ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Topology/LocallyConstant/Basic.lean
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝¹ : TopologicalSpace X
C₀ C₁ C₂ : Set X
h₀ : C₀ ⊆ C₁ ∪ C₂
h₁ : IsClosed C₁
h₂ : IsClosed C₂
f₁ : LocallyConstant (↑C₁) Z
f₂ : LocallyConstant (↑C₂) Z
inst✝ : DecidablePred fun x => x ∈ C₁
hf : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f₁ ⟨x, ⋯⟩ = f₂ ⟨x, ⋯⟩
x : ↑C₀
hx : ↑x ∈ C₂
this : (j : ↑C₀) → Decidable (j ∈ Subtype.val ⁻¹' C₁) := fun j => decidable_of_iff (↑j ∈ C₁) ⋯
\u22a2 (piecewise' h₀ h₁ h₂ f₁ f₂ hf) x = f₂ ⟨↑x, hx⟩
-- Mathlib/Topology/LocallyConstant/Basic.lean
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝¹ : TopologicalSpace X
C₀ C₁ C₂ : Set X
h₀ : C₀ ⊆ C₁ ∪ C₂
h₁ : IsClosed C₁
h₂ : IsClosed C₂
f₁ : LocallyConstant (↑C₁) Z
f₂ : LocallyConstant (↑C₂) Z
inst✝ : DecidablePred fun x => x ∈ C₁
hf : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f₁ ⟨x, ⋯⟩ = f₂ ⟨x, ⋯⟩
x : ↑C₀
hx : ↑x ∈ C₂
this : (j : ↑C₀) → Decidable (j ∈ Subtype.val ⁻¹' C₁) := fun j => decidable_of_iff (↑j ∈ C₁) ⋯
\u22a2 (comap { toFun := C₂.restrictPreimage Subtype.val, continuous_toFun := ⋯ } f₂) ⟨x, hx⟩ = f₂ ⟨↑x, hx⟩
-- Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
c : R
\u22a2 (c • A).det = ((diagonal fun x => c) * A).det
-- Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix n n R
c : R
\u22a2 (diagonal fun x => c).det * A.det = c ^ Fintype.card n * A.det
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : SeminormedAddCommGroup α
r C : ℝ
f : ℕ → α
hr : r < 1
hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n
a : α
ha : HasSum f a
n : ℕ
\u22a2 dist (∑ x ∈ Finset.range n, f x) a ≤ C * r ^ n / (1 - r)
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : SeminormedAddCommGroup α
r C : ℝ
f : ℕ → α
hr : r < 1
hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n
a : α
ha : HasSum f a
n : ℕ
\u22a2 Tendsto (fun n => ∑ i ∈ Finset.range n, f i) atTop (𝓝 a)
-- Mathlib/Topology/Algebra/InfiniteSum/Ring.lean
ι : Type u_1
κ : Type u_2
R : Type u_3
α : Type u_4
inst✝² : NonUnitalNonAssocSemiring α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSemiring α
f g : ι → α
a a₁ a₂✝ a₂ : α
h : HasSum f a₁
\u22a2 HasSum (fun i => a₂ * f i) (a₂ * a₁)
-- Mathlib/Topology/AlexandrovDiscrete.lean
ι : Sort u_1
κ : ι → Sort u_2
α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : AlexandrovDiscrete α
S : Set (Set α)
f : ι → Set α
hS : ∀ s ∈ S, IsClosed s
\u22a2 IsClosed (⋃₀ S)
-- Mathlib/Topology/AlexandrovDiscrete.lean
ι : Sort u_1
κ : ι → Sort u_2
α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : AlexandrovDiscrete α
S : Set (Set α)
f : ι → Set α
hS : ∀ s ∈ S, IsOpen sᶜ
\u22a2 IsOpen (⋂₀ (compl '' S))
-- Mathlib/Data/Complex/ExponentialBounds.lean
\u22a2 rexp (-1) < 0.3678794412
-- Mathlib/Data/Complex/ExponentialBounds.lean
\u22a2 0.3678794412⁻¹ < rexp 1
-- Mathlib/Data/Complex/ExponentialBounds.lean
\u22a2 0 < 0.3678794412
-- Mathlib/Analysis/InnerProductSpace/Projection.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : _root_.RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
n : ℕ
_i : Fact (finrank 𝕜 E = n + 1)
v : E
hv : v ≠ 0
\u22a2 finrank 𝕜 ↥(Submodule.span 𝕜 {v})ᗮ = n
-- Mathlib/Analysis/InnerProductSpace/Projection.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : _root_.RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
n : ℕ
_i : Fact (finrank 𝕜 E = n + 1)
v : E
hv : v ≠ 0
this : FiniteDimensional 𝕜 E
\u22a2 finrank 𝕜 ↥(Submodule.span 𝕜 {v})ᗮ = n
-- Mathlib/Algebra/Category/AlgebraCat/Basic.lean
R : Type u
inst✝ : CommRing R
M N U : ModuleCat R
X : Type u
\u22a2 { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map (𝟙 X) = 𝟙 ({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.obj X)
-- Mathlib/Algebra/Category/AlgebraCat/Basic.lean
R : Type u
inst✝ : CommRing R
M N U : ModuleCat R
X : Type u
\u22a2 ⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map (𝟙 X)) ∘ FreeAlgebra.ι R = ⇑(𝟙 ({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.obj X)) ∘ FreeAlgebra.ι R
-- Mathlib/Algebra/Category/AlgebraCat/Basic.lean
R : Type u
inst✝ : CommRing R
M N U : ModuleCat R
X : Type u
\u22a2 FreeAlgebra.ι R ∘ 𝟙 X = ⇑(𝟙 (mk (FreeAlgebra R X))) ∘ FreeAlgebra.ι R
-- Mathlib/Algebra/Category/AlgebraCat/Basic.lean
R : Type u
inst✝ : CommRing R
M N U : ModuleCat R
X✝ Y✝ Z✝ : Type u
f✝ : X✝ ⟶ Y✝
g✝ : Y✝ ⟶ Z✝
\u22a2 { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map (f✝ ≫ g✝) = { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map f✝ ≫ { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map g✝
-- Mathlib/Algebra/Category/AlgebraCat/Basic.lean
R : Type u
inst✝ : CommRing R
M N U : ModuleCat R
X✝ Y✝ Z✝ : Type u
f✝ : X✝ ⟶ Y✝
g✝ : Y✝ ⟶ Z✝
\u22a2 ⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map (f✝ ≫ g✝)) ∘ FreeAlgebra.ι R = ⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map f✝ ≫ { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map g✝) ∘ FreeAlgebra.ι R
-- Mathlib/Algebra/Category/AlgebraCat/Basic.lean
R : Type u
inst✝ : CommRing R
M N U : ModuleCat R
X✝ Y✝ Z✝ : Type u
f✝ : X✝ ⟶ Y✝
g✝ : Y✝ ⟶ Z✝
\u22a2 FreeAlgebra.ι R ∘ (f✝ ≫ g✝) = ⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝) ≫ (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ FreeAlgebra.ι R
-- Mathlib/Algebra/Category/AlgebraCat/Basic.lean
R : Type u
inst✝ : CommRing R
M N U : ModuleCat R
X✝ Y✝ Z✝ : Type u
f✝ : X✝ ⟶ Y✝
g✝ : Y✝ ⟶ Z✝
x✝ : X✝
\u22a2 (FreeAlgebra.ι R ∘ (f✝ ≫ g✝)) x✝ = (⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝) ≫ (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ FreeAlgebra.ι R) x✝
-- Mathlib/Algebra/Category/AlgebraCat/Basic.lean
R : Type u
inst✝ : CommRing R
M N U : ModuleCat R
X✝ Y✝ Z✝ : Type u
f✝ : X✝ ⟶ Y✝
g✝ : Y✝ ⟶ Z✝
x✝ : X✝
\u22a2 (FreeAlgebra.ι R ∘ (f✝ ≫ g✝)) x✝ = ((⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ ⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝))) ∘ FreeAlgebra.ι R) x✝
-- Mathlib/Algebra/Category/AlgebraCat/Basic.lean
R : Type u
inst✝ : CommRing R
M N U : ModuleCat R
X✝ Y✝ Z✝ : Type u
f✝ : X✝ ⟶ Y✝
g✝ : Y✝ ⟶ Z✝
x✝ : X✝
\u22a2 FreeAlgebra.ι R (g✝ (f✝ x✝)) = ((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) (((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝)) (FreeAlgebra.ι R x✝))
-- Mathlib/Algebra/Category/AlgebraCat/Basic.lean
R : Type u
inst✝ : CommRing R
M N U : ModuleCat R
X✝ Y✝ Z✝ : Type u
f✝ : X✝ ⟶ Y✝
g✝ : Y✝ ⟶ Z✝
x✝ : X✝
\u22a2 FreeAlgebra.ι R (g✝ (f✝ x✝)) = (FreeAlgebra.ι R ∘ g✝) (f✝ x✝)
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Restrict.lean
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : TopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : TopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : NonUnitalRing A
inst✝¹² : StarRing A
inst✝¹¹ : Module S A
inst✝¹⁰ : IsScalarTower S A A
inst✝⁹ : SMulCommClass S A A
inst✝⁸ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁷ : Algebra R S
inst✝⁶ : Module R A
inst✝⁵ : IsScalarTower R S A
inst✝⁴ : StarModule R S
inst✝³ : ContinuousSMul R S
inst✝² : CompleteSpace R
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(σₙ S a)
a : A
ha : p a
\u22a2 ∃ φ, ClosedEmbedding ⇑φ ∧ φ { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ } = a ∧ (∀ (f : C(↑(σₙ R a), R)₀), σₙ R (φ f) = range ⇑f) ∧ ∀ (f : C(↑(σₙ R a), R)₀), p (φ f)
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Restrict.lean
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : TopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : TopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : NonUnitalRing A
inst✝¹² : StarRing A
inst✝¹¹ : Module S A
inst✝¹⁰ : IsScalarTower S A A
inst✝⁹ : SMulCommClass S A A
inst✝⁸ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁷ : Algebra R S
inst✝⁶ : Module R A
inst✝⁵ : IsScalarTower R S A
inst✝⁴ : StarModule R S
inst✝³ : ContinuousSMul R S
inst✝² : CompleteSpace R
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(σₙ S a)
a : A
ha : p a
\u22a2 ClosedEmbedding ⇑(nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯)
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Restrict.lean
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : TopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : TopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : NonUnitalRing A
inst✝¹² : StarRing A
inst✝¹¹ : Module S A
inst✝¹⁰ : IsScalarTower S A A
inst✝⁹ : SMulCommClass S A A
inst✝⁸ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁷ : Algebra R S
inst✝⁶ : Module R A
inst✝⁵ : IsScalarTower R S A
inst✝⁴ : StarModule R S
inst✝³ : ContinuousSMul R S
inst✝² : CompleteSpace R
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(σₙ S a)
a : A
ha : p a
\u22a2 (nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯) { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ } = a
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Restrict.lean
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : TopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : TopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : NonUnitalRing A
inst✝¹² : StarRing A
inst✝¹¹ : Module S A
inst✝¹⁰ : IsScalarTower S A A
inst✝⁹ : SMulCommClass S A A
inst✝⁸ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁷ : Algebra R S
inst✝⁶ : Module R A
inst✝⁵ : IsScalarTower R S A
inst✝⁴ : StarModule R S
inst✝³ : ContinuousSMul R S
inst✝² : CompleteSpace R
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(σₙ S a)
a : A
ha : p a
\u22a2 ∀ (f_1 : C(↑(σₙ R a), R)₀), σₙ R ((nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯) f_1) = range ⇑f_1
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Restrict.lean
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : TopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : TopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : NonUnitalRing A
inst✝¹² : StarRing A
inst✝¹¹ : Module S A
inst✝¹⁰ : IsScalarTower S A A
inst✝⁹ : SMulCommClass S A A
inst✝⁸ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁷ : Algebra R S
inst✝⁶ : Module R A
inst✝⁵ : IsScalarTower R S A
inst✝⁴ : StarModule R S
inst✝³ : ContinuousSMul R S
inst✝² : CompleteSpace R
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(σₙ S a)
a : A
ha : p a
\u22a2 ∀ (f_1 : C(↑(σₙ R a), R)₀), p ((nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯) f_1)
-- Mathlib/Topology/Basic.lean
X : Type u
Y : Type v
ι : Sort w
α : Type u_1
β : Type u_2
x : X
s s₁ s₂ t : Set X
p p₁ p₂ : X → Prop
inst✝ t₁ t₂ : TopologicalSpace X
\u22a2 t₁ = t₂ ↔ ∀ (s : Set X), IsClosed s ↔ IsClosed s
-- Mathlib/Topology/Basic.lean
X : Type u
Y : Type v
ι : Sort w
α : Type u_1
β : Type u_2
x : X
s s₁ s₂ t : Set X
p p₁ p₂ : X → Prop
inst✝ t₁ t₂ : TopologicalSpace X
\u22a2 (∀ (x : Set X), IsOpen xᶜ ↔ IsOpen xᶜ) ↔ ∀ (s : Set X), IsClosed s ↔ IsClosed s
-- Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
h : ∀ d ∈ φ.support, (weight w) d ≠ n
\u22a2 (weightedHomogeneousComponent w n) φ = 0
-- Mathlib/CategoryTheory/Monoidal/Tor.lean
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : MonoidalCategory C
inst✝³ : Abelian C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasProjectiveResolutions C
X Y : C
inst✝ : Projective Y
n : ℕ
\u22a2 IsZero (((Tor C (n + 1)).obj X).obj Y)
-- Mathlib/Topology/UniformSpace/Basic.lean
α : Type ua
β : Type ub
γ : Type uc
δ : Type ud
ι : Sort u_1
V : Set (α × α)
\u22a2 SymmetricRel (symmetrizeRel V)
-- Mathlib/RingTheory/ChainOfDivisors.lean
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
\u22a2 m.card ≤ n + 1
-- Mathlib/LinearAlgebra/Matrix/Charpoly/FiniteField.lean
n : Type u_1
inst✝³ : DecidableEq n
inst✝² : Fintype n
K : Type u_2
inst✝¹ : Field K
inst✝ : Fintype K
M : Matrix n n K
\u22a2 (M ^ Fintype.card K).charpoly = M.charpoly
-- Mathlib/LinearAlgebra/Matrix/Charpoly/FiniteField.lean
n : Type u_1
inst✝³ : DecidableEq n
inst✝² : Fintype n
K : Type u_2
inst✝¹ : Field K
inst✝ : Fintype K
M : Matrix n n K
h✝ : Nonempty n
\u22a2 (M ^ Fintype.card K).charpoly = M.charpoly
-- Mathlib/LinearAlgebra/Matrix/Charpoly/FiniteField.lean
n : Type u_1
inst✝³ : DecidableEq n
inst✝² : Fintype n
K : Type u_2
inst✝¹ : Field K
inst✝ : Fintype K
M : Matrix n n K
h✝ : IsEmpty n
\u22a2 (M ^ Fintype.card K).charpoly = M.charpoly
-- Mathlib/LinearAlgebra/CliffordAlgebra/Fold.lean
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
Q : QuadraticForm R M
f : M →ₗ[R] N →ₗ[R] N
hf : ∀ (m : M) (x : N), (f m) ((f m) x) = Q m • x
n : N
\u22a2 ((foldl Q f hf) n) 1 = n
-- Mathlib/Algebra/Field/Subfield.lean
K : Type u
L : Type v
M : Type w
inst✝² : DivisionRing K
inst✝¹ : DivisionRing L
inst✝ : DivisionRing M
s : Set (Subfield K)
x : K
\u22a2 x ∈ (sInf s).toSubring ↔ x ∈ ⨅ t ∈ s, t.toSubring
-- Mathlib/Algebra/Field/Subfield.lean
K : Type u
L : Type v
M : Type w
inst✝² : DivisionRing K
inst✝¹ : DivisionRing L
inst✝ : DivisionRing M
s : Set (Subfield K)
x : K
\u22a2 (∀ p ∈ s, x ∈ p) ↔ x ∈ ⨅ t ∈ s, t.toSubring
-- Mathlib/Algebra/Field/Subfield.lean
K : Type u
L : Type v
M : Type w
inst✝² : DivisionRing K
inst✝¹ : DivisionRing L
inst✝ : DivisionRing M
s : Set (Subfield K)
x : K
\u22a2 (∀ p ∈ s, x ∈ p) ↔ ∀ p ∈ Set.range fun t => ⨅ (_ : t ∈ s), t.toSubring, x ∈ p
-- Mathlib/Algebra/MonoidAlgebra/Basic.lean
k✝ : Type u₁
G : Type u₂
H : Type u_1
R : Type u_2
k : Type u_3
A : Type u_4
G₁ : Type u_5
G₂ : Type u_6
G₃ : Type u_7
inst✝⁵ : CommSemiring k
inst✝⁴ : Semiring A
inst✝³ : Algebra k A
inst✝² : AddMonoid G₁
inst✝¹ : AddMonoid G₂
inst✝ : AddMonoid G₃
f : G₁ →+ G₂
g : G₂ →+ G₃
x✝ : A[G₁]
\u22a2 (mapDomainAlgHom k A (g.comp f)) x✝ = ((mapDomainAlgHom k A g).comp (mapDomainAlgHom k A f)) x✝
-- Mathlib/Analysis/NormedSpace/Pointwise.lean
𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : 0 ≤ r
x : E
\u22a2 x +ᵥ r • closedBall 0 1 = closedBall x r
-- Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.lean
C : Type u_1
D : Type u_2
H : Type u_3
inst✝² : Category.{?u.51683, u_1} C
inst✝¹ : Category.{?u.51687, u_2} D
inst✝ : Category.{?u.51691, u_3} H
L : C ⥤ D
F : C ⥤ H
E✝ E'✝ : L.RightExtension F
Y : D
E E' : L.RightExtension F
φ : E ⟶ E'
G : StructuredArrow Y L
\u22a2 φ.left.app Y ≫ E'.left.map G.hom ≫ E'.hom.app G.right = E.left.map G.hom ≫ E.hom.app G.right
-- Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.lean
C : Type u_1
D : Type u_2
H : Type u_3
inst✝² : Category.{?u.51683, u_1} C
inst✝¹ : Category.{?u.51687, u_2} D
inst✝ : Category.{?u.51691, u_3} H
L : C ⥤ D
F : C ⥤ H
E✝ E'✝ : L.RightExtension F
Y : D
E E' : L.RightExtension F
φ : E ⟶ E'
G : StructuredArrow Y L
\u22a2 φ.left.app Y ≫ E'.left.map G.hom ≫ E'.hom.app G.right = E.left.map G.hom ≫ (((whiskeringLeft C D H).obj L).map φ.left ≫ E'.hom).app G.right
-- Mathlib/Topology/Perfect.lean
X : Type u_1
inst✝ : TopologicalSpace X
\u22a2 PerfectSpace X ↔ ∀ (x : X), (𝓝[≠] x).NeBot
-- Mathlib/Order/CompleteLattice.lean
α : Type u_1
β : Type u_2
β₂ : Type u_3
γ : Type u_4
ι : Sort u_5
ι' : Sort u_6
κ : ι → Sort u_7
κ' : ι' → Sort u_8
inst✝¹ : CompleteLattice α
f✝ g s✝ t : ι → α
a b : α
inst✝ : CompleteLattice β
f : α ≃o β
s : Set α
\u22a2 f (sSup s) = sSup (⇑f.symm ⁻¹' s)
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
s : β → Set α
f : α → ℝ≥0∞
\u22a2 ∫⁻ (a : α) in ⋃ i, s i, f a ∂μ ≤ ∑' (i : β), ∫⁻ (a : α) in s i, f a ∂μ
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
s : β → Set α
f : α → ℝ≥0∞
\u22a2 ∫⁻ (a : α) in ⋃ i, s i, f a ∂μ ≤ ∫⁻ (a : α), f a ∂sum fun i => μ.restrict (s i)
-- Mathlib/RingTheory/RootsOfUnity/Basic.lean
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst✝⁵ : CommMonoid M
inst✝⁴ : CommMonoid N
inst✝³ : DivisionCommMonoid G
k✝ l : ℕ+
inst✝² : CommRing R
inst✝¹ : IsReduced R
p k : ℕ
m : ℕ+
inst✝ : ExpChar R p
ζ : Rˣ
\u22a2 ζ ∈ rootsOfUnity (⟨p, ⋯⟩ ^ k * m) R ↔ ζ ∈ rootsOfUnity m R
-- Mathlib/Analysis/Analytic/CPolynomial.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → F
p✝¹ pf pg : FormalMultilinearSeries 𝕜 E F
x✝ : E
r✝ r' : ℝ≥0∞
n✝ m : ℕ
p✝ : FormalMultilinearSeries 𝕜 E F
x y : E
r R : ℝ≥0
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
k l : ℕ
s : Finset (Fin (k + l))
hs : s.card = l
hkl : n ≤ k + l
\u22a2 p.changeOriginSeriesTerm k l s hs = 0
-- Mathlib/Topology/Bases.lean
α : Type u
β : Type u_1
t✝ : TopologicalSpace α
B : Set (Set α)
s t : Set α
hB : IsTopologicalBasis B
hs : IsOpen s
h : ∀ U ∈ B, U ⊆ s → U ⊆ t
\u22a2 s ⊆ t
-- Mathlib/Topology/Bases.lean
α : Type u
β : Type u_1
t✝ : TopologicalSpace α
B : Set (Set α)
s t : Set α
hB : IsTopologicalBasis B
hs : IsOpen s
h : ∀ U ∈ B, U ⊆ s → U ⊆ t
\u22a2 ⋃₀ {s_1 | s_1 ∈ B ∧ s_1 ⊆ s} ⊆ t
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
x : E
hx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}
\u22a2 x ∈ D f K
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
x : E
hx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}
\u22a2 ∀ (i : ℕ), x ∈ ⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ i)
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
x : E
hx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}
e : ℕ
this : 0 < (1 / 2) ^ e
\u22a2 x ∈ ⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
x : E
hx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)
\u22a2 x ∈ ⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
x : E
hx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
\u22a2 x ∈ ⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
x : E
hx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
\u22a2 ∃ i, ∀ i_1 ≥ i, ∀ i_3 ≥ i, ∃ i, ∃ (_ : i ∈ K), x ∈ A f i ((1 / 2) ^ i_1) ((1 / 2) ^ e) ∧ x ∈ A f i ((1 / 2) ^ i_3) ((1 / 2) ^ e)
-- Mathlib/MeasureTheory/Measure/VectorMeasure.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
j : Set α
a✝ : MeasurableSet j
\u22a2 ↑(v.restrict ∅) j ≤ ↑(w.restrict ∅) j
-- Mathlib/Data/Real/Sqrt.lean
x y : ℝ
h : 0 ≤ x
\u22a2 √(x ^ 2) = x
-- Mathlib/Data/Finset/Sigma.lean
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f g : ⦃i : ι⦄ → α i → β i → Finset (γ i)
a✝ : (i : ι) × α i
b✝ : (i : ι) × β i
h : ∀ ⦃i : ι⦄ ⦃a : α i⦄ ⦃b : β i⦄, f a b ⊆ g a b
a : (i : ι) × α i
b : (i : ι) × β i
x : Sigma γ
hx : x ∈ sigmaLift f a b
\u22a2 x ∈ sigmaLift g a b
-- Mathlib/Data/Finset/Sigma.lean
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f g : ⦃i : ι⦄ → α i → β i → Finset (γ i)
a✝ : (i : ι) × α i
b✝ : (i : ι) × β i
h : ∀ ⦃i : ι⦄ ⦃a : α i⦄ ⦃b : β i⦄, f a b ⊆ g a b
a : (i : ι) × α i
b : (i : ι) × β i
x : Sigma γ
hx : ∃ (ha : a.fst = x.fst) (hb : b.fst = x.fst), x.snd ∈ f (ha ▸ a.snd) (hb ▸ b.snd)
\u22a2 ∃ (ha : a.fst = x.fst) (hb : b.fst = x.fst), x.snd ∈ g (ha ▸ a.snd) (hb ▸ b.snd)
-- Mathlib/GroupTheory/Coset.lean
α : Type u_1
inst✝ : Group α
s : Subgroup α
x y : α
\u22a2 (∃ a, y * ↑a = x) ↔ ∃ a, x⁻¹ * y = ↑a⁻¹
-- Mathlib/GroupTheory/Coset.lean
α : Type u_1
inst✝ : Group α
s : Subgroup α
x y : α
\u22a2 (∃ a, x⁻¹ * y = ↑a⁻¹) ↔ x⁻¹ * y ∈ s