ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/Calculus/ContDiff/Basic.lean
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁹ : NormedAddCommGroup D
inst✝⁸ : NormedSpace 𝕜 D
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f✝ f₁ : E → F
g✝ : F → G
x✝ x₀ : E
c : F
b : E × F → G
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
g : G ≃ₗᵢ[𝕜] E
f : E → F
hs : UniqueDiffOn 𝕜 s
x : G
hx : g x ∈ s
i : ℕ
this : iteratedFDerivWithin 𝕜 i (f ∘ ⇑g) (⇑g ⁻¹' s) x = (iteratedFDerivWithin 𝕜 i f s (g x)).compContinuousLinearMap fun x => ↑{ toLinearEquiv := g.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }
\u22a2 ‖iteratedFDerivWithin 𝕜 i (f ∘ ⇑g) (⇑g ⁻¹' s) x‖ = ‖iteratedFDerivWithin 𝕜 i f s (g x)‖
-- Mathlib/Order/WellFoundedSet.lean
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r r' : α → α → Prop
f : β → α
s t : Set α
x y : α
hs : s.WellFoundedOn r
hx : x ∈ s
P : α → Prop
hP : ∀ y ∈ s, (∀ z ∈ s, r z y → P z) → P y
Q : ↑s → Prop := fun y => P ↑y
\u22a2 Q ⟨x, hx⟩
-- Mathlib/Order/WellFoundedSet.lean
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r r' : α → α → Prop
f : β → α
s t : Set α
x y : α
hs : s.WellFoundedOn r
hx : x ∈ s
P : α → Prop
hP : ∀ y ∈ s, (∀ z ∈ s, r z y → P z) → P y
Q : ↑s → Prop := fun y => P ↑y
\u22a2 ∀ (x : ↑s), (∀ (y : ↑s), r ↑y ↑x → Q y) → Q x
-- Mathlib/Dynamics/Ergodic/Ergodic.lean
α : Type u_1
m : MeasurableSpace α
f : α → α
s✝ : Set α
μ : Measure α
β : Type u_2
m' : MeasurableSpace β
μ' : Measure β
s' : Set β
g : α → β
hg : MeasurePreserving g μ μ'
hf : PreErgodic f μ
f' : β → β
h_comm : g ∘ f = f' ∘ g
s : Set β
hs₀ : MeasurableSet s
hs₁ : f' ⁻¹' s = s
\u22a2 s =ᶠ[ae μ'] ∅ ∨ s =ᶠ[ae μ'] univ
-- Mathlib/Dynamics/Ergodic/Ergodic.lean
α : Type u_1
m : MeasurableSpace α
f : α → α
s✝ : Set α
μ : Measure α
β : Type u_2
m' : MeasurableSpace β
μ' : Measure β
s' : Set β
g : α → β
hg : MeasurePreserving g μ μ'
hf : PreErgodic f μ
f' : β → β
h_comm : g ∘ f = f' ∘ g
s : Set β
hs₀ : MeasurableSet s
hs₁ : f ⁻¹' (g ⁻¹' s) = g ⁻¹' s
\u22a2 s =ᶠ[ae μ'] ∅ ∨ s =ᶠ[ae μ'] univ
-- Mathlib/Dynamics/Ergodic/Ergodic.lean
α : Type u_1
m : MeasurableSpace α
f : α → α
s✝ : Set α
μ : Measure α
β : Type u_2
m' : MeasurableSpace β
μ' : Measure β
s' : Set β
g : α → β
hg : MeasurePreserving g μ μ'
hf : PreErgodic f μ
f' : β → β
h_comm : g ∘ f = f' ∘ g
s : Set β
hs₀ : MeasurableSet s
hs₁ : f ⁻¹' (g ⁻¹' s) = g ⁻¹' s
hs₂ : g ⁻¹' s =ᶠ[ae μ] ∅
\u22a2 s =ᶠ[ae μ'] ∅
-- Mathlib/Dynamics/Ergodic/Ergodic.lean
α : Type u_1
m : MeasurableSpace α
f : α → α
s✝ : Set α
μ : Measure α
β : Type u_2
m' : MeasurableSpace β
μ' : Measure β
s' : Set β
g : α → β
hg : MeasurePreserving g μ μ'
hf : PreErgodic f μ
f' : β → β
h_comm : g ∘ f = f' ∘ g
s : Set β
hs₀ : MeasurableSet s
hs₁ : f ⁻¹' (g ⁻¹' s) = g ⁻¹' s
hs₂ : g ⁻¹' s =ᶠ[ae μ] univ
\u22a2 s =ᶠ[ae μ'] univ
-- Mathlib/Algebra/Order/Ring/Rat.lean
a b c p q : ℚ
\u22a2 a ≤ b ∨ b ≤ a
-- Mathlib/Algebra/Homology/ShortComplex/Homology.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
A : C
inst✝ : S.HasHomology
h : S.LeftHomologyData
\u22a2 h.leftHomologyIso.hom ≫ (S.leftHomologyIso.symm ≪≫ h.leftHomologyIso).inv = S.leftHomologyIso.hom
-- Mathlib/CategoryTheory/CofilteredSystem.lean
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this : ∀ (i : J'), Nonempty (F'.obj i)
\u22a2 F.sections.Nonempty
-- Mathlib/CategoryTheory/CofilteredSystem.lean
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝ : ∀ (i : J'), Nonempty (F'.obj i)
this : ∀ (i : J'), Finite (F'.obj i)
\u22a2 F.sections.Nonempty
-- Mathlib/CategoryTheory/CofilteredSystem.lean
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝ : ∀ (i : J'), Nonempty (F'.obj i)
this : ∀ (i : J'), Finite (F'.obj i)
h✝ : IsEmpty J
\u22a2 F.sections.Nonempty
-- Mathlib/CategoryTheory/CofilteredSystem.lean
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝ : ∀ (i : J'), Nonempty (F'.obj i)
this : ∀ (i : J'), Finite (F'.obj i)
h✝ : Nonempty J
\u22a2 F.sections.Nonempty
-- Mathlib/CategoryTheory/CofilteredSystem.lean
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝¹ : ∀ (i : J'), Nonempty (F'.obj i)
this✝ : ∀ (i : J'), Finite (F'.obj i)
h✝ : Nonempty J
this : IsCofiltered J
\u22a2 F.sections.Nonempty
-- Mathlib/CategoryTheory/CofilteredSystem.lean
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝¹ : ∀ (i : J'), Nonempty (F'.obj i)
this✝ : ∀ (i : J'), Finite (F'.obj i)
h✝ : Nonempty J
this : IsCofiltered J
u : (j : J') → F'.obj j
hu : u ∈ F'.sections
\u22a2 F.sections.Nonempty
-- Mathlib/CategoryTheory/CofilteredSystem.lean
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝¹ : ∀ (i : J'), Nonempty (F'.obj i)
this✝ : ∀ (i : J'), Finite (F'.obj i)
h✝ : Nonempty J
this : IsCofiltered J
u : (j : J') → F'.obj j
hu : u ∈ F'.sections
\u22a2 (fun j => (u { down := j }).down) ∈ F.sections
-- Mathlib/CategoryTheory/CofilteredSystem.lean
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝¹ : ∀ (i : J'), Nonempty (F'.obj i)
this✝ : ∀ (i : J'), Finite (F'.obj i)
h✝ : Nonempty J
this : IsCofiltered J
u : (j : J') → F'.obj j
hu : u ∈ F'.sections
j j' : J
f : j ⟶ j'
h : F'.map { down := f } (u { down := j }) = u { down := j' }
\u22a2 F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'
-- Mathlib/CategoryTheory/CofilteredSystem.lean
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝¹ : ∀ (i : J'), Nonempty (F'.obj i)
this✝ : ∀ (i : J'), Finite (F'.obj i)
h✝ : Nonempty J
this : IsCofiltered J
u : (j : J') → F'.obj j
hu : u ∈ F'.sections
j j' : J
f : j ⟶ j'
h : { down := F.map f (u { down := j }).down } = u { down := j' }
\u22a2 F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'
-- Mathlib/Order/Heyting/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : GeneralizedCoheytingAlgebra α
a b c d : α
h : a ⊔ c ≤ b ⊔ c
\u22a2 a \ c ≤ b \ c
-- Mathlib/Order/Heyting/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : GeneralizedCoheytingAlgebra α
a b c d : α
h : a ⊔ c ≤ b ⊔ c
\u22a2 (a ⊔ c) \ c ≤ (b ⊔ c) \ c
-- Mathlib/Topology/Separation.lean
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
inst✝ : T2Space X
x y : X
hd : Disjoint (𝓝 x) (𝓝 y)
he : x = y
\u22a2 False
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m x✝² x✝¹ : ℤ
hij : ¬(ComplexShape.up ℤ).Rel x✝² x✝¹
x✝ : ↑(AddCommGrp.of (Cochain F G x✝²))
\u22a2 ((fun i j => AddCommGrp.ofHom ↑(δ_hom ℤ F G i j)) x✝² x✝¹) x✝ = 0 x✝
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m x✝⁵ x✝⁴ x✝³ : ℤ
x✝² : (ComplexShape.up ℤ).Rel x✝⁵ x✝⁴
x✝¹ : (ComplexShape.up ℤ).Rel x✝⁴ x✝³
x✝ : ↑((fun i => AddCommGrp.of (Cochain F G i)) x✝⁵)
\u22a2 ((fun i j => AddCommGrp.ofHom ↑(δ_hom ℤ F G i j)) x✝⁵ x✝⁴ ≫ (fun i j => AddCommGrp.ofHom ↑(δ_hom ℤ F G i j)) x✝⁴ x✝³) x✝ = 0 x✝
-- Mathlib/Topology/Order/Basic.lean
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a : α
s : Set α
hs : s ∈ 𝓝[≤] a
\u22a2 ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s
-- Mathlib/Analysis/Convex/Body.lean
V : Type u_1
inst✝⁴ : TopologicalSpace V
inst✝³ : AddCommGroup V
inst✝² : Module ℝ V
inst✝¹ : ContinuousSMul ℝ V
inst✝ : ContinuousAdd V
K : ConvexBody V
h_zero : 0 ∈ K
a b : ℝ≥0
h : a ≤ b
\u22a2 a • K ≤ b • K
-- Mathlib/Analysis/Convex/Body.lean
V : Type u_1
inst✝⁴ : TopologicalSpace V
inst✝³ : AddCommGroup V
inst✝² : Module ℝ V
inst✝¹ : ContinuousSMul ℝ V
inst✝ : ContinuousAdd V
K : ConvexBody V
h_zero : 0 ∈ K
a b : ℝ≥0
h : a ≤ b
\u22a2 a • ↑K ⊆ b • ↑K
-- Mathlib/CategoryTheory/Sites/Spaces.lean
T : Type u
inst✝ : TopologicalSpace T
\u22a2 Pretopology.toGrothendieck (Opens T) (pretopology T) = grothendieckTopology T
-- Mathlib/CategoryTheory/Sites/Spaces.lean
T : Type u
inst✝ : TopologicalSpace T
\u22a2 Pretopology.toGrothendieck (Opens T) (Pretopology.ofGrothendieck (Opens T) (grothendieckTopology T)) = grothendieckTopology T
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α
inst✝ : MeasurableSpace α
\u22a2 Measurable fun p => (↑p).toReal
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
n : ℕ
hn : 0 < n
w z : ℂ
h : w.re ≤ z.re
\u22a2 ‖↑n ^ w‖ ≤ ‖↑n ^ z‖
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
n : ℕ
hn : 0 < n
w z : ℂ
h : w.re ≤ z.re
\u22a2 ↑n ^ w.re ≤ ↑n ^ z.re
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
J : Type u'
inst✝⁴ : Category.{v', u'} J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimitsOfShape J TopCat
inst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
s : Cocone F
U V : (Opens ↑↑s.pt)ᵒᵖ
i : U ⟶ V
\u22a2 s.pt.presheaf.map i ≫ limit.lift ((pushforwardDiagramToColimit F).leftOp ⋙ (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))) { pt := s.pt.presheaf.obj V, π := { app := fun j => (s.ι.app (unop j)).c.app V ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯), naturality := ⋯ } } ≫ (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))).inv = (limit.lift ((pushforwardDiagramToColimit F).leftOp ⋙ (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop U)))) { pt := s.pt.presheaf.obj U, π := { app := fun j => (s.ι.app (unop j)).c.app U ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯), naturality := ⋯ } } ≫ (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop U)))).inv) ≫ (limit (pushforwardDiagramToColimit F).leftOp).map ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).map i.unop).op
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
J : Type u'
inst✝⁴ : Category.{v', u'} J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimitsOfShape J TopCat
inst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
s : Cocone F
U V : (Opens ↑↑s.pt)ᵒᵖ
i : U ⟶ V
j : Jᵒᵖ
\u22a2 (s.pt.presheaf.map i ≫ limit.lift ((pushforwardDiagramToColimit F).leftOp ⋙ (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))) { pt := s.pt.presheaf.obj V, π := { app := fun j => (s.ι.app (unop j)).c.app V ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯), naturality := ⋯ } } ≫ (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))).inv) ≫ (limit.π (pushforwardDiagramToColimit F).leftOp j).app (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V))) = ((limit.lift ((pushforwardDiagramToColimit F).leftOp ⋙ (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop U)))) { pt := s.pt.presheaf.obj U, π := { app := fun j => (s.ι.app (unop j)).c.app U ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯), naturality := ⋯ } } ≫ (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop U)))).inv) ≫ (limit (pushforwardDiagramToColimit F).leftOp).map ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).map i.unop).op) ≫ (limit.π (pushforwardDiagramToColimit F).leftOp j).app (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
J : Type u'
inst✝⁴ : Category.{v', u'} J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimitsOfShape J TopCat
inst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
s : Cocone F
U V : (Opens ↑↑s.pt)ᵒᵖ
i : U ⟶ V
j : Jᵒᵖ
w : (Opens.map (colimit.ι (F ⋙ forget C) (unop j) ≫ colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).map i.unop = eqToHom ⋯ ≫ (Opens.map (((forget C).mapCocone s).ι.app (unop j))).map i.unop ≫ eqToHom ⋯
\u22a2 (s.pt.presheaf.map i ≫ limit.lift ((pushforwardDiagramToColimit F).leftOp ⋙ (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))) { pt := s.pt.presheaf.obj V, π := { app := fun j => (s.ι.app (unop j)).c.app V ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯), naturality := ⋯ } } ≫ (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))).inv) ≫ (limit.π (pushforwardDiagramToColimit F).leftOp j).app (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V))) = ((limit.lift ((pushforwardDiagramToColimit F).leftOp ⋙ (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop U)))) { pt := s.pt.presheaf.obj U, π := { app := fun j => (s.ι.app (unop j)).c.app U ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯), naturality := ⋯ } } ≫ (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop U)))).inv) ≫ (limit (pushforwardDiagramToColimit F).leftOp).map ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).map i.unop).op) ≫ (limit.π (pushforwardDiagramToColimit F).leftOp j).app (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
J : Type u'
inst✝⁴ : Category.{v', u'} J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimitsOfShape J TopCat
inst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
s : Cocone F
U V : (Opens ↑↑s.pt)ᵒᵖ
i : U ⟶ V
j : Jᵒᵖ
w : (Opens.map (colimit.ι (F ⋙ forget C) (unop j))).map ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).map i.unop) = eqToHom ⋯ ≫ (Opens.map (((forget C).mapCocone s).ι.app (unop j))).map i.unop ≫ eqToHom ⋯
\u22a2 (s.pt.presheaf.map i ≫ limit.lift ((pushforwardDiagramToColimit F).leftOp ⋙ (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))) { pt := s.pt.presheaf.obj V, π := { app := fun j => (s.ι.app (unop j)).c.app V ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯), naturality := ⋯ } } ≫ (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))).inv) ≫ (limit.π (pushforwardDiagramToColimit F).leftOp j).app (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V))) = ((limit.lift ((pushforwardDiagramToColimit F).leftOp ⋙ (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop U)))) { pt := s.pt.presheaf.obj U, π := { app := fun j => (s.ι.app (unop j)).c.app U ≫ (F.obj (unop j)).presheaf.map (eqToHom ⋯), naturality := ⋯ } } ≫ (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop U)))).inv) ≫ (limit (pushforwardDiagramToColimit F).leftOp).map ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).map i.unop).op) ≫ (limit.π (pushforwardDiagramToColimit F).leftOp j).app (op ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj (unop V)))
-- Mathlib/Geometry/Manifold/ChartedSpace.lean
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SecondCountableTopology H
s : Set M
hs : ⋃ x ∈ s, (chartAt H x).source = univ
hsc : s.Countable
\u22a2 SecondCountableTopology M
-- Mathlib/Geometry/Manifold/ChartedSpace.lean
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SecondCountableTopology H
s : Set M
hs : ⋃ x ∈ s, (chartAt H x).source = univ
hsc : s.Countable
this : ∀ (x : M), SecondCountableTopology ↑(chartAt H x).source
\u22a2 SecondCountableTopology M
-- Mathlib/Geometry/Manifold/ChartedSpace.lean
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SecondCountableTopology H
s : Set M
hs : ⋃ x ∈ s, (chartAt H x).source = univ
hsc : s.Countable
this✝ : ∀ (x : M), SecondCountableTopology ↑(chartAt H x).source
this : Encodable ↑s
\u22a2 SecondCountableTopology M
-- Mathlib/Geometry/Manifold/ChartedSpace.lean
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SecondCountableTopology H
s : Set M
hs : ⋃ x, (chartAt H ↑x).source = univ
hsc : s.Countable
this✝ : ∀ (x : M), SecondCountableTopology ↑(chartAt H x).source
this : Encodable ↑s
\u22a2 SecondCountableTopology M
-- Mathlib/Analysis/PSeries.lean
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
n : ℕ
\u22a2 ∑ k ∈ range (n + 1), 2 ^ k • f (2 ^ k) ≤ f 1 + 2 • ∑ k ∈ Ico 2 (2 ^ n + 1), f k
-- Mathlib/Analysis/PSeries.lean
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
n : ℕ
\u22a2 ∑ k ∈ range (n + 1), 2 ^ k • f (2 ^ k) = f 1 + 2 • ∑ k ∈ range n, 2 ^ k • f (2 ^ (k + 1))
-- Mathlib/LinearAlgebra/AffineSpace/Ordered.lean
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : LinearOrderedField k
inst✝² : OrderedAddCommGroup E
inst✝¹ : Module k E
inst✝ : OrderedSMul k E
f : k → E
a b r : k
hab : a < b
h₀ : 0 < r
h₁ : r < 1
\u22a2 f ((lineMap a b) r) ≤ (lineMap (f a) (f b)) r ↔ slope f a ((lineMap a b) r) ≤ slope f ((lineMap a b) r) b
-- Mathlib/LinearAlgebra/AffineSpace/Pointwise.lean
k : Type u_1
inst✝⁹ : Ring k
V : Type u_2
P : Type u_3
V₁ : Type u_4
P₁ : Type u_5
V₂ : Type u_6
P₂ : Type u_7
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AffineSpace V P
inst✝⁵ : AddCommGroup V₁
inst✝⁴ : Module k V₁
inst✝³ : AffineSpace V₁ P₁
inst✝² : AddCommGroup V₂
inst✝¹ : Module k V₂
inst✝ : AffineSpace V₂ P₂
v : V
s : AffineSubspace k P
\u22a2 (v +ᵥ s).direction = s.direction
-- Mathlib/LinearAlgebra/AffineSpace/Pointwise.lean
k : Type u_1
inst✝⁹ : Ring k
V : Type u_2
P : Type u_3
V₁ : Type u_4
P₁ : Type u_5
V₂ : Type u_6
P₂ : Type u_7
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AffineSpace V P
inst✝⁵ : AddCommGroup V₁
inst✝⁴ : Module k V₁
inst✝³ : AffineSpace V₁ P₁
inst✝² : AddCommGroup V₂
inst✝¹ : Module k V₂
inst✝ : AffineSpace V₂ P₂
v : V
s : AffineSubspace k P
\u22a2 Submodule.map (↑(AffineEquiv.constVAdd k P v)).linear s.direction = s.direction
-- Mathlib/Topology/Maps/Basic.lean
X : Type u_1
Y : Type u_2
Z : Type u_3
ι : Type u_4
f : X → Y
g : Y → Z
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
hf : OpenEmbedding f
s : Set Y
hs : s ⊆ range f
\u22a2 IsOpen s ↔ IsOpen (f ⁻¹' s)
-- Mathlib/Analysis/InnerProductSpace/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : _root_.RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : E
h : ⟪x, y⟫_𝕜 = 0
\u22a2 ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖
-- Mathlib/Analysis/InnerProductSpace/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : _root_.RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : E
h : ⟪x, y⟫_𝕜 = 0
\u22a2 2 = 0 ∨ re ⟪x, y⟫_𝕜 = 0
-- Mathlib/Analysis/InnerProductSpace/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : _root_.RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : E
h : ⟪x, y⟫_𝕜 = 0
\u22a2 re ⟪x, y⟫_𝕜 = 0
-- Mathlib/Algebra/Lie/IdealOperations.lean
R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
I : LieIdeal R L
J J₁ J₂ : LieIdeal R L'
h : f.IsIdealMorphism
\u22a2 comap f ⁅f.idealRange ⊓ J₁, f.idealRange ⊓ J₂⁆ = ⁅comap f J₁, comap f J₂⁆ ⊔ f.ker
-- Mathlib/Algebra/Lie/IdealOperations.lean
R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
I : LieIdeal R L
J J₁ J₂ : LieIdeal R L'
h : f.IsIdealMorphism
\u22a2 Submodule.comap (↑f) (Submodule.span R {m | ∃ x n, ⁅↑x, ↑n⁆ = m}) = Submodule.comap (↑f) (Submodule.span R (⇑↑f '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}))
-- Mathlib/Algebra/Lie/IdealOperations.lean
R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
I : LieIdeal R L
J J₁ J₂ : LieIdeal R L'
h : f.IsIdealMorphism
\u22a2 {m | ∃ x n, ⁅↑x, ↑n⁆ = m} = ⇑↑f '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}