ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Order/Interval/Finset/Basic.lean
ι : Type u_1
α : Type u_2
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrder α
a a₁ a₂ b b₁ b₂ c x : α
\u22a2 a ∈ Icc a b ↔ a ≤ b
-- Mathlib/Data/Nat/Factorization/Defs.lean
a b m n✝ p✝ n p : ℕ
pp : Prime p
hn : n ≠ 0
\u22a2 multiplicity p n = ↑(n.factorization p)
-- Mathlib/Order/Filter/SmallSets.lean
α : Type u_1
β : Type u_2
ι : Sort u_3
l l' la : Filter α
lb : Filter β
\u22a2 ⊥.smallSets = pure ∅
-- Mathlib/Order/Filter/SmallSets.lean
α : Type u_1
β : Type u_2
ι : Sort u_3
l l' la : Filter α
lb : Filter β
\u22a2 Monotone powerset
-- Mathlib/Algebra/Polynomial/Smeval.lean
R : Type u_1
inst✝³ : Semiring R
r : R
p : R[X]
S : Type u_2
inst✝² : AddCommMonoid S
inst✝¹ : Pow S ℕ
inst✝ : MulActionWithZero R S
x : S
\u22a2 (C r).smeval x = r • x ^ 0
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹³ : TopologicalSpace α✝
inst✝¹² : MeasurableSpace α✝
inst✝¹¹ : BorelSpace α✝
inst✝¹⁰ : TopologicalSpace β✝
inst✝⁹ : MeasurableSpace β✝
inst✝⁸ : BorelSpace β✝
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
α : Type u_6
β : Type u_7
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace β
inst✝ : TopologicalSpace β
hβ : BorelSpace β
e : α → β
h'e : MeasurableEmbedding e
h''e : Inducing e
\u22a2 inst✝³ = borel α
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹³ : TopologicalSpace α✝
inst✝¹² : MeasurableSpace α✝
inst✝¹¹ : BorelSpace α✝
inst✝¹⁰ : TopologicalSpace β✝
inst✝⁹ : MeasurableSpace β✝
inst✝⁸ : BorelSpace β✝
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
α : Type u_6
β : Type u_7
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace β
inst✝ : TopologicalSpace β
hβ : BorelSpace β
e : α → β
h'e : MeasurableEmbedding e
h''e : Inducing e
this : MeasurableSpace.comap e (borel β) = inst✝³
\u22a2 inst✝³ = borel α
-- Mathlib/SetTheory/Cardinal/Finite.lean
α : Type u_1
β : Type u_2
inst✝¹ : Finite α
inst✝ : Finite β
this✝ : Fintype α
this : Fintype β
\u22a2 Nat.card (α ⊕ β) = Nat.card α + Nat.card β
-- Mathlib/Combinatorics/Enumerative/Composition.lean
n : ℕ
c : CompositionAsSet n
\u22a2 c.length < c.boundaries.card
-- Mathlib/Combinatorics/Enumerative/Composition.lean
n : ℕ
c : CompositionAsSet n
\u22a2 c.length < c.length + 1
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
x : ↑(PrimeSpectrum.Top R)
f : R
s : ↥x.asIdeal.primeCompl
\u22a2 (toStalk R x) f = (toStalk R x) ↑s * ((structureSheaf R).presheaf.germ ⟨x, ⋯⟩) (const R f (↑s) (PrimeSpectrum.basicOpen ↑s) ⋯)
-- Mathlib/GroupTheory/Abelianization.lean
G : Type u
inst✝ : Group G
\u22a2 commutator G = Subgroup.normalClosure (commutatorSet G)
-- Mathlib/Algebra/Ring/Rat.lean
q : ℚ
\u22a2 q * ↑q.den = ↑q.num
-- Mathlib/Algebra/Ring/Rat.lean
q : ℚ
\u22a2 q.num /. ↑q.den * (↑q.den /. 1) = q.num /. 1
-- Mathlib/Algebra/Ring/Rat.lean
q : ℚ
this : ↑q.den ≠ 0
\u22a2 q.num /. ↑q.den * (↑q.den /. 1) = q.num /. 1
-- Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
F : Type u_1
R : Type u
A : Type v
B : Type w
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : IsScalarTower R A A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : NonUnitalNonAssocSemiring B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : FunLike F A B
inst✝ : NonUnitalAlgHomClass F R A B
s : Set A
\u22a2 (adjoin R s).toSubmodule = span R ↑(Subsemigroup.closure s)
-- Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
F : Type u_1
R : Type u
A : Type v
B : Type w
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : IsScalarTower R A A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : NonUnitalNonAssocSemiring B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : FunLike F A B
inst✝ : NonUnitalAlgHomClass F R A B
s : Set A
\u22a2 (adjoin R s).toSubmodule ≤ span R ↑(Subsemigroup.closure s)
-- Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
F : Type u_1
R : Type u
A : Type v
B : Type w
inst✝¹⁰ : CommSemiring R
inst✝⁹ : NonUnitalNonAssocSemiring A
inst✝⁸ : Module R A
inst✝⁷ : IsScalarTower R A A
inst✝⁶ : SMulCommClass R A A
inst✝⁵ : NonUnitalNonAssocSemiring B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : FunLike F A B
inst✝ : NonUnitalAlgHomClass F R A B
s : Set A
\u22a2 span R ↑(Subsemigroup.closure s) ≤ (adjoin R s).toSubmodule
-- Mathlib/Combinatorics/SimpleGraph/Finite.lean
V : Type u_1
G : SimpleGraph V
e : Sym2 V
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
v : V
\u22a2 G.minDegree ≤ G.degree v
-- Mathlib/Combinatorics/SimpleGraph/Finite.lean
V : Type u_1
G : SimpleGraph V
e : Sym2 V
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
v : V
t : ℕ
ht : (image (fun v => G.degree v) univ).min = ↑t
\u22a2 G.minDegree ≤ G.degree v
-- Mathlib/Combinatorics/SimpleGraph/Finite.lean
V : Type u_1
G : SimpleGraph V
e : Sym2 V
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
v : V
t : ℕ
ht : (image (fun v => G.degree v) univ).min = ↑t
this : t ≤ G.degree v
\u22a2 G.minDegree ≤ G.degree v
-- Mathlib/Data/Nat/Factorization/Basic.lean
a✝ b✝ m n p✝ a b p : ℕ
ha : a ≠ 0
hb : b ≠ 0
\u22a2 p ^ (a * b).factorization p = p ^ a.factorization p * p ^ b.factorization p
-- Mathlib/MeasureTheory/Measure/VectorMeasure.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
s : SignedMeasure α
i✝ j : Set α
hs : VectorMeasure.restrict 0 univ ≤ VectorMeasure.restrict s univ
i : Set α
hi : MeasurableSet i
\u22a2 ↑(s.toMeasureOfZeroLE univ ⋯ hs).toSignedMeasure i = ↑s i
-- Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean
R : Type u
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
\u22a2 S.Exact ↔ Function.Exact ⇑S.f ⇑S.g
-- Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean
R : Type u
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
\u22a2 LinearMap.range S.f = LinearMap.ker S.g ↔ LinearMap.ker S.g = LinearMap.range S.f
-- Mathlib/AlgebraicTopology/DoldKan/Projections.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q : ℕ
\u22a2 Q (q + 1) = Q q - P q ≫ Hσ q
-- Mathlib/AlgebraicTopology/DoldKan/Projections.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q : ℕ
\u22a2 𝟙 K[X] - (P q + P q ≫ Hσ q) = 𝟙 K[X] - P q - P q ≫ Hσ q
-- Mathlib/Analysis/MellinTransform.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → ℂ
s a : ℂ
\u22a2 mellin (fun t => f t / a) s = mellin f s / a
-- Mathlib/Data/Ordmap/Ordset.lean
α : Type u_1
l : Ordnode α
x : α
r : Ordnode α
\u22a2 (l.balanceR x r).dual = r.dual.balanceL x l.dual
-- Mathlib/Data/Seq/WSeq.lean
α : Type u
β : Type v
γ : Type w
f : α → β
s : WSeq α
\u22a2 (map (ret ∘ f) s).join ~ʷ map f s
-- Mathlib/Data/Seq/WSeq.lean
α : Type u
β : Type v
γ : Type w
f : α → β
s : WSeq α
\u22a2 (map ret (map f s)).join ~ʷ map f s
-- Mathlib/Topology/LocalAtTarget.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s✝ : Set β
ι : Type u_3
U : ι → Opens β
hU : iSup U = ⊤
H : IsClosedMap f
s : Set β
t : Set ↑(f ⁻¹' s)
\u22a2 IsClosed t → IsClosed (s.restrictPreimage f '' t)
-- Mathlib/Topology/LocalAtTarget.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s✝ : Set β
ι : Type u_3
U : ι → Opens β
hU : iSup U = ⊤
H : IsClosedMap f
s : Set β
t : Set ↑(f ⁻¹' s)
\u22a2 ∀ (u : Set α), IsClosed u → Subtype.val ⁻¹' u = t → ∃ v, IsClosed v ∧ Subtype.val ⁻¹' v = s.restrictPreimage f '' t
-- 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✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : 𝓢(E, F)
k n : ℕ
\u22a2 ∃ C, 0 < C ∧ ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ C
-- 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✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : 𝓢(E, F)
k n : ℕ
C : ℝ
hC : ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f.toFun x‖ ≤ C
\u22a2 ∃ C, 0 < C ∧ ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ C
-- Mathlib/Analysis/NormedSpace/Pointwise.lean
𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
x y z : E
δ ε : ℝ
hε : 0 ≤ ε
hδ : 0 < δ
a b : E
\u22a2 Metric.closedBall a ε - Metric.ball b δ = Metric.ball (a - b) (ε + δ)
-- Mathlib/Analysis/Normed/Group/Basic.lean
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝² : SeminormedGroup E
inst✝¹ : SeminormedGroup F
inst✝ : SeminormedGroup G
s : Set E
a✝ a₁ a₂ b b₁ b₂ : E
r r₁ r₂ : ℝ
a : E
\u22a2 dist 1 a = ‖a‖
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
\u22a2 ∃ j V, IsClopen V ∧ U = ⇑(C.π.app j) ⁻¹' V
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
hB : TopologicalSpace.IsTopologicalBasis {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}
\u22a2 ∃ j V, IsClopen V ∧ U = ⇑(C.π.app j) ⁻¹' V
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
\u22a2 ∀ (j : J), TopologicalSpace.IsTopologicalBasis ((fun j => {W | IsClopen W}) j)
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
\u22a2 ∀ (i j : J) (f : i ⟶ j), ∀ V ∈ (fun j => {W | IsClopen W}) j, ⇑((F ⋙ toTopCat).map f) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
hB : TopologicalSpace.IsTopologicalBasis {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS : S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}
h : U = ⋃₀ S
\u22a2 ∃ j V, IsClopen V ∧ U = ⇑(C.π.app j) ⁻¹' V
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS : S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}
h : U = ⋃₀ S
\u22a2 ∃ j V, IsClopen V ∧ U = ⇑(C.π.app j) ⁻¹' V
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS : S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}
h : U = ⋃₀ S
j : ↑S → J := fun s => Exists.choose ⋯
\u22a2 ∃ j V, IsClopen V ∧ U = ⇑(C.π.app j) ⁻¹' V
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS : S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}
h : U = ⋃₀ S
j : ↑S → J := fun s => Exists.choose ⋯
V : (s : ↑S) → Set ↑(F.obj (j s)).toTop := fun s => ⋯.choose
\u22a2 ∃ j V, IsClopen V ∧ U = ⇑(C.π.app j) ⁻¹' V
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS : S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}
h : U = ⋃₀ S
j : ↑S → J := fun s => Exists.choose ⋯
V : (s : ↑S) → Set ↑(F.obj (j s)).toTop := fun s => ⋯.choose
hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ⇑(C.π.app (j s)) ⁻¹' V s
\u22a2 ∃ j V, IsClopen V ∧ U = ⇑(C.π.app j) ⁻¹' V
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS : S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}
h : U = ⋃₀ S
j : ↑S → J := fun s => Exists.choose ⋯
V : (s : ↑S) → Set ↑(F.obj (j s)).toTop := fun s => ⋯.choose
hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ⇑(C.π.app (j s)) ⁻¹' V s
hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)
\u22a2 ∃ j V, IsClopen V ∧ U = ⇑(C.π.app j) ⁻¹' V
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS : S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}
h : U = ⋃₀ S
j : ↑S → J := fun s => Exists.choose ⋯
V : (s : ↑S) → Set ↑(F.obj (j s)).toTop := fun s => ⋯.choose
hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ⇑(C.π.app (j s)) ⁻¹' V s
hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)
hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i
\u22a2 ∃ j V, IsClopen V ∧ U = ⇑(C.π.app j) ⁻¹' V
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS : S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}
h : U = ⋃₀ S
j : ↑S → J := fun s => Exists.choose ⋯
V : (s : ↑S) → Set ↑(F.obj (j s)).toTop := fun s => ⋯.choose
hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ⇑(C.π.app (j s)) ⁻¹' V s
hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)
hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i
this : ∃ t, U ⊆ ⋃ i ∈ t, (fun s => ⇑(C.π.app (j s)) ⁻¹' V s) i
\u22a2 ∃ j V, IsClopen V ∧ U = ⇑(C.π.app j) ⁻¹' V
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS : S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}
h : U = ⋃₀ S
j : ↑S → J := fun s => Exists.choose ⋯
V : (s : ↑S) → Set ↑(F.obj (j s)).toTop := fun s => ⋯.choose
hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ⇑(C.π.app (j s)) ⁻¹' V s
hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)
hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i
G : Finset ↑S
hG : U ⊆ ⋃ i ∈ G, (fun s => ⇑(C.π.app (j s)) ⁻¹' V s) i
\u22a2 ∃ j V, IsClopen V ∧ U = ⇑(C.π.app j) ⁻¹' V
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS : S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}
h : U = ⋃₀ S
j : ↑S → J := fun s => Exists.choose ⋯
V : (s : ↑S) → Set ↑(F.obj (j s)).toTop := fun s => ⋯.choose
hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ⇑(C.π.app (j s)) ⁻¹' V s
hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)
hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i
G : Finset ↑S
hG : U ⊆ ⋃ i ∈ G, (fun s => ⇑(C.π.app (j s)) ⁻¹' V s) i
j0 : J
hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)
\u22a2 ∃ j V, IsClopen V ∧ U = ⇑(C.π.app j) ⁻¹' V
-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS : S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}
h : U = ⋃₀ S
j : ↑S → J := fun s => Exists.choose ⋯
V : (s : ↑S) → Set ↑(F.obj (j s)).toTop := fun s => ⋯.choose
hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ⇑(C.π.app (j s)) ⁻¹' V s
hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)
hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i
G : Finset ↑S
hG : U ⊆ ⋃ i ∈ G, (fun s => ⇑(C.π.app (j s)) ⁻¹' V s) i
j0 : J
hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)
f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => ⋯.some
\u22a2 ∃ j V, IsClopen V ∧ U = ⇑(C.π.app j) ⁻¹' V
-- Mathlib/Algebra/Polynomial/Mirror.lean
R : Type u_1
inst✝ : Semiring R
p q : R[X]
hp : ¬p = 0
a✝ : Nontrivial R
\u22a2 p.mirror.natDegree = p.natDegree
-- Mathlib/Algebra/Polynomial/Mirror.lean
R : Type u_1
inst✝ : Semiring R
p q : R[X]
hp : ¬p = 0
a✝ : Nontrivial R
\u22a2 p.reverse.leadingCoeff * (X ^ p.natTrailingDegree).leadingCoeff ≠ 0
-- Mathlib/Data/QPF/Univariate/Basic.lean
F : Type u → Type u
q : QPF F
α : Type u
p : α → Prop
x : F α
a : (P F).A
f : (P F).B a → α
h₀ : x = abs ⟨a, f⟩
h₁ : ∀ (i : (P F).B a), p (f i)
\u22a2 Liftp p x
-- Mathlib/Data/QPF/Univariate/Basic.lean
F : Type u → Type u
q : QPF F
α : Type u
p : α → Prop
x : F α
a : (P F).A
f : (P F).B a → α
h₀ : x = abs ⟨a, f⟩
h₁ : ∀ (i : (P F).B a), p (f i)
\u22a2 Subtype.val <$> abs ⟨a, fun i => ⟨f i, ⋯⟩⟩ = x
-- Mathlib/Data/QPF/Univariate/Basic.lean
F : Type u → Type u
q : QPF F
α : Type u
p : α → Prop
x : F α
a : (P F).A
f : (P F).B a → α
h₀ : x = abs ⟨a, f⟩
h₁ : ∀ (i : (P F).B a), p (f i)
\u22a2 abs ((P F).map Subtype.val ⟨a, fun i => ⟨f i, ⋯⟩⟩) = abs ⟨a, f⟩
-- Mathlib/Algebra/BigOperators/Finprod.lean
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f g : α → M
a b : α
s t : Set α
hst : s ⊆ t
ht : (t ∩ mulSupport f).Finite
\u22a2 (∏ᶠ (i : α) (_ : i ∈ s), f i) * ∏ᶠ (i : α) (_ : i ∈ t \ s), f i = ∏ᶠ (i : α) (_ : i ∈ t), f i
-- Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
S T : ℝ
hT : 0 < T
k : ℕ
\u22a2 Summable fun n => ↑|n| ^ k * rexp (-π * (T * ↑n ^ 2 - 2 * S * ↑|n|))
-- Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
S T : ℝ
hT : 0 < T
k : ℕ
\u22a2 Summable fun n => ↑n ^ k * rexp (-π * (T * ↑n ^ 2 - 2 * S * ↑n))
-- Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
S T : ℝ
hT : 0 < T
k : ℕ
\u22a2 (fun n => ↑n ^ k * rexp (-π * (T * ↑n ^ 2 - 2 * S * ↑n))) =O[atTop] fun n => ↑n ^ k * rexp (-1 * ↑n)
-- Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
S T : ℝ
hT : 0 < T
k : ℕ
\u22a2 (fun x => rexp (-π * (T * ↑x ^ 2 - 2 * S * ↑x))) =O[atTop] fun x => rexp (-1 * ↑x)
-- Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
S T : ℝ
hT : 0 < T
k : ℕ
\u22a2 Tendsto ((fun x => -π * (T * ↑x ^ 2 - 2 * S * ↑x)) - fun x => -1 * ↑x) atTop atBot
-- Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
S T : ℝ
hT : 0 < T
k : ℕ
\u22a2 Tendsto (fun x => -(-π * (T * ↑x ^ 2 - 2 * S * ↑x) - -1 * ↑x)) atTop atTop
-- Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
S T : ℝ
hT : 0 < T
k : ℕ
\u22a2 Tendsto (fun n => ↑n * (π * T * ↑n - (2 * π * S + 1))) atTop atTop
-- Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
S T : ℝ
hT : 0 < T
k : ℕ
\u22a2 Tendsto (fun n => π * T * ↑n) atTop atTop
-- Mathlib/Data/Nat/MaxPowDiv.lean
p n pow : ℕ
hp : 1 < p
hn : 0 < n
h : p ^ pow ∣ n
c : ℕ
hc : n = p ^ pow * c
\u22a2 pow ≤ p.maxPowDiv n
-- Mathlib/Data/Nat/MaxPowDiv.lean
p n pow : ℕ
hp : 1 < p
hn : 0 < n
h : p ^ pow ∣ n
c : ℕ
hc : n = p ^ pow * c
this : 0 < c
\u22a2 pow ≤ p.maxPowDiv n
-- Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
inst✝¹ : S₁.HasRightHomology
inst✝ : S₂.HasRightHomology
\u22a2 rightHomologyMap φ ≫ h₂.rightHomologyIso.hom = h₁.rightHomologyIso.hom ≫ γ.φH
-- Mathlib/Topology/Constructions.lean
X : Type u
Y : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : TopologicalSpace Z
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
x : X × Y
\u22a2 map Prod.snd (𝓝[Prod.fst ⁻¹' {x.1}] x) = 𝓝 x.2
-- Mathlib/Topology/Constructions.lean
X : Type u
Y : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : TopologicalSpace Z
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
x : X × Y
s : Set Y
hs : s ∈ map Prod.snd (𝓝[Prod.fst ⁻¹' {x.1}] x)
\u22a2 s ∈ 𝓝 x.2
-- Mathlib/Topology/Constructions.lean
X : Type u
Y : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : TopologicalSpace Z
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
s : Set Y
x : X
y : Y
hs : s ∈ map Prod.snd (𝓝[Prod.fst ⁻¹' {(x, y).1}] (x, y))
\u22a2 s ∈ 𝓝 (x, y).2
-- Mathlib/Topology/Constructions.lean
X : Type u
Y : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : TopologicalSpace Z
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
s : Set Y
x : X
y : Y
hs : ∃ u ∈ 𝓝 x, ∃ v ∈ 𝓝 y, u ×ˢ v ⊆ {x_1 | x_1 ∈ Prod.fst ⁻¹' {(x, y).1} → x_1 ∈ Prod.snd ⁻¹' s}
\u22a2 s ∈ 𝓝 (x, y).2
-- Mathlib/Topology/Constructions.lean
X : Type u
Y : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : TopologicalSpace Z
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
s : Set Y
x : X
y : Y
u : Set X
hu : u ∈ 𝓝 x
v : Set Y
hv : v ∈ 𝓝 y
H : u ×ˢ v ⊆ {x_1 | x_1 ∈ Prod.fst ⁻¹' {(x, y).1} → x_1 ∈ Prod.snd ⁻¹' s}
\u22a2 s ∈ 𝓝 (x, y).2
-- Mathlib/Topology/Constructions.lean
X : Type u
Y : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : TopologicalSpace Z
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
s : Set Y
x : X
y : Y
u : Set X
hu : u ∈ 𝓝 x
v : Set Y
hv : v ∈ 𝓝 y
H : ∀ x_1 ∈ u, ∀ y ∈ v, x_1 = x → y ∈ s
\u22a2 s ∈ 𝓝 (x, y).2
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
\u22a2 (range cos).Infinite
-- Mathlib/Data/Nat/Count.lean
p : ℕ → Prop
inst✝ : DecidablePred p
n : ℕ
\u22a2 count p n ≤ count p (n + 1)
-- Mathlib/Data/ZMod/Basic.lean
m n✝ n : ℕ
x : ZMod (n + 1)
\u22a2 x.valMinAbs = 0 ↔ x = 0
-- Mathlib/Data/ZMod/Basic.lean
m n✝ n : ℕ
x : ZMod (n + 1)
\u22a2 x.valMinAbs = valMinAbs 0 ↔ x = 0
-- Mathlib/NumberTheory/SumTwoSquares.lean
n : ℕ
\u22a2 (∃ x y, n = x ^ 2 + y ^ 2) ↔ ∀ {q : ℕ}, Prime q → q % 4 = 3 → Even (padicValNat q n)
-- Mathlib/NumberTheory/SumTwoSquares.lean
\u22a2 (∃ x y, 0 = x ^ 2 + y ^ 2) ↔ ∀ {q : ℕ}, Prime q → q % 4 = 3 → Even (padicValNat q 0)
-- Mathlib/NumberTheory/SumTwoSquares.lean
n : ℕ
hn₀ : n > 0
\u22a2 (∃ x y, n = x ^ 2 + y ^ 2) ↔ ∀ {q : ℕ}, Prime q → q % 4 = 3 → Even (padicValNat q n)
-- Mathlib/NumberTheory/SumTwoSquares.lean
n : ℕ
hn₀ : n > 0
\u22a2 (∃ a b, n = a ^ 2 * b ∧ IsSquare (-1)) ↔ ∀ {q : ℕ}, Prime q → q % 4 = 3 → Even (padicValNat q n)