ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/GroupTheory/MonoidLocalization/MonoidWithZero.lean
M : Type u_1
inst✝³ : CommMonoidWithZero M
S : Submonoid M
N : Type u_2
inst✝² : CommMonoidWithZero N
P : Type u_3
inst✝¹ : CommMonoidWithZero P
f : S.LocalizationWithZeroMap N
inst✝ : IsLeftCancelMulZero M
h : ∀ ⦃x : M⦄, x ∈ S → IsLeftRegular x
fl : S.LocalizationMap N := f.toLocalizationMap
g : M →* N := f.toMap
a z w : N
ha : a ≠ 0
hazw : a * z = a * w
b : M × ↥S
hb : a * fl.toMap ↑b.2 = fl.toMap b.1
x : M × ↥S
hx : z * fl.toMap ↑x.2 = fl.toMap x.1
y : M × ↥S
hy : w * fl.toMap ↑y.2 = fl.toMap y.1
b1ne0 : b.1 ≠ 0

⊢ ↑y.2 * x.1 = ↑x.2 * y.1
-- Mathlib/GroupTheory/MonoidLocalization/MonoidWithZero.lean
M : Type u_1
inst✝³ : CommMonoidWithZero M
S : Submonoid M
N : Type u_2
inst✝² : CommMonoidWithZero N
P : Type u_3
inst✝¹ : CommMonoidWithZero P
f : S.LocalizationWithZeroMap N
inst✝ : IsLeftCancelMulZero M
h : ∀ ⦃x : M⦄, x ∈ S → IsLeftRegular x
fl : S.LocalizationMap N := f.toLocalizationMap
g : M →* N := f.toMap
a z w : N
ha : a ≠ 0
hazw : a * z = a * w
b : M × ↥S
hb : a * fl.toMap ↑b.2 = fl.toMap b.1
x : M × ↥S
hx : z * fl.toMap ↑x.2 = fl.toMap x.1
y : M × ↥S
hy : w * fl.toMap ↑y.2 = fl.toMap y.1
b1ne0 : b.1 ≠ 0
main : g (b.1 * (↑x.2 * y.1)) = g (b.1 * (↑y.2 * x.1))

⊢ ↑y.2 * x.1 = ↑x.2 * y.1
-- Mathlib/Algebra/GeomSum.lean
α : Type u
inst✝ : Ring α
x y : α
h : Commute x y
n : ℕ

⊢ (∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ n
-- Mathlib/Algebra/GeomSum.lean
α : Type u
inst✝ : Ring α
x y : α
h : Commute x y
n : ℕ
this : (∑ i ∈ range n, (x - y + y) ^ i * y ^ (n - 1 - i)) * (x - y) + y ^ n = (x - y + y) ^ n

⊢ (∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ n
-- Mathlib/Algebra/GeomSum.lean
α : Type u
inst✝ : Ring α
x y : α
h : Commute x y
n : ℕ
this : (∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) * (x - y) + y ^ n = x ^ n

⊢ (∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ n
-- Mathlib/Order/CompleteLattice.lean
α : Type u_1
β : Type u_2
β₂ : Type u_3
γ : Type u_4
ι : Sort u_5
ι' : Sort u_6
κ : ι → Sort u_7
κ' : ι' → Sort u_8
inst✝ : SupSet α
f g✝ : ι → α
g : ι' → α
h : ι → ι'
h1 : Surjective h
h2 : ∀ (x : ι), g (h x) = f x

⊢ ⨆ x, f x = ⨆ y, g y
-- Mathlib/Order/CompleteLattice.lean
α : Type u_1
β : Type u_2
β₂ : Type u_3
γ : Type u_4
ι : Sort u_5
ι' : Sort u_6
κ : ι → Sort u_7
κ' : ι' → Sort u_8
inst✝ : SupSet α
f g✝ : ι → α
g : ι' → α
h : ι → ι'
h1 : Surjective h
h2 : ∀ (x : ι), g (h x) = f x
x✝ : ι

⊢ f x✝ = g (h x✝)
-- Mathlib/MeasureTheory/Decomposition/Jordan.lean
α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure

⊢ j₁ = j₂
-- Mathlib/MeasureTheory/Decomposition/Jordan.lean
α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : VectorMeasure.restrict j₁.toSignedMeasure S ≤ VectorMeasure.restrict 0 S
hS₃ : VectorMeasure.restrict 0 Sᶜ ≤ VectorMeasure.restrict j₁.toSignedMeasure Sᶜ
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0

⊢ j₁ = j₂
-- Mathlib/MeasureTheory/Decomposition/Jordan.lean
α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : VectorMeasure.restrict j₁.toSignedMeasure S ≤ VectorMeasure.restrict 0 S
hS₃ : VectorMeasure.restrict 0 Sᶜ ≤ VectorMeasure.restrict j₁.toSignedMeasure Sᶜ
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : VectorMeasure.restrict j₂.toSignedMeasure T ≤ VectorMeasure.restrict 0 T
hT₃ : VectorMeasure.restrict 0 Tᶜ ≤ VectorMeasure.restrict j₂.toSignedMeasure Tᶜ
hT₄ : j₂.posPart T = 0
hT₅ : j₂.negPart Tᶜ = 0

⊢ j₁ = j₂
-- Mathlib/MeasureTheory/Decomposition/Jordan.lean
α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : VectorMeasure.restrict j₁.toSignedMeasure S ≤ VectorMeasure.restrict 0 S
hS₃ : VectorMeasure.restrict 0 Sᶜ ≤ VectorMeasure.restrict j₁.toSignedMeasure Sᶜ
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : VectorMeasure.restrict j₁.toSignedMeasure T ≤ VectorMeasure.restrict 0 T
hT₃ : VectorMeasure.restrict 0 Tᶜ ≤ VectorMeasure.restrict j₁.toSignedMeasure Tᶜ
hT₄ : j₂.posPart T = 0
hT₅ : j₂.negPart Tᶜ = 0

⊢ j₁ = j₂
-- Mathlib/MeasureTheory/Decomposition/Jordan.lean
α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : VectorMeasure.restrict j₁.toSignedMeasure S ≤ VectorMeasure.restrict 0 S
hS₃ : VectorMeasure.restrict 0 Sᶜ ≤ VectorMeasure.restrict j₁.toSignedMeasure Sᶜ
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : VectorMeasure.restrict j₁.toSignedMeasure T ≤ VectorMeasure.restrict 0 T
hT₃ : VectorMeasure.restrict 0 Tᶜ ≤ VectorMeasure.restrict j₁.toSignedMeasure Tᶜ
hT₄ : j₂.posPart T = 0
hT₅ : j₂.negPart Tᶜ = 0
hST₁ : ↑j₁.toSignedMeasure (symmDiff Sᶜ Tᶜ) = 0

⊢ j₁ = j₂
-- Mathlib/MeasureTheory/Decomposition/Jordan.lean
α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : VectorMeasure.restrict j₁.toSignedMeasure S ≤ VectorMeasure.restrict 0 S
hS₃ : VectorMeasure.restrict 0 Sᶜ ≤ VectorMeasure.restrict j₁.toSignedMeasure Sᶜ
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : VectorMeasure.restrict j₁.toSignedMeasure T ≤ VectorMeasure.restrict 0 T
hT₃ : VectorMeasure.restrict 0 Tᶜ ≤ VectorMeasure.restrict j₁.toSignedMeasure Tᶜ
hT₄ : j₂.posPart T = 0
hT₅ : j₂.negPart Tᶜ = 0
hST₁ : ↑j₁.toSignedMeasure (symmDiff Sᶜ Tᶜ) = 0

⊢ j₁.posPart = j₂.posPart
-- Mathlib/MeasureTheory/Decomposition/Jordan.lean
α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : VectorMeasure.restrict j₁.toSignedMeasure S ≤ VectorMeasure.restrict 0 S
hS₃ : VectorMeasure.restrict 0 Sᶜ ≤ VectorMeasure.restrict j₁.toSignedMeasure Sᶜ
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : VectorMeasure.restrict j₁.toSignedMeasure T ≤ VectorMeasure.restrict 0 T
hT₃ : VectorMeasure.restrict 0 Tᶜ ≤ VectorMeasure.restrict j₁.toSignedMeasure Tᶜ
hT₄ : j₂.posPart T = 0
hT₅ : j₂.negPart Tᶜ = 0
hST₁ : ↑j₁.toSignedMeasure (symmDiff Sᶜ Tᶜ) = 0
i : Set α
hi : MeasurableSet i
hμ₁ : (j₁.posPart i).toReal = ↑j₁.toSignedMeasure (i ∩ Sᶜ)
hμ₂ : (j₂.posPart i).toReal = ↑j₂.toSignedMeasure (i ∩ Tᶜ)

⊢ j₁.posPart i = j₂.posPart i
-- Mathlib/MeasureTheory/Decomposition/Jordan.lean
α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : VectorMeasure.restrict j₁.toSignedMeasure S ≤ VectorMeasure.restrict 0 S
hS₃ : VectorMeasure.restrict 0 Sᶜ ≤ VectorMeasure.restrict j₁.toSignedMeasure Sᶜ
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : VectorMeasure.restrict j₁.toSignedMeasure T ≤ VectorMeasure.restrict 0 T
hT₃ : VectorMeasure.restrict 0 Tᶜ ≤ VectorMeasure.restrict j₁.toSignedMeasure Tᶜ
hT₄ : j₂.posPart T = 0
hT₅ : j₂.negPart Tᶜ = 0
hST₁ : ↑j₁.toSignedMeasure (symmDiff Sᶜ Tᶜ) = 0
i : Set α
hi : MeasurableSet i
hμ₁ : (j₁.posPart i).toReal = ↑j₁.toSignedMeasure (i ∩ Sᶜ)
hμ₂ : (j₂.posPart i).toReal = ↑j₂.toSignedMeasure (i ∩ Tᶜ)

⊢ ↑j₁.toSignedMeasure (i ∩ Sᶜ) = ↑j₁.toSignedMeasure (i ∩ Tᶜ)
-- Mathlib/Analysis/Calculus/FDeriv/Add.lean
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E

⊢ fderiv 𝕜 (fun y => -f y) x = -fderiv 𝕜 f x
-- Mathlib/Topology/Order/ScottTopology.lean
α : Type u_1
β : Type u_2
inst✝² : Preorder α
s✝ : Set α
inst✝¹ : TopologicalSpace α
inst✝ : IsScottHausdorff α
s : Set α

⊢ IsOpen s ↔ ∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (fun x x_1 => x ≤ x_1) d → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s → ∃ b ∈ d, Ici b ∩ d ⊆ s
-- Mathlib/RingTheory/PowerSeries/Inverse.lean
R : Type u_1
k : Type u_2
inst✝ : Field k
P : k[X]
hP : P ≠ 0

⊢ Multiset.count X (normalizedFactors ↑P) = Multiset.count Polynomial.X (normalizedFactors P)
-- Mathlib/RingTheory/PowerSeries/Inverse.lean
R : Type u_1
k : Type u_2
inst✝ : Field k
P : k[X]
hP : P ≠ 0

⊢ ∀ (c : ℕ), c ≤ Multiset.count X (normalizedFactors ↑P) ↔ c ≤ Multiset.count Polynomial.X (normalizedFactors P)
-- Mathlib/RingTheory/PowerSeries/Inverse.lean
R : Type u_1
k : Type u_2
inst✝ : Field k
P : k[X]
hP : P ≠ 0

⊢ ∀ (c : ℕ), ↑c ≤ ↑(Multiset.count X (normalizedFactors ↑P)) ↔ ↑c ≤ ↑(Multiset.count Polynomial.X (normalizedFactors P))
-- Mathlib/Topology/Bases.lean
α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s : Set α

⊢ IsSeparable (closure s) ↔ IsSeparable s
-- 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
A : SpecialLinearGroup n R

⊢ A ∈ center (SpecialLinearGroup n R) ↔ ∃ r, r ^ Fintype.card n = 1 ∧ (scalar n) r = ↑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
A : SpecialLinearGroup n R
hn : IsEmpty n

⊢ A ∈ center (SpecialLinearGroup n R) ↔ ∃ r, r ^ Fintype.card n = 1 ∧ (scalar n) r = ↑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
A : SpecialLinearGroup n R
i : n

⊢ A ∈ center (SpecialLinearGroup n R) ↔ ∃ r, r ^ Fintype.card n = 1 ∧ (scalar n) r = ↑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
A : SpecialLinearGroup n R
i : n
h : A ∈ center (SpecialLinearGroup n R)

⊢ ↑A i i ^ Fintype.card n = 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
A : SpecialLinearGroup n R
i : n
h : A ∈ center (SpecialLinearGroup n R)

⊢ (scalar n) (↑A i i) = ↑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
A : SpecialLinearGroup n R
i : n
x✝ : ∃ r, r ^ Fintype.card n = 1 ∧ (scalar n) r = ↑A
r : R
left✝ : r ^ Fintype.card n = 1
hr : (scalar n) r = ↑A
B : SpecialLinearGroup n R

⊢ B * A = A * B
-- Mathlib/Topology/Bornology/Constructions.lean
α : Type u_1
β : Type u_2
ι : Type u_3
π : ι → Type u_4
inst✝² : Bornology α
inst✝¹ : Bornology β
inst✝ : (i : ι) → Bornology (π i)
p : α → Prop

⊢ BoundedSpace (Subtype p) ↔ IsBounded {x | p x}
-- Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
v : E
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E → 𝕜 → F := fun x t => f (x + t • v)
hg : Continuous (Function.uncurry g)

⊢ MeasurableSet {x | LineDifferentiableAt 𝕜 f x v}
-- Mathlib/LinearAlgebra/Dimension/Free.lean
F : Type u
K : Type v
A✝ : Type w
inst✝¹⁵ : Ring F
inst✝¹⁴ : Ring K
inst✝¹³ : AddCommGroup A✝
inst✝¹² : Module F K
inst✝¹¹ : Module K A✝
inst✝¹⁰ : Module F A✝
inst✝⁹ : IsScalarTower F K A✝
inst✝⁸ : StrongRankCondition F
inst✝⁷ : StrongRankCondition K
inst✝⁶ : Module.Free F K
inst✝⁵ : Module.Free K A✝
A : Type v
inst✝⁴ : AddCommGroup A
inst✝³ : Module K A
inst✝² : Module F A
inst✝¹ : IsScalarTower F K A
inst✝ : Module.Free K A

⊢ Module.rank F K * Module.rank K A = Module.rank F A
-- Mathlib/Algebra/Jordan/Basic.lean
A : Type u_1
inst✝¹ : NonUnitalNonAssocCommRing A
inst✝ : IsCommJordan A
a b c : A

⊢ ⁅L a, L (b * b)⁆ + ⁅L b, L (a * a)⁆ + 2 • (⁅L a, L (a * b)⁆ + ⁅L b, L (a * b)⁆) + (⁅L a, L (c * c)⁆ + ⁅L c, L (a * a)⁆ + 2 • (⁅L a, L (c * a)⁆ + ⁅L c, L (c * a)⁆)) + (⁅L b, L (c * c)⁆ + ⁅L c, L (b * b)⁆ + 2 • (⁅L b, L (b * c)⁆ + ⁅L c, L (b * c)⁆)) + (2 • ⁅L a, L (b * c)⁆ + 2 • ⁅L b, L (c * a)⁆ + 2 • ⁅L c, L (a * b)⁆) = 2 • ⁅L a, L (b * c)⁆ + 2 • ⁅L b, L (c * a)⁆ + 2 • ⁅L c, L (a * b)⁆
-- Mathlib/Algebra/Jordan/Basic.lean
A : Type u_1
inst✝¹ : NonUnitalNonAssocCommRing A
inst✝ : IsCommJordan A
a b c : A

⊢ ⁅L a, L (b * b)⁆ + ⁅L b, L (a * a)⁆ + 2 • (⁅L a, L (a * b)⁆ + ⁅L b, L (a * b)⁆) + (⁅L a, L (c * c)⁆ + ⁅L c, L (a * a)⁆ + 2 • (⁅L a, L (c * a)⁆ + ⁅L c, L (c * a)⁆)) + (⁅L b, L (c * c)⁆ + ⁅L c, L (b * b)⁆ + 2 • (⁅L b, L (b * c)⁆ + ⁅L c, L (b * c)⁆)) = 0
-- Mathlib/Algebra/Jordan/Basic.lean
A : Type u_1
inst✝¹ : NonUnitalNonAssocCommRing A
inst✝ : IsCommJordan A
a b c : A

⊢ ⁅L a, L (b * b)⁆ + ⁅L b, L (a * a)⁆ + 2 • (⁅L a, L (a * b)⁆ + ⁅L b, L (b * a)⁆) + (⁅L a, L (c * c)⁆ + ⁅L c, L (a * a)⁆ + 2 • (⁅L a, L (c * a)⁆ + ⁅L c, L (c * a)⁆)) + (⁅L b, L (c * c)⁆ + ⁅L c, L (b * b)⁆ + 2 • (⁅L b, L (b * c)⁆ + ⁅L c, L (b * c)⁆)) = 0
-- Mathlib/Algebra/Jordan/Basic.lean
A : Type u_1
inst✝¹ : NonUnitalNonAssocCommRing A
inst✝ : IsCommJordan A
a b c : A

⊢ ⁅L a, L (b * b)⁆ + ⁅L b, L (a * a)⁆ + 2 • (⁅L a, L (a * b)⁆ + ⁅L b, L (b * a)⁆) + (⁅L a, L (c * c)⁆ + ⁅L c, L (a * a)⁆ + 2 • (⁅L a, L (a * c)⁆ + ⁅L c, L (c * a)⁆)) + (⁅L b, L (c * c)⁆ + ⁅L c, L (b * b)⁆ + 2 • (⁅L b, L (b * c)⁆ + ⁅L c, L (b * c)⁆)) = 0
-- Mathlib/Algebra/Jordan/Basic.lean
A : Type u_1
inst✝¹ : NonUnitalNonAssocCommRing A
inst✝ : IsCommJordan A
a b c : A

⊢ ⁅L a, L (b * b)⁆ + ⁅L b, L (a * a)⁆ + 2 • (⁅L a, L (a * b)⁆ + ⁅L b, L (b * a)⁆) + (⁅L a, L (c * c)⁆ + ⁅L c, L (a * a)⁆ + 2 • (⁅L a, L (a * c)⁆ + ⁅L c, L (c * a)⁆)) + (⁅L b, L (c * c)⁆ + ⁅L c, L (b * b)⁆ + 2 • (⁅L b, L (b * c)⁆ + ⁅L c, L (c * b)⁆)) = 0
-- Mathlib/Algebra/Jordan/Basic.lean
A : Type u_1
inst✝¹ : NonUnitalNonAssocCommRing A
inst✝ : IsCommJordan A
a b c : A

⊢ ⁅L a, L (b * b)⁆ + ⁅L b, L (a * a)⁆ + (⁅L (a * a), L b⁆ + ⁅L (b * b), L a⁆) + (⁅L a, L (c * c)⁆ + ⁅L c, L (a * a)⁆ + (⁅L (a * a), L c⁆ + ⁅L (c * c), L a⁆)) + (⁅L b, L (c * c)⁆ + ⁅L c, L (b * b)⁆ + (⁅L (b * b), L c⁆ + ⁅L (c * c), L b⁆)) = 0
-- Mathlib/Algebra/Jordan/Basic.lean
A : Type u_1
inst✝¹ : NonUnitalNonAssocCommRing A
inst✝ : IsCommJordan A
a b c : A

⊢ ⁅L a, L (b * b)⁆ + ⁅L b, L (a * a)⁆ + (-⁅L b, L (a * a)⁆ + -⁅L a, L (b * b)⁆) + (⁅L a, L (c * c)⁆ + ⁅L c, L (a * a)⁆ + (-⁅L c, L (a * a)⁆ + -⁅L a, L (c * c)⁆)) + (⁅L b, L (c * c)⁆ + ⁅L c, L (b * b)⁆ + (-⁅L c, L (b * b)⁆ + -⁅L b, L (c * c)⁆)) = 0
-- Mathlib/LinearAlgebra/FiniteDimensional.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
U W : Submodule K V
h : IsCompl U W

⊢ finrank K ↥U + finrank K ↥W = finrank K V
-- Mathlib/LinearAlgebra/FiniteDimensional.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
U W : Submodule K V
h : IsCompl U W

⊢ finrank K ↥⊤ = finrank K V
-- 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 : α → M
s : Set α
t : Finset α
h : s ∩ mulSupport f = ↑t ∩ mulSupport f
x : α
hxf : f x ≠ 1

⊢ x ∈ s ↔ x ∈ t
-- 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 : α → M
s : Set α
t : Finset α
h : s ∩ mulSupport f = ↑t ∩ mulSupport f
x : α
hxf : x ∈ mulSupport f

⊢ x ∈ s ↔ x ∈ t
-- Mathlib/RingTheory/Ideal/Operations.lean
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J✝ K L : Ideal R
x : R
I J : Ideal R

⊢ I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I
-- Mathlib/LinearAlgebra/Lagrange.lean
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
f g : R[X]
s : Finset R
degree_f_lt : f.degree < ↑s.card
degree_g_lt : g.degree < ↑s.card
eval_fg : ∀ x ∈ s, eval x f = eval x g

⊢ f = g
-- Mathlib/LinearAlgebra/Lagrange.lean
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
f g : R[X]
s : Finset R
degree_f_lt : f ∈ degreeLT R s.card
degree_g_lt : g ∈ degreeLT R s.card
eval_fg : ∀ x ∈ s, eval x f = eval x g

⊢ f = g
-- Mathlib/LinearAlgebra/Lagrange.lean
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
f g : R[X]
s : Finset R
degree_f_lt : f ∈ degreeLT R s.card
degree_g_lt : g ∈ degreeLT R s.card
eval_fg : ∀ x ∈ s, eval x f = eval x g

⊢ (f - g).degree < ↑s.card
-- Mathlib/LinearAlgebra/Lagrange.lean
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
f g : R[X]
s : Finset R
degree_f_lt : f ∈ degreeLT R s.card
degree_g_lt : g ∈ degreeLT R s.card
eval_fg : ∀ x ∈ s, eval x f = eval x g

⊢ f - g ∈ degreeLT R s.card
-- Mathlib/Algebra/Category/ModuleCat/Biproducts.lean
R : Type u
inst✝ : Ring R
J : Type w
f : J → ModuleCat R
s : Cone (Discrete.functor f)
m : s.pt ⟶ { pt := of R ((j : J) → ↑(f j)), π := Discrete.natTrans fun j => LinearMap.proj j.as }.pt
w : ∀ (j : Discrete J), m ≫ { pt := of R ((j : J) → ↑(f j)), π := Discrete.natTrans fun j => LinearMap.proj j.as }.π.app j = s.π.app j
x : ↑s.pt

⊢ m x = (lift f s) x
-- Mathlib/Algebra/Category/ModuleCat/Biproducts.lean
R : Type u
inst✝ : Ring R
J : Type w
f : J → ModuleCat R
s : Cone (Discrete.functor f)
m : s.pt ⟶ { pt := of R ((j : J) → ↑(f j)), π := Discrete.natTrans fun j => LinearMap.proj j.as }.pt
w : ∀ (j : Discrete J), m ≫ { pt := of R ((j : J) → ↑(f j)), π := Discrete.natTrans fun j => LinearMap.proj j.as }.π.app j = s.π.app j
x : ↑s.pt
j : J

⊢ m x j = (lift f s) x j
-- Mathlib/Order/WellFoundedSet.lean
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r : α → α → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
s : Set (Lex (α × β))
hα : ((fun x => (ofLex x).1) '' s).IsPWO
hβ : ∀ (a : α), {y | toLex (a, y) ∈ s}.IsPWO
f : ℕ → Lex (α × β)
hf : ∀ (n : ℕ), f n ∈ s

⊢ ∃ m n, m < n ∧ (fun x x_1 => x ≤ x_1) (f m) (f n)
-- Mathlib/Order/WellFoundedSet.lean
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r : α → α → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
s : Set (Lex (α × β))
hα : ∀ (f : ℕ → α), (∀ (n : ℕ), f n ∈ (fun x => (ofLex x).1) '' s) → ∃ g, Monotone (f ∘ ⇑g)
hβ : ∀ (a : α), {y | toLex (a, y) ∈ s}.IsPWO
f : ℕ → Lex (α × β)
hf : ∀ (n : ℕ), f n ∈ s

⊢ ∃ m n, m < n ∧ (fun x x_1 => x ≤ x_1) (f m) (f n)
-- Mathlib/Order/WellFoundedSet.lean
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r : α → α → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
s : Set (Lex (α × β))
hα : ∀ (f : ℕ → α), (∀ (n : ℕ), f n ∈ (fun x => (ofLex x).1) '' s) → ∃ g, Monotone (f ∘ ⇑g)
hβ : ∀ (a : α), {y | toLex (a, y) ∈ s}.IsPWO
f : ℕ → Lex (α × β)
hf : ∀ (n : ℕ), f n ∈ s
g : ℕ ↪o ℕ
hg : Monotone fun n => (ofLex f (g n)).1

⊢ ∃ m n, m < n ∧ (fun x x_1 => x ≤ x_1) (f m) (f n)
-- Mathlib/Order/WellFoundedSet.lean
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
π : ι → Type u_5
r : α → α → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
s : Set (Lex (α × β))
hα : ∀ (f : ℕ → α), (∀ (n : ℕ), f n ∈ (fun x => (ofLex x).1) '' s) → ∃ g, Monotone (f ∘ ⇑g)
hβ : ∀ (a : α), {y | toLex (a, y) ∈ s}.IsPWO
f : ℕ → Lex (α × β)
hf : ∀ (n : ℕ), f n ∈ s
g : ℕ ↪o ℕ
hg : Monotone fun n => (ofLex f (g n)).1
hhg : ∀ (n : ℕ), (ofLex f (g 0)).1 ≤ (ofLex f (g n)).1

⊢ ∃ m n, m < n ∧ (fun x x_1 => x ≤ x_1) (f m) (f n)
-- Mathlib/Analysis/BoundedVariation.lean
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a b c : α
hab : a ≤ b
hbc : b ≤ c
hb : b ∈ s

⊢ eVariationOn f (s ∩ Icc a b) + eVariationOn f (s ∩ Icc b c) = eVariationOn f (s ∩ Icc a c)
-- Mathlib/Analysis/BoundedVariation.lean
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a b c : α
hab : a ≤ b
hbc : b ≤ c
hb : b ∈ s
A : IsGreatest (s ∩ Icc a b) b

⊢ eVariationOn f (s ∩ Icc a b) + eVariationOn f (s ∩ Icc b c) = eVariationOn f (s ∩ Icc a c)
-- Mathlib/Analysis/BoundedVariation.lean
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a b c : α
hab : a ≤ b
hbc : b ≤ c
hb : b ∈ s
A : IsGreatest (s ∩ Icc a b) b
B : IsLeast (s ∩ Icc b c) b

⊢ eVariationOn f (s ∩ Icc a b) + eVariationOn f (s ∩ Icc b c) = eVariationOn f (s ∩ Icc a c)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i j : D.J
U : Opens ↑↑(D.U i)

⊢ ↑((Opens.map (D.ι j).base).obj (⋯.functor.obj U)) = ↑((opensFunctor (D.f j i)).obj ((Opens.map (D.t j i).base).obj ((Opens.map (D.f i j).base).obj U)))
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i j : D.J
U : Opens ↑↑(D.U i)

⊢ ⇑(D.ι j).base ⁻¹' (⇑(D.ι i).base '' ↑U) = ⇑(D.f j i).base '' (⇑(D.t j i).base ⁻¹' (⇑(D.f i j).base ⁻¹' ↑U))
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i j : D.J
U : Opens ↑↑(D.U i)

⊢ ⇑((D.mapGlueData (forget C)).ι j ≫ (D.gluedIso (forget C)).inv) ⁻¹' (⇑((D.mapGlueData (forget C)).ι i ≫ (D.gluedIso (forget C)).inv) '' ↑U) = ⇑(D.f j i).base '' (⇑(D.t j i).base ⁻¹' (⇑(D.f i j).base ⁻¹' ↑U))
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i j : D.J
U : Opens ↑↑(D.U i)

⊢ (⇑(preservesColimitIso (forget C) D.diagram.multispan).inv ∘ ⇑(HasColimit.isoOfNatIso (D.diagramIso (forget C))).inv) ∘ ⇑((D.mapGlueData (forget C)).ι j) ⁻¹' (⇑(D.gluedIso (forget C)).inv ∘ ⇑((D.mapGlueData (forget C)).ι i) '' ↑U) = ⇑(D.f j i).base '' (⇑(D.t j i).base ⁻¹' (⇑(D.f i j).base ⁻¹' ↑U))
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i j : D.J
U : Opens ↑↑(D.U i)

⊢ ⇑((D.mapGlueData (forget C)).ι j) ⁻¹' (⇑(preservesColimitIso (forget C) D.diagram.multispan).inv ∘ ⇑(HasColimit.isoOfNatIso (D.diagramIso (forget C))).inv ⁻¹' (⇑(D.gluedIso (forget C)).inv '' (⇑((D.mapGlueData (forget C)).ι i) '' ↑U))) = ⇑(D.f j i).base '' (⇑(D.t j i).base ⁻¹' (⇑(D.f i j).base ⁻¹' ↑U))
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i j : D.J
U : Opens ↑↑(D.U i)

⊢ ⇑((D.mapGlueData (forget C)).ι j) ⁻¹' (⇑((D.mapGlueData (forget C)).ι i) '' ↑U) = ⇑(D.f j i).base '' (⇑(D.t j i).base ⁻¹' (⇑(D.f i j).base ⁻¹' ↑U))
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i j : D.J
U : Opens ↑↑(D.U i)

⊢ Function.Injective (⇑(preservesColimitIso (forget C) D.diagram.multispan).inv ∘ ⇑(HasColimit.isoOfNatIso (D.diagramIso (forget C))).inv)
-- Mathlib/Topology/Instances/Irrational.lean

⊢ Dense {x | Irrational x}
-- Mathlib/Topology/Instances/Irrational.lean

⊢ ∀ o ∈ ⋃ a, ⋃ b, ⋃ (_ : a < b), {Ioo ↑a ↑b}, o.Nonempty → (o ∩ {x | Irrational x}).Nonempty
-- Mathlib/Topology/Instances/Irrational.lean

⊢ ∀ (o : Set ℝ) (x x_1 : ℚ), x < x_1 → o = Ioo ↑x ↑x_1 → o.Nonempty → (o ∩ {x | Irrational x}).Nonempty
-- Mathlib/Topology/Instances/Irrational.lean
a b : ℚ
hlt : a < b
a✝ : (Ioo ↑a ↑b).Nonempty

⊢ (Ioo ↑a ↑b ∩ {x | Irrational x}).Nonempty
-- Mathlib/Topology/Instances/Irrational.lean
a b : ℚ
hlt : a < b
a✝ : (Ioo ↑a ↑b).Nonempty

⊢ ({x | Irrational x} ∩ Ioo ↑a ↑b).Nonempty
-- Mathlib/Algebra/ContinuedFractions/Translations.lean
K : Type u_1
g : GenContFract K
n : ℕ
inst✝ : DivisionRing K
gp : Pair K
zeroth_s_eq : g.s.get? 0 = some gp

⊢ g.dens 1 = gp.b