ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Data/Finset/Union.lean
α : Type u_1
β : Type u_2
γ : Type u_3
s s₁ s₂ : Finset α
t t₁ t₂ : α → Finset β
inst✝ : DecidableEq β
h : ∀ a ∈ s, t₁ a ⊆ t₂ a
this : ∀ (b : β), ∀ a ∈ s, b ∈ t₁ a → ∃ a ∈ s, b ∈ t₂ a
\u22a2 s.biUnion t₁ ⊆ s.biUnion t₂
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
\u22a2 p.prod ≤ q.prod ↔ p ≤ q
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
\u22a2 p.prod ≤ q.prod → p ≤ q
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
\u22a2 p ≤ q
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
x : Associates α
hx : x ∈ p + factors c
\u22a2 Irreducible x
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
\u22a2 q.prod = (p + factors c).prod
-- Mathlib/NumberTheory/Padics/PadicIntegers.lean
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
n : ℕ
hx : ¬x = 0
\u22a2 ‖x‖ ≤ ↑p ^ (-↑n) ↔ x ∈ Ideal.span {↑p ^ n}
-- Mathlib/Algebra/Group/Support.lean
α : Type u_1
β : Type u_2
A : Type u_3
B : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
inst✝² : One M
inst✝¹ : One N
inst✝ : One P
f : α → M
g : α → N
x : α
\u22a2 (x ∈ mulSupport fun x => (f x, g x)) ↔ x ∈ mulSupport f ∪ mulSupport g
-- Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
A : Type v
inst✝ : CommRing A
φ : R →+* A
\u22a2 (W.map φ).Δ = φ W.Δ
-- Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
A : Type v
inst✝ : CommRing A
φ : R →+* A
\u22a2 -φ W.b₂ ^ 2 * φ W.b₈ - 8 * φ W.b₄ ^ 3 - 27 * φ W.b₆ ^ 2 + 9 * φ W.b₂ * φ W.b₄ * φ W.b₆ = φ (-W.b₂ ^ 2 * W.b₈ - 8 * W.b₄ ^ 3 - 27 * W.b₆ ^ 2 + 9 * W.b₂ * W.b₄ * W.b₆)
-- Mathlib/Analysis/Calculus/LHopital.lean
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : DifferentiableOn ℝ f (Iio a)
hg' : ∀ x ∈ Iio a, deriv g x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l
\u22a2 Tendsto (fun x => f x / g x) atBot l
-- Mathlib/Analysis/Calculus/LHopital.lean
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf✝ : DifferentiableOn ℝ f (Iio a)
hg' : ∀ x ∈ Iio a, deriv g x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l
hdf : ∀ x ∈ Iio a, DifferentiableAt ℝ f x
\u22a2 Tendsto (fun x => f x / g x) atBot l
-- Mathlib/Analysis/Calculus/LHopital.lean
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf✝ : DifferentiableOn ℝ f (Iio a)
hg' : ∀ x ∈ Iio a, deriv g x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l
hdf : ∀ x ∈ Iio a, DifferentiableAt ℝ f x
hdg : ∀ x ∈ Iio a, DifferentiableAt ℝ g x
\u22a2 Tendsto (fun x => f x / g x) atBot l
-- Mathlib/CategoryTheory/Comma/Presheaf.lean
C : Type u
inst✝ : Category.{v, u} C
A : Cᵒᵖ ⥤ Type v
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X : C
s : yoneda.obj X ⟶ A
x : F.obj (op (CostructuredArrow.mk s))
\u22a2 op (CostructuredArrow.mk s) = op (CostructuredArrow.mk (mk s x).fst)
-- Mathlib/Data/Rel.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : Rel α β
s : Rel β γ
t : Set γ
x : α
\u22a2 x ∈ (r • s).core t ↔ x ∈ r.core (s.core t)
-- Mathlib/Data/Rel.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : Rel α β
s : Rel β γ
t : Set γ
x : α
\u22a2 (∀ (y : γ) (x_1 : β), r x x_1 → s x_1 y → y ∈ t) ↔ ∀ (y : β), r x y → ∀ (y_1 : γ), s y y_1 → y_1 ∈ t
-- Mathlib/Data/Matroid/Constructions.lean
α : Type u_1
M : Matroid α
E B I X R J : Set α
\u22a2 M = loopyOn E ↔ M.E = E ∧ ∀ X ⊆ M.E, M.Indep X → X = ∅
-- Mathlib/Data/Matroid/Constructions.lean
α : Type u_1
M : Matroid α
E B I X R J : Set α
\u22a2 M.E = E → ((∀ I ⊆ M.E, M.Indep I ↔ I = ∅) ↔ ∀ X ⊆ M.E, M.Indep X → X = ∅)
-- Mathlib/Data/Matroid/Constructions.lean
α : Type u_1
M : Matroid α
B I X R J : Set α
\u22a2 (∀ I ⊆ M.E, M.Indep I ↔ I = ∅) ↔ ∀ X ⊆ M.E, M.Indep X → X = ∅
-- Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
d d' : Module.Dual R M
x : M
\u22a2 (contractLeft d) ((ι Q) x) = (algebraMap R (CliffordAlgebra Q)) (d x)
-- Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
d d' : Module.Dual R M
x : M
\u22a2 ∀ (m : M) (x fx : CliffordAlgebra Q), ((contractLeftAux Q d) m) ((ι Q) m * x, ((contractLeftAux Q d) m) (x, fx)) = Q m • fx
-- Mathlib/AlgebraicGeometry/AffineScheme.lean
X Y : Scheme
U : X.Opens
hU : IsAffineOpen U
f✝ : ↑Γ(X, U)
f : X ⟶ Y
H : IsOpenImmersion f
this : IsAffine ↑U
\u22a2 IsAffineOpen (f ''ᵁ U)
-- Mathlib/AlgebraicGeometry/AffineScheme.lean
X Y : Scheme
U : X.Opens
hU : IsAffineOpen U
f✝ : ↑Γ(X, U)
f : X ⟶ Y
H : IsOpenImmersion f
this : IsAffine ↑U
\u22a2 f ''ᵁ U = Scheme.Hom.opensRange (U.ι ≫ f)
-- Mathlib/AlgebraicGeometry/AffineScheme.lean
X Y : Scheme
U : X.Opens
hU : IsAffineOpen U
f✝ : ↑Γ(X, U)
f : X ⟶ Y
H : IsOpenImmersion f
this : IsAffine ↑U
\u22a2 ↑(f ''ᵁ U) = ↑(Scheme.Hom.opensRange (U.ι ≫ f))
-- Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
f f' : PartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜 E'
inst✝¹ : TopologicalSpace H'
inst✝ : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
s t : Set M
x : M
h : x ∈ f.source
ht : t ∈ 𝓝[s] x
\u22a2 ↑(f.extend I).symm ⁻¹' t ∈ 𝓝[↑(f.extend I).symm ⁻¹' s ∩ range ↑I] ↑(f.extend I) x
-- Mathlib/CategoryTheory/Sites/DenseSubsite.lean
C : Type u_1
inst✝⁵ : Category.{u_7, u_1} C
D : Type u_2
inst✝⁴ : Category.{u_6, u_2} D
E : Type u_3
inst✝³ : Category.{?u.119255, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
A : Type u_4
inst✝² : Category.{u_5, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ : Dᵒᵖ ⥤ A
ℱ' : Sheaf K A
α : ℱ ⟶ ℱ'.val
X : Dᵒᵖ
\u22a2 (sheafHom (whiskerLeft G.op α)).app X = α.app X
-- Mathlib/CategoryTheory/Sites/DenseSubsite.lean
C : Type u_1
inst✝⁵ : Category.{u_7, u_1} C
D : Type u_2
inst✝⁴ : Category.{u_6, u_2} D
E : Type u_3
inst✝³ : Category.{?u.119255, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
A : Type u_4
inst✝² : Category.{u_5, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ : Dᵒᵖ ⥤ A
ℱ' : Sheaf K A
α : ℱ ⟶ ℱ'.val
X : Dᵒᵖ
\u22a2 yoneda.map ((sheafHom (whiskerLeft G.op α)).app X) = yoneda.map (α.app X)
-- Mathlib/CategoryTheory/Sites/DenseSubsite.lean
C : Type u_1
inst✝⁵ : Category.{u_7, u_1} C
D : Type u_2
inst✝⁴ : Category.{u_6, u_2} D
E : Type u_3
inst✝³ : Category.{?u.119255, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
A : Type u_4
inst✝² : Category.{u_5, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ : Dᵒᵖ ⥤ A
ℱ' : Sheaf K A
α : ℱ ⟶ ℱ'.val
X : Dᵒᵖ
U : Aᵒᵖ
a✝ : (yoneda.obj (ℱ.obj X)).obj U
\u22a2 (yoneda.map ((sheafHom (whiskerLeft G.op α)).app X)).app U a✝ = (yoneda.map (α.app X)).app U a✝
-- Mathlib/CategoryTheory/Sites/DenseSubsite.lean
C : Type u_1
inst✝⁵ : Category.{u_7, u_1} C
D : Type u_2
inst✝⁴ : Category.{u_6, u_2} D
E : Type u_3
inst✝³ : Category.{?u.119255, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
A : Type u_4
inst✝² : Category.{u_5, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ : Dᵒᵖ ⥤ A
ℱ' : Sheaf K A
α : ℱ ⟶ ℱ'.val
X : Dᵒᵖ
U : Aᵒᵖ
a✝ : (yoneda.obj (ℱ.obj X)).obj U
\u22a2 ((sheafYonedaHom (whiskerLeft G.op α)).app X).app U a✝ = (yoneda.map (α.app X)).app U a✝
-- Mathlib/CategoryTheory/Sites/DenseSubsite.lean
C : Type u_1
inst✝⁵ : Category.{u_7, u_1} C
D : Type u_2
inst✝⁴ : Category.{u_6, u_2} D
E : Type u_3
inst✝³ : Category.{?u.119255, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
A : Type u_4
inst✝² : Category.{u_5, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ : Dᵒᵖ ⥤ A
ℱ' : Sheaf K A
α : ℱ ⟶ ℱ'.val
X : Dᵒᵖ
U : Aᵒᵖ
a✝ : (yoneda.obj (ℱ.obj X)).obj U
\u22a2 (let_fun this := (yoneda.map (α.app X)).app U a✝; this) = ((sheafYonedaHom (whiskerLeft G.op α)).app X).app U a✝
-- Mathlib/CategoryTheory/Sites/DenseSubsite.lean
C : Type u_1
inst✝⁵ : Category.{u_7, u_1} C
D : Type u_2
inst✝⁴ : Category.{u_6, u_2} D
E : Type u_3
inst✝³ : Category.{?u.119255, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
A : Type u_4
inst✝² : Category.{u_5, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ : Dᵒᵖ ⥤ A
ℱ' : Sheaf K A
α : ℱ ⟶ ℱ'.val
X : Dᵒᵖ
U : Aᵒᵖ
a✝ : (yoneda.obj (ℱ.obj X)).obj U
\u22a2 (pushforwardFamily (homOver (whiskerLeft G.op α) (unop U)) a✝).Compatible
-- Mathlib/CategoryTheory/Sites/DenseSubsite.lean
C : Type u_1
inst✝⁵ : Category.{u_7, u_1} C
D : Type u_2
inst✝⁴ : Category.{u_6, u_2} D
E : Type u_3
inst✝³ : Category.{?u.119255, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
A : Type u_4
inst✝² : Category.{u_5, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ : Dᵒᵖ ⥤ A
ℱ' : Sheaf K A
α : ℱ ⟶ ℱ'.val
X : Dᵒᵖ
U : Aᵒᵖ
a✝ : (yoneda.obj (ℱ.obj X)).obj U
\u22a2 (pushforwardFamily (homOver (whiskerLeft G.op α) (unop U)) a✝).IsAmalgamation (let_fun this := (yoneda.map (α.app X)).app U a✝; this)
-- Mathlib/CategoryTheory/Sites/DenseSubsite.lean
C : Type u_1
inst✝⁵ : Category.{u_7, u_1} C
D : Type u_2
inst✝⁴ : Category.{u_6, u_2} D
E : Type u_3
inst✝³ : Category.{?u.119255, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
A : Type u_4
inst✝² : Category.{u_5, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ : Dᵒᵖ ⥤ A
ℱ' : Sheaf K A
α : ℱ ⟶ ℱ'.val
X : Dᵒᵖ
U : Aᵒᵖ
a✝ : (yoneda.obj (ℱ.obj X)).obj U
Y : D
f : Y ⟶ unop X
hf : (Sieve.coverByImage G (unop X)).arrows f
\u22a2 (ℱ'.val ⋙ coyoneda.obj (op (unop U))).map f.op (let_fun this := (yoneda.map (α.app X)).app U a✝; this) = pushforwardFamily (homOver (whiskerLeft G.op α) (unop U)) a✝ f hf
-- Mathlib/CategoryTheory/Sites/DenseSubsite.lean
C : Type u_1
inst✝⁵ : Category.{u_7, u_1} C
D : Type u_2
inst✝⁴ : Category.{u_6, u_2} D
E : Type u_3
inst✝³ : Category.{?u.119255, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
A : Type u_4
inst✝² : Category.{u_5, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ : Dᵒᵖ ⥤ A
ℱ' : Sheaf K A
α : ℱ ⟶ ℱ'.val
X : Dᵒᵖ
U : Aᵒᵖ
a✝ : (yoneda.obj (ℱ.obj X)).obj U
Y : D
f : Y ⟶ unop X
hf : (Sieve.coverByImage G (unop X)).arrows f
\u22a2 (ℱ'.val ⋙ coyoneda.obj (op (unop U))).map ((Nonempty.some hf).lift ≫ (Nonempty.some hf).map).op (let_fun this := (yoneda.map (α.app X)).app U a✝; this) = pushforwardFamily (homOver (whiskerLeft G.op α) (unop U)) a✝ f hf
-- Mathlib/MeasureTheory/Measure/EverywherePos.lean
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hs : IsClosed s
\u22a2 IsClosed (μ.everywherePosSubset s)
-- Mathlib/MeasureTheory/Measure/EverywherePos.lean
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hs : IsClosed s
u : Set α
u_open : IsOpen u
hu : μ.everywherePosSubset s = s \ u
\u22a2 IsClosed (μ.everywherePosSubset s)
-- Mathlib/MeasureTheory/Measure/EverywherePos.lean
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s k : Set α
hs : IsClosed s
u : Set α
u_open : IsOpen u
hu : μ.everywherePosSubset s = s \ u
\u22a2 IsClosed (s \ u)
-- 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✝ : ι → α
s✝ t : Set α
f : α → β
s : Set β
\u22a2 f ⁻¹' (range f ∩ s) = f ⁻¹' s
-- Mathlib/Logic/IsEmpty.lean
α : Sort u_1
β : Sort u_2
γ : Sort u_3
\u22a2 IsEmpty (PProd α β) ↔ IsEmpty α ∨ IsEmpty β
-- Mathlib/GroupTheory/Perm/Sign.lean
α : Type u
inst✝¹ : DecidableEq α
β : Type v
inst✝ : Finite α
\u22a2 Submonoid.closure {σ | σ.IsSwap} = ⊤
-- Mathlib/GroupTheory/Perm/Sign.lean
α : Type u
inst✝¹ : DecidableEq α
β : Type v
inst✝ : Finite α
val✝ : Fintype α
\u22a2 Submonoid.closure {σ | σ.IsSwap} = ⊤
-- Mathlib/GroupTheory/Perm/Sign.lean
α : Type u
inst✝¹ : DecidableEq α
β : Type v
inst✝ : Finite α
val✝ : Fintype α
x : Perm α
x✝ : x ∈ ⊤
\u22a2 x ∈ Submonoid.closure {σ | σ.IsSwap}
-- Mathlib/GroupTheory/Perm/Sign.lean
α : Type u
inst✝¹ : DecidableEq α
β : Type v
inst✝ : Finite α
val✝ : Fintype α
x : Perm α
x✝ : x ∈ ⊤
h1 : (↑x.truncSwapFactors.out).prod = x
h2 : ∀ g ∈ ↑x.truncSwapFactors.out, g.IsSwap
\u22a2 x ∈ Submonoid.closure {σ | σ.IsSwap}
-- Mathlib/GroupTheory/Perm/Sign.lean
α : Type u
inst✝¹ : DecidableEq α
β : Type v
inst✝ : Finite α
val✝ : Fintype α
x : Perm α
x✝ : x ∈ ⊤
h1 : (↑x.truncSwapFactors.out).prod = x
h2 : ∀ g ∈ ↑x.truncSwapFactors.out, g.IsSwap
\u22a2 (↑x.truncSwapFactors.out).prod ∈ Submonoid.closure {σ | σ.IsSwap}
-- Mathlib/FieldTheory/RatFunc/Degree.lean
K : Type u
inst✝ : Field K
x : RatFunc K
hx : x ≠ 0
s : K[X]
hs : s ≠ 0
\u22a2 ↑(x.num * s).natDegree - ↑(s * x.denom).natDegree = x.intDegree
-- Mathlib/FieldTheory/RatFunc/Degree.lean
K : Type u
inst✝ : Field K
x : RatFunc K
hx : x ≠ 0
s : K[X]
hs : s ≠ 0
\u22a2 x.num * s * x.denom = x.num * (s * x.denom)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
x : ℂ
\u22a2 cos (↑π / 2 - x) = sin x
-- Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
\u22a2 fromSubtype f p ≫ toSubtype f p = 𝟙 (⨁ Subtype.restrict p f)
-- Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j : Subtype p
\u22a2 (fromSubtype f p ≫ toSubtype f p) ≫ π (Subtype.restrict p f) j = 𝟙 (⨁ Subtype.restrict p f) ≫ π (Subtype.restrict p f) j
-- Mathlib/Order/Filter/Basic.lean
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
s t : Set α
l : Filter α
\u22a2 s ∩ t =ᶠ[l] s ↔ ∀ᶠ (x : α) in l, x ∈ s → x ∈ t
-- Mathlib/GroupTheory/Congruence/Basic.lean
M✝ : Type u_1
N : Type u_2
P : Type u_3
α : Type u_4
M : Type u_5
inst✝² : MulOneClass M
inst✝¹ : SMul α M
inst✝ : IsScalarTower α M M
c : Con M
a : α
w x : M
h : c w x
\u22a2 c (a • w) (a • x)
-- Mathlib/MeasureTheory/Integral/Pi.lean
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
ι : Type u_3
inst✝² : Fintype ι
f : E → 𝕜
inst✝¹ : MeasureSpace E
inst✝ : SigmaFinite volume
\u22a2 ∫ (x : ι → E), ∏ i : ι, f (x i) = (∫ (x : E), f x) ^ card ι
-- Mathlib/Topology/Separation.lean
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
\u22a2 T2Space X ↔ IsClosed (diagonal X)
-- Mathlib/Combinatorics/Enumerative/Composition.lean
n : ℕ
c : Composition n
i : Fin (ones n).length
h : 0 < (ones n).blocksFun i
\u22a2 ↑(((ones n).embedding i) ⟨0, h⟩) = ↑⟨↑i, ⋯⟩
-- Mathlib/Probability/Martingale/Centering.lean
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
n : ℕ
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
\u22a2 Martingale (martingalePart f ℱ μ) ℱ μ
-- Mathlib/Probability/Martingale/Centering.lean
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
n : ℕ
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
\u22a2 μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ] martingalePart f ℱ μ i
-- Mathlib/Probability/Martingale/Centering.lean
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
n : ℕ
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum : μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ] f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
\u22a2 μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ] martingalePart f ℱ μ i
-- Mathlib/Probability/Martingale/Centering.lean
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
n : ℕ
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum : μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ] f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
\u22a2 f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i]) =ᶠ[ae μ] martingalePart f ℱ μ i
-- Mathlib/Probability/Martingale/Centering.lean
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
n : ℕ
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum : μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ] f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
h_ge : ∀ (k : ℕ), i ≤ k → μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] 0
\u22a2 f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i]) =ᶠ[ae μ] martingalePart f ℱ μ i
-- Mathlib/Probability/Martingale/Centering.lean
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
n : ℕ
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum : μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ] f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
h_ge : ∀ (k : ℕ), i ≤ k → μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] 0
h_lt : ∀ k < i, μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] f (k + 1) - f k - μ[f (k + 1) - f k|↑ℱ k]
\u22a2 f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i]) =ᶠ[ae μ] martingalePart f ℱ μ i
-- Mathlib/Probability/Martingale/Centering.lean
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
n : ℕ
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum : μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ] f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
h_ge : ∀ (k : ℕ), i ≤ k → μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] 0
h_lt : ∀ k < i, μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] f (k + 1) - f k - μ[f (k + 1) - f k|↑ℱ k]
\u22a2 f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i]) =ᶠ[ae μ] (fun n => f 0 + ∑ i ∈ Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i|↑ℱ i])) i
-- Mathlib/Probability/Martingale/Centering.lean
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
n : ℕ
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum : μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ] f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
h_ge : ∀ (k : ℕ), i ≤ k → μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] 0
h_lt : ∀ k < i, μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] f (k + 1) - f k - μ[f (k + 1) - f k|↑ℱ k]
\u22a2 ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i]) =ᶠ[ae μ] ∑ i ∈ Finset.range i, (f (i + 1) - f i - μ[f (i + 1) - f i|↑ℱ i])
-- Mathlib/Probability/Martingale/Centering.lean
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
n : ℕ
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum : μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ] f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
h_ge : ∀ (k : ℕ), i ≤ k → μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] 0
h_lt : ∀ k < i, μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] f (k + 1) - f k - μ[f (k + 1) - f k|↑ℱ k]
\u22a2 ∑ k ∈ Finset.range i, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i]) + ∑ k ∈ Finset.Ico i j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i]) =ᶠ[ae μ] ∑ i ∈ Finset.range i, (f (i + 1) - f i - μ[f (i + 1) - f i|↑ℱ i]) + 0
-- Mathlib/Probability/Martingale/Centering.lean
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
n : ℕ
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum : μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ] f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
h_ge : ∀ (k : ℕ), i ≤ k → μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] 0
h_lt : ∀ k < i, μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] f (k + 1) - f k - μ[f (k + 1) - f k|↑ℱ k]
\u22a2 ∑ k ∈ Finset.Ico i j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i]) =ᶠ[ae μ] 0
-- Mathlib/Probability/Martingale/Centering.lean
Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : ℕ → Ω → E
ℱ : Filtration ℕ m0
n : ℕ
hf : Adapted ℱ f
hf_int : ∀ (n : ℕ), Integrable (f n) μ
inst✝ : SigmaFiniteFiltration μ ℱ
i j : ℕ
hij : i ≤ j
h_eq_sum : μ[martingalePart f ℱ μ j|↑ℱ i] =ᶠ[ae μ] f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i])
h_ge : ∀ (k : ℕ), i ≤ k → μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] 0
h_lt : ∀ k < i, μ[f (k + 1) - f k|↑ℱ i] - μ[μ[f (k + 1) - f k|↑ℱ k]|↑ℱ i] =ᶠ[ae μ] f (k + 1) - f k - μ[f (k + 1) - f k|↑ℱ k]
\u22a2 ∑ i ∈ Finset.Ico i j, 0 =ᶠ[ae μ] 0