ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Group/Prod.lean
G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Group G
inst✝⁴ : MeasurableMul₂ G
μ ν : Measure G
inst✝³ : SFinite ν
inst✝² : SFinite μ
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulRightInvariant
g : G
this : QuasiMeasurePreserving (fun h => h * g⁻¹) μ μ
\u22a2 QuasiMeasurePreserving (fun h => g * h) μ μ
-- Mathlib/MeasureTheory/Group/Prod.lean
G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Group G
inst✝⁴ : MeasurableMul₂ G
μ ν : Measure G
inst✝³ : SFinite ν
inst✝² : SFinite μ
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulRightInvariant
g : G
this✝ : QuasiMeasurePreserving (fun h => h * g⁻¹) μ μ
this : QuasiMeasurePreserving (Inv.inv ∘ (fun h => h * g⁻¹) ∘ Inv.inv) μ μ
\u22a2 QuasiMeasurePreserving (fun h => g * h) μ μ
-- Mathlib/MeasureTheory/Group/Prod.lean
G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Group G
inst✝⁴ : MeasurableMul₂ G
μ ν : Measure G
inst✝³ : SFinite ν
inst✝² : SFinite μ
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulRightInvariant
g : G
this✝ : QuasiMeasurePreserving (fun h => h * g⁻¹) μ μ
this : QuasiMeasurePreserving (fun x => g * x) μ μ
\u22a2 QuasiMeasurePreserving (fun h => g * h) μ μ
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ A
\u22a2 ∃ U ⊇ A, IsOpen U ∧ μ U < r
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ A
HA : μ A < ⊤
\u22a2 ∃ U ⊇ A, IsOpen U ∧ μ U < r
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ A
HA : μ A < ⊤
hm : ∀ (n : ℕ), MeasurableSet (s n)
\u22a2 ∃ U ⊇ A, IsOpen U ∧ μ U < r
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
\u22a2 ∃ U ⊇ ⋃ n, A n, IsOpen U ∧ μ U < r
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < r - μ (⋃ n, A n)
\u22a2 ∃ U ⊇ ⋃ n, A n, IsOpen U ∧ μ U < r
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
\u22a2 ∃ U ⊇ ⋃ n, A n, IsOpen U ∧ μ U < r
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
this : ∀ (n : ℕ), ∃ U ⊇ A n, IsOpen U ∧ μ U < μ (A n) + δ n
\u22a2 ∃ U ⊇ ⋃ n, A n, IsOpen U ∧ μ U < r
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
U : ℕ → Set α
hAU : ∀ (n : ℕ), U n ⊇ A n
hUo : ∀ (n : ℕ), IsOpen (U n)
hU : ∀ (n : ℕ), μ (U n) < μ (A n) + δ n
\u22a2 ∃ U ⊇ ⋃ n, A n, IsOpen U ∧ μ U < r
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ⊤
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
U : ℕ → Set α
hAU : ∀ (n : ℕ), U n ⊇ A n
hUo : ∀ (n : ℕ), IsOpen (U n)
hU : ∀ (n : ℕ), μ (U n) < μ (A n) + δ n
\u22a2 μ (⋃ n, U n) < r
-- Mathlib/ModelTheory/Basic.lean
a₀ a₁ a₂ : Type u
i : ℕ
\u22a2 lift.{v, u} #(Sequence₂ a₀ a₁ a₂ i) = #(Sequence₂ (ULift.{v, u} a₀) (ULift.{v, u} a₁) (ULift.{v, u} a₂) i)
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
r₁ r₂ : ℝ
h : |r₁| < |r₂|
\u22a2 (fun x => ‖r₁ ^ x‖) =o[atTop] fun x => ‖r₂ ^ x‖
-- Mathlib/LinearAlgebra/DirectSum/Finsupp.lean
R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
ι : Type u_5
κ : Type u_6
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Semiring S
inst✝² : Algebra R S
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
i : ι × κ
r : R
\u22a2 (finsuppTensorFinsupp' R ι κ).symm (Finsupp.single i r) = Finsupp.single i.1 1 ⊗ₜ[R] Finsupp.single i.2 r
-- Mathlib/LinearAlgebra/DirectSum/Finsupp.lean
R : Type u_1
S : Type u_2
M : Type u_3
N : Type u_4
ι : Type u_5
κ : Type u_6
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Semiring S
inst✝² : Algebra R S
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
i : ι × κ
r : R
\u22a2 (finsuppTensorFinsupp' R ι κ).symm (Finsupp.single i (1 * r)) = Finsupp.single i.1 1 ⊗ₜ[R] Finsupp.single i.2 r
-- Mathlib/Data/Nat/Log.lean
b n : ℕ
hb : 1 < b
hn : 2 ≤ n
\u22a2 (n + b - 1) / b < n
-- Mathlib/Data/Nat/Log.lean
b n : ℕ
hb : 1 < b
hn : 2 ≤ n
\u22a2 n + b ≤ n * b
-- Mathlib/Data/Sum/Interval.lean
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : OrderTop α
inst✝² : OrderBot β
inst✝¹ : LocallyFiniteOrder α
inst✝ : LocallyFiniteOrder β
a a₁ a₂ : α
b b₁ b₂ : β
\u22a2 Ico (inlₗ a₁) (inlₗ a₂) = map (Embedding.inl.trans toLex.toEmbedding) (Ico a₁ a₂)
-- Mathlib/Data/Sum/Interval.lean
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : OrderTop α
inst✝² : OrderBot β
inst✝¹ : LocallyFiniteOrder α
inst✝ : LocallyFiniteOrder β
a a₁ a₂ : α
b b₁ b₂ : β
\u22a2 Ico (inlₗ a₁) (inlₗ a₂) = map toLex.toEmbedding (map Embedding.inl (Ico a₁ a₂))
-- Mathlib/Topology/MetricSpace/ThickenedIndicator.lean
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
f : α → ℝ≥0 × ℝ≥0∞ := fun x => (1, infEdist x E / ENNReal.ofReal δ)
sub : ℝ≥0 × ℝ≥0∞ → ℝ≥0∞ := fun p => ↑p.1 - p.2
\u22a2 Continuous (sub ∘ f)
-- Mathlib/Topology/MetricSpace/ThickenedIndicator.lean
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
f : α → ℝ≥0 × ℝ≥0∞ := fun x => (1, infEdist x E / ENNReal.ofReal δ)
sub : ℝ≥0 × ℝ≥0∞ → ℝ≥0∞ := fun p => ↑p.1 - p.2
\u22a2 Continuous fun x => (f x).2
-- Mathlib/Topology/MetricSpace/ThickenedIndicator.lean
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
f : α → ℝ≥0 × ℝ≥0∞ := fun x => (1, infEdist x E / ENNReal.ofReal δ)
sub : ℝ≥0 × ℝ≥0∞ → ℝ≥0∞ := fun p => ↑p.1 - p.2
\u22a2 ENNReal.ofReal δ ≠ 0
-- Mathlib/Algebra/Order/Floor.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : FloorRing α
z : ℤ
a : α
k : Type u_4
inst✝¹ : LinearOrderedField k
inst✝ : FloorRing k
b : k
m n : ℕ
\u22a2 fract (↑m / ↑n) = ↑(m % n) / ↑n
-- Mathlib/Algebra/Order/Floor.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : FloorRing α
z : ℤ
a : α
k : Type u_4
inst✝¹ : LinearOrderedField k
inst✝ : FloorRing k
b : k
m : ℕ
\u22a2 fract (↑m / ↑0) = ↑(m % 0) / ↑0
-- Mathlib/Algebra/Order/Floor.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : FloorRing α
z : ℤ
a : α
k : Type u_4
inst✝¹ : LinearOrderedField k
inst✝ : FloorRing k
b : k
m n : ℕ
hn : n > 0
\u22a2 fract (↑m / ↑n) = ↑(m % n) / ↑n
-- Mathlib/Algebra/Order/Floor.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : FloorRing α
z : ℤ
a : α
k : Type u_4
inst✝¹ : LinearOrderedField k
inst✝ : FloorRing k
b : k
m n : ℕ
hn : n > 0
hn' : 0 < ↑n
\u22a2 fract (↑m / ↑n) = ↑(m % n) / ↑n
-- Mathlib/Algebra/Order/Floor.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : FloorRing α
z : ℤ
a : α
k : Type u_4
inst✝¹ : LinearOrderedField k
inst✝ : FloorRing k
b : k
m n : ℕ
hn : n > 0
hn' : 0 < ↑n
\u22a2 0 ≤ ↑(m % n) / ↑n
-- Mathlib/Algebra/Order/Floor.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : FloorRing α
z : ℤ
a : α
k : Type u_4
inst✝¹ : LinearOrderedField k
inst✝ : FloorRing k
b : k
m n : ℕ
hn : n > 0
hn' : 0 < ↑n
\u22a2 ↑(m % n) / ↑n < 1
-- Mathlib/Algebra/Order/Floor.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : FloorRing α
z : ℤ
a : α
k : Type u_4
inst✝¹ : LinearOrderedField k
inst✝ : FloorRing k
b : k
m n : ℕ
hn : n > 0
hn' : 0 < ↑n
\u22a2 ↑m / ↑n - ↑(m % n) / ↑n = ↑(↑m / ↑n)
-- Mathlib/SetTheory/Cardinal/ENat.lean
m : ℕ∞
\u22a2 ↑m = 1 ↔ m = 1
-- Mathlib/Data/List/MinMax.lean
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
l : List α
a m : α
\u22a2 l.maximum = ↑m ↔ m ∈ l ∧ ∀ (a : α), a ∈ l → a ≤ m
-- Mathlib/Data/List/MinMax.lean
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
l : List α
a m : α
\u22a2 (m ∈ l ∧ (∀ (a : α), a ∈ l → id a ≤ id m) ∧ ∀ (a : α), a ∈ l → id m ≤ id a → indexOf m l ≤ indexOf a l) ↔ m ∈ l ∧ ∀ (a : α), a ∈ l → a ≤ m
-- Mathlib/Data/List/MinMax.lean
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
l : List α
a m : α
\u22a2 m ∈ l → (∀ (a : α), a ∈ l → a ≤ m) → ∀ (a : α), a ∈ l → m ≤ a → indexOf m l ≤ indexOf a l
-- Mathlib/Data/List/MinMax.lean
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
l : List α
a✝¹ m : α
a✝ : m ∈ l
h : ∀ (a : α), a ∈ l → a ≤ m
a : α
hal : a ∈ l
hma : m ≤ a
\u22a2 indexOf m l ≤ indexOf a l
-- Mathlib/Data/Multiset/Powerset.lean
α : Type u_1
l : List α
x : Multiset α × Multiset α
h : x ∈ (powersetAux l).revzip
\u22a2 x.1 + x.2 = ↑l
-- Mathlib/Data/Multiset/Powerset.lean
α : Type u_1
l : List α
x : Multiset α × Multiset α
h : ∃ a ∈ l.sublists.revzip, Prod.map ofList ofList a = x
\u22a2 x.1 + x.2 = ↑l
-- Mathlib/Data/Multiset/Powerset.lean
α : Type u_1
l : List α
x : Multiset α × Multiset α
h : ∃ a b, (a, b) ∈ l.sublists.revzip ∧ (↑a, ↑b) = x
\u22a2 x.1 + x.2 = ↑l
-- Mathlib/Data/Multiset/Powerset.lean
α : Type u_1
l l₁ l₂ : List α
h : (l₁, l₂) ∈ l.sublists.revzip
\u22a2 (↑l₁, ↑l₂).1 + (↑l₁, ↑l₂).2 = ↑l
-- .lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean
α : Type u_1
le : α → α → Bool
a✝ : α
n✝ : Nat
s✝ : HeapNode α
h✝ : WF le a✝ s✝ n✝
res : Heap α
a : α
c s : HeapNode α
n : Nat
left✝ : ∀ [inst : TotalBLE le], le a✝ a = true
h : WF le a c n
h' : WF le a✝ s n
hr : Heap.WF le (node a c s).rank res
\u22a2 WF le a c ((node a c s).rank - 1)
-- .lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean
α : Type u_1
le : α → α → Bool
a✝ : α
n✝ : Nat
s✝ : HeapNode α
h✝ : WF le a✝ s✝ n✝
res : Heap α
a : α
c s : HeapNode α
n : Nat
left✝ : ∀ [inst : TotalBLE le], le a✝ a = true
h : WF le a c s.rank
h' : WF le a✝ s n
hr : Heap.WF le (node a c s).rank res
\u22a2 WF le a c ((node a c s).rank - 1)
-- Mathlib/CategoryTheory/Limits/Elements.lean
C : Type u
inst✝⁴ : Category.{v, u} C
A : C ⥤ Type w
I : Type u₁
inst✝³ : Category.{v₁, u₁} I
inst✝² : Small.{w, u₁} I
F : I ⥤ A.Elements
inst✝¹ : HasLimitsOfShape I C
inst✝ : PreservesLimitsOfShape I A
i i' : I
f : i ⟶ i'
\u22a2 ↑(((Functor.const I).obj ⟨limit (F ⋙ π A), liftedConeElement F⟩).map f ≫ (fun i => ⟨limit.π (F ⋙ π A) i, ⋯⟩) i') = ↑((fun i => ⟨limit.π (F ⋙ π A) i, ⋯⟩) i ≫ F.map f)
-- 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 arccos (⟪x, -y⟫_ℝ / (‖x‖ * ‖-y‖)) = π - arccos (⟪x, y⟫_ℝ / (‖x‖ * ‖y‖))
-- Mathlib/RingTheory/AdicCompletion/Functoriality.lean
R : Type u_1
inst✝¹² : CommRing R
I : Ideal R
M✝ : Type u_2
inst✝¹¹ : AddCommGroup M✝
inst✝¹⁰ : Module R M✝
N : Type u_3
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
P : Type u_4
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R P
T : Type u_5
inst✝⁵ : AddCommGroup T
inst✝⁴ : Module (AdicCompletion I R) T
ι : Type u_6
inst✝³ : DecidableEq ι
M : ι → Type u_7
inst✝² : (i : ι) → AddCommGroup (M i)
inst✝¹ : (i : ι) → Module R (M i)
inst✝ : Fintype ι
j : ι
x : AdicCauchySequence I (M j)
\u22a2 ((sumInv I M ∘ₗ sum I M) ∘ₗ lof (AdicCompletion I R) ι (fun i => AdicCompletion I (M i)) j) ((mk I (M j)) x) = (LinearMap.id ∘ₗ lof (AdicCompletion I R) ι (fun i => AdicCompletion I (M i)) j) ((mk I (M j)) x)
-- Mathlib/RingTheory/AdicCompletion/Functoriality.lean
R : Type u_1
inst✝¹² : CommRing R
I : Ideal R
M✝ : Type u_2
inst✝¹¹ : AddCommGroup M✝
inst✝¹⁰ : Module R M✝
N : Type u_3
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
P : Type u_4
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R P
T : Type u_5
inst✝⁵ : AddCommGroup T
inst✝⁴ : Module (AdicCompletion I R) T
ι : Type u_6
inst✝³ : DecidableEq ι
M : ι → Type u_7
inst✝² : (i : ι) → AddCommGroup (M i)
inst✝¹ : (i : ι) → Module R (M i)
inst✝ : Fintype ι
j : ι
x : AdicCauchySequence I (M j)
i : ι
\u22a2 (component (AdicCompletion I R) ι (fun i => AdicCompletion I (M i)) i) (((sumInv I M ∘ₗ sum I M) ∘ₗ lof (AdicCompletion I R) ι (fun i => AdicCompletion I (M i)) j) ((mk I (M j)) x)) = (component (AdicCompletion I R) ι (fun i => AdicCompletion I (M i)) i) ((LinearMap.id ∘ₗ lof (AdicCompletion I R) ι (fun i => AdicCompletion I (M i)) j) ((mk I (M j)) x))
-- Mathlib/RingTheory/AdicCompletion/Functoriality.lean
R : Type u_1
inst✝¹² : CommRing R
I : Ideal R
M✝ : Type u_2
inst✝¹¹ : AddCommGroup M✝
inst✝¹⁰ : Module R M✝
N : Type u_3
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
P : Type u_4
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R P
T : Type u_5
inst✝⁵ : AddCommGroup T
inst✝⁴ : Module (AdicCompletion I R) T
ι : Type u_6
inst✝³ : DecidableEq ι
M : ι → Type u_7
inst✝² : (i : ι) → AddCommGroup (M i)
inst✝¹ : (i : ι) → Module R (M i)
inst✝ : Fintype ι
j : ι
x : AdicCauchySequence I (M j)
i : ι
n : ℕ
\u22a2 ↑((component (AdicCompletion I R) ι (fun i => AdicCompletion I (M i)) i) (((sumInv I M ∘ₗ sum I M) ∘ₗ lof (AdicCompletion I R) ι (fun i => AdicCompletion I (M i)) j) ((mk I (M j)) x))) n = ↑((component (AdicCompletion I R) ι (fun i => AdicCompletion I (M i)) i) ((LinearMap.id ∘ₗ lof (AdicCompletion I R) ι (fun i => AdicCompletion I (M i)) j) ((mk I (M j)) x))) n
-- Mathlib/RingTheory/AdicCompletion/Functoriality.lean
R : Type u_1
inst✝¹² : CommRing R
I : Ideal R
M✝ : Type u_2
inst✝¹¹ : AddCommGroup M✝
inst✝¹⁰ : Module R M✝
N : Type u_3
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
P : Type u_4
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R P
T : Type u_5
inst✝⁵ : AddCommGroup T
inst✝⁴ : Module (AdicCompletion I R) T
ι : Type u_6
inst✝³ : DecidableEq ι
M : ι → Type u_7
inst✝² : (i : ι) → AddCommGroup (M i)
inst✝¹ : (i : ι) → Module R (M i)
inst✝ : Fintype ι
j : ι
x : AdicCauchySequence I (M j)
i : ι
n : ℕ
\u22a2 Submodule.Quotient.mk ((component R ι M i) ((lof R ι M j) (↑x n))) = ↑((component (AdicCompletion I R) ι (fun i => AdicCompletion I (M i)) i) ((lof (AdicCompletion I R) ι (fun i => AdicCompletion I (M i)) j) ((mk I (M j)) x))) n
-- Mathlib/RingTheory/AdicCompletion/Functoriality.lean
R : Type u_1
inst✝¹² : CommRing R
I : Ideal R
M✝ : Type u_2
inst✝¹¹ : AddCommGroup M✝
inst✝¹⁰ : Module R M✝
N : Type u_3
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
P : Type u_4
inst✝⁷ : AddCommGroup P
inst✝⁶ : Module R P
T : Type u_5
inst✝⁵ : AddCommGroup T
inst✝⁴ : Module (AdicCompletion I R) T
ι : Type u_6
inst✝³ : DecidableEq ι
M : ι → Type u_7
inst✝² : (i : ι) → AddCommGroup (M i)
inst✝¹ : (i : ι) → Module R (M i)
inst✝ : Fintype ι
j : ι
x : AdicCauchySequence I (M j)
i : ι
n : ℕ
\u22a2 Submodule.Quotient.mk (if h : j = i then Eq.recOn h (↑x n) else 0) = ↑(if h : j = i then Eq.recOn h ((mk I (M j)) x) else 0) n
-- Mathlib/SetTheory/Game/PGame.lean
xl✝ xr✝ : Type u
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
h : mk xl xr xL xR ≤ mk yl yr yL yR
\u22a2 mk xl xr xL xR + mk zl zr zL zR ≤ mk yl yr yL yR + mk zl zr zL zR
-- Mathlib/SetTheory/Game/PGame.lean
xl✝ xr✝ : Type u
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
h : mk xl xr xL xR ≤ mk yl yr yL yR
i : xl
\u22a2 (∃ i', (mk xl xr xL xR + mk zl zr zL zR).moveLeft (Sum.inl i) ≤ (mk yl yr yL yR + mk zl zr zL zR).moveLeft i') ∨ ∃ j, ((mk xl xr xL xR + mk zl zr zL zR).moveLeft (Sum.inl i)).moveRight j ≤ mk yl yr yL yR + mk zl zr zL zR
-- Mathlib/SetTheory/Game/PGame.lean
xl✝ xr✝ : Type u
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
h : mk xl xr xL xR ≤ mk yl yr yL yR
i : zl
\u22a2 (∃ i', (mk xl xr xL xR + mk zl zr zL zR).moveLeft (Sum.inr i) ≤ (mk yl yr yL yR + mk zl zr zL zR).moveLeft i') ∨ ∃ j, ((mk xl xr xL xR + mk zl zr zL zR).moveLeft (Sum.inr i)).moveRight j ≤ mk yl yr yL yR + mk zl zr zL zR
-- Mathlib/SetTheory/Game/PGame.lean
xl✝ xr✝ : Type u
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
h : mk xl xr xL xR ≤ mk yl yr yL yR
i : yr
\u22a2 (∃ i_1, mk xl xr xL xR + mk zl zr zL zR ≤ ((mk yl yr yL yR + mk zl zr zL zR).moveRight (Sum.inl i)).moveLeft i_1) ∨ ∃ j', (mk xl xr xL xR + mk zl zr zL zR).moveRight j' ≤ (mk yl yr yL yR + mk zl zr zL zR).moveRight (Sum.inl i)
-- Mathlib/SetTheory/Game/PGame.lean
xl✝ xr✝ : Type u
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
h : mk xl xr xL xR ≤ mk yl yr yL yR
i : zr
\u22a2 (∃ i_1, mk xl xr xL xR + mk zl zr zL zR ≤ ((mk yl yr yL yR + mk zl zr zL zR).moveRight (Sum.inr i)).moveLeft i_1) ∨ ∃ j', (mk xl xr xL xR + mk zl zr zL zR).moveRight j' ≤ (mk yl yr yL yR + mk zl zr zL zR).moveRight (Sum.inr i)
-- Mathlib/Data/Finset/Pointwise/Interval.lean
α : Type u_1
inst✝⁵ : Mul α
inst✝⁴ : PartialOrder α
inst✝³ : DecidableEq α
inst✝² : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝¹ : CovariantClass α α (Function.swap HMul.hMul) LT.lt
inst✝ : LocallyFiniteOrderBot α
a b : α
\u22a2 ↑(Iic a * Iio b) ⊆ ↑(Iio (a * b))
-- Mathlib/Data/List/DropRight.lean
α : Type u_1
p : α → Bool
l : List α
n i j : ℕ
\u22a2 (l.rdrop i).rdrop j = l.rdrop (i + j)
-- Mathlib/Algebra/Homology/ExactSequence.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : HasZeroMorphisms C
n : ℕ
S : ComposableArrows C n
hS : S.IsComplex
i : ℕ
hk : autoParam (i + 1 + 1 ≤ n) _auto✝
\u22a2 S.map' i (i + 1) ⋯ ⋯ ≫ S.map' (i + 1) (i + 1 + 1) ⋯ hk = 0
-- Mathlib/Topology/Algebra/InfiniteSum/Group.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝² : CommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalGroup α
f g : β → α
a a₁ a₂ : α
h : HasProd f a
\u22a2 HasProd (fun b => (f b)⁻¹) a⁻¹
-- Mathlib/Algebra/Polynomial/Roots.lean
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
p✝ q p : R[X]
\u22a2 (-p).roots = p.roots
-- Mathlib/Data/Finset/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s✝ t u v : Finset α
a✝ b✝ a b : α
s : Finset α
x : α
\u22a2 x ∈ insert a (insert b s) ↔ x ∈ insert b (insert a s)
-- Mathlib/Algebra/Module/LocalizedModuleIntegers.lean
R : Type u_1
inst✝⁶ : CommSemiring R
S : Submonoid R
M : Type u_2
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
M' : Type u_3
inst✝³ : AddCommMonoid M'
inst✝² : Module R M'
f : M →ₗ[R] M'
inst✝¹ : IsLocalizedModule S f
inst✝ : DecidableEq M
s : Finset M'
\u22a2 ⇑f '' ↑(Finset.image (fun t => integerMultiple S f s id t) s.attach) = commonDenomOfFinset S f s • ↑s
-- Mathlib/Algebra/Module/LocalizedModuleIntegers.lean
R : Type u_1
inst✝⁶ : CommSemiring R
S : Submonoid R
M : Type u_2
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
M' : Type u_3
inst✝³ : AddCommMonoid M'
inst✝² : Module R M'
f : M →ₗ[R] M'
inst✝¹ : IsLocalizedModule S f
inst✝ : DecidableEq M
s : Finset M'
\u22a2 ⇑f '' ((fun t => integerMultiple S f s id t) '' ↑s.attach) = commonDenomOfFinset S f s • ↑s
-- Mathlib/MeasureTheory/Integral/Layercake.lean
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
s : Set α
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
\u22a2 ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t)
-- Mathlib/MeasureTheory/Integral/Layercake.lean
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
s : Set α
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᶠ[ae (volume.restrict (Ioi 0))] G
\u22a2 ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t)
-- Mathlib/MeasureTheory/Integral/Layercake.lean
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
s : Set α
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᶠ[ae (volume.restrict (Ioi 0))] G
g_eq_G_on : ∀ (t : ℝ), g =ᶠ[ae (volume.restrict (Ioc 0 t))] G
\u22a2 ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t)