ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/RingTheory/Ideal/Over.lean
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
f : R →+* S
I J : Ideal S
inst✝ : I.IsPrime
hIJ : I ≤ J
r : S
p : R[X]
p_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p ≠ 0
hpI : eval₂ f r p ∈ I
hrJ : r ∈ ↑J
hrI : r ∉ ↑I
rbar_ne_zero : (Quotient.mk I) r ≠ 0
rbar_mem_J : (Quotient.mk I) r ∈ map (Quotient.mk I) J
quotient_f : ∀ x ∈ comap f I, ((Quotient.mk I).comp f) x = 0
rbar_root : eval₂ (Quotient.lift (comap f I) ((Quotient.mk I).comp f) quotient_f) ((Quotient.mk I) r) (Polynomial.map (Quotient.mk (comap f I)) p) = 0
\u22a2 ∃ i, p.coeff i ∈ ↑(comap f J) \ ↑(comap f I)
-- Mathlib/RingTheory/Ideal/Over.lean
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
f : R →+* S
I J : Ideal S
inst✝ : I.IsPrime
hIJ : I ≤ J
r : S
p : R[X]
p_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p ≠ 0
hpI : eval₂ f r p ∈ I
hrJ : r ∈ ↑J
hrI : r ∉ ↑I
rbar_ne_zero : (Quotient.mk I) r ≠ 0
rbar_mem_J : (Quotient.mk I) r ∈ map (Quotient.mk I) J
quotient_f : ∀ x ∈ comap f I, ((Quotient.mk I).comp f) x = 0
rbar_root : eval₂ (Quotient.lift (comap f I) ((Quotient.mk I).comp f) quotient_f) ((Quotient.mk I) r) (Polynomial.map (Quotient.mk (comap f I)) p) = 0
i : ℕ
ne_zero : (Polynomial.map (Quotient.mk (comap f I)) p).coeff i ≠ 0
mem : (Polynomial.map (Quotient.mk (comap f I)) p).coeff i ∈ comap (Quotient.lift (comap f I) ((Quotient.mk I).comp f) quotient_f) (map (Quotient.mk I) J)
\u22a2 ∃ i, p.coeff i ∈ ↑(comap f J) \ ↑(comap f I)
-- Mathlib/RingTheory/Ideal/Over.lean
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
f : R →+* S
I J : Ideal S
inst✝ : I.IsPrime
hIJ : I ≤ J
r : S
p : R[X]
p_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p ≠ 0
hpI : eval₂ f r p ∈ I
hrJ : r ∈ ↑J
hrI : r ∉ ↑I
rbar_ne_zero : (Quotient.mk I) r ≠ 0
rbar_mem_J : (Quotient.mk I) r ∈ map (Quotient.mk I) J
quotient_f : ∀ x ∈ comap f I, ((Quotient.mk I).comp f) x = 0
rbar_root : eval₂ (Quotient.lift (comap f I) ((Quotient.mk I).comp f) quotient_f) ((Quotient.mk I) r) (Polynomial.map (Quotient.mk (comap f I)) p) = 0
i : ℕ
ne_zero : (Quotient.mk (comap f I)) (p.coeff i) ≠ 0
mem : (Quotient.mk (comap f I)) (p.coeff i) ∈ comap (Quotient.lift (comap f I) ((Quotient.mk I).comp f) quotient_f) (map (Quotient.mk I) J)
\u22a2 ∃ i, p.coeff i ∈ ↑(comap f J) \ ↑(comap f I)
-- Mathlib/RingTheory/Ideal/Over.lean
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
f : R →+* S
I J : Ideal S
inst✝ : I.IsPrime
hIJ : I ≤ J
r : S
p : R[X]
p_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p ≠ 0
hpI : eval₂ f r p ∈ I
hrJ : r ∈ ↑J
hrI : r ∉ ↑I
rbar_ne_zero : (Quotient.mk I) r ≠ 0
rbar_mem_J : (Quotient.mk I) r ∈ map (Quotient.mk I) J
quotient_f : ∀ x ∈ comap f I, ((Quotient.mk I).comp f) x = 0
rbar_root : eval₂ (Quotient.lift (comap f I) ((Quotient.mk I).comp f) quotient_f) ((Quotient.mk I) r) (Polynomial.map (Quotient.mk (comap f I)) p) = 0
i : ℕ
ne_zero : (Quotient.mk (comap f I)) (p.coeff i) ≠ 0
mem : (Quotient.mk (comap f I)) (p.coeff i) ∈ comap (Quotient.lift (comap f I) ((Quotient.mk I).comp f) quotient_f) (map (Quotient.mk I) J)
\u22a2 (Quotient.mk I) (f (p.coeff i)) ∈ map (Quotient.mk I) J
-- Mathlib/RingTheory/Ideal/Over.lean
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
f : R →+* S
I J : Ideal S
inst✝ : I.IsPrime
hIJ : I ≤ J
r : S
p : R[X]
p_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p ≠ 0
hpI : eval₂ f r p ∈ I
hrJ : r ∈ ↑J
hrI : r ∉ ↑I
rbar_ne_zero : (Quotient.mk I) r ≠ 0
rbar_mem_J : (Quotient.mk I) r ∈ map (Quotient.mk I) J
quotient_f : ∀ x ∈ comap f I, ((Quotient.mk I).comp f) x = 0
rbar_root : eval₂ (Quotient.lift (comap f I) ((Quotient.mk I).comp f) quotient_f) ((Quotient.mk I) r) (Polynomial.map (Quotient.mk (comap f I)) p) = 0
i : ℕ
ne_zero : (Quotient.mk (comap f I)) (p.coeff i) ≠ 0
mem : (Quotient.mk (comap f I)) (p.coeff i) ∈ comap (Quotient.lift (comap f I) ((Quotient.mk I).comp f) quotient_f) (map (Quotient.mk I) J)
\u22a2 p.coeff i ∈ ↑(comap f I) → (Quotient.mk (comap f I)) (p.coeff i) = 0
-- Mathlib/MeasureTheory/Measure/Trim.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s✝ : Set α
ν : Measure α
hm : m ≤ m0
s : Set α
hs : MeasurableSet s
\u22a2 ((μ + ν).trim hm) s = (μ.trim hm + ν.trim hm) s
-- Mathlib/Algebra/Ring/BooleanRing.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : BooleanRing α
inst✝¹ : BooleanRing β
inst✝ : BooleanRing γ
a b c : α
\u22a2 (a + b + a * b) * (a + c + a * c) + (a + b * c + a * (b * c)) + (a + b + a * b) * (a + c + a * c) * (a + b * c + a * (b * c)) = a + b * c + a * (b * c)
-- Mathlib/MeasureTheory/Function/LpSpace.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
β : Type u_5
inst✝ : MeasurableSpace β
μb : Measure β
f : α → β
s : Set β
hs : MeasurableSet s
hμs : μb s ≠ ⊤
c : E
hf : MeasurePreserving f μ μb
\u22a2 μ (f ⁻¹' s) ≠ ⊤
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
this : -(π / 2) < π / 2
\u22a2 Tendsto (fun x => tan ↑x) atBot atBot
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
this : -(π / 2) < π / 2
\u22a2 Tendsto tan (𝓝[>] (-(π / 2))) atBot
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
this : -(π / 2) < π / 2
\u22a2 Tendsto (fun x => tan ↑x) atTop atTop
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
this : -(π / 2) < π / 2
\u22a2 Tendsto tan (𝓝[<] (π / 2)) atTop
-- 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
f : α × β → M
hf : (mulSupport f).Finite
h₁ : ∀ (a : α × β), ∏ᶠ (_ : a ∈ hf.toFinset), f a = f a
h₂ : ∏ᶠ (a : α × β), f a = ∏ᶠ (a : α × β) (_ : a ∈ hf.toFinset), f a
\u22a2 ∏ᶠ (ab : α × β), f ab = ∏ᶠ (a : α) (b : β), f (a, b)
-- Mathlib/MeasureTheory/Measure/DiracProba.lean
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
\u22a2 ¬Tendsto diracProba L (𝓝 (diracProba x))
-- Mathlib/MeasureTheory/Measure/DiracProba.lean
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
\u22a2 ¬Tendsto diracProba L (𝓝 (diracProba x))
-- Mathlib/MeasureTheory/Measure/DiracProba.lean
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
\u22a2 ¬Tendsto diracProba L (𝓝 (diracProba x))
-- Mathlib/MeasureTheory/Measure/DiracProba.lean
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
\u22a2 ¬Tendsto diracProba L (𝓝 (diracProba x))
-- Mathlib/MeasureTheory/Measure/DiracProba.lean
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
\u22a2 ∃ x_1, ¬Tendsto (fun i => ∫⁻ (ω : X), ↑(x_1 ω) ∂↑(diracProba i)) L (𝓝 (∫⁻ (ω : X), ↑(x_1 ω) ∂↑(diracProba x)))
-- Mathlib/MeasureTheory/Measure/DiracProba.lean
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
\u22a2 ¬Tendsto (fun i => ∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba i)) L (𝓝 (∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba x)))
-- Mathlib/MeasureTheory/Measure/DiracProba.lean
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
\u22a2 ¬Tendsto (fun i => ↑(f i)) L (𝓝 ↑1)
-- Mathlib/MeasureTheory/Measure/DiracProba.lean
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
\u22a2 ∃ s ∈ 𝓝 ↑1, ∃ᶠ (x : X) in L, ↑(f x) ∉ s
-- Mathlib/MeasureTheory/Measure/DiracProba.lean
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
\u22a2 ∀ x ∉ U, ↑(f x) ∉ Ioi 0
-- Mathlib/MeasureTheory/Measure/DiracProba.lean
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x✝ : X
L : Filter X
h : ¬Tendsto id L (𝓝 x✝)
U : Set X
U_nhd : U ∈ 𝓝 x✝
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x✝
f : X →ᵇ ℝ≥0
fx_eq_one : f x✝ = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
x : X
x_notin_U : x ∉ U
\u22a2 ↑(f x) ∉ Ioi 0
-- Mathlib/MeasureTheory/Measure/DiracProba.lean
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x✝ : X
L : Filter X
h : ¬Tendsto id L (𝓝 x✝)
U : Set X
U_nhd : U ∈ 𝓝 x✝
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x✝
f : X →ᵇ ℝ≥0
fx_eq_one : f x✝ = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
x : X
x_notin_U : x ∉ U
\u22a2 ↑0 ∉ Ioi 0
-- Mathlib/Data/Fintype/BigOperators.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : CommMonoid α
s : Finset ι
f : ι → α
\u22a2 (∏ i : ι, if i ∈ s then f i else 1) = ∏ i ∈ s, f i
-- Mathlib/Topology/FiberBundle/IsHomeomorphicTrivialBundle.lean
B : Type u_1
F : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalSpace Z
proj : Z → B
h : IsHomeomorphicTrivialFiberBundle F proj
\u22a2 Continuous proj
-- Mathlib/Topology/FiberBundle/IsHomeomorphicTrivialBundle.lean
B : Type u_1
F : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalSpace Z
e : Z ≃ₜ B × F
h : IsHomeomorphicTrivialFiberBundle F (Prod.fst ∘ ⇑e)
\u22a2 Continuous (Prod.fst ∘ ⇑e)
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
R : Type u_2
inst✝³ : SMul R ℝ≥0
inst✝² : SMul R ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
μ : FiniteMeasure Ω
A : Set Ω
\u22a2 μ.restrict A = 0 ↔ μ A = 0
-- Mathlib/Algebra/Homology/DerivedCategory/Ext.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
\u22a2 (∀ (X Y : C), HasSmallLocalizedShiftedHom (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) ℤ ((CochainComplex.singleFunctor C 0).obj X) ((CochainComplex.singleFunctor C 0).obj Y)) ↔ ∀ (X Y : C) (n : ℤ), Small.{w, w'} ((DerivedCategory.singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((DerivedCategory.singleFunctor C 0).obj Y))
-- Mathlib/Algebra/Homology/DerivedCategory/Ext.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
\u22a2 (∀ (X Y : C) (a b : ℤ), Small.{w, w'} ((shiftFunctor (DerivedCategory C) a).obj (DerivedCategory.Q.obj ((CochainComplex.singleFunctor C 0).obj X)) ⟶ (shiftFunctor (DerivedCategory C) b).obj (DerivedCategory.Q.obj ((CochainComplex.singleFunctor C 0).obj Y)))) ↔ ∀ (X Y : C) (n : ℤ), Small.{w, w'} ((DerivedCategory.singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((DerivedCategory.singleFunctor C 0).obj Y))
-- Mathlib/MeasureTheory/Measure/Count.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
s : Set α
inst✝ : MeasurableSingletonClass α
hs' : s.Nonempty
\u22a2 count s ≠ 0
-- Mathlib/MeasureTheory/Measure/Count.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
s : Set α
inst✝ : MeasurableSingletonClass α
hs' : s.Nonempty
\u22a2 ¬s = ∅
-- Mathlib/Algebra/Regular/Basic.lean
R : Type u_1
inst✝ : MulZeroClass R
a b : R
hb : IsRightRegular b
\u22a2 a * b = 0 ↔ a = 0
-- Mathlib/Algebra/Regular/Basic.lean
R : Type u_1
inst✝ : MulZeroClass R
a b : R
hb : IsRightRegular b
\u22a2 a * b = 0 * b ↔ a = 0
-- Mathlib/Algebra/DirectSum/Algebra.lean
ι : Type uι
R : Type uR
A : ι → Type uA
B : Type uB
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommSemiring R
inst✝⁶ : (i : ι) → AddCommMonoid (A i)
inst✝⁵ : (i : ι) → Module R (A i)
inst✝⁴ : AddMonoid ι
inst✝³ : GSemiring A
inst✝² : Semiring B
inst✝¹ : GAlgebra R A
inst✝ : Algebra R B
f : (i : ι) → A i →ₗ[R] B
hone : (f 0) GradedMonoid.GOne.one = 1
hmul : ∀ {i j : ι} (ai : A i) (aj : A j), (f (i + j)) (GradedMonoid.GMul.mul ai aj) = (f i) ai * (f j) aj
r : R
\u22a2 (toModule R ι B f) ((algebraMap R (⨁ (i : ι), A i)) r) = (algebraMap R B) r
-- Mathlib/Analysis/InnerProductSpace/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : _root_.RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : Core 𝕜 F
x y : F
\u22a2 ⟪x - y, x - y⟫_𝕜 = ⟪x, x⟫_𝕜 - ⟪x, y⟫_𝕜 - ⟪y, x⟫_𝕜 + ⟪y, y⟫_𝕜
-- Mathlib/Analysis/InnerProductSpace/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : _root_.RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : Core 𝕜 F
x y : F
\u22a2 ⟪x, x⟫_𝕜 - ⟪y, x⟫_𝕜 - (⟪x, y⟫_𝕜 - ⟪y, y⟫_𝕜) = ⟪x, x⟫_𝕜 - ⟪x, y⟫_𝕜 - ⟪y, x⟫_𝕜 + ⟪y, y⟫_𝕜
-- Mathlib/Algebra/Group/Basic.lean
α : Type u_1
β : Type u_2
G : Type u_3
M : Type u_4
inst✝ : DivisionMonoid α
a✝ b c d a : α
n : ℕ
\u22a2 a⁻¹ ^ ↑n = (a ^ ↑n)⁻¹
-- Mathlib/Algebra/Group/Basic.lean
α : Type u_1
β : Type u_2
G : Type u_3
M : Type u_4
inst✝ : DivisionMonoid α
a✝ b c d a : α
n : ℕ
\u22a2 a⁻¹ ^ Int.negSucc n = (a ^ Int.negSucc n)⁻¹
-- Mathlib/LinearAlgebra/LinearIndependent.lean
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : AddCommGroup V'
inst✝¹ : Module K V
inst✝ : Module K V'
v : ι → V
s t : Set V
x y z : V
hs : LinearIndependent K fun (b : ↑s) => ↑b
hx : x ∉ span K s
\u22a2 LinearIndependent K fun (b : ↑(insert x s)) => ↑b
-- Mathlib/LinearAlgebra/LinearIndependent.lean
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : AddCommGroup V'
inst✝¹ : Module K V
inst✝ : Module K V'
v : ι → V
s t : Set V
x y z : V
hs : LinearIndependent K fun (b : ↑s) => ↑b
hx : x ∉ span K s
\u22a2 LinearIndependent K fun (b : ↑(s ∪ {x})) => ↑b
-- Mathlib/LinearAlgebra/LinearIndependent.lean
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : AddCommGroup V'
inst✝¹ : Module K V
inst✝ : Module K V'
v : ι → V
s t : Set V
x y z : V
hs : LinearIndependent K fun (b : ↑s) => ↑b
hx : x ∉ span K s
x0 : x ≠ 0
\u22a2 LinearIndependent K fun (b : ↑(s ∪ {x})) => ↑b
-- Mathlib/LinearAlgebra/LinearIndependent.lean
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : AddCommGroup V'
inst✝¹ : Module K V
inst✝ : Module K V'
v : ι → V
s t : Set V
x y z : V
hs : LinearIndependent K fun (b : ↑s) => ↑b
hx : x ∉ span K s
x0 : x ≠ 0
\u22a2 Disjoint (span K s) (span K {x})
-- Mathlib/LinearAlgebra/Matrix/Circulant.lean
α : Type u_1
β : Type u_2
m : Type u_3
n✝ : Type u_4
R : Type u_5
n : Type u_6
inst✝³ : Semiring α
inst✝² : DecidableEq n
inst✝¹ : AddGroup n
inst✝ : Fintype n
a : α
i j : n
\u22a2 circulant (Pi.single 0 a) i j = (scalar n) a i j
-- Mathlib/Data/Finset/NatAntidiagonal.lean
n k : ℕ
h : k ≤ n
aux₁ : (fun a => a.1 ≤ k) = (fun a => a.2 ≤ k) ∘ ⇑(Equiv.prodComm ℕ ℕ).symm
\u22a2 filter (fun a => a.1 ≤ k) (antidiagonal n) = map ((Embedding.refl ℕ).prodMap { toFun := fun x => x + (n - k), inj' := ⋯ }) (antidiagonal k)
-- Mathlib/Data/Finset/NatAntidiagonal.lean
n k : ℕ
h : k ≤ n
aux₁ : (fun a => a.1 ≤ k) = (fun a => a.2 ≤ k) ∘ ⇑(Equiv.prodComm ℕ ℕ).symm
aux₂ : ∀ (i j : ℕ), (∃ a b, a + b = k ∧ b = i ∧ a + (n - k) = j) ↔ ∃ a b, a + b = k ∧ a = i ∧ b + (n - k) = j
\u22a2 filter (fun a => a.1 ≤ k) (antidiagonal n) = map ((Embedding.refl ℕ).prodMap { toFun := fun x => x + (n - k), inj' := ⋯ }) (antidiagonal k)
-- Mathlib/Data/Finset/NatAntidiagonal.lean
n k : ℕ
h : k ≤ n
aux₁ : (fun a => a.1 ≤ k) = (fun a => a.2 ≤ k) ∘ ⇑(Equiv.prodComm ℕ ℕ).symm
aux₂ : ∀ (i j : ℕ), (∃ a b, a + b = k ∧ b = i ∧ a + (n - k) = j) ↔ ∃ a b, a + b = k ∧ a = i ∧ b + (n - k) = j
\u22a2 filter (fun a => a.1 ≤ k) (map (Equiv.prodComm ℕ ℕ).toEmbedding (antidiagonal n)) = map ((Embedding.refl ℕ).prodMap { toFun := fun x => x + (n - k), inj' := ⋯ }) (antidiagonal k)
-- Mathlib/Data/Finset/NatAntidiagonal.lean
n k : ℕ
h : k ≤ n
aux₁ : (fun a => a.1 ≤ k) = (fun a => a.2 ≤ k) ∘ ⇑(Equiv.prodComm ℕ ℕ).symm
aux₂ : ∀ (i j : ℕ), (∃ a b, a + b = k ∧ b = i ∧ a + (n - k) = j) ↔ ∃ a b, a + b = k ∧ a = i ∧ b + (n - k) = j
\u22a2 map (({ toFun := fun x => x + (n - k), inj' := ⋯ }.prodMap (Embedding.refl ℕ)).trans (Equiv.prodComm ℕ ℕ).toEmbedding) (antidiagonal k) = map ((Embedding.refl ℕ).prodMap { toFun := fun x => x + (n - k), inj' := ⋯ }) (antidiagonal k)
-- Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : CommMonoid α
inst✝ : TopologicalSpace α
f✝ g : β → α
a a₁ a₂ : α
f : β → α
s : Finset β
\u22a2 ∏ x ∈ s, f x = ∏' (x : β), (↑s).mulIndicator f x
-- Mathlib/Topology/Algebra/ProperAction.lean
G : Type u_1
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝⁷ : Group G
inst✝⁶ : MulAction G X
inst✝⁵ : MulAction G Y
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : T2Space X
\u22a2 ProperSMul G X ↔ ContinuousSMul G X ∧ ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X), Tendsto (fun gx => (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) → ∃ g, Tendsto fst (↑𝒰) (𝓝 g)
-- Mathlib/Topology/Algebra/ProperAction.lean
G : Type u_1
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝⁷ : Group G
inst✝⁶ : MulAction G X
inst✝⁵ : MulAction G Y
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : T2Space X
\u22a2 (ContinuousSMul G X ∧ ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X), Tendsto (fun gx => (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) → ∃ g, g • x₂ = x₁ ∧ Tendsto fst (↑𝒰) (𝓝 g)) ↔ ContinuousSMul G X ∧ ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X), Tendsto (fun gx => (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) → ∃ g, Tendsto fst (↑𝒰) (𝓝 g)
-- Mathlib/Topology/Algebra/ProperAction.lean
G : Type u_1
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝⁷ : Group G
inst✝⁶ : MulAction G X
inst✝⁵ : MulAction G Y
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : T2Space X
hc : ContinuousSMul G X
\u22a2 (∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X), Tendsto (fun gx => (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) → ∃ g, g • x₂ = x₁ ∧ Tendsto fst (↑𝒰) (𝓝 g)) ↔ ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X), Tendsto (fun gx => (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) → ∃ g, Tendsto fst (↑𝒰) (𝓝 g)
-- Mathlib/Topology/Algebra/ProperAction.lean
G : Type u_1
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝⁷ : Group G
inst✝⁶ : MulAction G X
inst✝⁵ : MulAction G Y
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : T2Space X
hc : ContinuousSMul G X
𝒰 : Ultrafilter (G × X)
x₁ x₂ : X
hxx : Tendsto (fun gx => (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂))
g : G
\u22a2 g • x₂ = x₁ ∧ Tendsto fst (↑𝒰) (𝓝 g) ↔ Tendsto fst (↑𝒰) (𝓝 g)
-- Mathlib/Topology/Separation.lean
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
s : Set X
inst✝ : DiscreteTopology ↑s
x : X
hx : x ∈ s
V : Set X
h : V ∈ 𝓝 x
h' : V ∩ s = {x}
\u22a2 {x}ᶜ ∩ V ∩ s = ∅
-- .lake/packages/batteries/Batteries/Data/List/Basic.lean
\u22a2 @takeList = @takeListTR
-- .lake/packages/batteries/Batteries/Data/List/Basic.lean
α : Type u_1
xs : List α
ns : List Nat
\u22a2 xs.takeList ns = xs.takeListTR ns
-- Mathlib/Algebra/Polynomial/Eval.lean
R : Type u
S : Type v
T : Type w
ι : Type y
a b : R
m n : ℕ
inst✝¹ : CommSemiring R
p✝ q✝ : R[X]
x : R
inst✝ : CommSemiring S
f : R →+* S
p q : R[X]
H : p.IsRoot a
\u22a2 (p * q).IsRoot a
-- Mathlib/Algebra/Algebra/Spectrum.lean
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : Aˣ
\u22a2 0 ∈ resolventSet R ↑a
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
β : Type u_2
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), m (s (k + 1) \ s k) ≠ ⊤
\u22a2 m (⋃ n, s n) = ⨆ n, m (s n)
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
β : Type u_2
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), m (s (k + 1) \ s k) ≠ ⊤
\u22a2 Tendsto (fun k => m ((⋃ n, s n) \ s k)) atTop (𝓝 0)
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
β : Type u_2
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), m (s (k + 1) \ s k) ≠ ⊤
n : ℕ
\u22a2 m ((⋃ n, s n) \ s n) ≤ ∑' (k : ℕ), m (s (k + n + 1) \ s (k + n))
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
β : Type u_2
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), m (s (k + 1) \ s k) ≠ ⊤
n : ℕ
\u22a2 (⋃ n, s n) \ s n ⊆ ⋃ i, s (i + n + 1) \ s (i + n)
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
β : Type u_2
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), m (s (k + 1) \ s k) ≠ ⊤
n : ℕ
h' : Monotone s
\u22a2 (⋃ n, s n) \ s n ⊆ ⋃ i, s (i + n + 1) \ s (i + n)
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
β : Type u_2
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), m (s (k + 1) \ s k) ≠ ⊤
n : ℕ
h' : Monotone s
\u22a2 ∀ (i : ℕ), s i ⊆ s n ∪ ⋃ i, s (i + n + 1) \ s (i + n)
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
β : Type u_2
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), m (s (k + 1) \ s k) ≠ ⊤
n : ℕ
h' : Monotone s
x : α
this : ∃ i, x ∈ s i
j : ℕ
hj : x ∈ s j
hlt : ∀ m < j, x ∉ s m
\u22a2 x ∈ s n ∪ ⋃ i, s (i + n + 1) \ s (i + n)
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
β : Type u_2
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), m (s (k + 1) \ s k) ≠ ⊤
n : ℕ
h' : Monotone s
x : α
this : ∃ i, x ∈ s i
j : ℕ
hj : x ∈ s j
hlt : ∀ m < j, x ∉ s m
hjn : j ≤ n
\u22a2 x ∈ s n ∪ ⋃ i, s (i + n + 1) \ s (i + n)
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
β : Type u_2
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), m (s (k + 1) \ s k) ≠ ⊤
n : ℕ
h' : Monotone s
x : α
this : ∃ i, x ∈ s i
j : ℕ
hj : x ∈ s j
hlt : ∀ m < j, x ∉ s m
hnj : n < j
\u22a2 x ∈ s n ∪ ⋃ i, s (i + n + 1) \ s (i + n)