ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Algebra/Lie/Basic.lean
R : Type u
L : Type v
M : Type w
N : Type w₁
P : Type w₂
inst✝¹⁴ : CommRing R
inst✝¹³ : LieRing L
inst✝¹² : LieAlgebra R L
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : AddCommGroup N
inst✝⁹ : AddCommGroup P
inst✝⁸ : Module R M
inst✝⁷ : Module R N
inst✝⁶ : Module R P
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieRingModule L N
inst✝³ : LieRingModule L P
inst✝² : LieModule R L M
inst✝¹ : LieModule R L N
inst✝ : LieModule R L P
f_inv : N → M
f : M → N
map_add'✝¹ : ∀ (x y : M), f (x + y) = f x + f y
map_smul'✝¹ : ∀ (m : R) (x : M), { toFun := f, map_add' := map_add'✝¹ }.toFun (m • x) = (RingHom.id R) m • { toFun := f, map_add' := map_add'✝¹ }.toFun x
map_lie'✝¹ : ∀ {x : L} {m : M}, { toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ }.toFun ⁅x, m⁆ = ⁅x, { toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ }.toFun m⁆
left_inv✝¹ : LeftInverse f_inv (↑{ toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹, map_lie' := map_lie'✝¹ }).toFun
right_inv✝¹ : Function.RightInverse f_inv (↑{ toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹, map_lie' := map_lie'✝¹ }).toFun
g_inv : N → M
g : M → N
map_add'✝ : ∀ (x y : M), g (x + y) = g x + g y
map_smul'✝ : ∀ (m : R) (x : M), { toFun := g, map_add' := map_add'✝ }.toFun (m • x) = (RingHom.id R) m • { toFun := g, map_add' := map_add'✝ }.toFun x
map_lie'✝ : ∀ {x : L} {m : M}, { toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝ }.toFun ⁅x, m⁆ = ⁅x, { toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝ }.toFun m⁆
left_inv✝ : LeftInverse g_inv (↑{ toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝, map_lie' := map_lie'✝ }).toFun
right_inv✝ : Function.RightInverse g_inv (↑{ toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝, map_lie' := map_lie'✝ }).toFun
h : f = g ∧ f_inv = g_inv
\u22a2 f_inv = g_inv
-- Mathlib/Logic/Equiv/PartialEquiv.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
e' : PartialEquiv β γ
s : Set α
\u22a2 ((e.restr s).trans e').source = ((e.trans e').restr s).source
-- Mathlib/MeasureTheory/Function/L1Space.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedAddCommGroup γ
f : α → ℝ≥0∞
hf : Measurable f
hflt : ∀ᵐ (x : α) ∂μ, f x < ⊤
g : α → ℝ
\u22a2 Integrable g (μ.withDensity f) ↔ Integrable (fun x => g x * (f x).toReal) μ
-- Mathlib/MeasureTheory/Function/L1Space.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedAddCommGroup γ
f : α → ℝ≥0∞
hf : Measurable f
hflt : ∀ᵐ (x : α) ∂μ, f x < ⊤
g : α → ℝ
this : (fun x => g x * (f x).toReal) = fun x => (f x).toReal • g x
\u22a2 Integrable g (μ.withDensity f) ↔ Integrable (fun x => g x * (f x).toReal) μ
-- Mathlib/MeasureTheory/Function/L1Space.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedAddCommGroup γ
f : α → ℝ≥0∞
hf : Measurable f
hflt : ∀ᵐ (x : α) ∂μ, f x < ⊤
g : α → ℝ
this : (fun x => g x * (f x).toReal) = fun x => (f x).toReal • g x
\u22a2 Integrable g (μ.withDensity f) ↔ Integrable (fun x => (f x).toReal • g x) μ
-- Mathlib/GroupTheory/NoncommCoprod.lean
M : Type u_1
N : Type u_2
P : Type u_3
inst✝² : MulOneClass M
inst✝¹ : MulOneClass N
inst✝ : Monoid P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
\u22a2 (fun mn => f mn.1 * g mn.2) 1 = 1
-- Mathlib/Algebra/AlgebraicCard.lean
R A : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : IsDomain A
inst✝¹ : Algebra R A
inst✝ : NoZeroSMulDivisors R A
\u22a2 #{ x // IsAlgebraic R x } ≤ #R[X] * ℵ₀
-- Mathlib/Algebra/AlgebraicCard.lean
R A : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : IsDomain A
inst✝¹ : Algebra R A
inst✝ : NoZeroSMulDivisors R A
\u22a2 lift.{u, u} #{ x // IsAlgebraic R x } ≤ lift.{u, u} #R[X] * ℵ₀
-- Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₁ : InfinitePlace K
B : ℕ
hB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B
n : ℕ
\u22a2 ((Algebra.norm ℤ) ↑(seq K w₁ hB n)).natAbs ≤ B
-- Mathlib/Topology/Compactification/OnePoint.lean
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
Y : Type u_2
inst✝ : TopologicalSpace Y
f : OnePoint X → Y
\u22a2 Tendsto (f ∘ some) (coclosedCompact X) (𝓝 (f ∞)) ↔ ∀ s ∈ 𝓝 (f ∞), ∃ t, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ some) tᶜ s
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X : Cᵒᵖ
\u22a2 limMap (whiskerLeft F (restriction R (𝟙 X))) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map (𝟙 X))) (F ⋙ evaluation R X)).inv = (ModuleCat.restrictScalarsId'App (R.map (𝟙 X)) ⋯ (limit (F ⋙ evaluation R X))).inv
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X : Cᵒᵖ
\u22a2 limMap (whiskerLeft F (restriction R (𝟙 X))) = (ModuleCat.restrictScalarsId'App (R.map (𝟙 X)) ⋯ (limit (F ⋙ evaluation R X))).inv ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map (𝟙 X))) (F ⋙ evaluation R X)).hom
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X : Cᵒᵖ
\u22a2 ∀ (j : J), limMap (whiskerLeft F (restriction R (𝟙 X))) ≫ limit.π (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (R.map (𝟙 X))) j = ((ModuleCat.restrictScalarsId'App (R.map (𝟙 X)) ⋯ (limit (F ⋙ evaluation R X))).inv ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map (𝟙 X))) (F ⋙ evaluation R X)).hom) ≫ limit.π (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (R.map (𝟙 X))) j
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X : Cᵒᵖ
j : J
\u22a2 limMap (whiskerLeft F (restriction R (𝟙 X))) ≫ limit.π (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (R.map (𝟙 X))) j = ((ModuleCat.restrictScalarsId'App (R.map (𝟙 X)) ⋯ (limit (F ⋙ evaluation R X))).inv ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map (𝟙 X))) (F ⋙ evaluation R X)).hom) ≫ limit.π (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (R.map (𝟙 X))) j
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X : Cᵒᵖ
j : J
\u22a2 limit.π (F ⋙ evaluation R X) j ≫ restrictionApp (𝟙 X) (F.obj j) = (ModuleCat.restrictScalarsId'App (R.map (𝟙 X)) ⋯ (limit (F ⋙ evaluation R X))).inv ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map (𝟙 X))) (F ⋙ evaluation R X)).hom ≫ limit.π (F ⋙ evaluation R X ⋙ ModuleCat.restrictScalars (R.map (𝟙 X))) j
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X : Cᵒᵖ
j : J
\u22a2 limit.π (F ⋙ evaluation R X) j ≫ restrictionApp (𝟙 X) (F.obj j) = (ModuleCat.restrictScalarsId'App (R.map (𝟙 X)) ⋯ (limit (F ⋙ evaluation R X))).inv ≫ (ModuleCat.restrictScalars (R.map (𝟙 X))).map (limit.π (F ⋙ evaluation R X) j)
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X : Cᵒᵖ
j : J
\u22a2 limit.π (F ⋙ evaluation R X) j ≫ (ModuleCat.restrictScalarsId' (R.map (𝟙 X)) ⋯).inv.app ((F.obj j).obj X) = limit.π (F ⋙ evaluation R X) j ≫ (ModuleCat.restrictScalarsId'App (R.map (𝟙 X)) ⋯ ((F ⋙ evaluation R X).obj j)).inv
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
\u22a2 limMap (whiskerLeft F (restriction R (f ≫ g))) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map (f ≫ g))) (F ⋙ evaluation R Z)).inv = (limMap (whiskerLeft F (restriction R f)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map f)) (F ⋙ evaluation R Y)).inv) ≫ (ModuleCat.restrictScalars (R.map f)).map (limMap (whiskerLeft F (restriction R g)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map g)) (F ⋙ evaluation R Z)).inv) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ (limit (F ⋙ evaluation R Z))).inv
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
\u22a2 limMap (whiskerLeft F (restriction R (f ≫ g))) = limMap (whiskerLeft F (restriction R f)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map f)) (F ⋙ evaluation R Y)).inv ≫ (ModuleCat.restrictScalars (R.map f)).map (limMap (whiskerLeft F (restriction R g)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map g)) (F ⋙ evaluation R Z)).inv) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ (limit (F ⋙ evaluation R Z))).inv ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map (f ≫ g))) (F ⋙ evaluation R Z)).hom
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
\u22a2 ∀ (j : J), limMap (whiskerLeft F (restriction R (f ≫ g))) ≫ limit.π (F ⋙ evaluation R Z ⋙ ModuleCat.restrictScalars (R.map (f ≫ g))) j = (limMap (whiskerLeft F (restriction R f)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map f)) (F ⋙ evaluation R Y)).inv ≫ (ModuleCat.restrictScalars (R.map f)).map (limMap (whiskerLeft F (restriction R g)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map g)) (F ⋙ evaluation R Z)).inv) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ (limit (F ⋙ evaluation R Z))).inv ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map (f ≫ g))) (F ⋙ evaluation R Z)).hom) ≫ limit.π (F ⋙ evaluation R Z ⋙ ModuleCat.restrictScalars (R.map (f ≫ g))) j
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
j : J
\u22a2 limMap (whiskerLeft F (restriction R (f ≫ g))) ≫ limit.π (F ⋙ evaluation R Z ⋙ ModuleCat.restrictScalars (R.map (f ≫ g))) j = (limMap (whiskerLeft F (restriction R f)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map f)) (F ⋙ evaluation R Y)).inv ≫ (ModuleCat.restrictScalars (R.map f)).map (limMap (whiskerLeft F (restriction R g)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map g)) (F ⋙ evaluation R Z)).inv) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ (limit (F ⋙ evaluation R Z))).inv ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map (f ≫ g))) (F ⋙ evaluation R Z)).hom) ≫ limit.π (F ⋙ evaluation R Z ⋙ ModuleCat.restrictScalars (R.map (f ≫ g))) j
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
j : J
\u22a2 limit.π (F ⋙ evaluation R X) j ≫ restrictionApp f (F.obj j) ≫ (ModuleCat.restrictScalars (R.map f)).map (restrictionApp g (F.obj j)) ≫ (ModuleCat.restrictScalarsComp' (R.map f) (R.map g) (R.map (f ≫ g)) ⋯).inv.app ((F.obj j).obj Z) = limMap (whiskerLeft F (restriction R f)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map f)) (F ⋙ evaluation R Y)).inv ≫ (ModuleCat.restrictScalars (R.map f)).map (limMap (whiskerLeft F (restriction R g))) ≫ (ModuleCat.restrictScalars (R.map f)).map (preservesLimitIso (ModuleCat.restrictScalars (R.map g)) (F ⋙ evaluation R Z)).inv ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ (limit (F ⋙ evaluation R Z))).inv ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map (f ≫ g))) (F ⋙ evaluation R Z)).hom ≫ limit.π (F ⋙ evaluation R Z ⋙ ModuleCat.restrictScalars (R.map (f ≫ g))) j
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
j : J
\u22a2 limit.π (F ⋙ evaluation R X) j ≫ restrictionApp f (F.obj j) ≫ (ModuleCat.restrictScalars (R.map f)).map (restrictionApp g (F.obj j)) ≫ (ModuleCat.restrictScalarsComp' (R.map f) (R.map g) (R.map (f ≫ g)) ⋯).inv.app ((F.obj j).obj Z) = limMap (whiskerLeft F (restriction R f)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map f)) (F ⋙ evaluation R Y)).inv ≫ (ModuleCat.restrictScalars (R.map f)).map (limMap (whiskerLeft F (restriction R g))) ≫ (ModuleCat.restrictScalars (R.map f)).map (preservesLimitIso (ModuleCat.restrictScalars (R.map g)) (F ⋙ evaluation R Z)).inv ≫ (ModuleCat.restrictScalars (R.map f)).map ((ModuleCat.restrictScalars (R.map g)).map (limit.π (F ⋙ evaluation R Z) j)) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ ((F ⋙ evaluation R Z).obj j)).inv
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
j : J
\u22a2 limit.π (F ⋙ evaluation R X) j ≫ restrictionApp f (F.obj j) ≫ (ModuleCat.restrictScalars (R.map f)).map (restrictionApp g (F.obj j)) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ ((F.obj j).obj Z)).inv = limMap (whiskerLeft F (restriction R f)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map f)) (F ⋙ evaluation R Y)).inv ≫ (ModuleCat.restrictScalars (R.map f)).map (limMap (whiskerLeft F (restriction R g))) ≫ (ModuleCat.restrictScalars (R.map f)).map (preservesLimitIso (ModuleCat.restrictScalars (R.map g)) (F ⋙ evaluation R Z)).inv ≫ (ModuleCat.restrictScalars (R.map f)).map ((ModuleCat.restrictScalars (R.map g)).map (limit.π (F ⋙ evaluation R Z) j)) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ ((F.obj j).obj Z)).inv
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
j : J
\u22a2 limit.π (F ⋙ evaluation R X) j ≫ restrictionApp f (F.obj j) ≫ (ModuleCat.restrictScalars (R.map f)).map (restrictionApp g (F.obj j)) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ ((F.obj j).obj Z)).inv = limMap (whiskerLeft F (restriction R f)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map f)) (F ⋙ evaluation R Y)).inv ≫ (ModuleCat.restrictScalars (R.map f)).map (limMap (whiskerLeft F (restriction R g)) ≫ limit.π ((F ⋙ evaluation R Z) ⋙ ModuleCat.restrictScalars (R.map g)) j) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ ((F.obj j).obj Z)).inv
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
j : J
\u22a2 limit.π (F ⋙ evaluation R X) j ≫ restrictionApp f (F.obj j) ≫ (ModuleCat.restrictScalars (R.map f)).map (restrictionApp g (F.obj j)) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ ((F.obj j).obj Z)).inv = limMap (whiskerLeft F (restriction R f)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map f)) (F ⋙ evaluation R Y)).inv ≫ (ModuleCat.restrictScalars (R.map f)).map (limit.π (F ⋙ evaluation R Y) j ≫ (whiskerLeft F (restriction R g)).app j) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ ((F.obj j).obj Z)).inv
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
j : J
\u22a2 limit.π (F ⋙ evaluation R X) j ≫ restrictionApp f (F.obj j) ≫ (ModuleCat.restrictScalars (R.map f)).map (restrictionApp g (F.obj j)) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ ((F.obj j).obj Z)).inv = limMap (whiskerLeft F (restriction R f)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map f)) (F ⋙ evaluation R Y)).inv ≫ (ModuleCat.restrictScalars (R.map f)).map (limit.π (F ⋙ evaluation R Y) j ≫ restrictionApp g (F.obj j)) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ ((F.obj j).obj Z)).inv
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
F : J ⥤ PresheafOfModules R
inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
j : J
\u22a2 limit.π (F ⋙ evaluation R X) j ≫ restrictionApp f (F.obj j) ≫ (ModuleCat.restrictScalars (R.map f)).map (restrictionApp g (F.obj j)) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ ((F.obj j).obj Z)).inv = limMap (whiskerLeft F (restriction R f)) ≫ limit.π ((F ⋙ evaluation R Y) ⋙ ModuleCat.restrictScalars (R.map f)) j ≫ (ModuleCat.restrictScalars (R.map f)).map (restrictionApp g (F.obj j)) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ ((F.obj j).obj Z)).inv
-- Mathlib/LinearAlgebra/DFinsupp.lean
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
inst✝³ : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
p : ι → Submodule R N
h : Function.Injective ⇑((lsum ℕ) fun i => (p i).subtype)
\u22a2 Independent p
-- Mathlib/LinearAlgebra/DFinsupp.lean
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
inst✝³ : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
p : ι → Submodule R N
h : Function.Injective ⇑((lsum ℕ) fun i => (p i).subtype)
\u22a2 ∀ (i : ι) (x : ↥(p i)) (v : Π₀ (i : ι), ↥(p i)), ((lsum ℕ) fun i => (p i).subtype) (erase i v) = ↑x → x = 0
-- Mathlib/LinearAlgebra/DFinsupp.lean
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
inst✝³ : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
p : ι → Submodule R N
h : Function.Injective ⇑((lsum ℕ) fun i => (p i).subtype)
i : ι
x : ↥(p i)
v : Π₀ (i : ι), ↥(p i)
hv : ((lsum ℕ) fun i => (p i).subtype) (erase i v) = ↑x
\u22a2 x = 0
-- Mathlib/LinearAlgebra/DFinsupp.lean
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
inst✝³ : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
p : ι → Submodule R N
h : Function.Injective ⇑((lsum ℕ) fun i => (p i).subtype)
i : ι
x : ↥(p i)
v : Π₀ (i : ι), ↥(p i)
hv : ((lsum ℕ) fun i => (p i).subtype) (erase i v) = ((lsum ℕ) fun i => (p i).subtype) (single i x)
\u22a2 x = 0
-- Mathlib/LinearAlgebra/DFinsupp.lean
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
inst✝³ : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
p : ι → Submodule R N
h : Function.Injective ⇑((lsum ℕ) fun i => (p i).subtype)
i : ι
x : ↥(p i)
v : Π₀ (i : ι), ↥(p i)
hv : ((lsum ℕ) fun i => (p i).subtype) (erase i v) = ((lsum ℕ) fun i => (p i).subtype) (single i x)
this : (erase i v) i = (single i x) i
\u22a2 x = 0
-- Mathlib/Algebra/Homology/HomologicalComplex.lean
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
i j : ι
r : c.Rel i j
\u22a2 C.dFrom i = C.d i j ≫ (C.xNextIso r).inv
-- Mathlib/Algebra/Homology/HomologicalComplex.lean
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
i : ι
r : c.Rel i (c.next i)
\u22a2 C.dFrom i = C.d i (c.next i) ≫ (C.xNextIso r).inv
-- Mathlib/Order/Interval/Set/Disjoint.lean
ι : Sort u
α : Type v
β : Type w
inst✝ : LinearOrder α
s : Set α
a : α
f : ι → α
\u22a2 ⋂ i, Iic (f i) = ∅ ↔ ¬BddBelow (range f)
-- Mathlib/Analysis/Convex/EGauge.lean
𝕜 : Type u_1
inst✝² : NormedField 𝕜
α : Type u_2
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
s t : Set E
x y : E
r : ℝ≥0
hc : 1 < ‖c‖
h₀ : r ≠ 0 ∨ x ≠ 0
\u22a2 egauge 𝕜 (ball 0 ↑r) x ≤ ↑‖c‖₊ * ↑‖x‖₊ / ↑r
-- Mathlib/Analysis/Convex/EGauge.lean
𝕜 : Type u_1
inst✝² : NormedField 𝕜
α : Type u_2
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
s t : Set E
x y : E
hc : 1 < ‖c‖
h₀ : 0 ≠ 0 ∨ x ≠ 0
\u22a2 egauge 𝕜 (ball 0 ↑0) x ≤ ↑‖c‖₊ * ↑‖x‖₊ / ↑0
-- Mathlib/Analysis/Convex/EGauge.lean
𝕜 : Type u_1
inst✝² : NormedField 𝕜
α : Type u_2
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
s t : Set E
x y : E
r : ℝ≥0
hc : 1 < ‖c‖
h₀ : r ≠ 0 ∨ x ≠ 0
hr : 0 < r
\u22a2 egauge 𝕜 (ball 0 ↑r) x ≤ ↑‖c‖₊ * ↑‖x‖₊ / ↑r
-- Mathlib/Combinatorics/Additive/ETransform.lean
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Group α
e : α
x : Finset α × Finset α
\u22a2 x.1.card + (op e • x.1).card = 2 * x.1.card
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
p : ι → Prop
s : ι → Set α
𝓕 : Filter ι
a : α
\u22a2 limsup s cofinite = {x | {n | x ∈ s n}.Infinite}
-- Mathlib/NumberTheory/Bernoulli.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
\u22a2 (coeff A (n + 1)) (bernoulli'PowerSeries A * (exp A - 1)) = (coeff A (n + 1)) (X * exp A)
-- Mathlib/NumberTheory/Bernoulli.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
\u22a2 (coeff A (n + 1, 0).1) (PowerSeries.mk fun n => (algebraMap ℚ A) (bernoulli' n / ↑n !)) * (coeff A (n + 1, 0).2) (exp A - 1) + ∑ p ∈ antidiagonal n, (coeff A (p.1, p.2 + 1).1) (PowerSeries.mk fun n => (algebraMap ℚ A) (bernoulli' n / ↑n !)) * (coeff A (p.1, p.2 + 1).2) (exp A - 1) = (coeff A (n + 1)) (exp A * X)
-- Mathlib/NumberTheory/Bernoulli.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
\u22a2 ∑ p ∈ antidiagonal n, bernoulli' p.1 / ↑p.1! * ((↑p.2 + 1) * ↑p.2!)⁻¹ = (↑n !)⁻¹