ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.2869, u_2} D
inst✝³ : ConcreteCategory D
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
h : Presheaf.IsLocallySurjective (extensiveTopology C) f
X : C
x : G.obj (op X)
\u22a2 ∃ a, f.app (op X) a = x
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.2869, u_2} D
inst✝³ : ConcreteCategory D
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
h : Presheaf.imageSieve f x ∈ (extensiveTopology C).sieves X
\u22a2 ∃ a, f.app (op X) a = x
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.2869, u_2} D
inst✝³ : ConcreteCategory D
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
h : ∃ α, ∃ (_ : Finite α), ∃ Y π, Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), (Presheaf.imageSieve f x).arrows (π a)
\u22a2 ∃ a, f.app (op X) a = x
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.2869, u_2} D
inst✝³ : ConcreteCategory D
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), (Presheaf.imageSieve f x).arrows (π a)
y : (a : α) → F.obj (op (Y a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
ht : IsLimit (Types.productLimitCone fun a => F.obj (op (Y a))).cone := (Types.productLimitCone fun a => F.obj (op (Y a))).isLimit
ht' : IsLimit (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))) := (Functor.Initial.isLimitWhiskerEquiv (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))).symm h.some.op
i : ((a : α) → F.obj (op (Y a))) ≅ F.obj (op X) := ht.conePointsIsoOfNatIso (isLimitOfPreserves F ht') (Discrete.natIso fun x => Iso.refl (F.obj (op (Y x.as))))
\u22a2 f.app (op X) (i.hom y) = x
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.2869, u_2} D
inst✝³ : ConcreteCategory D
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), (Presheaf.imageSieve f x).arrows (π a)
y : (a : α) → F.obj (op (Y a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
ht : IsLimit (Types.productLimitCone fun a => F.obj (op (Y a))).cone := (Types.productLimitCone fun a => F.obj (op (Y a))).isLimit
ht' : IsLimit (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))) := (Functor.Initial.isLimitWhiskerEquiv (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))).symm h.some.op
i : ((a : α) → F.obj (op (Y a))) ≅ F.obj (op X) := ht.conePointsIsoOfNatIso (isLimitOfPreserves F ht') (Discrete.natIso fun x => Iso.refl (F.obj (op (Y x.as))))
\u22a2 ∀ (j : Discrete α), ((G.mapCone (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π)))).π.app j) (f.app (op X) (i.hom y)) = ((G.mapCone (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π)))).π.app j) x
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.2869, u_2} D
inst✝³ : ConcreteCategory D
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), (Presheaf.imageSieve f x).arrows (π a)
y : (a : α) → F.obj (op (Y a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
ht : IsLimit (Types.productLimitCone fun a => F.obj (op (Y a))).cone := (Types.productLimitCone fun a => F.obj (op (Y a))).isLimit
ht' : IsLimit (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))) := (Functor.Initial.isLimitWhiskerEquiv (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))).symm h.some.op
i : ((a : α) → F.obj (op (Y a))) ≅ F.obj (op X) := ht.conePointsIsoOfNatIso (isLimitOfPreserves F ht') (Discrete.natIso fun x => Iso.refl (F.obj (op (Y x.as))))
a : α
\u22a2 ((G.mapCone (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π)))).π.app { as := a }) (f.app (op X) (i.hom y)) = ((G.mapCone (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π)))).π.app { as := a }) x
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.2869, u_2} D
inst✝³ : ConcreteCategory D
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), (Presheaf.imageSieve f x).arrows (π a)
y : (a : α) → F.obj (op (Y a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
ht : IsLimit (Types.productLimitCone fun a => F.obj (op (Y a))).cone := (Types.productLimitCone fun a => F.obj (op (Y a))).isLimit
ht' : IsLimit (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))) := (Functor.Initial.isLimitWhiskerEquiv (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))).symm h.some.op
i : ((a : α) → F.obj (op (Y a))) ≅ F.obj (op X) := ht.conePointsIsoOfNatIso (isLimitOfPreserves F ht') (Discrete.natIso fun x => Iso.refl (F.obj (op (Y x.as))))
a : α
\u22a2 (G.map (π a).op) (f.app (op X) (i.hom y)) = (G.map (π a).op) x
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.2869, u_2} D
inst✝³ : ConcreteCategory D
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), (Presheaf.imageSieve f x).arrows (π a)
y : (a : α) → F.obj (op (Y a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
ht : IsLimit (Types.productLimitCone fun a => F.obj (op (Y a))).cone := (Types.productLimitCone fun a => F.obj (op (Y a))).isLimit
ht' : IsLimit (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))) := (Functor.Initial.isLimitWhiskerEquiv (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))).symm h.some.op
i : ((a : α) → F.obj (op (Y a))) ≅ F.obj (op X) := ht.conePointsIsoOfNatIso (isLimitOfPreserves F ht') (Discrete.natIso fun x => Iso.refl (F.obj (op (Y x.as))))
a : α
this : f.app (op (Y a)) (y a) = G.map (π a).op x
\u22a2 (G.map (π a).op) (f.app (op X) (i.hom y)) = (G.map (π a).op) x
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.2869, u_2} D
inst✝³ : ConcreteCategory D
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), (Presheaf.imageSieve f x).arrows (π a)
y : (a : α) → F.obj (op (Y a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
ht : IsLimit (Types.productLimitCone fun a => F.obj (op (Y a))).cone := (Types.productLimitCone fun a => F.obj (op (Y a))).isLimit
ht' : IsLimit (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))) := (Functor.Initial.isLimitWhiskerEquiv (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))).symm h.some.op
i : ((a : α) → F.obj (op (Y a))) ≅ F.obj (op X) := ht.conePointsIsoOfNatIso (isLimitOfPreserves F ht') (Discrete.natIso fun x => Iso.refl (F.obj (op (Y x.as))))
a : α
this : f.app (op (Y a)) (y a) = G.map (π a).op x
\u22a2 (G.map (π a).op) (f.app (op X) (i.hom y)) = G.map (π a).op x
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.2869, u_2} D
inst✝³ : ConcreteCategory D
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), (Presheaf.imageSieve f x).arrows (π a)
y : (a : α) → F.obj (op (Y a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
ht : IsLimit (Types.productLimitCone fun a => F.obj (op (Y a))).cone := (Types.productLimitCone fun a => F.obj (op (Y a))).isLimit
ht' : IsLimit (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))) := (Functor.Initial.isLimitWhiskerEquiv (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))).symm h.some.op
i : ((a : α) → F.obj (op (Y a))) ≅ F.obj (op X) := ht.conePointsIsoOfNatIso (isLimitOfPreserves F ht') (Discrete.natIso fun x => Iso.refl (F.obj (op (Y x.as))))
a : α
this : f.app (op (Y a)) (y a) = G.map (π a).op x
\u22a2 (f.app (op (Y a))) ((F.map (π a).op) (i.hom y)) = f.app (op (Y a)) (y a)
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.2869, u_2} D
inst✝³ : ConcreteCategory D
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), (Presheaf.imageSieve f x).arrows (π a)
y : (a : α) → F.obj (op (Y a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
ht : IsLimit (Types.productLimitCone fun a => F.obj (op (Y a))).cone := (Types.productLimitCone fun a => F.obj (op (Y a))).isLimit
ht' : IsLimit (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))) := (Functor.Initial.isLimitWhiskerEquiv (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))).symm h.some.op
i : ((a : α) → F.obj (op (Y a))) ≅ F.obj (op X) := ht.conePointsIsoOfNatIso (isLimitOfPreserves F ht') (Discrete.natIso fun x => Iso.refl (F.obj (op (Y x.as))))
a : α
this : f.app (op (Y a)) (y a) = G.map (π a).op x
\u22a2 (F.map (π a).op) (i.hom y) = y a
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.2869, u_2} D
inst✝³ : ConcreteCategory D
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), (Presheaf.imageSieve f x).arrows (π a)
y : (a : α) → F.obj (op (Y a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
ht : IsLimit (Types.productLimitCone fun a => F.obj (op (Y a))).cone := (Types.productLimitCone fun a => F.obj (op (Y a))).isLimit
ht' : IsLimit (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))) := (Functor.Initial.isLimitWhiskerEquiv (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))).symm h.some.op
i : ((a : α) → F.obj (op (Y a))) ≅ F.obj (op X) := ht.conePointsIsoOfNatIso (isLimitOfPreserves F ht') (Discrete.natIso fun x => Iso.refl (F.obj (op (Y x.as))))
a : α
this : f.app (op (Y a)) (y a) = G.map (π a).op x
\u22a2 (i.hom ≫ F.map (π a).op) y = y a
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.2869, u_2} D
inst✝³ : ConcreteCategory D
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), (Presheaf.imageSieve f x).arrows (π a)
y : (a : α) → F.obj (op (Y a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
ht : IsLimit (Types.productLimitCone fun a => F.obj (op (Y a))).cone := (Types.productLimitCone fun a => F.obj (op (Y a))).isLimit
ht' : IsLimit (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))) := (Functor.Initial.isLimitWhiskerEquiv (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))).symm h.some.op
i : ((a : α) → F.obj (op (Y a))) ≅ F.obj (op X) := ht.conePointsIsoOfNatIso (isLimitOfPreserves F ht') (Discrete.natIso fun x => Iso.refl (F.obj (op (Y x.as))))
a : α
this : f.app (op (Y a)) (y a) = G.map (π a).op x
\u22a2 ((Types.productLimitCone fun a => F.obj (op (Y a))).cone.π.app { as := a } ≫ (Discrete.natIso fun x => Iso.refl (F.obj (op (Y x.as)))).hom.app { as := a }) y = y a
-- Mathlib/FieldTheory/RatFunc/Basic.lean
K : Type u
inst✝ : Field K
p : K[X]
\u22a2 ((algebraMap K[X] (RatFunc K)) p).num = p
-- Mathlib/Data/Finset/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s✝ t✝ u v : Finset α
a✝ b : α
s t : Finset α
a : α
\u22a2 s.erase a \ t = (s \ t).erase a
-- Mathlib/Data/PFunctor/Multivariate/W.lean
n : ℕ
P : MvPFunctor.{u} (n + 1)
α : TypeVec.{u} n
C : Type u_1
g : (a : P.A) → P.drop.B a ⟹ α → (P.last.B a → P.W α) → (P.last.B a → C) → C
a : P.A
f' : P.drop.B a ⟹ α
f : P.last.B a → P.W α
\u22a2 P.wpRec (fun a f h h' => g a (P.wPathDestLeft h) (fun i => ⟨f i, P.wPathDestRight h i⟩) h') (WType.mk a fun i => (f i).fst) (P.wPathCasesOn f' fun i => (f i).snd) = g a f' f fun i => P.wRec g (f i)
-- Mathlib/Data/PFunctor/Multivariate/W.lean
n : ℕ
P : MvPFunctor.{u} (n + 1)
α : TypeVec.{u} n
C : Type u_1
g : (a : P.A) → P.drop.B a ⟹ α → (P.last.B a → P.W α) → (P.last.B a → C) → C
a : P.A
f' : P.drop.B a ⟹ α
f : P.last.B a → P.W α
\u22a2 (g a (P.wPathDestLeft (P.wPathCasesOn f' fun i => (f i).snd)) (fun i => ⟨(f i).fst, P.wPathDestRight (P.wPathCasesOn f' fun i => (f i).snd) i⟩) fun i => P.wpRec (fun a f h h' => g a (P.wPathDestLeft h) (fun i => ⟨f i, P.wPathDestRight h i⟩) h') (f i).fst (P.wPathDestRight (P.wPathCasesOn f' fun i => (f i).snd) i)) = g a f' f fun i => P.wRec g (f i)
-- Mathlib/Data/PFunctor/Multivariate/W.lean
n : ℕ
P : MvPFunctor.{u} (n + 1)
α : TypeVec.{u} n
C : Type u_1
g : (a : P.A) → P.drop.B a ⟹ α → (P.last.B a → P.W α) → (P.last.B a → C) → C
a : P.A
f' : P.drop.B a ⟹ α
f : P.last.B a → P.W α
\u22a2 (g a f' (fun i => ⟨(f i).fst, (f i).snd⟩) fun i => P.wpRec (fun a f h h' => g a (P.wPathDestLeft h) (fun i => ⟨f i, P.wPathDestRight h i⟩) h') (f i).fst (f i).snd) = g a f' f fun i => P.wRec g (f i)
-- 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✝ : Zero M
a a' : α
b : M
f : α →₀ M
\u22a2 f.support.card = 1 ↔ ∃ a b, b ≠ 0 ∧ f = single a b
-- Mathlib/RingTheory/RootsOfUnity/Basic.lean
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst✝⁴ : CommMonoid M
inst✝³ : CommMonoid N
inst✝² : DivisionCommMonoid G
k✝ l : ℕ
inst✝¹ : CommRing R
ζ✝ : Rˣ
h : IsPrimitiveRoot ζ✝ k✝
inst✝ : IsDomain R
k : ℕ
hk : 0 < k
ζ : R
hζ : IsPrimitiveRoot ζ k
ξ : Rˣ
hξ : ξ ∈ rootsOfUnity ⟨k, hk⟩ R
\u22a2 ∃ i < k, ζ ^ i = ↑ξ
-- Mathlib/RingTheory/RootsOfUnity/Basic.lean
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst✝⁴ : CommMonoid M
inst✝³ : CommMonoid N
inst✝² : DivisionCommMonoid G
k✝ l : ℕ
inst✝¹ : CommRing R
ζ✝ : Rˣ
h : IsPrimitiveRoot ζ✝ k✝
inst✝ : IsDomain R
k : ℕ
hk : 0 < k
ζ : R
hζ : IsPrimitiveRoot ζ k
ξ : Rˣ
hξ : ξ ∈ rootsOfUnity ⟨k, hk⟩ R
hζ' : IsPrimitiveRoot ⋯.unit ↑⟨k, hk⟩
\u22a2 ∃ i < k, ζ ^ i = ↑ξ
-- Mathlib/RingTheory/RootsOfUnity/Basic.lean
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst✝⁴ : CommMonoid M
inst✝³ : CommMonoid N
inst✝² : DivisionCommMonoid G
k✝ l : ℕ
inst✝¹ : CommRing R
ζ✝ : Rˣ
h : IsPrimitiveRoot ζ✝ k✝
inst✝ : IsDomain R
k : ℕ
hk : 0 < k
ζ : R
hζ : IsPrimitiveRoot ζ k
ξ : Rˣ
hξ : ξ ∈ rootsOfUnity ⟨k, hk⟩ R
hζ' : IsPrimitiveRoot ⋯.unit ↑⟨k, hk⟩
i : ℕ
hi₁ : i < ↑⟨k, hk⟩
hi₂ : ⋯.unit ^ i = ξ
\u22a2 ∃ i < k, ζ ^ i = ↑ξ
-- Mathlib/Data/Set/Pointwise/BigOperators.lean
ι : Type u_1
α : Type u_2
β : Type u_3
F : Type u_4
inst✝⁴ : FunLike F α β
inst✝³ : CommMonoid α
inst✝² : CommMonoid β
inst✝¹ : MonoidHomClass F α β
inst✝ : Fintype ι
S : ι → Set α
\u22a2 (fun f => ∏ i : ι, f i) '' univ.pi S = ∏ i : ι, S i
-- Mathlib/Data/Finset/Lattice.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
ι : Type u_5
κ : Type u_6
inst✝ : LinearOrder α
s : Finset α
H : s.Nonempty
x : α
h₂ : 1 < s.card
\u22a2 s.min' ⋯ < s.max' ⋯
-- Mathlib/Data/Finset/Lattice.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
ι : Type u_5
κ : Type u_6
inst✝ : LinearOrder α
s : Finset α
H : s.Nonempty
x : α
h₂ : 1 < s.card
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
\u22a2 s.min' ⋯ < s.max' ⋯
-- Mathlib/Probability/Process/Stopping.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
\u22a2 MeasurableSet {ω | τ ω = π ω}
-- Mathlib/Probability/Process/Stopping.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
\u22a2 ∀ (i : ι), MeasurableSet ({ω | τ ω = π ω} ∩ {ω | τ ω ≤ i})
-- Mathlib/Probability/Process/Stopping.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
this : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j}
\u22a2 MeasurableSet ({ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j})
-- Mathlib/Probability/Process/Stopping.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
this : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j}
\u22a2 MeasurableSet {ω | min (τ ω) j = min (π ω) j}
-- Mathlib/Probability/Process/Stopping.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
this : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j}
\u22a2 Measurable fun x => min (τ x) j
-- Mathlib/Probability/Process/Stopping.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
this : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j}
\u22a2 Measurable fun x => min (π x) j
-- Mathlib/Algebra/Group/Basic.lean
α : Type u_1
β : Type u_2
G : Type u_3
M : Type u_4
inst✝ : CommGroup G
a b c d : G
\u22a2 a = b / c ↔ c * a = b
-- Mathlib/Tactic/ComputeDegree.lean
R : Type u_1
inst✝ : Semiring R
n : ℕ
f g : R[X]
\u22a2 (f + g).coeff n = f.coeff n + g.coeff n
-- Mathlib/Analysis/Normed/Ring/Units.lean
R : Type u_1
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : Rˣ
\u22a2 (fun t => inverse (↑x + t) - ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖
-- Mathlib/LinearAlgebra/Trace.lean
R : Type u_1
inst✝¹² : CommRing R
M : Type u_2
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_3
P : Type u_4
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R P
ι : Type u_5
inst✝⁵ : Module.Free R M
inst✝⁴ : Module.Finite R M
inst✝³ : Module.Free R N
inst✝² : Module.Finite R N
inst✝¹ : Module.Free R P
inst✝ : Module.Finite R P
\u22a2 (mapBilinear R M N M N).compr₂ (trace R (M ⊗[R] N)) = (lsmul R R).compl₁₂ (trace R M) (trace R N)
-- Mathlib/LinearAlgebra/Trace.lean
R : Type u_1
inst✝¹² : CommRing R
M : Type u_2
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_3
P : Type u_4
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R P
ι : Type u_5
inst✝⁵ : Module.Free R M
inst✝⁴ : Module.Finite R M
inst✝³ : Module.Free R N
inst✝² : Module.Finite R N
inst✝¹ : Module.Free R P
inst✝ : Module.Finite R P
\u22a2 ((mapBilinear R M N M N).compr₂ (trace R (M ⊗[R] N))).compl₁₂ (dualTensorHom R M M) (dualTensorHom R N N) = ((lsmul R R).compl₁₂ (trace R M) (trace R N)).compl₁₂ (dualTensorHom R M M) (dualTensorHom R N N)
-- Mathlib/LinearAlgebra/Trace.lean
R : Type u_1
inst✝¹² : CommRing R
M : Type u_2
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
N : Type u_3
P : Type u_4
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R P
ι : Type u_5
inst✝⁵ : Module.Free R M
inst✝⁴ : Module.Finite R M
inst✝³ : Module.Free R N
inst✝² : Module.Finite R N
inst✝¹ : Module.Free R P
inst✝ : Module.Finite R P
f : Module.Dual R M
m : M
g : Module.Dual R N
n : N
\u22a2 ((AlgebraTensorModule.curry (((AlgebraTensorModule.curry (((mapBilinear R M N M N).compr₂ (trace R (M ⊗[R] N))).compl₁₂ (dualTensorHom R M M) (dualTensorHom R N N))) f) m)) g) n = ((AlgebraTensorModule.curry (((AlgebraTensorModule.curry (((lsmul R R).compl₁₂ (trace R M) (trace R N)).compl₁₂ (dualTensorHom R M M) (dualTensorHom R N N))) f) m)) g) n
-- Mathlib/MeasureTheory/Measure/AEMeasurable.lean
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
R : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
f✝ g : α → β
μ✝ ν : Measure α
s : Set α
hs : MeasurableSet s
μ : Measure α
f : α → β
\u22a2 AEMeasurable f (μ.restrict s) ↔ AEMeasurable (f ∘ Subtype.val) (Measure.comap Subtype.val μ)
-- Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean
α : Type u_1
β : Type u_2
inst✝⁵ : SemilatticeSup α
inst✝⁴ : OrderTop α
inst✝³ : DecidableRel fun x x_1 => x ≤ x_1
inst✝² : SemilatticeSup β
inst✝¹ : BoundedOrder β
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
s t : Finset α
a✝ b✝ b a : α
\u22a2 (if h : a ≤ b then (filter (fun b => a ≤ b) {b}).sup' ⋯ fun x => x else ⊤) = if a ≤ b then b else ⊤