ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Probability/Independence/Kernel.lean
α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsMarkovKernel κ
hf : Measurable f
hg : Measurable g

⊢ (∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)) ↔ ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ
-- Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
R : Type u
inst✝² : CommRing R
W' : Jacobian R
F : Type v
inst✝¹ : Field F
W : Jacobian F
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hQz : Q z ≠ 0
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy : P y * Q z ^ 3 = Q y * P z ^ 3
hy' : P y * Q z ^ 3 = W'.negY Q * P z ^ 3

⊢ W'.dblZ P = 0
-- Mathlib/Computability/Primrec.lean
n : ℕ
f : Vector ℕ n → ℕ
hf : Primrec' f
v : Vector ℕ n.succ

⊢ f (ofFn fun i => v.get i.succ) = f v.tail
-- Mathlib/Computability/Primrec.lean
n : ℕ
f : Vector ℕ n → ℕ
hf : Primrec' f
v : Vector ℕ n.succ

⊢ f (ofFn fun i => v.get i.succ) = f (ofFn v.tail.get)
-- Mathlib/Computability/Primrec.lean
n : ℕ
f : Vector ℕ n → ℕ
hf : Primrec' f
v : Vector ℕ n.succ

⊢ (fun i => v.get i.succ) = v.tail.get
-- Mathlib/Computability/Primrec.lean
n : ℕ
f : Vector ℕ n → ℕ
hf : Primrec' f
v : Vector ℕ n.succ
i : Fin n

⊢ v.get i.succ = v.tail.get i
-- Mathlib/CategoryTheory/Monad/Limits.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
T : Monad C
J : Type u
inst✝² : Category.{v, u} J
D : J ⥤ T.Algebra
c : Cocone (D ⋙ T.forget)
t : IsColimit c
inst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor
inst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor

⊢ T.η.app c.pt ≫ lambda c t = 𝟙 c.pt
-- Mathlib/CategoryTheory/Monad/Limits.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
T : Monad C
J : Type u
inst✝² : Category.{v, u} J
D : J ⥤ T.Algebra
c : Cocone (D ⋙ T.forget)
t : IsColimit c
inst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor
inst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor

⊢ ∀ (j : J), c.ι.app j ≫ T.η.app c.pt ≫ lambda c t = c.ι.app j ≫ 𝟙 c.pt
-- Mathlib/CategoryTheory/Monad/Limits.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
T : Monad C
J : Type u
inst✝² : Category.{v, u} J
D : J ⥤ T.Algebra
c : Cocone (D ⋙ T.forget)
t : IsColimit c
inst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor
inst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor
j : J

⊢ c.ι.app j ≫ T.η.app c.pt ≫ lambda c t = c.ι.app j ≫ 𝟙 c.pt
-- Mathlib/CategoryTheory/Monad/Limits.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
T : Monad C
J : Type u
inst✝² : Category.{v, u} J
D : J ⥤ T.Algebra
c : Cocone (D ⋙ T.forget)
t : IsColimit c
inst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor
inst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor
j : J

⊢ c.ι.app j = c.ι.app j ≫ 𝟙 c.pt
-- Mathlib/CategoryTheory/Monad/Limits.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
T : Monad C
J : Type u
inst✝² : Category.{v, u} J
D : J ⥤ T.Algebra
c : Cocone (D ⋙ T.forget)
t : IsColimit c
inst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor
inst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor

⊢ T.μ.app c.pt ≫ lambda c t = T.map (lambda c t) ≫ lambda c t
-- Mathlib/CategoryTheory/Monad/Limits.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
T : Monad C
J : Type u
inst✝² : Category.{v, u} J
D : J ⥤ T.Algebra
c : Cocone (D ⋙ T.forget)
t : IsColimit c
inst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor
inst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor
j : J

⊢ (T.mapCocone (T.mapCocone c)).ι.app j ≫ T.μ.app c.pt ≫ lambda c t = (T.mapCocone (T.mapCocone c)).ι.app j ≫ T.map (lambda c t) ≫ lambda c t
-- Mathlib/CategoryTheory/Monad/Limits.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
T : Monad C
J : Type u
inst✝² : Category.{v, u} J
D : J ⥤ T.Algebra
c : Cocone (D ⋙ T.forget)
t : IsColimit c
inst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor
inst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor
j : J

⊢ T.μ.app ((D ⋙ T.forget).obj j) ≫ (D.obj j).a ≫ c.ι.app j = T.map (D.obj j).a ≫ (D.obj j).a ≫ c.ι.app j
-- Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
ι : Type u_1
inst✝³ : Fintype ι
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : DecidableEq ι
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f

⊢ ∫⁻ (x : (i : ι) → A i), ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤ (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ p
-- Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
ι : Type u_1
inst✝³ : Fintype ι
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : DecidableEq ι
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι

⊢ ∫⁻ (x : (i : ι) → A i), ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤ (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ p
-- Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
ι : Type u_1
inst✝³ : Fintype ι
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : DecidableEq ι
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι

⊢ ∫⁻ (x : (i : ι) → A i), ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤ (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ p
-- Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
ι : Type u_1
inst✝³ : Fintype ι
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : DecidableEq ι
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
h3 : (↑#ι - 1) * (1 / (↑#ι - 1)) ≤ 1

⊢ ∫⁻ (x : (i : ι) → A i), ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤ (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ p
-- Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
ι : Type u_1
inst✝³ : Fintype ι
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : DecidableEq ι
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
h3 : (↑#ι - 1) * (1 / (↑#ι - 1)) ≤ 1
h4 : p = 1 + 1 / (↑#ι - 1)

⊢ ∫⁻ (x : (i : ι) → A i), ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤ (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ p
-- Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
ι : Type u_1
inst✝³ : Fintype ι
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : DecidableEq ι
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
h3 : (↑#ι - 1) * (1 / (↑#ι - 1)) ≤ 1
h4 : p = 1 + 1 / (↑#ι - 1)

⊢ ∫⁻ (x : (i : ι) → A i), ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤ (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ (1 + 1 / (↑#ι - 1))
-- Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
ι : Type u_1
inst✝³ : Fintype ι
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : DecidableEq ι
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
h3 : (↑#ι - 1) * (1 / (↑#ι - 1)) ≤ 1
h4 : p = 1 + 1 / (↑#ι - 1)

⊢ (fun x => ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1))) = fun x => f x ^ (1 - (↑#ι - 1) * (1 / (↑#ι - 1))) * ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1))
-- Mathlib/Order/Interval/Multiset.lean
α : Type u_1
inst✝² : Preorder α
inst✝¹ : LocallyFiniteOrder α
a b c : α
inst✝ : DecidablePred fun x => c ≤ x
hac : a ≤ c

⊢ filter (fun x => c ≤ x) (Ico a b) = Ico c b
-- Mathlib/Order/Interval/Multiset.lean
α : Type u_1
inst✝² : Preorder α
inst✝¹ : LocallyFiniteOrder α
a b c : α
inst✝ : DecidablePred fun x => c ≤ x
hac : a ≤ c

⊢ (Finset.Ico c b).val = Ico c b
-- Mathlib/Data/Real/GoldenRatio.lean
α : Type u_1
inst✝ : CommSemiring α

⊢ fibRec.IsSolution fun x => ψ ^ x
-- Mathlib/Data/Real/GoldenRatio.lean
α : Type u_1
inst✝ : CommSemiring α

⊢ (X ^ 2 - (X + 1)).IsRoot ψ
-- Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
b : Set (Set α)
bct : b.Countable
hbm : ∀ s ∈ b, MeasurableSet s
hb : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ b, x ∈ s ↔ y ∈ s) → x = y

⊢ ∃ m', CountablyGenerated α ∧ SeparatesPoints α ∧ m' ≤ m
-- Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
b : Set (Set α)
bct : b.Countable
hbm : ∀ s ∈ b, MeasurableSet s
hb : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ b, x ∈ s ↔ y ∈ s) → x = y

⊢ CountablyGenerated α
-- Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
b : Set (Set α)
bct : b.Countable
hbm : ∀ s ∈ b, MeasurableSet s
hb : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ b, x ∈ s ↔ y ∈ s) → x = y

⊢ SeparatesPoints α
-- Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
b : Set (Set α)
bct : b.Countable
hbm : ∀ s ∈ b, MeasurableSet s
hb : ∀ x ∈ univ, ∀ y ∈ univ, (∀ s ∈ b, x ∈ s ↔ y ∈ s) → x = y

⊢ ∀ (x y : α), (∀ (s : Set α), MeasurableSet s → (x ∈ s ↔ y ∈ s)) → x = y
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X

⊢ (NatTrans.leftDerivedToHomotopyCategory α).app X = (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex) ≫ (P.isoLeftDerivedToHomotopyCategoryObj G).inv
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X

⊢ (NatTrans.leftDerivedToHomotopyCategory α).app X ≫ (P.isoLeftDerivedToHomotopyCategoryObj G).hom = (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X

⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫ (G.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫ 𝟙 ((HomotopyCategory.quotient D (ComplexShape.down ℕ)).obj ((G.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) = ((F.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫ 𝟙 ((HomotopyCategory.quotient D (ComplexShape.down ℕ)).obj ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex))) ≫ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X

⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫ (G.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫ 𝟙 ((HomotopyCategory.quotient D (ComplexShape.down ℕ)).obj ((G.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) = (F.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫ 𝟙 ((HomotopyCategory.quotient D (ComplexShape.down ℕ)).obj ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) ≫ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X

⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫ (G.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom = (F.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
β : (projectiveResolution X).complex ⟶ P.complex
hβ : (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β = P.iso.hom

⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫ (G.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom = (F.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
β : (projectiveResolution X).complex ⟶ P.complex
hβ : (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β = P.iso.hom

⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((G.mapHomologicalComplex (ComplexShape.down ℕ)).map β) = (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map β) ≫ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
β : (projectiveResolution X).complex ⟶ P.complex
hβ : (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β = P.iso.hom

⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as ≫ (G.mapHomologicalComplex (ComplexShape.down ℕ)).map β) = (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app (projectiveResolution X).complex ≫ (G.mapHomologicalComplex (ComplexShape.down ℕ)).map β)
-- Mathlib/Algebra/Polynomial/Degree/Lemmas.lean
R : Type u
S : Type v
ι : Type w
a b : R
m n : ℕ
inst✝ : Semiring R
p q r : R[X]
pn : p.natDegree < n

⊢ (p + q).coeff n = q.coeff n
-- Mathlib/Algebra/Polynomial/Degree/Lemmas.lean
R : Type u
S : Type v
ι : Type w
a b : R
m n : ℕ
inst✝ : Semiring R
p q r : R[X]
pn : p.natDegree < n

⊢ (q + p).coeff n = q.coeff n
-- Mathlib/Data/QPF/Univariate/Basic.lean
F : Type u → Type u
q : QPF F
α : Type u
g₁ g₂ : Fix F → α
h : ∀ (x : F (Fix F)), g₁ <$> x = g₂ <$> x → g₁ (mk x) = g₂ (mk x)
x✝ : Fix F
a : (P F).A
f : (P F).B a → WType (P F).B
ih : ∀ (a : (P F).B a), g₁ (Quot.mk Setoid.r (f a)) = g₂ (Quot.mk Setoid.r (f a))

⊢ g₁ ⟦WType.mk a f⟧ = g₂ ⟦WType.mk a f⟧
-- Mathlib/Data/QPF/Univariate/Basic.lean
F : Type u → Type u
q : QPF F
α : Type u
g₁ g₂ : Fix F → α
h : ∀ (x : F (Fix F)), g₁ <$> x = g₂ <$> x → g₁ (mk x) = g₂ (mk x)
x✝ : Fix F
a : (P F).A
f : (P F).B a → WType (P F).B
ih : ∀ (a : (P F).B a), g₁ (Quot.mk Setoid.r (f a)) = g₂ (Quot.mk Setoid.r (f a))

⊢ g₁ (mk (abs ⟨a, fun x => ⟦f x⟧⟩)) = g₂ (mk (abs ⟨a, fun x => ⟦f x⟧⟩))
-- Mathlib/Data/QPF/Univariate/Basic.lean
F : Type u → Type u
q : QPF F
α : Type u
g₁ g₂ : Fix F → α
h : ∀ (x : F (Fix F)), g₁ <$> x = g₂ <$> x → g₁ (mk x) = g₂ (mk x)
x✝ : Fix F
a : (P F).A
f : (P F).B a → WType (P F).B
ih : ∀ (a : (P F).B a), g₁ (Quot.mk Setoid.r (f a)) = g₂ (Quot.mk Setoid.r (f a))

⊢ g₁ <$> abs ⟨a, fun x => ⟦f x⟧⟩ = g₂ <$> abs ⟨a, fun x => ⟦f x⟧⟩
-- Mathlib/Data/QPF/Univariate/Basic.lean
F : Type u → Type u
q : QPF F
α : Type u
g₁ g₂ : Fix F → α
h : ∀ (x : F (Fix F)), g₁ <$> x = g₂ <$> x → g₁ (mk x) = g₂ (mk x)
x✝ : Fix F
a : (P F).A
f : (P F).B a → WType (P F).B
ih : ∀ (a : (P F).B a), g₁ (Quot.mk Setoid.r (f a)) = g₂ (Quot.mk Setoid.r (f a))

⊢ abs ⟨a, g₁ ∘ fun x => ⟦f x⟧⟩ = abs ⟨a, g₂ ∘ fun x => ⟦f x⟧⟩
-- Mathlib/Algebra/QuadraticDiscriminant.lean
K : Type u_1
inst✝ : LinearOrderedField K
a b c : K
ha : a ≠ 0
h : ∀ (x : K), 0 < a * x * x + b * x + c

⊢ discrim a b c < 0
-- Mathlib/Algebra/QuadraticDiscriminant.lean
K : Type u_1
inst✝ : LinearOrderedField K
a b c : K
ha : a ≠ 0
h : ∀ (x : K), 0 < a * x * x + b * x + c
this : ∀ (x : K), 0 ≤ a * x * x + b * x + c

⊢ discrim a b c < 0
-- Mathlib/Algebra/QuadraticDiscriminant.lean
K : Type u_1
inst✝ : LinearOrderedField K
a b c : K
ha : a ≠ 0
h : ∀ (x : K), 0 < a * x * x + b * x + c
this : ∀ (x : K), 0 ≤ a * x * x + b * x + c
h' : discrim a b c = 0

⊢ False
-- Mathlib/Algebra/QuadraticDiscriminant.lean
K : Type u_1
inst✝ : LinearOrderedField K
a b c : K
ha : a ≠ 0
h : ∀ (x : K), 0 < a * x * x + b * x + c
this✝ : ∀ (x : K), 0 ≤ a * x * x + b * x + c
h' : discrim a b c = 0
this : 0 < a * (-b / (2 * a)) * (-b / (2 * a)) + b * (-b / (2 * a)) + c

⊢ False
-- Mathlib/Algebra/QuadraticDiscriminant.lean
K : Type u_1
inst✝ : LinearOrderedField K
a b c : K
ha : a ≠ 0
h : ∀ (x : K), 0 < a * x * x + b * x + c
this✝¹ : ∀ (x : K), 0 ≤ a * x * x + b * x + c
h' : discrim a b c = 0
this✝ : 0 < a * (-b / (2 * a)) * (-b / (2 * a)) + b * (-b / (2 * a)) + c
this : a * (-b / (2 * a)) * (-b / (2 * a)) + b * (-b / (2 * a)) + c = 0

⊢ False
-- Mathlib/Topology/PartialHomeomorph.lean
X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
s : Opens X
hs : Nonempty ↥s

⊢ (e.subtypeRestr hs).source = Subtype.val ⁻¹' e.source
-- Mathlib/MeasureTheory/Integral/SetToL1.lean
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hfi : Integrable f μ
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i) μ
hfs : Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0)

⊢ Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 (setToFun μ T hT f))
-- Mathlib/LinearAlgebra/Finsupp.lean
α : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
R : Type u_5
S : Type u_6
inst✝⁷ : Semiring R
inst✝⁶ : Semiring S
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
a a' : α
h : a ≠ a'
x✝ : M

⊢ (lapply a ∘ₗ lsingle a') x✝ = 0 x✝
-- Mathlib/Algebra/Group/Subgroup/Basic.lean
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
s : Set G
N : Subgroup G
tn : N.Normal
a✝ : G
h : a✝ ∈ N
a : G
hc : a ∈ conjugatesOf a✝

⊢ a ∈ ↑N
-- Mathlib/Algebra/Group/Subgroup/Basic.lean
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
s : Set G
N : Subgroup G
tn : N.Normal
a : G
h : a ∈ N
c : G
hc : c * a * c⁻¹ ∈ conjugatesOf a

⊢ c * a * c⁻¹ ∈ ↑N
-- Mathlib/RingTheory/HahnSeries/PowerSeries.lean
Γ : Type u_1
R : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
r : R
n : ℕ

⊢ (PowerSeries.coeff A n) (__src✝.toFun ((algebraMap R (HahnSeries ℕ A)) r)) = (PowerSeries.coeff A n) ((algebraMap R (PowerSeries A)) r)
-- Mathlib/Data/Fin/Basic.lean
n✝ m n : ℕ
a : Fin n
b : Fin (n + 1)

⊢ a.castSucc.predAbove b.castSucc = (a.predAbove b).castSucc
-- Mathlib/Data/Fin/Basic.lean
n✝ m n : ℕ
a : Fin n
b : Fin (n + 1)
h : a.castSucc < b

⊢ a.castSucc.predAbove b.castSucc = (a.predAbove b).castSucc
-- Mathlib/Data/Fin/Basic.lean
n✝ m n : ℕ
a : Fin n
b : Fin (n + 1)
h : b ≤ a.castSucc

⊢ a.castSucc.predAbove b.castSucc = (a.predAbove b).castSucc
-- Mathlib/Data/ENNReal/Basic.lean
α : Type u_1
a b c d : ℝ≥0∞
r p q : ℝ≥0
x : ℝ≥0∞

⊢ x ∈ ⋃ n, Iio ↑n ↔ x ∈ {⊤}ᶜ
-- Mathlib/Data/ENNReal/Basic.lean
α : Type u_1
a b c d : ℝ≥0∞
r p q : ℝ≥0
x : ℝ≥0∞

⊢ (∃ i, x ∈ Iio ↑i) ↔ x ∈ {⊤}ᶜ
-- Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{?u.39244, u_1} C
inst✝³ : Category.{?u.39248, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
d e : ℤ
γ : Cochain F K d
he : 1 + d = e

⊢ -1 + e = d
-- Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{?u.39248, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
d e : ℤ
γ : Cochain F K d
he : 1 + d = e

⊢ (inl φ).comp ((↑(fst φ)).comp γ he) ⋯ = γ
-- Mathlib/RingTheory/DedekindDomain/Ideal.lean
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r X : R
hr : r ≠ 0
hX₁ : normUnit X = 1
hX : Prime X

⊢ Multiset.count (span {X}) (normalizedFactors (span {r})) = Multiset.count X (normalizedFactors r)