMonoid R
inst✝² : Fintype R
R' : Type u_2
inst✝¹ : CommRing R'
inst✝ : DecidableEq R
\u22a2 ↑(Finset.filter IsUnit Finset.univ).card = ↑(Finset.map { toFun := Units.val, inj' := ⋯ } Finset.univ).card
-- Mathlib/Topology/Order/IntermediateValue.lean
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a✝ b✝ a b : α
s : Set α
hs : IsClosed (s ∩ Icc a b)
ha : a ∈ s
hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x
\u22a2 Icc a b ⊆ s
-- Mathlib/Topology/Order/IntermediateValue.lean
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a✝ b✝ a b : α
s : Set α
hs : IsClosed (s ∩ Icc a b)
ha : a ∈ s
hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x
\u22a2 ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).Nonempty
-- Mathlib/Topology/Order/IntermediateValue.lean
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a✝ b✝ a b : α
s : Set α
hs : IsClosed (s ∩ Icc a b)
ha : a ∈ s
hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x
x : α
hxs : x ∈ s
hxab : x ∈ Ico a b
y : α
hyxb : y ∈ Ioi x
\u22a2 (s ∩ Ioc x y).Nonempty
-- Mathlib/Topology/Order/IntermediateValue.lean
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a✝ b✝ a b : α
s : Set α
hs : IsClosed (s ∩ Icc a b)
ha : a ∈ s
hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x
x : α
hxs : x ∈ s
hxab : x ∈ Ico a b
y : α
hyxb : y ∈ Ioi x
this : s ∩ Ioc x y ∈ 𝓝[>] x
\u22a2 (s ∩ Ioc x y).Nonempty
-- Mathlib/Data/ENNReal/Real.lean
ι : Sort u_1
f g : ι → ℝ≥0∞
a b c d : ℝ≥0∞
r p q : ℝ≥0
s : Set ℝ≥0∞
\u22a2 sInf s + a = ⨅ b ∈ s, b + a
-- Mathlib/SetTheory/Game/PGame.lean
xl xr : Type u
x : PGame
\u22a2 0 ⧏ x ↔ ∃ i, 0 ≤ x.moveLeft i
-- Mathlib/SetTheory/Game/PGame.lean
xl xr : Type u
x : PGame
\u22a2 ((∃ i, 0 ≤ x.moveLeft i) ∨ ∃ j, moveRight 0 j ≤ x) ↔ ∃ i, 0 ≤ x.moveLeft i
-- Mathlib/Algebra/Group/UniqueProds.lean
G : Type u_1
H : Type u_2
inst✝² : Mul G
inst✝¹ : Mul H
A B : Finset G
a0 b0 : G
inst✝ : DecidableEq G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
\u22a2 UniqueMul A B a0 b0 ↔ (filter (fun p => p.1 * p.2 = a0 * b0) (A ×ˢ B)).card ≤ 1
-- Mathlib/Algebra/Group/UniqueProds.lean
G : Type u_1
H : Type u_2
inst✝² : Mul G
inst✝¹ : Mul H
A B : Finset G
a0 b0 : G
inst✝ : DecidableEq G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
\u22a2 UniqueMul A B a0 b0 ↔ ∀ {a b : G × G}, (a.1 ∈ A ∧ a.2 ∈ B) ∧ a.1 * a.2 = a0 * b0 → (b.1 ∈ A ∧ b.2 ∈ B) ∧ b.1 * b.2 = a0 * b0 → a = b
-- Mathlib/RingTheory/Adjoin/Basic.lean
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
\u22a2 adjoin S ↑(adjoin R s) = adjoin S s
-- Mathlib/RingTheory/Adjoin/Basic.lean
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
\u22a2 adjoin S s ≤ adjoin S ↑(adjoin R s)
-- Mathlib/RingTheory/Adjoin/Basic.lean
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
\u22a2 ↑(adjoin R s) ⊆ ↑(adjoin S s)
-- Mathlib/Combinatorics/SimpleGraph/Finite.lean
V : Type u_1
G : SimpleGraph V
e : Sym2 V
v : V
inst✝¹ : Fintype ↑(G.neighborSet v)
inst✝ : DecidableEq V
\u22a2 Fintype.card ↑(G.incidenceSet v) = G.degree v
-- Mathlib/Combinatorics/SimpleGraph/Finite.lean
V : Type u_1
G : SimpleGraph V
e : Sym2 V
v : V
inst✝¹ : Fintype ↑(G.neighborSet v)
inst✝ : DecidableEq V
\u22a2 Fintype.card ↑(G.neighborSet v) = G.degree v
-- Mathlib/CategoryTheory/Sites/SheafHom.lean
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
Y : C
g : Y ⟶ X
\u22a2 ∃ φ, ∀ {Z : C} (p : Z ⟶ Y) (hp : S.arrows (p ≫ g)), φ ≫ G.map p.op = F.map p.op ≫ (x (p ≫ g) hp).app (op (Over.mk (𝟙 Z)))
-- Mathlib/CategoryTheory/Sites/SheafHom.lean
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
Y : C
g : Y ⟶ X
c : Cone ((Sieve.pullback g S).arrows.diagram.op ⋙ G) := { pt := F.obj (op Y), π := { app := fun x_1 => match x_1 with | op { obj := Z, property := hZ } => F.map Z.hom.op ≫ (x (Z.hom ≫ g) hZ).app (op (Over.mk (𝟙 (unop (op ((𝟭 C).obj Z.left)))))), naturality := ⋯ } }
\u22a2 ∃ φ, ∀ {Z : C} (p : Z ⟶ Y) (hp : S.arrows (p ≫ g)), φ ≫ G.map p.op = F.map p.op ≫ (x (p ≫ g) hp).app (op (Over.mk (𝟙 Z)))
-- Mathlib/CategoryTheory/Sites/SheafHom.lean
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
Y : C
g : Y ⟶ X
c : Cone ((Sieve.pullback g S).arrows.diagram.op ⋙ G) := { pt := F.obj (op Y), π := { app := fun x_1 => match x_1 with | op { obj := Z, property := hZ } => F.map Z.hom.op ≫ (x (Z.hom ≫ g) hZ).app (op (Over.mk (𝟙 (unop (op ((𝟭 C).obj Z.left)))))), naturality := ⋯ } }
\u22a2 ∀ {Z : C} (p : Z ⟶ Y) (hp : S.arrows (p ≫ g)), (hG g).lift c ≫ G.map p.op = F.map p.op ≫ (x (p ≫ g) hp).app (op (Over.mk (𝟙 Z)))
-- Mathlib/CategoryTheory/Sites/SheafHom.lean
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
Y : C
g : Y ⟶ X
c : Cone ((Sieve.pullback g S).arrows.diagram.op ⋙ G) := { pt := F.obj (op Y), π := { app := fun x_1 => match x_1 with | op { obj := Z, property := hZ } => F.map Z.hom.op ≫ (x (Z.hom ≫ g) hZ).app (op (Over.mk (𝟙 (unop (op ((𝟭 C).obj Z.left)))))), naturality := ⋯ } }
Z : C
p : Z ⟶ Y
hp : S.arrows (p ≫ g)
\u22a2 (hG g).lift c ≫ G.map p.op = F.map p.op ≫ (x (p ≫ g) hp).app (op (Over.mk (𝟙 Z)))
-- Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
α : Type u_1
x : ℂ
h : x ∈ slitPlane
\u22a2 ContinuousAt log x
-- Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
α : Type u_1
x : ℂ
h : x ∈ slitPlane
\u22a2 ContinuousAt (fun x => ↑(Real.log (Complex.abs x))) x
-- Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
α : Type u_1
x : ℂ
h : x ∈ slitPlane
\u22a2 ContinuousAt (fun x => ↑x.arg * I) x
-- Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
\u22a2 Tendsto (fun x => log x * x) (𝓝[<] 0) (𝓝 0)
-- Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
h : Tendsto (fun x => log x * x ^ 1) (𝓝[>] 0) (𝓝 0)
\u22a2 Tendsto (fun x => log x * x) (𝓝[<] 0) (𝓝 0)
-- Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
h : Tendsto (fun x => log x * x) (𝓝[>] 0) (𝓝 0)
\u22a2 Tendsto (fun x => log x * x) (𝓝[<] 0) (𝓝 0)
-- Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
h : Tendsto (fun x => log x * x) (𝓝[>] 0) (𝓝 0)
h_eq : ∀ x ∈ Set.Iio 0, (-(fun x => log x * x) ∘ fun x => |x|) x = log x * x
\u22a2 Tendsto (fun x => log x * x) (𝓝[<] 0) (𝓝 0)
-- Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
h : Tendsto (fun x => log x * x) (𝓝[>] 0) (𝓝 0)
h_eq : ∀ x ∈ Set.Iio 0, (-(fun x => log x * x) ∘ fun x => |x|) x = log x * x
\u22a2 Tendsto (-(fun x => log x * x) ∘ fun x => |x|) (𝓝[<] 0) (𝓝 0)
-- Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
h : Tendsto (fun x => log x * x) (𝓝[>] 0) (𝓝 0)
h_eq : ∀ x ∈ Set.Iio 0, (-(fun x => log x * x) ∘ fun x => |x|) x = log x * x
\u22a2 Tendsto (-(fun x => log x * x) ∘ fun x => |x|) (𝓝[<] 0) (𝓝 (-0))
-- Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
h : Tendsto (fun x => log x * x) (𝓝[>] 0) (𝓝 0)
h_eq : ∀ x ∈ Set.Iio 0, (-(fun x => log x * x) ∘ fun x => |x|) x = log x * x
\u22a2 𝓝[<] 0 ≤ 𝓝[≠] 0
-- Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
h : Tendsto (fun x => log x * x) (𝓝[>] 0) (𝓝 0)
h_eq : ∀ x ∈ Set.Iio 0, (-(fun x => log x * x) ∘ fun x => |x|) x = log x * x
x : ℝ
hx : x ∈ Set.Iio 0
\u22a2 x ∈ {0}ᶜ
-- Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
h : Tendsto (fun x => log x * x) (𝓝[>] 0) (𝓝 0)
h_eq : ∀ x ∈ Set.Iio 0, (-(fun x => log x * x) ∘ fun x => |x|) x = log x * x
x : ℝ
hx : x < 0
\u22a2 x ∈ {0}ᶜ
-- Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
P : WeakFEPair E
s : ℂ
\u22a2 P.symm.Λ₀ s = P.symm.Λ s + (1 / s) • P.g₀ + (P.ε⁻¹ / (↑P.k - s)) • P.f₀
-- Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
P : WeakFEPair E
s : ℂ
\u22a2 P.symm.Λ s + (1 / s) • P.symm.f₀ + (P.symm.ε / (↑P.symm.k - s)) • P.symm.g₀ = P.symm.Λ s + (1 / s) • P.g₀ + (P.ε⁻¹ / (↑P.k - s)) • P.f₀
-- Mathlib/Algebra/AddTorsor.lean
G : Type u_1
P : Type u_2
inst✝¹ : AddCommGroup G
inst✝ : AddTorsor G P
v₁ v₂ : G
p₁ p₂ : P
\u22a2 v₁ +ᵥ p₁ = v₂ +ᵥ p₂ ↔ v₂ - v₁ = p₁ -ᵥ p₂
-- Mathlib/Topology/MetricSpace/Infsep.lean
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Set α
inst✝ : Finite ↑s
\u22a2 0 < s.infsep ↔ s.Nontrivial
-- Mathlib/Topology/MetricSpace/Infsep.lean
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Set α
inst✝ : Finite ↑s
\u22a2 s.Nontrivial → 0 < s.einfsep
-- Mathlib/CategoryTheory/Galois/Prorepresentability.lean
C : Type u₁
inst✝³ : Category.{u₂, u₁} C
inst✝² : GaloisCategory C
F : C ⥤ FintypeCat
inst✝¹ : FiberFunctor F
f : Aut F
A : C
inst✝ : IsGalois A
a : ↑(F.obj A)
\u22a2 F.map ((AutGalois.π F { obj := A, pt := a, isGalois := ⋯ }) ((endEquivAutGalois F) ↑((Aut.unitsEndEquivAut F).symm f))).hom a = f.hom.app A a
-- Mathlib/CategoryTheory/Galois/Prorepresentability.lean
C : Type u₁
inst✝³ : Category.{u₂, u₁} C
inst✝² : GaloisCategory C
F : C ⥤ FintypeCat
inst✝¹ : FiberFunctor F
f : Aut F
A : C
inst✝ : IsGalois A
a : ↑(F.obj A)
\u22a2 (↑((Aut.unitsEndEquivAut F).symm f)).app { obj := A, pt := a, isGalois := ⋯ }.obj { obj := A, pt := a, isGalois := ⋯ }.pt = f.hom.app A a
-- Mathlib/Analysis/SpecialFunctions/Gamma/Deligne.lean
s : ℂ
hs : s ≠ 0
\u22a2 (s + 1).Gammaℂ = s.Gammaℂ * s / 2 / ↑π
-- Mathlib/Analysis/SpecialFunctions/Gamma/Deligne.lean
s : ℂ
hs : s ≠ 0
\u22a2 2 * ((2 * ↑π) ^ (-s) * (2 * ↑π)⁻¹) * (s * Gamma s) = 2 * (2 * ↑π) ^ (-s) * Gamma s * s / 2 / ↑π
-- Mathlib/Analysis/SpecialFunctions/Gamma/Deligne.lean
s : ℂ
hs : s ≠ 0
\u22a2 2 * (2 * ↑π) ^ (-s) * (s * Gamma s) = 2 * (2 * ↑π) ^ (-s) * Gamma s * s
-- Mathlib/Geometry/Manifold/VectorBundle/Basic.lean
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁹ : NontriviallyNormedField 𝕜
inst✝¹⁸ : NormedAddCommGroup F
inst✝¹⁷ : NormedSpace 𝕜 F
inst✝¹⁶ : TopologicalSpace (TotalSpace F E)
inst✝¹⁵ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁴ : NormedAddCommGroup EB
inst✝¹³ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹² : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
E' : B → Type u_9
inst✝¹¹ : (x : B) → Zero (E' x)
EM : Type u_10
inst✝¹⁰ : NormedAddCommGroup EM
inst✝⁹ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁸ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁵ : TopologicalSpace B
inst✝⁴ : ChartedSpace HB B
inst✝³ : FiberBundle F E
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module 𝕜 (E x)
inst✝ : VectorBundle 𝕜 F E
x : B
\u22a2 ContMDiffAt IB (IB.prod 𝓘(𝕜, F)) ⊤ (fun x => { proj := x, snd := 0 }) x
-- Mathlib/Geometry/Manifold/VectorBundle/Basic.lean
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁹ : NontriviallyNormedField 𝕜
inst✝¹⁸ : NormedAddCommGroup F
inst✝¹⁷ : NormedSpace 𝕜 F
inst✝¹⁶ : TopologicalSpace (TotalSpace F E)
inst✝¹⁵ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁴ : NormedAddCommGroup EB
inst✝¹³ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹² : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
E' : B → Type u_9
inst✝¹¹ : (x : B) → Zero (E' x)
EM : Type u_10
inst✝¹⁰ : NormedAddCommGroup EM
inst✝⁹ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁸ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁵ : TopologicalSpace B
inst✝⁴ : ChartedSpace HB B
inst✝³ : FiberBundle F E
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module 𝕜 (E x)
inst✝ : VectorBundle 𝕜 F E
x : B
\u22a2 ContMDiffAt IB 𝓘(𝕜, F) ⊤ (fun x_1 => (↑(trivializationAt F E x) { proj := x_1, snd := 0 }).2) x
-- Mathlib/Geometry/Manifold/VectorBundle/Basic.lean
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁹ : NontriviallyNormedField 𝕜
inst✝¹⁸ : NormedAddCommGroup F
inst✝¹⁷ : NormedSpace 𝕜 F
inst✝¹⁶ : TopologicalSpace (TotalSpace F E)
inst✝¹⁵ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁴ : NormedAddCommGroup EB
inst✝¹³ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹² : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
E' : B → Type u_9
inst✝¹¹ : (x : B) → Zero (E' x)
EM : Type u_10
inst✝¹⁰ : NormedAddCommGroup EM
inst✝⁹ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁸ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁵ : TopologicalSpace B
inst✝⁴ : ChartedSpace HB B
inst✝³ : FiberBundle F E
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module 𝕜 (E x)
inst✝ : VectorBundle 𝕜 F E
x : B
\u22a2 (fun x_1 => (↑(trivializationAt F E x) { proj := x_1, snd := 0 }).2) =ᶠ[𝓝 x] fun x => 0
-- Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
α : Type u_1
a b : ℂ
ha : a ≠ 0
\u22a2 ContinuousAt (fun x => a ^ x) b
-- Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
α : Type u_1
a b : ℂ
ha : a ≠ 0
cpow_eq : (fun x => a ^ x) = fun x => cexp (Complex.log a * x)
\u22a2 ContinuousAt (fun x => a ^ x) b
-- Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
α : Type u_1
a b : ℂ
ha : a ≠ 0
cpow_eq : (fun x => a ^ x) = fun x => cexp (Complex.log a * x)
\u22a2 ContinuousAt (fun x => cexp (Complex.log a * x)) b
-- Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean
C : Type u
inst✝² : Category.{v, u} C
I : MulticospanIndex C
inst✝¹ : HasProduct I.left
inst✝ : HasProduct I.right
K : Multifork I
\u22a2 ∀ (j : WalkingMulticospan I.fstTo I.sndTo), ((𝟭 (Multifork I)).obj K).π.app j = (Iso.refl ((𝟭 (Multifork I)).obj K).pt).hom ≫ ((I.toPiForkFunctor ⋙ I.ofPiForkFunctor).obj K).π.app j
-- Mathlib/MeasureTheory/Group/Prod.lean
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
\u22a2 MeasurePreserving (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod ν) (μ.prod ν)
-- Mathlib/MeasureTheory/Group/Prod.lean
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
\u22a2 (fun z => (z.2 * z.1, z.1⁻¹)) = (fun z => (z.2, z.2⁻¹ * z.1)) ∘ fun z => (z.2, z.2 * z.1)
-- Mathlib/MeasureTheory/Group/Prod.lean
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
x y : G
\u22a2 ((x, y).2 * (x, y).1, (x, y).1⁻¹) = ((fun z => (z.2, z.2⁻¹ * z.1)) ∘ fun z => (z.2, z.2 * z.1)) (x, y)
-- Mathlib/NumberTheory/DirichletCharacter/GaussSum.lean
N : ℕ
inst✝² : NeZero N
R : Type u_1
inst✝¹ : CommRing R
e : AddChar (ZMod N) R
inst✝ : IsDomain R
χ : DirichletCharacter R N
hχ : χ.IsPrimitive
he : ¬e.IsPrimitive
\u22a2 gaussSum χ e = 0
-- Mathlib/NumberTheory/DirichletCharacter/GaussSum.lean
N : ℕ
inst✝² : NeZero N
R : Type u_1
inst✝¹ : CommRing R
e : AddChar (ZMod N) R
inst✝ : IsDomain R
χ : DirichletCharacter R N
he : ¬e.IsPrimitive
hχ : gaussSum χ e ≠ 0
\u22a2 ¬χ.IsPrimitive
-- Mathlib/NumberTheory/DirichletCharacter/GaussSum.lean
N : ℕ
inst✝² : NeZero N
R : Type u_1
inst✝¹ : CommRing R
e : AddChar (ZMod N) R
inst✝ : IsDomain R
χ : DirichletCharacter R N
he : ¬e.IsPrimitive
hχ : gaussSum χ e ≠ 0
d : ℕ
hd₁ : d ∣ N
hd₂ : d < N
hed : e.mulShift ↑d = 1
\u22a2 ¬χ.IsPrimitive
-- Mathlib/NumberTheory/DirichletCharacter/GaussSum.lean
N : ℕ
inst✝² : NeZero N
R : Type u_1
inst✝¹ : CommRing R
e : AddChar (ZMod N) R
inst✝ : IsDomain R
χ : DirichletCharacter R N
he : ¬e.IsPrimitive
hχ : gaussSum χ e ≠ 0
d : ℕ
hd₁ : d ∣ N
hd₂ : d < N
hed : e.mulShift ↑d = 1
this : χ.conductor ≤ d
\u22a2 ¬χ.IsPrimitive
-- Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
θ : Real.Angle
\u22a2 ∀ (x : V), ((-o).rotation θ) x = (o.rotation (-θ)) x
-- Mathlib/GroupTheory/Coset.lean
α : Type u_1
inst✝ : Group α
s t : Subgroup α
ι : Type u_2
f : ι → Subgroup α
H : Subgroup α
\u22a2 ∀ (a₁ a₂ : ↥H), (fun q i => quotientSubgroupOfMapOfLE H ⋯ q) (Quotient.mk'' a₁) = (fun q i => quotientSubgroupOfMapOfLE H ⋯ q) (Quotient.mk'' a₂) → Quotient.mk'' a₁ = Quotient.mk'' a₂
-- Mathlib/Algebra/Jordan/Basic.lean
A : Type u_1
inst✝¹ : NonUnitalNonAssocCommRing A
inst✝ : IsCommJordan A
a b c : A
\u22a2 0 = 2 • (⁅L a, L (b * c)⁆ + ⁅L b, L (c * a)⁆ + ⁅L c, L (a * b)⁆)
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝¹³ : CommRing R✝
inst✝¹² : CommRing S✝
M✝ : Submonoid R✝
N : Submonoid S✝
R'✝ S'✝ : Type u
inst✝¹¹ : CommRing R'✝
inst✝¹⁰ : CommRing S'✝
f✝ : R✝ →+* S✝
inst✝⁹ : Algebra R✝ R'✝
inst✝⁸ : Algebra S✝ S'✝
R S : Type u_1
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
M : Submonoid R
R' S' : Type u_1
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization M R'
inst✝ : IsLocalization (Submonoid.map f M) S'
hf : f.FiniteType
this✝ : Algebra R S := f.toAlgebra
this : Algebra R S' := ((algebraMap S S').comp f).toAlgebra
\u22a2 (IsLocalization.map S' f ⋯).FiniteType