ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/Convex/Deriv.lean
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'_mono : MonotoneOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
hxzD : Icc x z ⊆ D
hxyD : Icc x y ⊆ D
hxyD' : Ioo x y ⊆ interior D
hyzD : Icc y z ⊆ D
hyzD' : Ioo y z ⊆ interior D
a : ℝ
ha : deriv f a = (f y - f x) / (y - x)
hxa : x < a
hay : a < y
b : ℝ
hb : deriv f b = (f z - f y) / (z - y)
hyb : y < b
hbz : b < z
\u22a2 (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)
-- Mathlib/Order/Filter/ENNReal.lean
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u v : α → ℝ≥0∞
\u22a2 limsup (u * v) f ≤ limsup (fun x => limsup u f * v x) f
-- Mathlib/Order/Filter/ENNReal.lean
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u v : α → ℝ≥0∞
\u22a2 u * v ≤ᶠ[f] fun x => limsup u f * v x
-- Mathlib/CategoryTheory/Localization/SmallHom.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
W : MorphismProperty C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
D' : Type u₃
inst✝² : Category.{v₃, u₃} D'
X Y Z T : C
inst✝¹ : HasSmallLocalizedHom W X Y
inst✝ : HasSmallLocalizedHom W X X
α : SmallHom W X Y
\u22a2 (equiv W W.Q) ((mk W (𝟙 X)).comp α) = (equiv W W.Q) α
-- Mathlib/SetTheory/Cardinal/Ordinal.lean
a b : Cardinal.{u_1}
hb : ℵ₀ ≤ b
ha : a ≤ b
ha' : a ≠ 0
\u22a2 a * b = b
-- Mathlib/Algebra/Ring/Parity.lean
F : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
inst✝¹ : Semiring α
inst✝ : Semiring β
a b : α
m n : ℕ
\u22a2 Odd (a + (a + 1))
-- Mathlib/Geometry/Manifold/MFDeriv/UniqueDifferential.lean
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
inst✝ : SmoothManifoldWithCorners I' M'
s : Set M
x✝ : M
hs : UniqueMDiffOn I s
x : M
\u22a2 UniqueMDiffOn 𝓘(𝕜, E) ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s)
-- Mathlib/Geometry/Manifold/MFDeriv/UniqueDifferential.lean
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
inst✝⁶ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
inst✝ : SmoothManifoldWithCorners I' M'
s : Set M
x✝ : M
hs : UniqueMDiffOn I s
x : M
\u22a2 UniqueMDiffOn 𝓘(𝕜, E) (↑(extChartAt I x) '' (s ∩ (chartAt H x).source))
-- Mathlib/Topology/Sets/Opens.lean
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
U V : Opens α
i : U ≤ V
\u22a2 IsOpen (range (inclusion ⋯))
-- Mathlib/Topology/Sets/Opens.lean
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
U V : Opens α
i : U ≤ V
\u22a2 IsOpen {x | ↑x ∈ ↑U}
-- Mathlib/GroupTheory/Perm/Cycle/Type.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
h : σ.IsThreeCycle
\u22a2 σ.IsCycle
-- Mathlib/Order/Interval/Set/UnorderedInterval.lean
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : LinearOrder β
f : α → β
s : Set α
a a₁ a₂ b b₁ b₂ c d x : α
\u22a2 (∀ (a b c : α), c ∈ [[a, b]] → f c ∈ [[f a, f b]]) → Monotone f ∨ Antitone f
-- Mathlib/Order/Interval/Set/UnorderedInterval.lean
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : LinearOrder β
f : α → β
s : Set α
a a₁ a₂ b b₁ b₂ c d x : α
\u22a2 ¬Monotone f ∧ ¬Antitone f → ∃ a b, ∃ c ∈ [[a, b]], f c ∉ [[f a, f b]]
-- Mathlib/Order/Interval/Set/UnorderedInterval.lean
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : LinearOrder β
f : α → β
s : Set α
a a₁ a₂ b b₁ b₂ c d x : α
\u22a2 (∃ a b c, a ≤ b ∧ b ≤ c ∧ (f a < f b ∧ f c < f b ∨ f b < f a ∧ f b < f c)) → ∃ a b, ∃ c ∈ [[a, b]], f c ∉ [[f a, f b]]
-- Mathlib/Data/Set/Prod.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
s s₁ s₂ : Set α
t t₁ t₂ : Set β
a : α
b : β
\u22a2 s ×ˢ t = ∅ ↔ s = ∅ ∨ t = ∅
-- Mathlib/Order/Filter/Basic.lean
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
l f✝ f₁ f₂ : Filter α
g g₁ g₂ : Filter β
m : α → β
m' : β → γ
s : Set α
t : Set β
f : Filter α
c : β → α
h : range c ∈ f
W : Set β
W_in : W ∈ comap c f
\u22a2 c '' W ∈ f
-- Mathlib/Order/Filter/Basic.lean
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
l f✝ f₁ f₂ : Filter α
g g₁ g₂ : Filter β
m : α → β
m' : β → γ
s : Set α
t : Set β
f : Filter α
c : β → α
h : range c ∈ f
W : Set β
W_in : W ∈ comap c f
\u22a2 c '' W ∈ map c (comap c f)
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hnm : m ≤ m0
hfint : Integrable f μ
\u22a2 ∫ (x : α) in s, |(μ[f|m]) x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hnm : m ≤ m0
hfint : Integrable f μ
this : ∫ (x : α) in s, |(μ[f|m]) x| ∂μ = ∫ (x : α), |(μ[s.indicator f|m]) x| ∂μ
\u22a2 ∫ (x : α) in s, |(μ[f|m]) x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hnm : m ≤ m0
hfint : Integrable f μ
this : ∫ (x : α) in s, |(μ[f|m]) x| ∂μ = ∫ (x : α), |(μ[s.indicator f|m]) x| ∂μ
\u22a2 ∫ (x : α), |(μ[s.indicator f|m]) x| ∂μ ≤ ∫ (x : α), s.indicator (fun x => |f x|) x ∂μ
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hnm : m ≤ m0
hfint : Integrable f μ
this : ∫ (x : α) in s, |(μ[f|m]) x| ∂μ = ∫ (x : α), |(μ[s.indicator f|m]) x| ∂μ
x : α
\u22a2 (fun x => |s.indicator f x|) x = s.indicator (fun x => |f x|) x
-- Mathlib/Analysis/Normed/Order/Lattice.lean
α : Type u_1
inst✝ : NormedLatticeAddCommGroup α
a b c d : α
\u22a2 ‖a ⊔ b - c ⊔ d‖ ≤ ‖a - c‖ + ‖b - d‖
-- Mathlib/Analysis/Normed/Order/Lattice.lean
α : Type u_1
inst✝ : NormedLatticeAddCommGroup α
a b c d : α
\u22a2 ‖a ⊔ b - c ⊔ d‖ ≤ ‖|a - c|‖ + ‖|b - d|‖
-- Mathlib/Analysis/Normed/Order/Lattice.lean
α : Type u_1
inst✝ : NormedLatticeAddCommGroup α
a b c d : α
\u22a2 |a ⊔ b - c ⊔ d| ≤ ||a - c| + |b - d||
-- Mathlib/Analysis/Normed/Order/Lattice.lean
α : Type u_1
inst✝ : NormedLatticeAddCommGroup α
a b c d : α
\u22a2 |a ⊔ b - c ⊔ d| ≤ |a - c| + |b - d|
-- Mathlib/GroupTheory/Transfer.lean
G : Type u_1
inst✝² : Group G
H : Subgroup G
A : Type u_2
inst✝¹ : CommGroup A
ϕ : ↥H →* A
R S T : ↑(leftTransversals ↑H)
inst✝ : H.FiniteIndex
g : G
x✝¹ : Fintype (G ⧸ H) := H.fintypeQuotientOfFiniteIndex
x✝ : G ⧸ H
\u22a2 ϕ ⟨(↑((MemLeftTransversals.toEquiv ⋯) x✝))⁻¹ * ↑((MemLeftTransversals.toEquiv ⋯) x✝), ⋯⟩ = ϕ ⟨(↑((MemLeftTransversals.toEquiv ⋯) ((Equiv.symm (toPerm g)) x✝)))⁻¹ * ↑((MemLeftTransversals.toEquiv ⋯) ((Equiv.symm (toPerm g)) x✝)), ⋯⟩
-- Mathlib/Order/RelSeries.lean
α : Type u_1
r : Rel α α
β : Type u_2
s : Rel β β
p q : RelSeries r
h : p.last = q.head
i : Fin q.length
\u22a2 (p.smash q h).toFun (Fin.natAdd p.length i).castSucc = q.toFun i.castSucc
-- Mathlib/Order/RelSeries.lean
α : Type u_1
r : Rel α α
β : Type u_2
s : Rel β β
p q : RelSeries r
h : p.last = q.head
i : Fin q.length
\u22a2 q.toFun ⟨↑(Fin.natAdd p.length i).castSucc - p.length, ⋯⟩ = q.toFun i.castSucc
-- Mathlib/Order/RelSeries.lean
α : Type u_1
r : Rel α α
β : Type u_2
s : Rel β β
p q : RelSeries r
h : p.last = q.head
i : Fin q.length
\u22a2 ↑(Fin.natAdd p.length i).castSucc - p.length = ↑i
-- Mathlib/Data/DFinsupp/Multiset.lean
α : Type u_1
β : α → Type u_2
inst✝ : DecidableEq α
s t : Multiset α
a : α
n : ℕ
i : α
\u22a2 count i (replicate n a) = (DFinsupp.single a n) i
-- Mathlib/Data/ZMod/Quotient.lean
n : ℕ
A : Type u_1
R : Type u_2
inst✝³ : AddGroup A
inst✝² : Ring R
α : Type u_3
β : Type u_4
inst✝¹ : Group α
a : α
inst✝ : MulAction α β
b : β
k : ℤ
\u22a2 (orbitZPowersEquiv a b).symm ↑k = ⟨a, ⋯⟩ ^ k • ⟨b, ⋯⟩
-- Mathlib/Data/ZMod/Quotient.lean
n : ℕ
A : Type u_1
R : Type u_2
inst✝³ : AddGroup A
inst✝² : Ring R
α : Type u_3
β : Type u_4
inst✝¹ : Group α
a : α
inst✝ : MulAction α β
b : β
k : ℤ
\u22a2 ⟨a, ⋯⟩ ^ (k % ↑(minimalPeriod (fun x => a • x) b)) • ⟨b, ⋯⟩ = ⟨a, ⋯⟩ ^ k • ⟨b, ⋯⟩
-- Mathlib/Topology/Connected/LocallyConnected.lean
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t u v : Set α
inst✝ : LocallyConnectedSpace α
F : Set α
x : α
hF : IsOpen F
\u22a2 IsOpen (connectedComponentIn F x)
-- Mathlib/Topology/Connected/LocallyConnected.lean
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t u v : Set α
inst✝ : LocallyConnectedSpace α
F : Set α
x : α
hF : IsOpen F
\u22a2 ∀ x_1 ∈ connectedComponentIn F x, connectedComponentIn F x ∈ 𝓝 x_1
-- Mathlib/Topology/Connected/LocallyConnected.lean
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t u v : Set α
inst✝ : LocallyConnectedSpace α
F : Set α
x : α
hF : IsOpen F
y : α
hy : y ∈ connectedComponentIn F x
\u22a2 connectedComponentIn F x ∈ 𝓝 y
-- Mathlib/Topology/Connected/LocallyConnected.lean
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t u v : Set α
inst✝ : LocallyConnectedSpace α
F : Set α
x : α
hF : IsOpen F
y : α
hy : y ∈ connectedComponentIn F x
\u22a2 connectedComponentIn F y ∈ 𝓝 y
-- Mathlib/Analysis/Calculus/Gradient/Basic.lean
𝕜 : Type u_1
F : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : CompleteSpace F
f : F → 𝕜
f' x : F
s : Set F
L : Filter F
c : 𝕜
\u22a2 ∇ (fun x => c) x = 0
-- Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean
R : Type u
S : Type v
a b : R
n✝ m : ℕ
inst✝ : Semiring R
p q r : R[X]
n : ℕ
ha : a ≠ 0
\u22a2 (C a * X ^ n).trailingDegree = ↑n
-- Mathlib/Algebra/CubicDiscriminant.lean
R : Type u_1
S : Type u_2
F : Type u_3
K : Type u_4
P Q : Cubic R
a b c d a' b' c' d' : R
inst✝ : Semiring R
ha : P.a = 0
hb : P.b ≠ 0
\u22a2 P.toPoly.leadingCoeff = P.b
-- Mathlib/FieldTheory/KummerExtension.lean
K : Type u
inst✝³ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
hn : 0 < n
a : K
H : Irreducible (X ^ n - C a)
L : Type u_1
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsSplittingField K L (X ^ n - C a)
α : L
hα : α ^ n = (algebraMap K L) a
\u22a2 (adjoinRootXPowSubCEquiv hζ H hα).symm α = root (X ^ n - C a)
-- Mathlib/FieldTheory/KummerExtension.lean
K : Type u
inst✝³ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
hn : 0 < n
a : K
H : Irreducible (X ^ n - C a)
L : Type u_1
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsSplittingField K L (X ^ n - C a)
α : L
hα : α ^ n = (algebraMap K L) a
\u22a2 (adjoinRootXPowSubCEquiv hζ H hα) ((adjoinRootXPowSubCEquiv hζ H hα).symm α) = (adjoinRootXPowSubCEquiv hζ H hα) (root (X ^ n - C a))
-- Mathlib/Algebra/EuclideanDomain/Basic.lean
R : Type u
inst✝ : EuclideanDomain R
x z : R
hz : ¬z = 0
p : R
\u22a2 x * (z * p) / z = x * (z * p / z)
-- Mathlib/Data/Set/Pointwise/Interval.lean
α : Type u_1
inst✝ : LinearOrderedField α
a✝ a b c : α
h : 0 < c
\u22a2 (fun x => x * c) ⁻¹' Icc a b = Icc (a / c) (b / c)
-- Mathlib/Data/Set/Prod.lean
α : Type u_1
s t : Set α
x✝ : α × α
a : α
x : α × α
\u22a2 x ∈ (s ∩ t).offDiag ↔ x ∈ s.offDiag ∩ t.offDiag
-- Mathlib/Data/Set/Prod.lean
α : Type u_1
s t : Set α
x✝ : α × α
a : α
x : α × α
\u22a2 (x.1 ∈ s ∧ x.1 ∈ t) ∧ (x.2 ∈ s ∧ x.2 ∈ t) ∧ x.1 ≠ x.2 ↔ (x.1 ∈ s ∧ x.2 ∈ s ∧ x.1 ≠ x.2) ∧ x.1 ∈ t ∧ x.2 ∈ t ∧ x.1 ≠ x.2
-- Mathlib/Data/Set/Image.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : α → β
s✝ t✝ : Set α
f : α → β
s : Set α
t : Set β
\u22a2 f '' (s \ f ⁻¹' t) = f '' s \ t
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/Zero.lean
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
E : Type u₃
inst✝⁵ : Category.{v₃, u₃} E
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
inst✝² : HasZeroMorphisms E
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : F.Faithful
X Y : C
\u22a2 F.map 0 = 0
-- Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : SnakeInput C
\u22a2 S.L₀.g ≫ S.δ = 0
-- Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean
α : Type u
β : Type u_1
inst✝⁵ : Semiring α
inst✝⁴ : PartialOrder α
a b c d : α
inst✝³ : ExistsAddOfLE α
inst✝² : MulPosStrictMono α
inst✝¹ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x < x_1
inst✝ : ContravariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x < x_1
hb : b < 0
h : 1 < a
\u22a2 a * b < b
-- Mathlib/Order/Bounded.lean
α : Type u_1
r : α → α → Prop
s t : Set α
H : ∀ (a b : α), ∃ m, ∀ (c : α), r c a ∨ r c b → r c m
a : α
\u22a2 Bounded r (s ∩ {b | ¬r b a}) ↔ Bounded r s
-- Mathlib/Order/Bounded.lean
α : Type u_1
r : α → α → Prop
s t : Set α
H : ∀ (a b : α), ∃ m, ∀ (c : α), r c a ∨ r c b → r c m
a : α
\u22a2 Bounded r (s ∩ {b | ¬r b a}) → Bounded r s
-- Mathlib/Order/Bounded.lean
α : Type u_1
r : α → α → Prop
s t : Set α
H : ∀ (a b : α), ∃ m, ∀ (c : α), r c a ∨ r c b → r c m
a b : α
hb : ∀ b_1 ∈ s ∩ {b | ¬r b a}, r b_1 b
m : α
hm : ∀ (c : α), r c a ∨ r c b → r c m
\u22a2 Bounded r s
-- 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
U : Opens ↑↑(Limits.colimit F)
j : J
\u22a2 ((sheafIsoOfIso (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm).app (op U) ≪≫ limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp ((Opens.map (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj (op U)) ≪≫ Limits.lim.mapIso (NatIso.ofComponents (fun X => mapIso (F.obj (unop X)).presheaf (eqToIso ⋯)) ⋯)).inv ≫ (colimit.ι F j).c.app (op U) = limit.π (componentwiseDiagram F U) (op j)
-- 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
U : Opens ↑↑(Limits.colimit F)
j : J
\u22a2 (((Limits.lim.mapIso (NatIso.ofComponents (fun X => mapIso (F.obj (unop X)).presheaf (eqToIso ⋯)) ⋯)).inv ≫ (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp ((Opens.map (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj (op U))).inv) ≫ ((colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.c.app (op ((Opens.map ((forget C).mapIso (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm).symm.inv).obj (unop (op U)))) ≫ (Limits.colimit F).presheaf.map (eqToHom ⋯)) ≫ (Limits.colimit F).presheaf.map (eqToHom ⋯)) ≫ (colimit.ι F j).c.app (op U) = limit.π (componentwiseDiagram F U) (op j)
-- 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
U : Opens ↑↑(Limits.colimit F)
j : J
\u22a2 ((limMap (NatIso.ofComponents (fun X => mapIso (F.obj (unop X)).presheaf (eqToIso ⋯)) ⋯).inv ≫ (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp (op ((Opens.map (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj U))).inv) ≫ ((colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.c.app (op ((Opens.map (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj U)) ≫ (Limits.colimit F).presheaf.map (𝟙 (op ((Opens.map (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base).obj ((Opens.map (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj U))))) ≫ (Limits.colimit F).presheaf.map (eqToHom ⋯)) ≫ (colimit.ι F j).c.app (op U) = limit.π (componentwiseDiagram F U) (op j)
-- 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
U : Opens ↑↑(Limits.colimit F)
j : J
\u22a2 limMap (NatIso.ofComponents (fun X => mapIso (F.obj (unop X)).presheaf (eqToIso ⋯)) ⋯).inv ≫ (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp (op ((Opens.map (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj U))).inv ≫ (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.c.app (op ((Opens.map (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj U)) ≫ (colimit.ι F j).c.app (op ((Opens.map (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base).obj ((Opens.map (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj U))) ≫ ((pushforward C (colimit.ι F j).base).obj (F.obj j).presheaf).map (eqToHom ⋯) = limit.π (componentwiseDiagram F U) (op j)
-- 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
U : Opens ↑↑(Limits.colimit F)
j : J
\u22a2 limMap (NatIso.ofComponents (fun X => mapIso (F.obj (unop X)).presheaf (eqToIso ⋯)) ⋯).inv ≫ (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp (op ((Opens.map (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj U))).inv ≫ (colimit.ι F j ≫ (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom).c.app (op ((Opens.map (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj U)) ≫ ((pushforward C (colimit.ι F j).base).obj (F.obj j).presheaf).map (eqToHom ⋯) = limit.π (componentwiseDiagram F U) (op j)
-- 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
U : Opens ↑↑(Limits.colimit F)
j : J
\u22a2 limMap (NatIso.ofComponents (fun X => mapIso (F.obj (unop X)).presheaf (eqToIso ⋯)) ⋯).inv ≫ (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp (op ((Opens.map (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj U))).inv ≫ ({ cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.ι.app j).c.app (op ((Opens.map (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj U)) ≫ (F.obj j).presheaf.map (eqToHom ⋯) ≫ ((pushforward C (colimit.ι F j).base).obj (F.obj j).presheaf).map (eqToHom ⋯) = limit.π (componentwiseDiagram F U) (op j)
-- 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
U : Opens ↑↑(Limits.colimit F)
j : J
f : (componentwiseDiagram F U).obj (op j) ⟶ (F.obj j).presheaf.obj (op ((Opens.map (colimit.ι F j).base).obj U)) := (NatIso.ofComponents (fun X => mapIso (F.obj (unop X)).presheaf (eqToIso ⋯)) ⋯).inv.app (op j) ≫ (F.obj j).presheaf.map (eqToHom ⋯) ≫ ((pushforward C (colimit.ι F j).base).obj (F.obj j).presheaf).map (eqToHom ⋯)
\u22a2 limit.π (componentwiseDiagram F U) (op j) ≫ f = limit.π (componentwiseDiagram F U) (op j)
-- 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
U : Opens ↑↑(Limits.colimit F)
j : J
f : (componentwiseDiagram F U).obj (op j) ⟶ (F.obj j).presheaf.obj (op ((Opens.map (colimit.ι F j).base).obj U)) := (NatIso.ofComponents (fun X => mapIso (F.obj (unop X)).presheaf (eqToIso ⋯)) ⋯).inv.app (op j) ≫ (F.obj j).presheaf.map (eqToHom ⋯) ≫ ((pushforward C (colimit.ι F j).base).obj (F.obj j).presheaf).map (eqToHom ⋯)
\u22a2 f = 𝟙 ((componentwiseDiagram F U).obj (op j))
-- 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
U : Opens ↑↑(Limits.colimit F)
j : J
f : (componentwiseDiagram F U).obj (op j) ⟶ (F.obj j).presheaf.obj (op ((Opens.map (colimit.ι F j).base).obj U)) := (NatIso.ofComponents (fun X => mapIso (F.obj (unop X)).presheaf (eqToIso ⋯)) ⋯).inv.app (op j) ≫ (F.obj j).presheaf.map (eqToHom ⋯) ≫ ((pushforward C (colimit.ι F j).base).obj (F.obj j).presheaf).map (eqToHom ⋯)
\u22a2 f = (F.obj j).presheaf.map (𝟙 (op ((Opens.map (colimit.ι F (unop (op j))).base).obj U)))
-- 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
U : Opens ↑↑(Limits.colimit F)
j : J
f : (componentwiseDiagram F U).obj (op j) ⟶ (F.obj j).presheaf.obj (op ((Opens.map (colimit.ι F j).base).obj U)) := (NatIso.ofComponents (fun X => mapIso (F.obj (unop X)).presheaf (eqToIso ⋯)) ⋯).inv.app (op j) ≫ (F.obj j).presheaf.map (eqToHom ⋯) ≫ ((pushforward C (colimit.ι F j).base).obj (F.obj j).presheaf).map (eqToHom ⋯)
\u22a2 (F.obj j).presheaf.map (eqToIso ⋯).inv ≫ (F.obj j).presheaf.map (eqToHom ⋯) ≫ ((pushforward C (colimit.ι F j).base).obj (F.obj j).presheaf).map (eqToHom ⋯) = (F.obj j).presheaf.map (𝟙 (op ((Opens.map (colimit.ι F (unop (op j))).base).obj U)))
-- 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
U : Opens ↑↑(Limits.colimit F)
j : J
f : (componentwiseDiagram F U).obj (op j) ⟶ (F.obj j).presheaf.obj (op ((Opens.map (colimit.ι F j).base).obj U)) := (NatIso.ofComponents (fun X => mapIso (F.obj (unop X)).presheaf (eqToIso ⋯)) ⋯).inv.app (op j) ≫ (F.obj j).presheaf.map (eqToHom ⋯) ≫ ((pushforward C (colimit.ι F j).base).obj (F.obj j).presheaf).map (eqToHom ⋯)
\u22a2 (F.obj j).presheaf.map ((eqToIso ⋯).inv ≫ eqToHom ⋯ ≫ (Opens.map (colimit.ι F j).base).op.map (eqToHom ⋯)) = (F.obj j).presheaf.map (𝟙 (op ((Opens.map (colimit.ι F (unop (op j))).base).obj U)))