ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})
this : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : FiniteDimensional.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i
aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n
hintsum : IsIntegral R (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
r : R
hr : (algebraMap R K) r = (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))

⊢ (algebraMap R K) (Q.coeff 0) ^ n.succ * ((-1) ^ (n.succ * n) * (algebraMap R K) ((minpoly R B.gen).coeff 0) ^ n) = (algebraMap R K) p ^ n.succ * (algebraMap R K) r
-- Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})
this : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : FiniteDimensional.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i
aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n
hintsum : IsIntegral R (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
r : R
hr : (algebraMap R K) r = (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))

⊢ (algebraMap R K) (Q.coeff 0) ^ n.succ * ((-1) ^ (n.succ * n) * (algebraMap R K) ((minpoly R B.gen).coeff 0) ^ n) = (Algebra.norm K) (Q.coeff 0 • B.gen ^ n)
-- Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})
this : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : FiniteDimensional.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i
aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n
hintsum : IsIntegral R (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
r : R
hr : (algebraMap R K) r = (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))

⊢ Q.coeff 0 • B.gen ^ n + ∑ x ∈ (range (Q.natDegree + 1)).erase 0, p • Q.coeff x • f (x + n) = p • (z * B.gen ^ n)
-- Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})
this : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : FiniteDimensional.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i
aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n
hintsum : IsIntegral R (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
r : R
hr : (algebraMap R K) r = (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))

⊢ (Algebra.norm K) (p • (z * B.gen ^ n) - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, p • Q.coeff x • f (x + n)) = (algebraMap R K) p ^ n.succ * (algebraMap R K) r
-- Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})
this : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : FiniteDimensional.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i
aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n
hintsum : IsIntegral R (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
r : R
hr : (algebraMap R K) r = (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))

⊢ Q.coeff 0 • B.gen ^ n + ∑ x ∈ (range (Q.natDegree + 1)).erase 0, p • Q.coeff x • f (x + n) = (Q.coeff 0 • 1 + ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • B.gen ^ x) * B.gen ^ n
-- Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})
this : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : FiniteDimensional.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i
aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n
hintsum : IsIntegral R (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
r : R
hr : (algebraMap R K) r = (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))

⊢ (Q.coeff 0 • B.gen ^ 0 + ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • B.gen ^ x) * B.gen ^ n = (aeval B.gen) Q * B.gen ^ n
-- Mathlib/Probability/StrongLaw.lean
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f✝ f : α → ℝ
A : ℝ
x : α

⊢ |truncation f A x| ≤ |A|
-- Mathlib/Probability/StrongLaw.lean
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f✝ f : α → ℝ
A : ℝ
x : α

⊢ |if f x ∈ Set.Ioc (-A) A then f x else 0| ≤ |A|
-- Mathlib/CategoryTheory/Sites/Preserves.lean
C : Type u
inst✝³ : Category.{v, u} C
I : C
F : Cᵒᵖ ⥤ Type w
hF : IsSheafFor F (ofArrows Empty.elim fun a => instIsEmptyEmpty.elim a)
hI : IsInitial I
α : Type
X : α → C
c : Cofan X
hc : IsColimit c
inst✝² : (ofArrows X c.inj).hasPullbacks
inst✝¹ : HasInitial C
inst✝ : ∀ (i : α), Mono (c.inj i)
hd : Pairwise fun i j => IsPullback (initial.to (X i)) (initial.to (X j)) (c.inj i) (c.inj j)
a : Equalizer.Presieve.Arrows.FirstObj F X
i j : α

⊢ Pi.π (fun ij => F.obj (op (pullback (c.inj ij.1) (c.inj ij.2)))) (i, j) (Equalizer.Presieve.Arrows.firstMap F X c.inj a) = Pi.π (fun ij => F.obj (op (pullback (c.inj ij.1) (c.inj ij.2)))) (i, j) (Equalizer.Presieve.Arrows.secondMap F X c.inj a)
-- Mathlib/CategoryTheory/Sites/Preserves.lean
C : Type u
inst✝³ : Category.{v, u} C
I : C
F : Cᵒᵖ ⥤ Type w
hF : IsSheafFor F (ofArrows Empty.elim fun a => instIsEmptyEmpty.elim a)
hI : IsInitial I
α : Type
X : α → C
c : Cofan X
hc : IsColimit c
inst✝² : (ofArrows X c.inj).hasPullbacks
inst✝¹ : HasInitial C
inst✝ : ∀ (i : α), Mono (c.inj i)
hd : Pairwise fun i j => IsPullback (initial.to (X i)) (initial.to (X j)) (c.inj i) (c.inj j)
a : Equalizer.Presieve.Arrows.FirstObj F X
i j : α

⊢ F.map (pullback.fst (c.inj i) (c.inj j)).op (Pi.π (fun i => F.obj (op (X i))) i a) = F.map (pullback.snd (c.inj i) (c.inj j)).op (Pi.π (fun i => F.obj (op (X i))) j a)
-- Mathlib/SetTheory/Game/Nim.lean
o : Ordinal.{u_1}

⊢ (nim o).LeftMoves = (Quotient.out o).α
-- Mathlib/SetTheory/Game/Nim.lean
o : Ordinal.{u_1}

⊢ (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)).LeftMoves = (Quotient.out o).α
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
n p : ℕ

⊢ (↑p + 1) * ∑ k ∈ range n, ↑k ^ p = eval (↑n) (bernoulli p.succ) - _root_.bernoulli p.succ
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
n p : ℕ

⊢ ∑ i ∈ range (p + 1), _root_.bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) = ∑ i ∈ range (p.succ + 1), eval (↑n) ((monomial i) (_root_.bernoulli (p.succ - i) * ↑(p.succ.choose i))) - _root_.bernoulli p.succ
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
n p : ℕ

⊢ ↑p + 1 ≠ 0
-- 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

⊢ P.toPoly.degree ≤ 1
-- Mathlib/SetTheory/Cardinal/ENat.lean
x : Cardinal.{v}
m : ℕ∞

⊢ ↑m < lift.{u, v} x ↔ ↑m < x
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : LocallyRingedSpace
r : ↑(Γ.obj (op X))

⊢ IsUnit ((X.toToΓSpecMapBasicOpen r) r)
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : LocallyRingedSpace
r : ↑(Γ.obj (op X))

⊢ (X.toToΓSpecMapBasicOpen r) r = (X.presheaf.map (eqToHom ⋯).op) ((X.toRingedSpace.presheaf.map (homOfLE ⋯).op) r)
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : LocallyRingedSpace
r : ↑(Γ.obj (op X))

⊢ (X.toToΓSpecMapBasicOpen r) r = (X.toRingedSpace.presheaf.map ((homOfLE ⋯).op ≫ (eqToHom ⋯).op)) r
-- Mathlib/Analysis/MellinTransform.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : ℝ → E
s : ℂ
hf : MellinConvergent f s
hg : MellinConvergent g s

⊢ MellinConvergent (fun t => f t + g t) s
-- Mathlib/Analysis/MellinTransform.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : ℝ → E
s : ℂ
hf : MellinConvergent f s
hg : MellinConvergent g s

⊢ mellin (fun t => f t + g t) s = mellin f s + mellin g s
-- Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean
α : Type u_1
β : Type u_2
n d k N : ℕ
x✝ x : Fin n → ℕ
hx : x ∈ ↑(sphere n d k)

⊢ x ∈ (fun x => (WithLp.equiv 2 (Fin n → ℝ)).symm (Nat.cast ∘ x)) ⁻¹' Metric.sphere 0 √↑k
-- Mathlib/Dynamics/PeriodicPts.lean
α : Type u_1
β : Type u_2
f fa : α → α
fb : β → β
x y : α
m✝ n : ℕ
hf : IsPeriodicPt f n x
m : ℕ

⊢ IsPeriodicPt f^[m] n x
-- Mathlib/Dynamics/PeriodicPts.lean
α : Type u_1
β : Type u_2
f fa : α → α
fb : β → β
x y : α
m✝ n : ℕ
hf : IsPeriodicPt f n x
m : ℕ

⊢ IsFixedPt f^[n]^[m] x
-- Mathlib/GroupTheory/Coxeter/Inversion.lean
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
t : W
ht : t ∈ cs.leftInvSeq ω

⊢ cs.IsReflection t
-- Mathlib/GroupTheory/Coxeter/Inversion.lean
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
t : W
ht : t ∈ cs.rightInvSeq ω.reverse

⊢ cs.IsReflection t
-- Mathlib/AlgebraicTopology/CechNerve.lean
C : Type u
inst✝¹ : Category.{v, u} C
f : Arrow C
inst✝ : ∀ (n : ℕ), HasWidePushout f.left (fun x => f.right) fun x => f.hom
x y : SimplexCategory
g : x ⟶ y
j : Fin (x.len + 1)

⊢ f.hom ≫ (fun i => WidePushout.ι (fun x => f.hom) ((SimplexCategory.Hom.toOrderHom g) i)) j = WidePushout.head fun x => f.hom
-- Mathlib/CategoryTheory/ClosedUnderIsomorphisms.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
P Q : C → Prop
inst✝ : ClosedUnderIsomorphisms Q
h : P ≤ Q

⊢ isoClosure Q ≤ Q
-- Mathlib/Data/Finsupp/Defs.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝ : AddGroup G
f : α →₀ G
a a' : α

⊢ (erase a f) a' = (f - single a (f a)) a'
-- Mathlib/Data/Finsupp/Defs.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝ : AddGroup G
f : α →₀ G
a : α

⊢ (erase a f) a = (f - single a (f a)) a
-- Mathlib/Data/Finsupp/Defs.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝ : AddGroup G
f : α →₀ G
a a' : α
h : a ≠ a'

⊢ (erase a f) a' = (f - single a (f a)) a'
-- Mathlib/FieldTheory/PurelyInseparable.lean
F : Type u
E : Type v
inst✝⁵ : Field F
inst✝⁴ : Field E
inst✝³ : Algebra F E
K : Type w
inst✝² : Field K
inst✝¹ : Algebra F K
q : ℕ
inst✝ : ExpChar F q
x : E

⊢ x ∈ perfectClosure F E ↔ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
-- Mathlib/CategoryTheory/Monoidal/Category.lean
C✝ : Type u
𝒞 : Category.{v, u} C✝
inst✝² : MonoidalCategory C✝
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : MonoidalCategory C
U V W X✝ Y Z X X' : C
f : X ⟶ X'

⊢ f = (λ_ X).inv ≫ 𝟙_ C ◁ f ≫ (λ_ X').hom
-- Mathlib/Probability/Martingale/Basic.lean
Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁵ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
f g : ι → Ω → E
ℱ : Filtration ι m0
inst✝¹ : Preorder E
inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
hf : Submartingale f ℱ μ
hg : Supermartingale g ℱ μ

⊢ Submartingale (f - g) ℱ μ
-- Mathlib/Probability/Martingale/Basic.lean
Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁵ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
f g : ι → Ω → E
ℱ : Filtration ι m0
inst✝¹ : Preorder E
inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
hf : Submartingale f ℱ μ
hg : Supermartingale g ℱ μ

⊢ Submartingale (f + -g) ℱ μ
-- Mathlib/Topology/UniformSpace/Basic.lean
α✝ : Type ua
β : Type ub
γ✝ : Type uc
δ : Type ud
ι✝ : Sort u_1
ι : Sort u_2
α : Type u_3
γ : Type u_4
u : ι → UniformSpace γ
f : α → γ

⊢ 𝓤 α = 𝓤 α
-- Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{u_4, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ ψ : W.LeftFraction₂ X Y

⊢ φ.fst.map L ⋯ = ψ.fst.map L ⋯ ∧ φ.snd.map L ⋯ = ψ.snd.map L ⋯ ↔ LeftFraction₂Rel φ ψ
-- Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{u_4, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ ψ : W.LeftFraction₂ X Y

⊢ LeftFractionRel φ.fst ψ.fst ∧ LeftFractionRel φ.snd ψ.snd ↔ LeftFraction₂Rel φ ψ
-- Mathlib/Analysis/Distribution/SchwartzSpace.lean
𝕜 : Type u_1
𝕜' : Type u_2
D : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
V : Type u_7
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedAddCommGroup D
inst✝⁴ : NormedSpace ℝ D
inst✝³ : MeasurableSpace D
inst✝² : BorelSpace D
inst✝¹ : SecondCountableTopology D
inst✝ : FiniteDimensional ℝ D
C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂

⊢ x ^ k * f ≤ 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l)
-- Mathlib/Analysis/Distribution/SchwartzSpace.lean
𝕜 : Type u_1
𝕜' : Type u_2
D : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
V : Type u_7
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedAddCommGroup D
inst✝⁴ : NormedSpace ℝ D
inst✝³ : MeasurableSpace D
inst✝² : BorelSpace D
inst✝¹ : SecondCountableTopology D
inst✝ : FiniteDimensional ℝ D
C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this : 0 ≤ C₂

⊢ x ^ k * f ≤ 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l)
-- Mathlib/Analysis/Distribution/SchwartzSpace.lean
𝕜 : Type u_1
𝕜' : Type u_2
D : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
V : Type u_7
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedAddCommGroup D
inst✝⁴ : NormedSpace ℝ D
inst✝³ : MeasurableSpace D
inst✝² : BorelSpace D
inst✝¹ : SecondCountableTopology D
inst✝ : FiniteDimensional ℝ D
C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)

⊢ x ^ k * f ≤ 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l)
-- Mathlib/Analysis/Distribution/SchwartzSpace.lean
𝕜 : Type u_1
𝕜' : Type u_2
D : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
V : Type u_7
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedAddCommGroup D
inst✝⁴ : NormedSpace ℝ D
inst✝³ : MeasurableSpace D
inst✝² : BorelSpace D
inst✝¹ : SecondCountableTopology D
inst✝ : FiniteDimensional ℝ D
C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)

⊢ x ^ k * f ≤ ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
-- Mathlib/Analysis/Distribution/SchwartzSpace.lean
𝕜 : Type u_1
𝕜' : Type u_2
D : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
V : Type u_7
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedAddCommGroup D
inst✝⁴ : NormedSpace ℝ D
inst✝³ : MeasurableSpace D
inst✝² : BorelSpace D
inst✝¹ : SecondCountableTopology D
inst✝ : FiniteDimensional ℝ D
C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
h'x : x ≤ 1

⊢ x ^ k * f ≤ ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
-- Mathlib/Analysis/Distribution/SchwartzSpace.lean
𝕜 : Type u_1
𝕜' : Type u_2
D : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
V : Type u_7
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedAddCommGroup D
inst✝⁴ : NormedSpace ℝ D
inst✝³ : MeasurableSpace D
inst✝² : BorelSpace D
inst✝¹ : SecondCountableTopology D
inst✝ : FiniteDimensional ℝ D
C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
h'x : 1 ≤ x

⊢ x ^ k * f ≤ ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
-- Mathlib/Data/List/Basic.lean
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ l : List α
a : α
h : a ∈ l

⊢ ⟨a, h⟩ ∈ l.attach
-- Mathlib/Data/List/Basic.lean
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ l : List α
a : α
h : a ∈ l
this : ∃ a_1, a_1 ∈ l.attach ∧ ↑a_1 = a

⊢ ⟨a, h⟩ ∈ l.attach
-- Mathlib/Probability/Distributions/Geometric.lean
p : ℝ
n : ℕ
hp_pos : 0 < p
hp_lt_one : p < 1

⊢ 0 < (1 - p) ^ n * p
-- Mathlib/Probability/Distributions/Geometric.lean
p : ℝ
n : ℕ
hp_pos : 0 < p
hp_lt_one : p < 1
this : 0 < 1 - p

⊢ 0 < (1 - p) ^ n * p
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
S : Set ↑↑(D.V (i, j))

⊢ ⇑(pullback.snd (D.f i j) (D.f i k)).base '' (⇑(pullback.fst (D.f i j) (D.f i k)).base ⁻¹' S) = ⇑(D.f i k).base ⁻¹' (⇑(D.f i j).base '' S)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
S : Set ↑↑(D.V (i, j))
eq₁ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.fst ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.fst (D.f i j) (D.f i k)).base

⊢ ⇑(pullback.snd (D.f i j) (D.f i k)).base '' (⇑(pullback.fst (D.f i j) (D.f i k)).base ⁻¹' S) = ⇑(D.f i k).base ⁻¹' (⇑(D.f i j).base '' S)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
S : Set ↑↑(D.V (i, j))
eq₁ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.fst ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.fst (D.f i j) (D.f i k)).base
eq₂ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.snd ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.snd (D.f i j) (D.f i k)).base

⊢ ⇑(pullback.snd (D.f i j) (D.f i k)).base '' (⇑(pullback.fst (D.f i j) (D.f i k)).base ⁻¹' S) = ⇑(D.f i k).base ⁻¹' (⇑(D.f i j).base '' S)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
S : Set ↑↑(D.V (i, j))
eq₁ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.fst ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.fst (D.f i j) (D.f i k)).base
eq₂ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.snd ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.snd (D.f i j) (D.f i k)).base

⊢ ⇑((PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.snd ((forget C).map (D.f i j)) ((forget C).map (D.f i k))) '' (⇑((PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.fst ((forget C).map (D.f i j)) ((forget C).map (D.f i k))) ⁻¹' S) = ⇑(D.f i k).base ⁻¹' (⇑(D.f i j).base '' S)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
S : Set ↑↑(D.V (i, j))
eq₁ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.fst ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.fst (D.f i j) (D.f i k)).base
eq₂ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.snd ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.snd (D.f i j) (D.f i k)).base

⊢ ⇑(pullback.snd ((forget C).map (D.f i j)) ((forget C).map (D.f i k))) ∘ ⇑(PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom '' (⇑((PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.fst ((forget C).map (D.f i j)) ((forget C).map (D.f i k))) ⁻¹' S) = ⇑(D.f i k).base ⁻¹' (⇑(D.f i j).base '' S)