ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/GroupTheory/HNNExtension.lean
G : Type u_1
inst✝² : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝¹ : Group H
M : Type u_3
inst✝ : Monoid M
w₁ w₂ : ReducedWord G A B
hprod : ReducedWord.prod φ w₁ = ReducedWord.prod φ w₂
d : TransversalPair G A B
w₁' : NormalWord d
hw₁'1 : ReducedWord.prod φ w₁'.toReducedWord = ReducedWord.prod φ w₁
hw₁'2 : List.map Prod.fst w₁'.toList = List.map Prod.fst w₁.toList
hw₁'3 : ∀ u ∈ Option.map Prod.fst w₁.toList.head?, w₁'.head⁻¹ * w₁.head ∈ toSubgroup A B (-u)

⊢ List.map Prod.fst w₁.toList = List.map Prod.fst w₂.toList ∧ ∀ u ∈ Option.map Prod.fst w₁.toList.head?, w₁.head⁻¹ * w₂.head ∈ toSubgroup A B (-u)
-- Mathlib/GroupTheory/HNNExtension.lean
G : Type u_1
inst✝² : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝¹ : Group H
M : Type u_3
inst✝ : Monoid M
w₁ w₂ : ReducedWord G A B
hprod : ReducedWord.prod φ w₁ = ReducedWord.prod φ w₂
d : TransversalPair G A B
w₁' : NormalWord d
hw₁'1 : ReducedWord.prod φ w₁'.toReducedWord = ReducedWord.prod φ w₁
hw₁'2 : List.map Prod.fst w₁'.toList = List.map Prod.fst w₁.toList
hw₁'3 : ∀ u ∈ Option.map Prod.fst w₁.toList.head?, w₁'.head⁻¹ * w₁.head ∈ toSubgroup A B (-u)
w₂' : NormalWord d
hw₂'1 : ReducedWord.prod φ w₂'.toReducedWord = ReducedWord.prod φ w₂
hw₂'2 : List.map Prod.fst w₂'.toList = List.map Prod.fst w₂.toList
hw₂'3 : ∀ u ∈ Option.map Prod.fst w₂.toList.head?, w₂'.head⁻¹ * w₂.head ∈ toSubgroup A B (-u)

⊢ List.map Prod.fst w₁.toList = List.map Prod.fst w₂.toList ∧ ∀ u ∈ Option.map Prod.fst w₁.toList.head?, w₁.head⁻¹ * w₂.head ∈ toSubgroup A B (-u)
-- Mathlib/GroupTheory/HNNExtension.lean
G : Type u_1
inst✝² : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝¹ : Group H
M : Type u_3
inst✝ : Monoid M
w₁ w₂ : ReducedWord G A B
hprod : ReducedWord.prod φ w₁ = ReducedWord.prod φ w₂
d : TransversalPair G A B
w₁' : NormalWord d
hw₁'1 : ReducedWord.prod φ w₁'.toReducedWord = ReducedWord.prod φ w₁
hw₁'2 : List.map Prod.fst w₁'.toList = List.map Prod.fst w₁.toList
hw₁'3 : ∀ u ∈ Option.map Prod.fst w₁.toList.head?, w₁'.head⁻¹ * w₁.head ∈ toSubgroup A B (-u)
w₂' : NormalWord d
hw₂'1 : ReducedWord.prod φ w₂'.toReducedWord = ReducedWord.prod φ w₂
hw₂'2 : List.map Prod.fst w₂'.toList = List.map Prod.fst w₂.toList
hw₂'3 : ∀ u ∈ Option.map Prod.fst w₂.toList.head?, w₂'.head⁻¹ * w₂.head ∈ toSubgroup A B (-u)
this : w₁' = w₂'

⊢ List.map Prod.fst w₁.toList = List.map Prod.fst w₂.toList ∧ ∀ u ∈ Option.map Prod.fst w₁.toList.head?, w₁.head⁻¹ * w₂.head ∈ toSubgroup A B (-u)
-- Mathlib/GroupTheory/HNNExtension.lean
G : Type u_1
inst✝² : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝¹ : Group H
M : Type u_3
inst✝ : Monoid M
w₁ w₂ : ReducedWord G A B
hprod : ReducedWord.prod φ w₁ = ReducedWord.prod φ w₂
d : TransversalPair G A B
w₁' : NormalWord d
hw₁'1 : ReducedWord.prod φ w₁'.toReducedWord = ReducedWord.prod φ w₁
hw₁'2 : List.map Prod.fst w₁'.toList = List.map Prod.fst w₁.toList
hw₁'3 : ∀ u ∈ Option.map Prod.fst w₁.toList.head?, w₁'.head⁻¹ * w₁.head ∈ toSubgroup A B (-u)
hw₂'1 : ReducedWord.prod φ w₁'.toReducedWord = ReducedWord.prod φ w₂
hw₂'2 : List.map Prod.fst w₁'.toList = List.map Prod.fst w₂.toList
hw₂'3 : ∀ u ∈ Option.map Prod.fst w₂.toList.head?, w₁'.head⁻¹ * w₂.head ∈ toSubgroup A B (-u)

⊢ ∀ u ∈ Option.map Prod.fst w₁.toList.head?, w₁.head⁻¹ * w₂.head ∈ toSubgroup A B (-u)
-- Mathlib/GroupTheory/HNNExtension.lean
G : Type u_1
inst✝² : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝¹ : Group H
M : Type u_3
inst✝ : Monoid M
w₁ w₂ : ReducedWord G A B
hprod : ReducedWord.prod φ w₁ = ReducedWord.prod φ w₂
d : TransversalPair G A B
w₁' : NormalWord d
hw₁'1 : ReducedWord.prod φ w₁'.toReducedWord = ReducedWord.prod φ w₁
hw₁'2 : List.map Prod.fst w₁'.toList = List.map Prod.fst w₁.toList
hw₂'1 : ReducedWord.prod φ w₁'.toReducedWord = ReducedWord.prod φ w₂
hw₂'2 : List.map Prod.fst w₁'.toList = List.map Prod.fst w₂.toList
hw₁'3 : ∀ u ∈ Option.map Prod.fst w₁.toList.head?, w₁'.head • ↑(toSubgroup A B (-u)) = w₁.head • ↑(toSubgroup A B (-u))
hw₂'3 : ∀ u ∈ Option.map Prod.fst w₂.toList.head?, w₁'.head • ↑(toSubgroup A B (-u)) = w₂.head • ↑(toSubgroup A B (-u))

⊢ ∀ u ∈ Option.map Prod.fst w₁.toList.head?, w₁.head • ↑(toSubgroup A B (-u)) = w₂.head • ↑(toSubgroup A B (-u))
-- Mathlib/GroupTheory/HNNExtension.lean
G : Type u_1
inst✝² : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝¹ : Group H
M : Type u_3
inst✝ : Monoid M
w₁ w₂ : ReducedWord G A B
hprod : ReducedWord.prod φ w₁ = ReducedWord.prod φ w₂
d : TransversalPair G A B
w₁' : NormalWord d
hw₁'1 : ReducedWord.prod φ w₁'.toReducedWord = ReducedWord.prod φ w₁
hw₁'2 : List.map Prod.fst w₁'.toList = List.map Prod.fst w₁.toList
hw₂'1 : ReducedWord.prod φ w₁'.toReducedWord = ReducedWord.prod φ w₂
hw₂'2 : List.map Prod.fst w₁'.toList = List.map Prod.fst w₂.toList
hw₁'3 : ∀ u ∈ Option.map Prod.fst w₁.toList.head?, w₁'.head • ↑(toSubgroup A B (-u)) = w₁.head • ↑(toSubgroup A B (-u))
hw₂'3 : ∀ u ∈ Option.map Prod.fst w₂.toList.head?, w₁'.head • ↑(toSubgroup A B (-u)) = w₂.head • ↑(toSubgroup A B (-u))
u : ℤˣ
hu : u ∈ Option.map Prod.fst w₁.toList.head?

⊢ u ∈ Option.map Prod.fst w₂.toList.head?
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
inst✝ : ConditionallyCompleteLattice α
f : Filter β
u : β → α
p : β → Prop

⊢ blimsup u f p = limsup (u ∘ Subtype.val) (comap Subtype.val f)
-- Mathlib/Analysis/Convex/EGauge.lean
𝕜 : Type u_1
inst✝² : NormedDivisionRing 𝕜
α : Type u_2
E : Type u_3
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
c : 𝕜
s✝ t : Set E
x✝ y : E
r : ℝ≥0∞
hc : c ≠ 0
s : Set E
x : E

⊢ egauge 𝕜 (c • s) x = egauge 𝕜 s x / ↑‖c‖₊
-- Mathlib/Analysis/Convex/EGauge.lean
𝕜 : Type u_1
inst✝² : NormedDivisionRing 𝕜
α : Type u_2
E : Type u_3
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
c : 𝕜
s✝ t : Set E
x✝ y : E
r : ℝ≥0∞
hc : c ≠ 0
s : Set E
x : E

⊢ egauge 𝕜 (c • s) x ≤ egauge 𝕜 s x / ↑‖c‖₊
-- Mathlib/Analysis/Convex/EGauge.lean
𝕜 : Type u_1
inst✝² : NormedDivisionRing 𝕜
α : Type u_2
E : Type u_3
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
c : 𝕜
s✝ t : Set E
x✝ y : E
r : ℝ≥0∞
hc : c ≠ 0
s : Set E
x : E

⊢ egauge 𝕜 (c • s) x * ↑‖c‖₊ ≤ egauge 𝕜 s x
-- Mathlib/NumberTheory/Pell.lean
d : ℤ
a : Solution₁ d
h : IsFundamental a
n : ℤ

⊢ a ^ n = 1 ↔ n = 0
-- Mathlib/NumberTheory/Pell.lean
d : ℤ
a : Solution₁ d
h : IsFundamental a
n : ℤ

⊢ a ^ n = a ^ 0 ↔ n = 0
-- Mathlib/Algebra/GroupWithZero/NeZero.lean
M₀ : Type u_1
M₀' : Type u_2
inst✝² : MulZeroOneClass M₀
inst✝¹ : Nontrivial M₀
G₀ : Type u_3
inst✝ : GroupWithZero G₀
a : G₀
h : a ≠ 0
a_eq_0 : a⁻¹ = 0

⊢ False
-- Mathlib/Algebra/GroupWithZero/NeZero.lean
M₀ : Type u_1
M₀' : Type u_2
inst✝² : MulZeroOneClass M₀
inst✝¹ : Nontrivial M₀
G₀ : Type u_3
inst✝ : GroupWithZero G₀
a : G₀
h : a ≠ 0
a_eq_0 : a⁻¹ = 0
this : a * a⁻¹ = 1

⊢ False
-- Mathlib/Analysis/SpecialFunctions/Log/Base.lean
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
h : 0 < x
h₁ : 0 < y

⊢ logb b x ≤ logb b y ↔ y ≤ x
-- Mathlib/Dynamics/PeriodicPts.lean
α : Type u_1
β : Type u_2
f fa : α → α
fb : β → β
x y : α
m n : ℕ
hx : IsPeriodicPt f n x
hy : IsPeriodicPt f n y
hn : 0 < n
h : f x = f y

⊢ x = y
-- Mathlib/Probability/Cdf.lean
f : StieltjesFunction
hf0 : Tendsto (↑f) atBot (𝓝 0)
hf1 : Tendsto (↑f) atTop (𝓝 1)

⊢ cdf f.measure = f
-- Mathlib/Probability/Cdf.lean
f : StieltjesFunction
hf0 : Tendsto (↑f) atBot (𝓝 0)
hf1 : Tendsto (↑f) atTop (𝓝 1)

⊢ (cdf f.measure).measure = f.measure
-- Mathlib/Probability/Cdf.lean
f : StieltjesFunction
hf0 : Tendsto (↑f) atBot (𝓝 0)
hf1 : Tendsto (↑f) atTop (𝓝 1)
h_prob : IsProbabilityMeasure f.measure

⊢ (cdf f.measure).measure = f.measure
-- Mathlib/Topology/PartitionOfUnity.lean
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f : BumpCovering ι X s
i : ι

⊢ Continuous (f.toPOUFun i)
-- Mathlib/Topology/PartitionOfUnity.lean
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f : BumpCovering ι X s
i : ι

⊢ LocallyFinite fun i => mulSupport fun x => 1 - (f i) x
-- Mathlib/Topology/PartitionOfUnity.lean
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f : BumpCovering ι X s
i : ι

⊢ LocallyFinite fun i => support ⇑(f i)
-- Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean
K : Type u_1
n✝ : ℕ
inst✝ : LinearOrderedField K
c : ContFract K
n : ℕ

⊢ (↑↑c).convs.get n = Stream'.get (↑↑c).convs' n
-- Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean
K : Type u_1
n✝ : ℕ
inst✝ : LinearOrderedField K
c : ContFract K
n : ℕ

⊢ ∀ {gp : Pair K} {m : ℕ}, m < n → (↑↑c).s.get? m = some gp → 0 < gp.a ∧ 0 < gp.b
-- Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean
K : Type u_1
n✝ : ℕ
inst✝ : LinearOrderedField K
c : ContFract K
n : ℕ
gp : Pair K
m : ℕ
a✝ : m < n
s_nth_eq : (↑↑c).s.get? m = some gp

⊢ 0 < gp.a ∧ 0 < gp.b
-- Mathlib/NumberTheory/FLT/Four.lean
r s : ℤ
h2 : IsCoprime s r

⊢ IsCoprime (r * r + s * s) r
-- Mathlib/RingTheory/Coprime/Basic.lean
R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R

⊢ IsCoprime (y + z * x) x
-- Mathlib/Data/Nat/Choose/Sum.lean
R : Type u_1
m i : ℕ
hi : i ∈ range (m + 1)

⊢ i ≤ 2 * m + 1
-- Mathlib/Data/Nat/Choose/Sum.lean
R : Type u_1
m : ℕ
this : ∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i) = ∑ i ∈ range (m + 1), (2 * m + 1).choose i

⊢ 2 * ∑ i ∈ range (m + 1), (2 * m + 1).choose i = ∑ i ∈ range (m + 1), (2 * m + 1).choose i + ∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i)
-- Mathlib/Data/Nat/Choose/Sum.lean
R : Type u_1
m : ℕ
this : ∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i) = ∑ i ∈ range (m + 1), (2 * m + 1).choose i

⊢ ∑ i ∈ range (m + 1), (2 * m + 1).choose i + ∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i) = ∑ i ∈ range (m + 1), (2 * m + 1).choose i + ∑ i ∈ Ico (m + 1) (2 * m + 2), (2 * m + 1).choose i
-- Mathlib/Data/Nat/Choose/Sum.lean
R : Type u_1
m : ℕ
this : ∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i) = ∑ i ∈ range (m + 1), (2 * m + 1).choose i

⊢ ∑ i ∈ Ico 0 (m + 1), (2 * m + 1).choose i + ∑ j ∈ Ico (2 * m + 1 + 1 - (m + 1)) (2 * m + 1 + 1 - 0), (2 * m + 1).choose j = ∑ i ∈ Ico 0 (m + 1), (2 * m + 1).choose i + ∑ i ∈ Ico (m + 1) (2 * m + 2), (2 * m + 1).choose i
-- Mathlib/Data/Nat/Choose/Sum.lean
R : Type u_1
m : ℕ
this : ∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i) = ∑ i ∈ range (m + 1), (2 * m + 1).choose i

⊢ m + 1 ≤ 2 * m + 1 + 1
-- Mathlib/Data/Nat/Choose/Sum.lean
R : Type u_1
m : ℕ
this : ∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i) = ∑ i ∈ range (m + 1), (2 * m + 1).choose i

⊢ 2 ^ (2 * m + 1) = 2 * 4 ^ m
-- Mathlib/Data/Nat/Choose/Sum.lean
R : Type u_1
m : ℕ
this : ∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i) = ∑ i ∈ range (m + 1), (2 * m + 1).choose i

⊢ 2 * (2 ^ 2) ^ m = 2 * 4 ^ m
-- Mathlib/Data/Setoid/Basic.lean
α : Type u_1
β : Type u_2
r✝ : Setoid α
f✝ f : α → β
r : Setoid β
x y : α

⊢ Setoid.r x y ↔ ⟦f x⟧ = ⟦f y⟧
-- Mathlib/Data/Setoid/Basic.lean
α : Type u_1
β : Type u_2
r✝ : Setoid α
f✝ f : α → β
r : Setoid β
x y : α

⊢ Setoid.r x y ↔ f x ≈ f y
-- Mathlib/LinearAlgebra/Matrix/ToLin.lean
R : Type u_1
inst✝⁹ : CommSemiring R
l : Type u_2
m : Type u_3
n : Type u_4
inst✝⁸ : Fintype n
inst✝⁷ : Fintype m
inst✝⁶ : DecidableEq n
M₁ : Type u_5
M₂ : Type u_6
inst✝⁵ : AddCommMonoid M₁
inst✝⁴ : AddCommMonoid M₂
inst✝³ : Module R M₁
inst✝² : Module R M₂
v₁ : Basis n R M₁
v₂ : Basis m R M₂
M₃ : Type u_7
inst✝¹ : AddCommMonoid M₃
inst✝ : Module R M₃
v₃ : Basis l R M₃
M : Matrix n n R
v : M₁

⊢ v₁.equivFun.symm ((toLinAlgEquiv' M) ⇑(v₁.repr v)) = ∑ j : n, (M *ᵥ ⇑(v₁.repr v)) j • v₁ j
-- Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hx : x₁ = x₂
hy : y₁ = W.negY x₂ y₂

⊢ some h₁ + some h₂ = 0
-- Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hx : x₁ = x₂
hy : y₁ = W.negY x₂ y₂

⊢ (if h : x₁ = x₂ ∧ y₁ = W.negY x₂ y₂ then 0 else some ⋯) = 0
-- Mathlib/Data/Complex/Module.lean
A : Type u_1
inst✝³ : AddCommGroup A
inst✝² : Module ℂ A
inst✝¹ : StarAddMonoid A
inst✝ : StarModule ℂ A
x : A
hx : IsSelfAdjoint x

⊢ ℑ x = 0
-- Mathlib/Data/Nat/Log.lean
b n : ℕ

⊢ 0 < log b n ↔ b ≤ n ∧ 1 < b
-- Mathlib/Topology/Order/DenselyOrdered.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
h : a < b

⊢ frontier (Ioo a b) = {a, b}
-- Mathlib/MeasureTheory/SetSemiring.lean
α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C

⊢ Disjoint I (hC.diffFinset₀ hs hI)
-- Mathlib/MeasureTheory/SetSemiring.lean
α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
h : ¬Disjoint I (hC.diffFinset₀ hs hI)

⊢ False
-- Mathlib/MeasureTheory/SetSemiring.lean
α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
h : ∃ a ∈ I, a ∈ hC.diffFinset₀ hs hI

⊢ False
-- Mathlib/MeasureTheory/SetSemiring.lean
α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
u : Set α
huI : u ∈ I
hu_diffFinset₀ : u ∈ hC.diffFinset₀ hs hI

⊢ False
-- Mathlib/MeasureTheory/SetSemiring.lean
α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
u : Set α
huI : u ∈ I
hu_diffFinset₀ : u ∈ hC.diffFinset₀ hs hI
h_disj : u ≤ ⊥

⊢ False
-- Mathlib/MeasureTheory/SetSemiring.lean
α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
u : Set α
huI : u ∈ I
hu_diffFinset₀ : u ∈ hC.diffFinset₀ hs hI
h_disj : u = ∅

⊢ False
-- Mathlib/MeasureTheory/SetSemiring.lean
α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
u : Set α
huI : u ∈ I
hu_diffFinset₀ : u ∈ hC.diffFinset₀ hs hI
h_disj : u = ∅

⊢ ∅ ∈ hC.diffFinset₀ hs hI
-- Mathlib/Topology/MetricSpace/GromovHausdorff.lean
X : Type u
inst✝² : MetricSpace X
inst✝¹ : CompactSpace X
inst✝ : Nonempty X
p : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)

⊢ ⟦p⟧ = toGHSpace X ↔ ∃ Ψ, Isometry Ψ ∧ range Ψ = ↑p
-- Mathlib/Topology/MetricSpace/GromovHausdorff.lean
X : Type u
inst✝² : MetricSpace X
inst✝¹ : CompactSpace X
inst✝ : Nonempty X
p : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)

⊢ p ≈ NonemptyCompacts.kuratowskiEmbedding X ↔ ∃ Ψ, Isometry Ψ ∧ range Ψ = ↑p
-- Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean
α : Type u_1
β : Type u_2
l : Filter α
k f g g' : α → β
inst✝² : TopologicalSpace β
inst✝¹ : LinearOrderedField β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
n : ℕ

⊢ SuperpolynomialDecay l k (f * k ^ n) ↔ SuperpolynomialDecay l k f
-- Mathlib/Topology/Order/Monotone.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : Nonempty γ
ι : Sort u_4
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : ContinuousAt f (iInf g)
Mf : Monotone f
bdd : autoParam (BddBelow (range g)) _auto✝

⊢ f (⨅ i, g i) = ⨅ i, f (g i)
-- Mathlib/Topology/Order/Monotone.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : Nonempty γ
ι : Sort u_4
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : ContinuousAt f (iInf g)
Mf : Monotone f
bdd : autoParam (BddBelow (range g)) _auto✝

⊢ sInf (range (f ∘ g)) = sInf (range fun i => f (g i))
-- Mathlib/Combinatorics/SimpleGraph/Triangle/Removal.lean
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
s t : Finset α
P : Finpartition univ
ε : ℝ
hε : ε ≤ 0

⊢ triangleRemovalBound ε ≤ 0
-- Mathlib/Probability/Distributions/Gamma.lean
y z : ℝ
f : ℝ → ℝ≥0∞
hzy : z ≤ y

⊢ ∫⁻ (x : ℝ) in Iic y, f x = (∫⁻ (x : ℝ) in Iio z, f x) + ∫⁻ (x : ℝ) in Icc z y, f x
-- Mathlib/Probability/Distributions/Gamma.lean
y z : ℝ
f : ℝ → ℝ≥0∞
hzy : z ≤ y

⊢ Disjoint (Iio z) (Icc z y)
-- Mathlib/Probability/Distributions/Gamma.lean
y z : ℝ
f : ℝ → ℝ≥0∞
hzy : z ≤ y

⊢ Iio z ∩ Icc z y ⊆ ∅
-- Mathlib/Analysis/SpecialFunctions/Gaussian/PoissonSummation.lean
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
ha : 0 < a

⊢ ∑' (n : ℤ), rexp (-π * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), rexp (-π / a * ↑n ^ 2)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean

⊢ cos 0 = 1
-- Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean
R : Type u
inst✝ : CommRing R
X Y : ModuleCat R
f : X ⟶ Y
Z : ModuleCat R

⊢ f ▷ Z ≫ (Y.braiding Z).hom = (X.braiding Z).hom ≫ Z ◁ f
-- Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean
R : Type u
inst✝ : CommRing R
X Y : ModuleCat R
f : X ⟶ Y
Z : ModuleCat R

⊢ f ▷ Z ≫ (Y.braiding Z).hom = (X.braiding Z).hom ≫ (𝟙 Z ⊗ f)
-- Mathlib/Data/Finset/NatAntidiagonal.lean
n : ℕ

⊢ antidiagonal n = map { toFun := fun i => (n - i, i), inj' := ⋯ } (range (n + 1))
-- Mathlib/Data/Finset/NatAntidiagonal.lean
n : ℕ

⊢ map ({ toFun := fun i => (i, n - i), inj' := ⋯ }.trans { toFun := Prod.swap, inj' := ⋯ }) (range (n + 1)) = map { toFun := fun i => (n - i, i), inj' := ⋯ } (range (n + 1))
-- Mathlib/RingTheory/Adjoin/Basic.lean
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t✝ s : Set A
t : Submodule R A

⊢ Subalgebra.toSubmodule (adjoin R s) ≤ t ↔ ↑(Submonoid.closure s) ⊆ ↑t
-- Mathlib/Data/Finsupp/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝ : AddCommMonoid M
v v₁ v₂ : α →₀ M
f : α → β
x : α →₀ M
a : β
h : a ∉ Set.range f

⊢ (mapDomain f x) a = 0
-- Mathlib/Data/Finsupp/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝ : AddCommMonoid M
v v₁ v₂ : α →₀ M
f : α → β
x : α →₀ M
a : β
h : a ∉ Set.range f

⊢ ∑ a_1 ∈ x.support, (single (f a_1) (x a_1)) a = 0
-- Mathlib/Analysis/InnerProductSpace/TwoDim.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o✝ : Orientation ℝ E (Fin 2)
w z : ℂ
o : Orientation ℝ ℂ (Fin 2) := Complex.orientation

⊢ (Complex.orientation.areaForm w) z = ((starRingEnd ℂ) w * z).im
-- Mathlib/Analysis/InnerProductSpace/TwoDim.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o✝ : Orientation ℝ E (Fin 2)
w z : ℂ
o : Orientation ℝ ℂ (Fin 2) := Complex.orientation

⊢ w.re * z.im - z.re * w.im = w.re * z.im + -w.im * z.re
-- Mathlib/RingTheory/WittVector/MulCoeff.lean
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ

⊢ (polyOfInterest p n).vars ⊆ univ ×ˢ range (n + 1)
-- Mathlib/RingTheory/WittVector/MulCoeff.lean
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ

⊢ (↑p ^ (n + 1) * (wittMul p (n + 1) + ↑p ^ (n + 1) * X (0, n + 1) * X (1, n + 1) - X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) - X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)))).vars ⊆ univ ×ˢ range (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
v : ι → M
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid M'
inst✝³ : AddCommMonoid M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hf : ∀ (l : ι →₀ R), (Finsupp.total ι M R v) l = 0 → l = 0
s : Finset ι
g : ι → R
hg : ∑ i ∈ s, g i • v i = 0
i : ι
his : i ∈ s

⊢ (Finsupp.total ι M R v) (∑ i ∈ s, Finsupp.single i (g i)) = 0
-- 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
v : ι → M
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid M'
inst✝³ : AddCommMonoid M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hf : ∀ (l : ι →₀ R), (Finsupp.total ι M R v) l = 0 → l = 0
s : Finset ι
g : ι → R
hg : ∑ i ∈ s, g i • v i = 0
i : ι
his : i ∈ s
h : ∑ i ∈ s, Finsupp.single i (g i) = 0

⊢ g i = (Finsupp.lapply i) (Finsupp.single i (g i))
-- 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
v : ι → M
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid M'
inst✝³ : AddCommMonoid M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hf : ∀ (l : ι →₀ R), (Finsupp.total ι M R v) l = 0 → l = 0
s : Finset ι
g : ι → R
hg : ∑ i ∈ s, g i • v i = 0
i : ι
his : i ∈ s
h : ∑ i ∈ s, Finsupp.single i (g i) = 0
j : ι
_hjs : j ∈ s
hji : j ≠ i

⊢ (Finsupp.lapply i) (Finsupp.single j (g j)) = 0
-- Mathlib/RingTheory/HahnSeries/Basic.lean
Γ : Type u_1
R : Type u_2
inst✝² : PartialOrder Γ
inst✝¹ : Zero R
a b : Γ
r : R
Γ' : Type u_3
inst✝ : PartialOrder Γ'
f : Γ ↪o Γ'
x y : HahnSeries Γ R
xy : embDomain f x = embDomain f y
g : Γ

⊢ x.coeff g = y.coeff g
-- Mathlib/RingTheory/HahnSeries/Basic.lean
Γ : Type u_1
R : Type u_2
inst✝² : PartialOrder Γ
inst✝¹ : Zero R
a b : Γ
r : R
Γ' : Type u_3
inst✝ : PartialOrder Γ'
f : Γ ↪o Γ'
x y : HahnSeries Γ R
xy : ∀ (a : Γ'), (embDomain f x).coeff a = (embDomain f y).coeff a
g : Γ

⊢ x.coeff g = y.coeff g
-- Mathlib/RingTheory/HahnSeries/Basic.lean
Γ : Type u_1
R : Type u_2
inst✝² : PartialOrder Γ
inst✝¹ : Zero R
a b : Γ
r : R
Γ' : Type u_3
inst✝ : PartialOrder Γ'
f : Γ ↪o Γ'
x y : HahnSeries Γ R
xy : ∀ (a : Γ'), (embDomain f x).coeff a = (embDomain f y).coeff a
g : Γ
xyg : (embDomain f x).coeff (f g) = (embDomain f y).coeff (f g)

⊢ x.coeff g = y.coeff g
-- Mathlib/Data/Num/Lemmas.lean
α : Type u_1
inst✝ : AddGroupWithOne α

⊢ ↑↑0 = ↑0
-- Mathlib/Data/Num/Lemmas.lean
α : Type u_1
inst✝ : AddGroupWithOne α
p : PosNum

⊢ ↑↑(pos p) = ↑(pos p)
-- Mathlib/Data/Num/Lemmas.lean
α : Type u_1
inst✝ : AddGroupWithOne α
p : PosNum

⊢ ↑↑(neg p) = ↑(neg p)
-- Mathlib/Order/BooleanAlgebra.lean
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α

⊢ x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z
-- Mathlib/Order/BooleanAlgebra.lean
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α

⊢ x \ (y \ z) = z ⊓ x ⊔ x \ y
-- Mathlib/Order/BooleanAlgebra.lean
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α

⊢ x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = x
-- Mathlib/Order/BooleanAlgebra.lean
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α

⊢ x ⊓ y \ z ⊓ (z ⊓ x ⊔ x \ y) = ⊥
-- Mathlib/LinearAlgebra/Lagrange.lean
F : Type u_1
inst✝ : Field F
x y : F

⊢ (basisDivisor x x).natDegree = 0
-- Mathlib/SetTheory/Cardinal/Ordinal.lean
a b c : Cardinal.{u_1}
h : a + b = c + b
hb : b < ℵ₀

⊢ a = c
-- Mathlib/SetTheory/Cardinal/Ordinal.lean
a b c : Cardinal.{u_1}
h : b + a = b + c
hb : b < ℵ₀

⊢ a = c
-- Mathlib/Algebra/Polynomial/Module/Basic.lean
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
f : R[X]
g : PolynomialModule R M
n : ℕ

⊢ (f • g) n = ∑ x ∈ Finset.antidiagonal n, f.coeff x.1 • g x.2
-- Mathlib/Algebra/Polynomial/Module/Basic.lean
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
g : PolynomialModule R M
n : ℕ
p q : R[X]
hp : (p • g) n = ∑ x ∈ Finset.antidiagonal n, p.coeff x.1 • g x.2
hq : (q • g) n = ∑ x ∈ Finset.antidiagonal n, q.coeff x.1 • g x.2

⊢ ((p + q) • g) n = ∑ x ∈ Finset.antidiagonal n, (p + q).coeff x.1 • g x.2