ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a : ℤ
b : ℕ
inst✝ : NeZero b
\u22a2 (∃ a_1, ∃ (h : a_1 ∈ b.primeFactorsList), legendreSym a_1 a = 0) ↔ ∃ p, Nat.Prime p ∧ p ∣ a.natAbs ∧ p ∣ (↑b).natAbs
-- Mathlib/Topology/Sheaves/Stalks.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
U✝ : Opens ↑X
hxU✝ : (𝟙 X) x ∈ U✝
\u22a2 ((pushforward C (𝟙 X)).obj ℱ).germ ⟨(𝟙 X) x, hxU✝⟩ ≫ stalkPushforward C (𝟙 X) ℱ x = ((pushforward C (𝟙 X)).obj ℱ).germ ⟨(𝟙 X) x, hxU✝⟩ ≫ (stalkFunctor C x).map (Pushforward.id ℱ).hom
-- Mathlib/Topology/Sheaves/Stalks.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
U✝ : Opens ↑X
hxU✝ : (𝟙 X) x ∈ U✝
\u22a2 ℱ.map ((NatTrans.op (OpenNhds.inclusionMapIso (𝟙 X) x).inv).app (op { obj := U✝, property := ⋯ })) ≫ colimit.ι ((OpenNhds.map (𝟙 X) x).op ⋙ ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj ℱ) (op { obj := U✝, property := ⋯ }) ≫ colimit.pre (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj ℱ) (OpenNhds.map (𝟙 X) x).op = colimit.ι ((OpenNhds.inclusion ((𝟙 X) x)).op ⋙ (pushforward C (𝟙 X)).obj ℱ) (op { obj := U✝, property := ⋯ }) ≫ (stalkFunctor C x).map (Pushforward.id ℱ).hom
-- Mathlib/Topology/Sheaves/Stalks.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
U✝ : Opens ↑X
hxU✝ : (𝟙 X) x ∈ U✝
\u22a2 𝟙 (ℱ.obj ((OpenNhds.inclusion ((𝟙 X) x) ⋙ Opens.map (𝟙 X)).op.obj (op { obj := U✝, property := ⋯ }))) ≫ colimit.ι ((OpenNhds.map (𝟙 X) x).op ⋙ ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj ℱ) (op { obj := U✝, property := ⋯ }) ≫ colimit.pre (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj ℱ) (OpenNhds.map (𝟙 X) x).op = colimit.ι ((OpenNhds.inclusion ((𝟙 X) x)).op ⋙ (pushforward C (𝟙 X)).obj ℱ) (op { obj := U✝, property := ⋯ }) ≫ (stalkFunctor C x).map (Pushforward.id ℱ).hom
-- Mathlib/Algebra/Ring/Prod.lean
α : Type u_1
β : Type u_2
R✝ : Type u_3
R' : Type u_4
S✝ : Type u_5
S' : Type u_6
T : Type u_7
T' : Type u_8
R : Type u_9
S : Type u_10
inst✝⁴ : Ring R
inst✝³ : Ring S
inst✝² : IsDomain (R × S)
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
\u22a2 False
-- Mathlib/Algebra/Ring/Prod.lean
α : Type u_1
β : Type u_2
R✝ : Type u_3
R' : Type u_4
S✝ : Type u_5
S' : Type u_6
T : Type u_7
T' : Type u_8
R : Type u_9
S : Type u_10
inst✝⁴ : Ring R
inst✝³ : Ring S
inst✝² : IsDomain (R × S)
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
this : (0, 1) = 0 ∨ (1, 0) = 0
\u22a2 False
-- Mathlib/Algebra/Ring/Prod.lean
α : Type u_1
β : Type u_2
R✝ : Type u_3
R' : Type u_4
S✝ : Type u_5
S' : Type u_6
T : Type u_7
T' : Type u_8
R : Type u_9
S : Type u_10
inst✝⁴ : Ring R
inst✝³ : Ring S
inst✝² : IsDomain (R × S)
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
this : 0 = 0 ∧ 1 = 0 ∨ 1 = 0 ∧ 0 = 0
\u22a2 False
-- Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ := { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }
this✝¹ : Epi α.τ₁
this✝ : IsIso α.τ₂
this : Mono α.τ₃
\u22a2 (ShortComplex.mk (d f) f ⋯).Exact
-- Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ := { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }
this✝¹ : Epi α.τ₁
this✝ : IsIso α.τ₂
this : Mono α.τ₃
\u22a2 (ShortComplex.mk (kernel.ι f) f ⋯).Exact
-- Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ := { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }
this✝¹ : Epi α.τ₁
this✝ : IsIso α.τ₂
this : Mono α.τ₃
\u22a2 IsLimit (KernelFork.ofι (ShortComplex.mk (kernel.ι f) f ⋯).f ⋯)
-- 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 Ioo a b ⊆ Ico a b
-- 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 Set.Ioo a b ⊆ Set.Ico a b
-- Mathlib/Algebra/Group/Subgroup/Basic.lean
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁴ : Group G
inst✝³ : Group G'
inst✝² : Group G''
A : Type u_4
inst✝¹ : AddGroup A
N : Type u_5
inst✝ : Group N
f✝ f : G →* N
H : Subgroup G
h : f.ker ≤ H
\u22a2 comap f (map f H) = H
-- Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Module.Oriented ℝ V (Fin 2)
p₁ p₂ p₃ : P
h : ∡ p₁ p₂ p₃ = ↑(π / 2)
\u22a2 ∡ p₃ p₁ p₂ = ↑(Real.arccos (dist p₁ p₂ / dist p₁ p₃))
-- Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Module.Oriented ℝ V (Fin 2)
p₁ p₂ p₃ : P
h : ∡ p₁ p₂ p₃ = ↑(π / 2)
hs : (∡ p₃ p₁ p₂).sign = 1
\u22a2 ∡ p₃ p₁ p₂ = ↑(Real.arccos (dist p₁ p₂ / dist p₁ p₃))
-- Mathlib/Algebra/MonoidAlgebra/Basic.lean
k : Type u₁
G : Type u₂
H : Type u_1
R : Type u_2
inst✝¹ : Semiring k
inst✝ : Group G
f g : MonoidAlgebra k G
x : G
\u22a2 (f * g) x = sum f fun a b => (single a b * g) x
-- Mathlib/Algebra/MonoidAlgebra/Basic.lean
k : Type u₁
G : Type u₂
H : Type u_1
R : Type u_2
inst✝¹ : Semiring k
inst✝ : Group G
f g : MonoidAlgebra k G
x : G
\u22a2 (sum f fun a b => (single a b * g) x) = sum f fun a b => b * g (a⁻¹ * x)
-- Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : DecidableEq ι
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
inst✝ : GradedAlgebra 𝒜
x : Submonoid A
f : HomogeneousLocalization 𝒜 x
\u22a2 f.den • f.val = (algebraMap A (at x)) f.num
-- Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : DecidableEq ι
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ι → Submodule R A
inst✝ : GradedAlgebra 𝒜
x : Submonoid A
f : HomogeneousLocalization 𝒜 x
\u22a2 IsLocalization.mk' (at x) (f.den * f.num) ⟨f.den, ⋯⟩ = (algebraMap A (at x)) f.num
-- Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean
b : ℂ
hb : 0 < b.re
f : ℂ → ℝ → ℂ := fun c x => cexp (-c * ↑x ^ 2)
\u22a2 ContinuousAt (fun c => ∫ (x : ℝ), cexp (-c * ↑x ^ 2)) b
-- Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean
b : ℂ
hb : 0 < b.re
f : ℂ → ℝ → ℂ := fun c x => cexp (-c * ↑x ^ 2)
d : ℝ
hd : 0 < d
hd' : d < b.re
\u22a2 ContinuousAt (fun c => ∫ (x : ℝ), cexp (-c * ↑x ^ 2)) b
-- Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean
b : ℂ
hb : 0 < b.re
f : ℂ → ℝ → ℂ := fun c x => cexp (-c * ↑x ^ 2)
d : ℝ
hd : 0 < d
hd' : d < b.re
f_meas : ∀ (c : ℂ), AEStronglyMeasurable (f c) volume
\u22a2 ContinuousAt (fun c => ∫ (x : ℝ), cexp (-c * ↑x ^ 2)) b
-- Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean
b : ℂ
hb : 0 < b.re
f : ℂ → ℝ → ℂ := fun c x => cexp (-c * ↑x ^ 2)
d : ℝ
hd : 0 < d
hd' : d < b.re
f_meas : ∀ (c : ℂ), AEStronglyMeasurable (f c) volume
f_cts : ∀ (x : ℝ), ContinuousAt (fun c => f c x) b
\u22a2 ContinuousAt (fun c => ∫ (x : ℝ), cexp (-c * ↑x ^ 2)) b
-- Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean
b : ℂ
hb : 0 < b.re
f : ℂ → ℝ → ℂ := fun c x => cexp (-c * ↑x ^ 2)
d : ℝ
hd : 0 < d
hd' : d < b.re
f_meas : ∀ (c : ℂ), AEStronglyMeasurable (f c) volume
f_cts : ∀ (x : ℝ), ContinuousAt (fun c => f c x) b
f_le_bd : ∀ᶠ (c : ℂ) in 𝓝 b, ∀ᵐ (x : ℝ), ‖f c x‖ ≤ rexp (-d * x ^ 2)
\u22a2 ContinuousAt (fun c => ∫ (x : ℝ), cexp (-c * ↑x ^ 2)) b
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → μ G ≤ liminf (fun i => (μs i) G) atTop
\u22a2 ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (x : Ω), f x ∂μs i) atTop
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → μ G ≤ liminf (fun i => (μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
\u22a2 ∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (x : Ω), f x ∂μs i) atTop
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → μ G ≤ liminf (fun i => (μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
\u22a2 (∫⁻ (a : Ω), ENNReal.ofReal (f a) ∂μ).toReal ≤ liminf (fun i => ∫ (x : Ω), f x ∂μs i) atTop
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → μ G ≤ liminf (fun i => (μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
\u22a2 liminf (fun i => ∫ (x : Ω), f x ∂μs i) atTop = (liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop).toReal
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → μ G ≤ liminf (fun i => (μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
\u22a2 ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≠ ⊤
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → μ G ≤ liminf (fun i => (μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
\u22a2 liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop ≠ ⊤
-- Mathlib/GroupTheory/Finiteness.lean
M : Type u_1
N : Type u_2
inst✝⁴ : Monoid M
inst✝³ : AddMonoid N
G : Type u_3
H : Type u_4
inst✝² : Group G
inst✝¹ : AddGroup H
s : Set G
inst✝ : Finite ↑s
this : Fintype ↑s
\u22a2 Group.rank ↥(closure s) ≤ Nat.card ↑s
-- Mathlib/GroupTheory/Finiteness.lean
M : Type u_1
N : Type u_2
inst✝⁴ : Monoid M
inst✝³ : AddMonoid N
G : Type u_3
H : Type u_4
inst✝² : Group G
inst✝¹ : AddGroup H
s : Set G
inst✝ : Finite ↑s
this : Fintype ↑s
\u22a2 Group.rank ↥(closure ↑s.toFinset) ≤ s.toFinset.card
-- Mathlib/MeasureTheory/Group/Arithmetic.lean
α : Type u_1
G : Type u_2
inst✝⁴ : MeasurableSpace G
inst✝³ : MeasurableSpace α
inst✝² : CommGroup G
inst✝¹ : MeasurableMul₂ G
inst✝ : MeasurableInv G
f g : α → G
hf : Measurable f
h : Measurable (f * g)
\u22a2 g = f * g * f⁻¹
-- Mathlib/Data/Sign.lean
α : Type u_1
inst✝ : LinearOrderedRing α
a b x : α
\u22a2 |x| * ↑(sign x) = x
-- Mathlib/SetTheory/Surreal/Basic.lean
x y : PGame
ox : x.Numeric
oy : y.Numeric
\u22a2 x < y ↔ (∃ i, x ≤ y.moveLeft i) ∨ ∃ j, x.moveRight j ≤ y
-- Mathlib/Analysis/Normed/Lp/PiLp.lean
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → PseudoEMetricSpace (β i)
x y : WithLp ⊤ ((i : ι) → β i)
\u22a2 edist ((WithLp.equiv ⊤ ((i : ι) → β i)) x) ((WithLp.equiv ⊤ ((i : ι) → β i)) y) ≤ edist x y
-- Mathlib/Analysis/Normed/Lp/PiLp.lean
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → PseudoEMetricSpace (β i)
x y : WithLp ⊤ ((i : ι) → β i)
\u22a2 edist x y ≤ edist ((WithLp.equiv ⊤ ((i : ι) → β i)) x) ((WithLp.equiv ⊤ ((i : ι) → β i)) y)
-- Mathlib/MeasureTheory/Function/SimpleFunc.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m✝ : MeasurableSpace α
μ✝ ν : Measure α
m : MeasurableSpace α
f : α →ₛ ℝ≥0∞
s : Set α
μ : Measure α
\u22a2 f.lintegral (μ.restrict s) = ∑ y ∈ f.range, y * μ (↑f ⁻¹' {y} ∩ s)
-- Mathlib/GroupTheory/Finiteness.lean
M : Type u_1
N : Type u_2
inst✝¹ : Monoid M
inst✝ : AddMonoid N
P : Submonoid M
h : P.FG
S : Set M
hS : closure S = P
hf : S.Finite
\u22a2 AddSubmonoid.closure (⇑Additive.toMul ⁻¹' S) = toAddSubmonoid P
-- Mathlib/GroupTheory/Finiteness.lean
M : Type u_1
N : Type u_2
inst✝¹ : Monoid M
inst✝ : AddMonoid N
P : Submonoid M
h : (toAddSubmonoid P).FG
T : Set (Additive M)
hT : AddSubmonoid.closure T = toAddSubmonoid P
hf : T.Finite
\u22a2 closure (⇑Multiplicative.ofAdd ⁻¹' T) = P
-- Mathlib/NumberTheory/NumberField/Units/Regulator.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
w₁ w₂ : InfinitePlace K
e₁ : { w // w ≠ w₁ } ≃ Fin (rank K)
e₂ : { w // w ≠ w₂ } ≃ Fin (rank K)
f : Fin (rank K + 1) ≃ InfinitePlace K := (finSuccEquiv (rank K)).trans ↑((Equiv.optionSubtype w₁).symm e₁.symm)
g : { w // w ≠ w₂ } ≃ Fin (rank K) := (f.symm.subtypeEquiv ⋯).trans (finSuccAboveEquiv (f.symm w₂)).symm
\u22a2 |(Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₁ i))))).det| = |(Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).det|
-- Mathlib/NumberTheory/NumberField/Units/Regulator.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
w₁ w₂ : InfinitePlace K
e₁ : { w // w ≠ w₁ } ≃ Fin (rank K)
e₂ : { w // w ≠ w₂ } ≃ Fin (rank K)
f : Fin (rank K + 1) ≃ InfinitePlace K := (finSuccEquiv (rank K)).trans ↑((Equiv.optionSubtype w₁).symm e₁.symm)
g : { w // w ≠ w₂ } ≃ Fin (rank K) := (f.symm.subtypeEquiv ⋯).trans (finSuccAboveEquiv (f.symm w₂)).symm
h_col : |((Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).submatrix (⇑(g.trans e₂.symm)) id).det| = |↑↑(Equiv.Perm.sign (g.trans e₂.symm)) * (Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).det|
\u22a2 |(Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₁ i))))).det| = |(Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).det|
-- Mathlib/NumberTheory/NumberField/Units/Regulator.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
w₁ w₂ : InfinitePlace K
e₁ : { w // w ≠ w₁ } ≃ Fin (rank K)
e₂ : { w // w ≠ w₂ } ≃ Fin (rank K)
f : Fin (rank K + 1) ≃ InfinitePlace K := (finSuccEquiv (rank K)).trans ↑((Equiv.optionSubtype w₁).symm e₁.symm)
g : { w // w ≠ w₂ } ≃ Fin (rank K) := (f.symm.subtypeEquiv ⋯).trans (finSuccAboveEquiv (f.symm w₂)).symm
h_col : |((Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).submatrix (⇑(g.trans e₂.symm)) id).det| = |(Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).det|
\u22a2 |(Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₁ i))))).det| = |(Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).det|
-- Mathlib/NumberTheory/NumberField/Units/Regulator.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
w₁ w₂ : InfinitePlace K
e₁ : { w // w ≠ w₁ } ≃ Fin (rank K)
e₂ : { w // w ≠ w₂ } ≃ Fin (rank K)
f : Fin (rank K + 1) ≃ InfinitePlace K := (finSuccEquiv (rank K)).trans ↑((Equiv.optionSubtype w₁).symm e₁.symm)
g : { w // w ≠ w₂ } ≃ Fin (rank K) := (f.symm.subtypeEquiv ⋯).trans (finSuccAboveEquiv (f.symm w₂)).symm
h_col : |((Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).submatrix (⇑(g.trans e₂.symm)) id).det| = |(Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).det|
\u22a2 |(Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₁ i))))).det| = |((Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).submatrix (⇑(g.trans e₂.symm)) id).det|
-- Mathlib/NumberTheory/NumberField/Units/Regulator.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
w₁ w₂ : InfinitePlace K
e₁ : { w // w ≠ w₁ } ≃ Fin (rank K)
e₂ : { w // w ≠ w₂ } ≃ Fin (rank K)
f : Fin (rank K + 1) ≃ InfinitePlace K := (finSuccEquiv (rank K)).trans ↑((Equiv.optionSubtype w₁).symm e₁.symm)
g : { w // w ≠ w₂ } ≃ Fin (rank K) := (f.symm.subtypeEquiv ⋯).trans (finSuccAboveEquiv (f.symm w₂)).symm
h_col : |((Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).submatrix (⇑(g.trans e₂.symm)) id).det| = |(Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).det|
h : |((Matrix.of fun i w => ↑(f w).mult * Real.log ((f w) ((algebraMap (𝓞 K) K) ↑(u i)))).submatrix id (Fin.succAbove 0)).det| = |(↑↑0 - ↑↑(f.symm w₂)).negOnePow • ((Matrix.of fun i w => ↑(f w).mult * Real.log ((f w) ((algebraMap (𝓞 K) K) ↑(u i)))).submatrix id (f.symm w₂).succAbove).det|
\u22a2 |(Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₁ i))))).det| = |((Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).submatrix (⇑(g.trans e₂.symm)) id).det|
-- Mathlib/NumberTheory/NumberField/Units/Regulator.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
w₁ w₂ : InfinitePlace K
e₁ : { w // w ≠ w₁ } ≃ Fin (rank K)
e₂ : { w // w ≠ w₂ } ≃ Fin (rank K)
f : Fin (rank K + 1) ≃ InfinitePlace K := (finSuccEquiv (rank K)).trans ↑((Equiv.optionSubtype w₁).symm e₁.symm)
g : { w // w ≠ w₂ } ≃ Fin (rank K) := (f.symm.subtypeEquiv ⋯).trans (finSuccAboveEquiv (f.symm w₂)).symm
h_col : |((Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).submatrix (⇑(g.trans e₂.symm)) id).det| = |(Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e₂ i))))).det|
\u22a2 ∀ (i : Fin (rank K)), ∑ j : Fin (rank K + 1), Matrix.of (fun i w => ↑(f w).mult * Real.log ((f w) ((algebraMap (𝓞 K) K) ↑(u i)))) i j = 0
-- Mathlib/Data/Fintype/Card.lean
α : Type u_1
β : Type u_2
γ : Type u_3
p : Set α
s : Finset α
H : ∀ (x : α), x ∈ s ↔ x ∈ p
inst✝ : Fintype ↑p
\u22a2 card ↑p = s.card
-- Mathlib/Data/Fintype/Card.lean
α : Type u_1
β : Type u_2
γ : Type u_3
p : Set α
s : Finset α
H : ∀ (x : α), x ∈ s ↔ x ∈ p
inst✝ : Fintype ↑p
\u22a2 card ↑p = card ↑p
-- Mathlib/SetTheory/Ordinal/CantorNormalForm.lean
b o : Ordinal.{u_1}
\u22a2 foldr (fun p r => b ^ p.1 * p.2 + r) 0 (CNF b 0) = 0
-- Mathlib/SetTheory/Ordinal/CantorNormalForm.lean
b o : Ordinal.{u_1}
\u22a2 foldr (fun p r => b ^ p.1 * p.2 + r) 0 [] = 0
-- Mathlib/SetTheory/Ordinal/CantorNormalForm.lean
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : foldr (fun p r => b ^ p.1 * p.2 + r) 0 (CNF b (o % b ^ log b o)) = o % b ^ log b o
\u22a2 foldr (fun p r => b ^ p.1 * p.2 + r) 0 (CNF b o) = o
-- Mathlib/LinearAlgebra/Matrix/ZPow.lean
n' : Type u_1
inst✝² : DecidableEq n'
inst✝¹ : Fintype n'
R : Type u_2
inst✝ : CommRing R
A X Y : M
hx : IsUnit X.det
hy : IsUnit Y.det
h : SemiconjBy A X Y
n : ℕ
\u22a2 SemiconjBy A (X ^ ↑n) (Y ^ ↑n)
-- Mathlib/LinearAlgebra/Matrix/ZPow.lean
n' : Type u_1
inst✝² : DecidableEq n'
inst✝¹ : Fintype n'
R : Type u_2
inst✝ : CommRing R
A X Y : M
hx : IsUnit X.det
hy : IsUnit Y.det
h : SemiconjBy A X Y
n : ℕ
\u22a2 SemiconjBy A (X ^ -[n+1]) (Y ^ -[n+1])
-- Mathlib/LinearAlgebra/Matrix/ZPow.lean
n' : Type u_1
inst✝² : DecidableEq n'
inst✝¹ : Fintype n'
R : Type u_2
inst✝ : CommRing R
A X Y : M
hx : IsUnit X.det
hy : IsUnit Y.det
h : SemiconjBy A X Y
n : ℕ
hx' : IsUnit (X ^ n.succ).det
\u22a2 SemiconjBy A (X ^ -[n+1]) (Y ^ -[n+1])
-- Mathlib/LinearAlgebra/Matrix/ZPow.lean
n' : Type u_1
inst✝² : DecidableEq n'
inst✝¹ : Fintype n'
R : Type u_2
inst✝ : CommRing R
A X Y : M
hx : IsUnit X.det
hy : IsUnit Y.det
h : SemiconjBy A X Y
n : ℕ
hx' : IsUnit (X ^ n.succ).det
hy' : IsUnit (Y ^ n.succ).det
\u22a2 SemiconjBy A (X ^ -[n+1]) (Y ^ -[n+1])
-- Mathlib/LinearAlgebra/Matrix/ZPow.lean
n' : Type u_1
inst✝² : DecidableEq n'
inst✝¹ : Fintype n'
R : Type u_2
inst✝ : CommRing R
A X Y : M
hx : IsUnit X.det
hy : IsUnit Y.det
h : SemiconjBy A X Y
n : ℕ
hx' : IsUnit (X ^ n.succ).det
hy' : IsUnit (Y ^ n.succ).det
\u22a2 A * ↑hx'.unit⁻¹ • (X ^ n.succ).adjugate = ↑hy'.unit⁻¹ • (Y ^ n.succ).adjugate * A
-- Mathlib/LinearAlgebra/Matrix/ZPow.lean
n' : Type u_1
inst✝² : DecidableEq n'
inst✝¹ : Fintype n'
R : Type u_2
inst✝ : CommRing R
A X Y : M
hx : IsUnit X.det
hy : IsUnit Y.det
h : SemiconjBy A X Y
n : ℕ
hx' : IsUnit (X ^ n.succ).det
hy' : IsUnit (Y ^ n.succ).det
\u22a2 (fun x => Y ^ n.succ * x) (A * ↑hx'.unit⁻¹ • (X ^ n.succ).adjugate) = (fun x => Y ^ n.succ * x) (↑hy'.unit⁻¹ • (Y ^ n.succ).adjugate * A)
-- Mathlib/LinearAlgebra/Matrix/ZPow.lean
n' : Type u_1
inst✝² : DecidableEq n'
inst✝¹ : Fintype n'
R : Type u_2
inst✝ : CommRing R
A X Y : M
hx : IsUnit X.det
hy : IsUnit Y.det
h : SemiconjBy A X Y
n : ℕ
hx' : IsUnit (X ^ n.succ).det
hy' : IsUnit (Y ^ n.succ).det
\u22a2 Y ^ n.succ * (A * ↑hx'.unit⁻¹ • (X ^ n.succ).adjugate) = Y ^ n.succ * (↑hy'.unit⁻¹ • (Y ^ n.succ).adjugate * A)
-- Mathlib/Topology/Order/Basic.lean
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : LinearOrder α
inst✝⁵ : OrderTopology α
ι : Type u_1
π : ι → Type u_2
inst✝⁴ : Finite ι
inst✝³ : (i : ι) → LinearOrder (π i)
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : ∀ (i : ι), OrderTopology (π i)
a b x : (i : ι) → π i
a' b' x' : ι → α
inst✝ : Nonempty ι
ha : ∀ (i : ι), a i < x i
hb : ∀ (i : ι), x i < b i
\u22a2 Ico a b ∈ 𝓝 x
-- Mathlib/Topology/Order/Basic.lean
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : LinearOrder α
inst✝⁵ : OrderTopology α
ι : Type u_1
π : ι → Type u_2
inst✝⁴ : Finite ι
inst✝³ : (i : ι) → LinearOrder (π i)
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : ∀ (i : ι), OrderTopology (π i)
a b x : (i : ι) → π i
a' b' x' : ι → α
inst✝ : Nonempty ι
ha : ∀ (i : ι), a i < x i
hb : ∀ (i : ι), x i < b i
i : ι
x✝ : i ∈ univ
\u22a2 Ico (a i) (b i) ∈ 𝓝 (x i)
-- Mathlib/MeasureTheory/Group/Prod.lean
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
f : G → ℝ≥0∞
hf : Measurable f
\u22a2 μ s * ∫⁻ (y : G), f y ∂ν = ∫⁻ (x : G), ν ((fun z => z * x) ⁻¹' s) * f x⁻¹ ∂μ
-- Mathlib/MeasureTheory/Group/Prod.lean
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
f : G → ℝ≥0∞
hf : Measurable f
\u22a2 ∫⁻ (x : G), ∫⁻ (y : G), s.indicator (fun x => 1) (y * x) * f x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ν ((fun z => z * x) ⁻¹' s) * f x⁻¹ ∂μ