ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
\u22a2 ∃ c, (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧ Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
\u22a2 ∃ c, (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧ Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
\u22a2 ∀ᶠ (n : ℕ) in atTop, ↑((fun n => ⌊c k ^ n⌋₊) (n + 1)) ≤ a * ↑((fun n => ⌊c k ^ n⌋₊) n)
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
\u22a2 ∀ᶠ (n : ℕ) in atTop, ↑((fun n => ⌊c k ^ n⌋₊) (n + 1)) ≤ a * ↑((fun n => ⌊c k ^ n⌋₊) n)
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
A : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k / (↑⌊c k ^ n⌋₊ / c k ^ n)) atTop (𝓝 (1 * c k / 1))
\u22a2 ∀ᶠ (n : ℕ) in atTop, ↑((fun n => ⌊c k ^ n⌋₊) (n + 1)) ≤ a * ↑((fun n => ⌊c k ^ n⌋₊) n)
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
A : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k / (↑⌊c k ^ n⌋₊ / c k ^ n)) atTop (𝓝 (1 * c k / 1))
B : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 (c k))
\u22a2 ∀ᶠ (n : ℕ) in atTop, ↑((fun n => ⌊c k ^ n⌋₊) (n + 1)) ≤ a * ↑((fun n => ⌊c k ^ n⌋₊) n)
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
A : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k / (↑⌊c k ^ n⌋₊ / c k ^ n)) atTop (𝓝 (1 * c k / 1))
B : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 (c k))
n : ℕ
hn : ↑⌊c k ^ (n + 1)⌋₊ / ↑⌊c k ^ n⌋₊ < a
\u22a2 ↑⌊c k ^ (n + 1)⌋₊ ≤ a * ↑⌊c k ^ n⌋₊
-- Mathlib/AlgebraicTopology/DoldKan/PInfty.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
\u22a2 (PInfty ≫ QInfty).f n = HomologicalComplex.Hom.f 0 n
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝² : Group α
inst✝¹ : Finite α
inst✝ : IsCyclic α
\u22a2 ∃ x, ∀ (y : α), y ∈ Submonoid.powers x
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝² : Group α
inst✝¹ : Finite α
inst✝ : IsCyclic α
\u22a2 ∃ x, ∀ (y : α), y ∈ zpowers x
-- Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x✝ y✝ x y : V
\u22a2 sin (arccos (⟪x, y⟫_ℝ / (‖x‖ * ‖y‖))) * (‖x‖ * ‖y‖) = √(⟪x, x⟫_ℝ * ⟪y, y⟫_ℝ - ⟪x, y⟫_ℝ * ⟪x, y⟫_ℝ)
-- Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x✝ y✝ x y : V
\u22a2 √((1 - ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) * (⟪x, y⟫_ℝ / (‖x‖ * ‖y‖))) * (‖x‖ * ‖y‖ * (‖x‖ * ‖y‖))) = √(‖x‖ * ‖x‖ * (‖y‖ * ‖y‖) - ⟪x, y⟫_ℝ * ⟪x, y⟫_ℝ)
-- Mathlib/Algebra/Order/UpperLower.lean
α : Type u_1
inst✝ : OrderedCommGroup α
s t : Set α
a : α
\u22a2 ↑(lowerClosure (s * t)) = ↑(lowerClosure s * lowerClosure t)
-- Mathlib/GroupTheory/QuotientGroup.lean
G : Type u
inst✝² : Group G
N : Subgroup G
nN : N.Normal
H : Type v
inst✝¹ : Group H
M : Type x
inst✝ : Monoid M
φ : G →* H
a✝ b✝ : G ⧸ φ.ker
a b : G
h : φ a = φ b
\u22a2 Setoid.r a b
-- Mathlib/Analysis/Calculus/ContDiff/Bounds.lean
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s✝ s₁ t u : Set E
f : E → F →L[𝕜] G
g : E → F
s : Set E
x : E
N : ℕ∞
n : ℕ
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hn : ↑n ≤ N
B : (F →L[𝕜] G) →L[𝕜] F →L[𝕜] G := (ContinuousLinearMap.apply 𝕜 G).flip
\u22a2 ‖iteratedFDerivWithin 𝕜 n (fun y => (f y) (g y)) s x‖ ≤ ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
-- Mathlib/Analysis/Calculus/ContDiff/Bounds.lean
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s✝ s₁ t u : Set E
f : E → F →L[𝕜] G
g : E → F
s : Set E
x : E
N : ℕ∞
n : ℕ
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hn : ↑n ≤ N
B : (F →L[𝕜] G) →L[𝕜] F →L[𝕜] G := (ContinuousLinearMap.apply 𝕜 G).flip
hB : ‖B‖ ≤ 1
\u22a2 ‖iteratedFDerivWithin 𝕜 n (fun y => (f y) (g y)) s x‖ ≤ ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
-- Mathlib/Topology/Algebra/ConstMulAction.lean
M : Type u_1
α✝ : Type u_2
β✝ : Type u_3
inst✝⁹ : TopologicalSpace α✝
inst✝⁸ : Monoid M
inst✝⁷ : MulAction M α✝
inst✝⁶ : ContinuousConstSMul M α✝
N : Type u_4
inst✝⁵ : Monoid N
α : Type u_5
β : Type u_6
inst✝⁴ : MulAction M α
inst✝³ : MulAction N β
inst✝² : TopologicalSpace β
inst✝¹ : T2Space β
inst✝ : ContinuousConstSMul N β
σ : M → N
\u22a2 IsClosed {f | ∀ (c : M) (x : α), f (c • x) = σ c • f x}
-- Mathlib/Topology/Algebra/ConstMulAction.lean
M : Type u_1
α✝ : Type u_2
β✝ : Type u_3
inst✝⁹ : TopologicalSpace α✝
inst✝⁸ : Monoid M
inst✝⁷ : MulAction M α✝
inst✝⁶ : ContinuousConstSMul M α✝
N : Type u_4
inst✝⁵ : Monoid N
α : Type u_5
β : Type u_6
inst✝⁴ : MulAction M α
inst✝³ : MulAction N β
inst✝² : TopologicalSpace β
inst✝¹ : T2Space β
inst✝ : ContinuousConstSMul N β
σ : M → N
\u22a2 IsClosed (⋂ i, ⋂ i_1, {x | x (i • i_1) = σ i • x i_1})
-- Mathlib/FieldTheory/RatFunc/Basic.lean
K : Type u
inst✝ : Field K
x y : RatFunc K
hxy : x + y ≠ 0
h_zero : x.num * y.denom + x.denom * y.num = 0
\u22a2 False
-- Mathlib/FieldTheory/RatFunc/Basic.lean
K : Type u
inst✝ : Field K
x y : RatFunc K
hxy : x + y ≠ 0
h_zero : x.num * y.denom + x.denom * y.num = 0
h : (x + y).num * (x.denom * y.denom) = (x.num * y.denom + x.denom * y.num) * (x + y).denom
\u22a2 False
-- Mathlib/FieldTheory/RatFunc/Basic.lean
K : Type u
inst✝ : Field K
x y : RatFunc K
hxy : x + y ≠ 0
h_zero : x.num * y.denom + x.denom * y.num = 0
h : (x + y).num * (x.denom * y.denom) = 0
\u22a2 False
-- Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : ℝ → H →L[𝕜] E
x₀ : H
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))
h_lip : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → LipschitzOnWith (Real.nnabs (bound t)) (fun x => F x t) (ball x₀ ε)
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → HasFDerivAt (fun x => F x t) (F' t) x₀
\u22a2 IntervalIntegrable F' μ a b ∧ HasFDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀
-- Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : ℝ → H →L[𝕜] E
x₀ : H
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))
h_lip : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasFDerivAt (fun x_1 => F x_1 x) (F' x) x₀
\u22a2 IntervalIntegrable F' μ a b ∧ HasFDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀
-- Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : ℝ → H →L[𝕜] E
x₀ : H
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))
h_lip : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasFDerivAt (fun x_1 => F x_1 x) (F' x) x₀
hF_int : IntegrableOn (F x₀) (Ι a b) μ
bound_integrable : IntegrableOn bound (Ι a b) μ
\u22a2 IntegrableOn F' (Ι a b) μ ∧ HasFDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀
-- Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : ℝ → H →L[𝕜] E
x₀ : H
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))
h_lip : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasFDerivAt (fun x_1 => F x_1 x) (F' x) x₀
hF_int : IntegrableOn (F x₀) (Ι a b) μ
bound_integrable : IntegrableOn bound (Ι a b) μ
\u22a2 IntegrableOn F' (Ι a b) μ ∧ HasFDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F x t ∂μ) ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F' t ∂μ) x₀
-- Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : ℝ → H →L[𝕜] E
x₀ : H
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))
h_lip : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasFDerivAt (fun x_1 => F x_1 x) (F' x) x₀
hF_int : IntegrableOn (F x₀) (Ι a b) μ
bound_integrable : IntegrableOn bound (Ι a b) μ
this : Integrable F' (μ.restrict (Ι a b)) ∧ HasFDerivAt (fun x => ∫ (a : ℝ) in Ι a b, F x a ∂μ) (∫ (a : ℝ) in Ι a b, F' a ∂μ) x₀
\u22a2 IntegrableOn F' (Ι a b) μ ∧ HasFDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F x t ∂μ) ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F' t ∂μ) x₀
-- Mathlib/GroupTheory/Nilpotent.lean
G✝ : Type u_1
inst✝² : Group G✝
H : Subgroup G✝
inst✝¹ : H.Normal
G : Type u_2
inst✝ : CommGroup G
\u22a2 Group.nilpotencyClass G ≤ 1
-- Mathlib/GroupTheory/Nilpotent.lean
G✝ : Type u_1
inst✝² : Group G✝
H : Subgroup G✝
inst✝¹ : H.Normal
G : Type u_2
inst✝ : CommGroup G
\u22a2 center G = ⊤
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β : Type u
x : Cardinal.{u_1}
\u22a2 x ∈ range Nat.cast ↔ x ∈ Iio ℵ₀
-- Mathlib/Topology/IndicatorConstPointwise.lean
α : Type u_1
A : Set α
β : Type u_2
inst✝¹ : Zero β
inst✝ : TopologicalSpace β
ι : Type u_3
L : Filter ι
As : ι → Set α
b : β
nhd_b : {0}ᶜ ∈ 𝓝 b
nhd_o : {b}ᶜ ∈ 𝓝 0
\u22a2 Tendsto (fun i => (As i).indicator fun x => b) L (𝓝 (A.indicator fun x => b)) ↔ ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
-- Mathlib/Topology/IndicatorConstPointwise.lean
α : Type u_1
A : Set α
β : Type u_2
inst✝¹ : Zero β
inst✝ : TopologicalSpace β
ι : Type u_3
L : Filter ι
As : ι → Set α
b : β
nhd_b : {0}ᶜ ∈ 𝓝 b
nhd_o : {b}ᶜ ∈ 𝓝 0
\u22a2 (∀ (x : α), Tendsto (fun i => (As i).indicator (fun x => b) x) L (𝓝 (A.indicator (fun x => b) x))) ↔ ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
-- Mathlib/Topology/IndicatorConstPointwise.lean
α : Type u_1
A : Set α
β : Type u_2
inst✝¹ : Zero β
inst✝ : TopologicalSpace β
ι : Type u_3
L : Filter ι
As : ι → Set α
b : β
nhd_b : {0}ᶜ ∈ 𝓝 b
nhd_o : {b}ᶜ ∈ 𝓝 0
\u22a2 ∀ (a : α), Tendsto (fun i => (As i).indicator (fun x => b) a) L (𝓝 (A.indicator (fun x => b) a)) ↔ ∀ᶠ (i : ι) in L, a ∈ As i ↔ a ∈ A
-- Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean
C : Type u_1
D : Type u_2
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Category.{u_4, u_2} D
inst✝⁷ : HasZeroMorphisms C
inst✝⁶ : HasZeroMorphisms D
S : ShortComplex C
h₁✝ : S.LeftHomologyData
h₂✝ : S.RightHomologyData
F✝ : C ⥤ D
inst✝⁵ : F✝.PreservesZeroMorphisms
hl : S.LeftHomologyData
hr : S.RightHomologyData
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
F : C ⥤ D
inst✝⁴ : F.PreservesZeroMorphisms
inst✝³ : S₁.HasRightHomology
inst✝² : S₂.HasRightHomology
inst✝¹ : F.PreservesRightHomologyOf S₁
inst✝ : F.PreservesRightHomologyOf S₂
\u22a2 F.map (opcyclesMap φ) ≫ (S₂.mapOpcyclesIso F).inv = (S₁.mapOpcyclesIso F).inv ≫ opcyclesMap (F.mapShortComplex.map φ)
-- Mathlib/Algebra/Ring/CentroidHom.lean
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsScalarTower N α α
inst✝² : Module R α
inst✝¹ : SMulCommClass R α α
inst✝ : IsScalarTower R α α
T S : CentroidHom α
a b : α
\u22a2 (⇑T ∘ ⇑S) (a * b) = (⇑S ∘ ⇑T) (a * b)
-- Mathlib/Algebra/Ring/CentroidHom.lean
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsScalarTower N α α
inst✝² : Module R α
inst✝¹ : SMulCommClass R α α
inst✝ : IsScalarTower R α α
T S : CentroidHom α
a b : α
\u22a2 T (S (a * b)) = S (T (a * b))
-- Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
R : Type v
inst✝¹ : CommRing R
S : Type u_1
inst✝ : CommRing S
i : n
A : ↥(center (SpecialLinearGroup n R))
this : Nonempty n
\u22a2 ↑↑A i i ^ ↑(Fintype.card n).toPNat' = 1
-- Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
R : Type v
inst✝¹ : CommRing R
S : Type u_1
inst✝ : CommRing S
i : n
A : ↥(center (SpecialLinearGroup n R))
this : Nonempty n
r : R
hr : r ^ Fintype.card n = 1
hr' : (scalar n) r = ↑↑A
\u22a2 ↑↑A i i ^ ↑(Fintype.card n).toPNat' = 1
-- Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
R : Type v
inst✝¹ : CommRing R
S : Type u_1
inst✝ : CommRing S
i : n
a : ↥(rootsOfUnity (Fintype.card n).toPNat' R)
B : SpecialLinearGroup n R
\u22a2 ↑(B * ⟨a • 1, ⋯⟩) = ↑(⟨a • 1, ⋯⟩ * B)
-- Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
R : Type v
inst✝¹ : CommRing R
S : Type u_1
inst✝ : CommRing S
i : n
A : ↥(center (SpecialLinearGroup n R))
\u22a2 (fun a => ⟨⟨a • 1, ⋯⟩, ⋯⟩) ((fun A => rootsOfUnity.mkOfPowEq (↑↑A i i) ⋯) A) = A
-- Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
R : Type v
inst✝¹ : CommRing R
S : Type u_1
inst✝ : CommRing S
i : n
A : ↥(center (SpecialLinearGroup n R))
\u22a2 ↑↑((fun a => ⟨⟨a • 1, ⋯⟩, ⋯⟩) ((fun A => rootsOfUnity.mkOfPowEq (↑↑A i i) ⋯) A)) = ↑↑A
-- Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
R : Type v
inst✝¹ : CommRing R
S : Type u_1
inst✝ : CommRing S
i : n
A : ↥(center (SpecialLinearGroup n R))
r : R
left✝ : r ^ Fintype.card n = 1
hr : (scalar n) r = ↑↑A
\u22a2 ↑↑((fun a => ⟨⟨a • 1, ⋯⟩, ⋯⟩) ((fun A => rootsOfUnity.mkOfPowEq (↑↑A i i) ⋯) A)) = ↑↑A
-- Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
R : Type v
inst✝¹ : CommRing R
S : Type u_1
inst✝ : CommRing S
i : n
a inv✝ : R
val_inv✝ : a * inv✝ = 1
inv_val✝ : inv✝ * a = 1
ha : { val := a, inv := inv✝, val_inv := val_inv✝, inv_val := inv_val✝ } ∈ rootsOfUnity (Fintype.card n).toPNat' R
\u22a2 (fun A => rootsOfUnity.mkOfPowEq (↑↑A i i) ⋯) ((fun a => ⟨⟨a • 1, ⋯⟩, ⋯⟩) ⟨{ val := a, inv := inv✝, val_inv := val_inv✝, inv_val := inv_val✝ }, ha⟩) = ⟨{ val := a, inv := inv✝, val_inv := val_inv✝, inv_val := inv_val✝ }, ha⟩
-- Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
R : Type v
inst✝¹ : CommRing R
S : Type u_1
inst✝ : CommRing S
i : n
A B : ↥(center (SpecialLinearGroup n R))
\u22a2 ↑↑(rootsOfUnity.mkOfPowEq ((↑↑A * ↑↑B) i i) ⋯) = ↑↑(rootsOfUnity.mkOfPowEq (↑↑A i i) ⋯ * rootsOfUnity.mkOfPowEq (↑↑B i i) ⋯)
-- Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
R : Type v
inst✝¹ : CommRing R
S : Type u_1
inst✝ : CommRing S
i : n
A B : ↥(center (SpecialLinearGroup n R))
\u22a2 (↑↑A * ↑↑B) i i = ↑↑A i i * ↑↑B i i
-- Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
R : Type v
inst✝¹ : CommRing R
S : Type u_1
inst✝ : CommRing S
i : n
A B : ↥(center (SpecialLinearGroup n R))
\u22a2 ((scalar n) (↑↑A i i) * (scalar n) (↑↑B i i)) i i = (scalar n) (↑↑A i i) i i * (scalar n) (↑↑B i i) i i
-- Mathlib/Topology/VectorBundle/Hom.lean
𝕜₁ : Type u_1
inst✝²² : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝²¹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝²⁰ : NormedAddCommGroup F₁
inst✝¹⁹ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁵ : NormedAddCommGroup F₂
inst✝¹⁴ : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹³ : (x : B) → AddCommGroup (E₂ x)
inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x)
inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁰ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁹ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁸ : FiberBundle F₁ E₁
inst✝⁷ : (x : B) → TopologicalSpace (E₂ x)
ita : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝⁶ : FiberBundle F₂ E₂
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
\u22a2 ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))
-- Mathlib/Topology/VectorBundle/Hom.lean
𝕜₁ : Type u_1
inst✝²² : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝²¹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝²⁰ : NormedAddCommGroup F₁
inst✝¹⁹ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁵ : NormedAddCommGroup F₂
inst✝¹⁴ : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹³ : (x : B) → AddCommGroup (E₂ x)
inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x)
inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁰ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁹ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁸ : FiberBundle F₁ E₁
inst✝⁷ : (x : B) → TopologicalSpace (E₂ x)
ita : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝⁶ : FiberBundle F₂ E₂
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
h₁ : Continuous ⇑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂))
\u22a2 ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))
-- Mathlib/Topology/VectorBundle/Hom.lean
𝕜₁ : Type u_1
inst✝²² : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝²¹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝²⁰ : NormedAddCommGroup F₁
inst✝¹⁹ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁵ : NormedAddCommGroup F₂
inst✝¹⁴ : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹³ : (x : B) → AddCommGroup (E₂ x)
inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x)
inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁰ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁹ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁸ : FiberBundle F₁ E₁
inst✝⁷ : (x : B) → TopologicalSpace (E₂ x)
ita : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝⁶ : FiberBundle F₂ E₂
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
h₁ : Continuous ⇑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂))
h₂ : Continuous ⇑(compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ).flip
\u22a2 ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))