ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : IsFiniteMeasure μ
g : α → ℝ
hint : Integrable g μ
ℱ : ι → MeasurableSpace α
hℱ : ∀ (i : ι), ℱ i ≤ m0
A : MeasurableSpace α := m0
hmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ≤ ‖(μ[g|ℱ n]) x‖₊}
hg : Memℒp g 1 μ
ε : ℝ
hε : 0 < ε
hne : ¬eLpNorm g 1 μ = 0
δ : ℝ
hδ : 0 < δ
h : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator g) 1 μ ≤ ENNReal.ofReal ε
C : ℝ≥0 := ⟨δ, ⋯⟩⁻¹ * (eLpNorm g 1 μ).toNNReal
hC : C = ⟨δ, ⋯⟩⁻¹ * (eLpNorm g 1 μ).toNNReal

⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖(μ[g|ℱ i]) x‖₊}.indicator (μ[g|ℱ i])) 1 μ ≤ ENNReal.ofReal ε
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : IsFiniteMeasure μ
g : α → ℝ
hint : Integrable g μ
ℱ : ι → MeasurableSpace α
hℱ : ∀ (i : ι), ℱ i ≤ m0
A : MeasurableSpace α := m0
hmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ≤ ‖(μ[g|ℱ n]) x‖₊}
hg : Memℒp g 1 μ
ε : ℝ
hε : 0 < ε
hne : ¬eLpNorm g 1 μ = 0
δ : ℝ
hδ : 0 < δ
h : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator g) 1 μ ≤ ENNReal.ofReal ε
C : ℝ≥0 := ⟨δ, ⋯⟩⁻¹ * (eLpNorm g 1 μ).toNNReal
hC : C = ⟨δ, ⋯⟩⁻¹ * (eLpNorm g 1 μ).toNNReal
hCpos : 0 < C

⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖(μ[g|ℱ i]) x‖₊}.indicator (μ[g|ℱ i])) 1 μ ≤ ENNReal.ofReal ε
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : IsFiniteMeasure μ
g : α → ℝ
hint : Integrable g μ
ℱ : ι → MeasurableSpace α
hℱ : ∀ (i : ι), ℱ i ≤ m0
A : MeasurableSpace α := m0
hmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ≤ ‖(μ[g|ℱ n]) x‖₊}
hg : Memℒp g 1 μ
ε : ℝ
hε : 0 < ε
hne : ¬eLpNorm g 1 μ = 0
δ : ℝ
hδ : 0 < δ
h : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator g) 1 μ ≤ ENNReal.ofReal ε
C : ℝ≥0 := ⟨δ, ⋯⟩⁻¹ * (eLpNorm g 1 μ).toNNReal
hC : C = ⟨δ, ⋯⟩⁻¹ * (eLpNorm g 1 μ).toNNReal
hCpos : 0 < C
this : ∀ (n : ι), μ {x | C ≤ ‖(μ[g|ℱ n]) x‖₊} ≤ ENNReal.ofReal δ

⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖(μ[g|ℱ i]) x‖₊}.indicator (μ[g|ℱ i])) 1 μ ≤ ENNReal.ofReal ε
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : IsFiniteMeasure μ
g : α → ℝ
hint : Integrable g μ
ℱ : ι → MeasurableSpace α
hℱ : ∀ (i : ι), ℱ i ≤ m0
A : MeasurableSpace α := m0
hmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ≤ ‖(μ[g|ℱ n]) x‖₊}
hg : Memℒp g 1 μ
ε : ℝ
hε : 0 < ε
hne : ¬eLpNorm g 1 μ = 0
δ : ℝ
hδ : 0 < δ
h : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator g) 1 μ ≤ ENNReal.ofReal ε
C : ℝ≥0 := ⟨δ, ⋯⟩⁻¹ * (eLpNorm g 1 μ).toNNReal
hC : C = ⟨δ, ⋯⟩⁻¹ * (eLpNorm g 1 μ).toNNReal
hCpos : 0 < C
this : ∀ (n : ι), μ {x | C ≤ ‖(μ[g|ℱ n]) x‖₊} ≤ ENNReal.ofReal δ
n : ι

⊢ eLpNorm ({x | C ≤ ‖(μ[g|ℱ n]) x‖₊}.indicator (μ[g|ℱ n])) 1 μ ≤ eLpNorm ({x | C ≤ ‖(μ[g|ℱ n]) x‖₊}.indicator g) 1 μ
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : IsFiniteMeasure μ
g : α → ℝ
hint : Integrable g μ
ℱ : ι → MeasurableSpace α
hℱ : ∀ (i : ι), ℱ i ≤ m0
A : MeasurableSpace α := m0
hmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ≤ ‖(μ[g|ℱ n]) x‖₊}
hg : Memℒp g 1 μ
ε : ℝ
hε : 0 < ε
hne : ¬eLpNorm g 1 μ = 0
δ : ℝ
hδ : 0 < δ
h : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator g) 1 μ ≤ ENNReal.ofReal ε
C : ℝ≥0 := ⟨δ, ⋯⟩⁻¹ * (eLpNorm g 1 μ).toNNReal
hC : C = ⟨δ, ⋯⟩⁻¹ * (eLpNorm g 1 μ).toNNReal
hCpos : 0 < C
this : ∀ (n : ι), μ {x | C ≤ ‖(μ[g|ℱ n]) x‖₊} ≤ ENNReal.ofReal δ
n : ι
hmeasℱ : MeasurableSet {x | C ≤ ‖(μ[g|ℱ n]) x‖₊}

⊢ eLpNorm ({x | C ≤ ‖(μ[g|ℱ n]) x‖₊}.indicator (μ[g|ℱ n])) 1 μ ≤ eLpNorm ({x | C ≤ ‖(μ[g|ℱ n]) x‖₊}.indicator g) 1 μ
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : IsFiniteMeasure μ
g : α → ℝ
hint : Integrable g μ
ℱ : ι → MeasurableSpace α
hℱ : ∀ (i : ι), ℱ i ≤ m0
A : MeasurableSpace α := m0
hmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ≤ ‖(μ[g|ℱ n]) x‖₊}
hg : Memℒp g 1 μ
ε : ℝ
hε : 0 < ε
hne : ¬eLpNorm g 1 μ = 0
δ : ℝ
hδ : 0 < δ
h : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator g) 1 μ ≤ ENNReal.ofReal ε
C : ℝ≥0 := ⟨δ, ⋯⟩⁻¹ * (eLpNorm g 1 μ).toNNReal
hC : C = ⟨δ, ⋯⟩⁻¹ * (eLpNorm g 1 μ).toNNReal
hCpos : 0 < C
this : ∀ (n : ι), μ {x | C ≤ ‖(μ[g|ℱ n]) x‖₊} ≤ ENNReal.ofReal δ
n : ι
hmeasℱ : MeasurableSet {x | C ≤ ‖(μ[g|ℱ n]) x‖₊}

⊢ eLpNorm (μ[{x | C ≤ ‖(μ[g|ℱ n]) x‖₊}.indicator g|ℱ n]) 1 μ ≤ eLpNorm ({x | C ≤ ‖(μ[g|ℱ n]) x‖₊}.indicator g) 1 μ
-- Mathlib/Algebra/Order/Rearrangement.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : LinearOrderedAddCommGroup β
inst✝¹ : Module α β
inst✝ : OrderedSMul α β
s : Finset ι
σ : Perm ι
f : ι → α
g : ι → β
hfg : AntivaryOn f g ↑s
hσ : {x | σ x ≠ x} ⊆ ↑s

⊢ ∑ i ∈ s, f i • g i ≤ ∑ i ∈ s, f (σ i) • g i
-- Mathlib/Algebra/Order/Rearrangement.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : LinearOrderedAddCommGroup β
inst✝¹ : Module α β
inst✝ : OrderedSMul α β
s : Finset ι
σ : Perm ι
f : ι → α
g : ι → β
hfg : AntivaryOn f g ↑s
hσ : {x | σ x ≠ x} ⊆ ↑s

⊢ ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g (σ⁻¹ i)
-- Mathlib/RingTheory/Polynomial/Tower.lean
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring A
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra A B
inst✝³ : Algebra R B
inst✝² : IsScalarTower R A B
inst✝¹ : NoZeroSMulDivisors A B
inst✝ : Nontrivial B
x : A
p : R[X]

⊢ (aeval ((algebraMap A B) x)) p = 0 ↔ (aeval x) p = 0
-- Mathlib/Data/Fintype/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : Fintype α
s t : Finset α
inst✝ : DecidableEq α
a : α

⊢ a ∉ sᶜ ↔ a ∈ s
-- Mathlib/Combinatorics/SimpleGraph/Basic.lean
ι : Sort u_1
V : Type u
G : SimpleGraph V
a b c u v w : V
e : Sym2 V
G₁ G₂ : SimpleGraph V
x✝ : Sym2 V
x y : V

⊢ Quot.mk (Sym2.Rel V) (x, y) ∈ (G₁ ⊔ G₂).edgeSet ↔ Quot.mk (Sym2.Rel V) (x, y) ∈ G₁.edgeSet ∪ G₂.edgeSet
-- Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)

⊢ ‖y‖ / (o.oangle x (x + y)).tan = ‖x‖
-- Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
hs : (o.oangle x (x + y)).sign = 1

⊢ ‖y‖ / (o.oangle x (x + y)).tan = ‖x‖
-- Mathlib/RingTheory/FreeCommRing.lean
α : Type u
x : FreeRing α

⊢ ↑(-x) = -↑x
-- Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)

⊢ o.oangle x (x + y) = ↑(Real.arcsin (‖y‖ / ‖x + y‖))
-- Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
hs : (o.oangle x (x + y)).sign = 1

⊢ o.oangle x (x + y) = ↑(Real.arcsin (‖y‖ / ‖x + y‖))
-- Mathlib/CategoryTheory/Bicategory/LocallyDiscrete.lean
B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F : PrelaxFunctor B C
a b : B
f : a ⟶ b

⊢ F.map₂ (eqToHom ⋯) = eqToHom ⋯
-- Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ : Measure α
inst✝ : NormedAddCommGroup E
hm : m ≤ m0
f : α → ℝ≥0∞
hf : Measurable f

⊢ limsup f (ae (μ.trim hm)) = limsup f (ae μ)
-- Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ : Measure α
inst✝ : NormedAddCommGroup E
hm : m ≤ m0
f : α → ℝ≥0∞
hf : Measurable f

⊢ sInf {a | ∀ᵐ (n : α) ∂μ.trim hm, f n ≤ a} = sInf {a | ∀ᵐ (n : α) ∂μ, f n ≤ a}
-- Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ : Measure α
inst✝ : NormedAddCommGroup E
hm : m ≤ m0
f : α → ℝ≥0∞
hf : Measurable f
a : ℝ≥0∞

⊢ a ∈ {a | ∀ᵐ (n : α) ∂μ.trim hm, f n ≤ a} ↔ a ∈ {a | ∀ᵐ (n : α) ∂μ, f n ≤ a}
-- Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ : Measure α
inst✝ : NormedAddCommGroup E
hm : m ≤ m0
f : α → ℝ≥0∞
hf : Measurable f
a : ℝ≥0∞

⊢ μ {x | ¬f x ≤ a} = (μ.trim hm) {x | ¬f x ≤ a}
-- Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ : Measure α
inst✝ : NormedAddCommGroup E
hm : m ≤ m0
f : α → ℝ≥0∞
hf : Measurable f
a : ℝ≥0∞

⊢ MeasurableSet {x | ¬f x ≤ a}
-- Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ : Measure α
inst✝ : NormedAddCommGroup E
hm : m ≤ m0
f : α → ℝ≥0∞
hf : Measurable f
a : ℝ≥0∞

⊢ Measurable fun x => a
-- Mathlib/Topology/MetricSpace/PiNat.lean
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x : (n : ℕ) → E n
n : ℕ

⊢ IsOpen (cylinder x n)
-- Mathlib/Topology/MetricSpace/PiNat.lean
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x : (n : ℕ) → E n
n : ℕ

⊢ IsOpen ((↑(Finset.range n)).pi fun i => {x i})
-- Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean
α : Type u
β : Type u_1
inst✝⁴ : Semiring α
inst✝³ : Preorder α
a b c d : α
inst✝² : ZeroLEOneClass α
inst✝¹ : MulPosMono α
inst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
a2 : 2 ≤ a
b0 : 0 ≤ b

⊢ a + (a + a * b) ≤ a * (2 + b)
-- Mathlib/Data/PNat/Factors.lean
l : List ℕ
h : ∀ p ∈ l, Nat.Prime p

⊢ ↑(ofNatList l h).prod = l.prod
-- Mathlib/Data/PNat/Factors.lean
l : List ℕ
h : ∀ p ∈ l, Nat.Prime p
this : ↑(ofNatMultiset (↑l) h).prod = (↑l).prod

⊢ ↑(ofNatList l h).prod = l.prod
-- Mathlib/Data/PNat/Factors.lean
l : List ℕ
h : ∀ p ∈ l, Nat.Prime p
this : ↑(ofNatMultiset (↑l) h).prod = l.prod

⊢ ↑(ofNatList l h).prod = l.prod
-- Mathlib/MeasureTheory/Group/FundamentalDomain.lean
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝¹⁰ : Group G
inst✝⁹ : Group H
inst✝⁸ : MulAction G α
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MulAction H β
inst✝⁵ : MeasurableSpace β
inst✝⁴ : NormedAddCommGroup E
s t✝ : Set α
μ : Measure α
inst✝³ : MeasurableSpace G
inst✝² : MeasurableSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
ν : Measure α
h : IsFundamentalDomain G s μ
f : α → ℝ≥0∞
t : Set α

⊢ ∑' (g : G), ∫⁻ (x : α) in g • s, f x ∂μ.restrict t = ∑' (g : G), ∫⁻ (x : α) in t ∩ g • s, f x ∂μ
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g

⊢ padicNorm p (↑f (stationaryPoint hf)) = padicNorm p (↑g (stationaryPoint hg))
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : ¬padicNorm p (↑f (stationaryPoint hf)) = padicNorm p (↑g (stationaryPoint hg))

⊢ False
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : ¬padicNorm p (↑f (stationaryPoint hf)) = padicNorm p (↑g (stationaryPoint hg))
hlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))

⊢ False
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : ¬padicNorm p (↑f (stationaryPoint hf)) = padicNorm p (↑g (stationaryPoint hg))
hnlt : ¬padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))

⊢ False
-- Mathlib/Data/List/Basic.lean
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ x y : List α

⊢ y = x ++ y ↔ x = []
-- Mathlib/Algebra/Homology/ShortComplex/Homology.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology

⊢ (homologyMap' φ S₁.homologyData S₂.homologyData).op = (S₂.op.leftHomologyIso.symm ≪≫ S₂.leftHomologyOpIso ≪≫ S₂.rightHomologyIso.symm.op).inv ≫ homologyMap' (opMap φ) S₂.op.homologyData S₁.op.homologyData ≫ (S₁.op.leftHomologyIso.symm ≪≫ S₁.leftHomologyOpIso ≪≫ S₁.rightHomologyIso.symm.op).hom
-- Mathlib/Algebra/Homology/ShortComplex/Homology.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology

⊢ S₂.homologyData.iso.inv.op ≫ homologyMap' (opMap φ) S₂.homologyData.op S₁.homologyData.op ≫ S₁.homologyData.iso.hom.op = (S₂.op.leftHomologyIso.symm ≪≫ S₂.leftHomologyOpIso ≪≫ S₂.rightHomologyIso.symm.op).inv ≫ homologyMap' (opMap φ) S₂.op.homologyData S₁.op.homologyData ≫ (S₁.op.leftHomologyIso.symm ≪≫ S₁.leftHomologyOpIso ≪≫ S₁.rightHomologyIso.symm.op).hom
-- Mathlib/Algebra/Homology/ShortComplex/Homology.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology

⊢ S₂.homologyData.iso.inv.op ≫ leftHomologyMap' (opMap φ) S₂.homologyData.op.left S₁.homologyData.op.left ≫ S₁.homologyData.iso.hom.op = (((rightHomologyMap' (𝟙 S₂) S₂.rightHomologyData S₂.homologyData.right ≫ S₂.homologyData.iso.inv).op ≫ leftHomologyMap' (𝟙 S₂.op) S₂.rightHomologyData.op S₂.op.leftHomologyData) ≫ leftHomologyMap' (𝟙 S₂.op) S₂.op.leftHomologyData S₂.op.homologyData.left) ≫ leftHomologyMap' (opMap φ) S₂.op.homologyData.left S₁.op.homologyData.left ≫ leftHomologyMap' (𝟙 S₁.op) S₁.op.homologyData.left S₁.op.leftHomologyData ≫ leftHomologyMap' (𝟙 S₁.op) S₁.op.leftHomologyData S₁.rightHomologyData.op ≫ (S₁.homologyData.iso.hom ≫ rightHomologyMap' (𝟙 S₁) S₁.homologyData.right S₁.rightHomologyData).op
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
n : Nat

⊢ "".drop n = ""
-- Mathlib/Algebra/Order/Field/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedField α
a b c d : α
n : ℤ
hc : c < 0

⊢ a ≤ b / c ↔ b ≤ c * a
-- Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
H : ∀ ε > 0, ∃ s, s.Countable ∧ ∀ (x : α), ∃ y ∈ s, dist x y ≤ ε

⊢ SecondCountableTopology α
-- Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
H : ∀ ε > 0, ∃ s, s.Countable ∧ ∀ (x : α), ∃ y ∈ s, dist x y ≤ ε
ε : ℝ≥0∞
ε0 : ε > 0

⊢ ∃ t, t.Countable ∧ ⋃ x ∈ t, EMetric.closedBall x ε = univ
-- Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
H : ∀ ε > 0, ∃ s, s.Countable ∧ ∀ (x : α), ∃ y ∈ s, dist x y ≤ ε
ε : ℝ≥0∞
ε0 : ε > 0
ε' : ℝ≥0
ε'0 : 0 < ↑ε'
ε'ε : ↑ε' < ε

⊢ ∃ t, t.Countable ∧ ⋃ x ∈ t, EMetric.closedBall x ε = univ
-- Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
H : ∀ ε > 0, ∃ s, s.Countable ∧ ∀ (x : α), ∃ y ∈ s, dist x y ≤ ε
ε : ℝ≥0∞
ε0 : ε > 0
ε' : ℝ≥0
ε'0 : 0 < ↑ε'
ε'ε : ↑ε' < ε
s : Set α
hsc : s.Countable
y : α → α
hys : ∀ (x : α), y x ∈ s
hyx : ∀ (x : α), dist x (y x) ≤ ↑ε'

⊢ ∃ t, t.Countable ∧ ⋃ x ∈ t, EMetric.closedBall x ε = univ
-- Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
H : ∀ ε > 0, ∃ s, s.Countable ∧ ∀ (x : α), ∃ y ∈ s, dist x y ≤ ε
ε : ℝ≥0∞
ε0 : ε > 0
ε' : ℝ≥0
ε'0 : 0 < ↑ε'
ε'ε : ↑ε' < ε
s : Set α
hsc : s.Countable
y : α → α
hys : ∀ (x : α), y x ∈ s
hyx : ∀ (x : α), dist x (y x) ≤ ↑ε'
x : α

⊢ edist x (y x) ≤ ↑ε'
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ✝ : Kernel α (γ × β)
ν✝ : Kernel α γ
κ : Kernel α (γ × β)
ν : Kernel α γ
x : γ
s : Set β
hs : MeasurableSet s

⊢ Measurable ((fun p => κ.density ν p.1 p.2 s) ∘ fun a => (a, x))
-- Mathlib/FieldTheory/RatFunc/Defs.lean
K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
p q : K[X]
hq : q ≠ 0

⊢ RatFunc.mk p q = { toFractionRing := Localization.mk p ⟨q, ⋯⟩ }
-- Mathlib/SetTheory/Game/PGame.lean
xl xr : Type u
x y z : PGame
h₁ : x ⧏ y
h₂ : y ≤ z

⊢ x ⧏ z
-- Mathlib/SetTheory/Game/PGame.lean
xl xr : Type u
x y z : PGame
h₁ : ¬y ≤ x
h₂ : y ≤ z

⊢ ¬z ≤ x
-- Mathlib/Analysis/Calculus/FDeriv/Mul.lean
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type u_4
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
G' : Type u_5
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
f✝ f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
𝕜' : Type u_6
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : E → 𝕜'
c' : E →L[𝕜] 𝕜'
hc : HasFDerivAt c c' x
f : F

⊢ HasFDerivAt (fun y => c y • f) (c'.smulRight f) x
-- Mathlib/Probability/Process/HittingTime.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁶ : ConditionallyCompleteLinearOrder ι
inst✝⁵ : IsWellOrder ι fun x x_1 => x < x_1
inst✝⁴ : Countable ι
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
f : Filtration ι m
u : ι → Ω → β
s : Set β
n n' : ι
hu : Adapted f u
hs : MeasurableSet s
i : ι

⊢ MeasurableSet {ω | hitting u s n n' ω ≤ i}
-- Mathlib/Probability/Process/HittingTime.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁶ : ConditionallyCompleteLinearOrder ι
inst✝⁵ : IsWellOrder ι fun x x_1 => x < x_1
inst✝⁴ : Countable ι
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
f : Filtration ι m
u : ι → Ω → β
s : Set β
n n' : ι
hu : Adapted f u
hs : MeasurableSet s
i : ι
hi : n' ≤ i

⊢ MeasurableSet {ω | hitting u s n n' ω ≤ i}
-- Mathlib/Probability/Process/HittingTime.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁶ : ConditionallyCompleteLinearOrder ι
inst✝⁵ : IsWellOrder ι fun x x_1 => x < x_1
inst✝⁴ : Countable ι
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
f : Filtration ι m
u : ι → Ω → β
s : Set β
n n' : ι
hu : Adapted f u
hs : MeasurableSet s
i : ι
hi : i < n'

⊢ MeasurableSet {ω | hitting u s n n' ω ≤ i}
-- Mathlib/RingTheory/Idempotents.lean
R : Type u_1
S : Type u_2
inst✝¹ : Ring R
inst✝ : Ring S
f : R →+* S
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
e₁ : S
he : e₁ ∈ f.range
he₁ : IsIdempotentElem e₁
e₂ : R
he₂ : IsIdempotentElem e₂
he₁e₂ : e₁ * f e₂ = 0
he₂e₁ : f e₂ * e₁ = 0

⊢ ∃ e', IsIdempotentElem e' ∧ f e' = e₁ ∧ e' * e₂ = 0 ∧ e₂ * e' = 0
-- Mathlib/RingTheory/Idempotents.lean
R : Type u_1
S : Type u_2
inst✝¹ : Ring R
inst✝ : Ring S
f : R →+* S
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
e₂ : R
he₂ : IsIdempotentElem e₂
e' : R
h₁ : IsIdempotentElem e'
h₂ : e' * e₂ = 0
he : f e' ∈ f.range
he₁ : IsIdempotentElem (f e')
he₁e₂ : f e' * f e₂ = 0
he₂e₁ : f e₂ * f e' = 0

⊢ ∃ e'_1, IsIdempotentElem e'_1 ∧ f e'_1 = f e' ∧ e'_1 * e₂ = 0 ∧ e₂ * e'_1 = 0
-- Mathlib/MeasureTheory/OuterMeasure/Induced.lean
α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
β : Type u_2
inst✝ : Countable β
f : β → Set α
hd : Pairwise (Disjoint on f)
hm : ∀ (i : β), P (f i)

⊢ extend m (⋃ i, f i) = ∑' (i : β), extend m (f i)
-- Mathlib/MeasureTheory/OuterMeasure/Induced.lean
α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
β : Type u_2
inst✝ : Countable β
f : β → Set α
hd : Pairwise (Disjoint on f)
hm : ∀ (i : β), P (f i)
val✝ : Encodable β

⊢ extend m (⋃ i, f i) = ∑' (i : β), extend m (f i)
-- Mathlib/MeasureTheory/OuterMeasure/Induced.lean
α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
β : Type u_2
inst✝ : Countable β
f : β → Set α
hd : Pairwise (Disjoint on f)
hm : ∀ (i : β), P (f i)
val✝ : Encodable β

⊢ extend m (⋃ i, ⋃ b ∈ Encodable.decode₂ β i, f b) = ∑' (i : ℕ), extend m (⋃ b ∈ Encodable.decode₂ β i, f b)
-- Mathlib/MeasureTheory/OuterMeasure/Induced.lean
α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
β : Type u_2
inst✝ : Countable β
f : β → Set α
hd : Pairwise (Disjoint on f)
hm : ∀ (i : β), P (f i)
val✝ : Encodable β

⊢ extend m ∅ = 0
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
𝕜 : Type u_4
inst✝ : NormedDivisionRing 𝕜
m : ℤ
hm : m < 0

⊢ Tendsto (fun x => ‖x ^ m‖) (𝓝[≠] 0) atTop
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
𝕜 : Type u_4
inst✝ : NormedDivisionRing 𝕜
m : ℤ
hm : -m < 0

⊢ Tendsto (fun x => ‖x ^ (-m)‖) (𝓝[≠] 0) atTop
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
𝕜 : Type u_4
inst✝ : NormedDivisionRing 𝕜
m : ℕ
hm : 0 < ↑m

⊢ Tendsto (fun x => ‖x ^ (-↑m)‖) (𝓝[≠] 0) atTop
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
𝕜 : Type u_4
inst✝ : NormedDivisionRing 𝕜
m : ℕ
hm : 0 < m

⊢ Tendsto (fun x => ‖x ^ (-↑m)‖) (𝓝[≠] 0) atTop
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
𝕜 : Type u_4
inst✝ : NormedDivisionRing 𝕜
m : ℕ
hm : 0 < m

⊢ Tendsto (fun x => ‖x⁻¹‖ ^ m) (𝓝[≠] 0) atTop
-- Mathlib/LinearAlgebra/FiniteDimensional.lean
K : Type u
V : Type v
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Ring E
inst✝ : Algebra F E
hr : finrank F E = 2
i : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))
h : ⊥ = ⊤

⊢ False
-- Mathlib/LinearAlgebra/FiniteDimensional.lean
K : Type u
V : Type v
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Ring E
inst✝ : Algebra F E
hr : finrank F E = 2
i : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))
S : Subalgebra F E

⊢ S = ⊥ ∨ S = ⊤
-- Mathlib/LinearAlgebra/FiniteDimensional.lean
K : Type u
V : Type v
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Ring E
inst✝ : Algebra F E
hr : finrank F E = 2
i : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))
S : Subalgebra F E
this : FiniteDimensional F E

⊢ S = ⊥ ∨ S = ⊤