ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : (interior V).Nonempty
s : Finset G
h1s : ↑K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' ↑K₀
h2s : s.card = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' V
h2t : t.card = index (↑K₀) V

⊢ ∀ i ∈ s, (fun h => i * h) ⁻¹' ↑K₀ ⊆ ⋃ g ∈ t * s, (fun h => g * h) ⁻¹' V
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : (interior V).Nonempty
s : Finset G
h1s : ↑K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' ↑K₀
h2s : s.card = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' V
h2t : t.card = index (↑K₀) V
g₁ : G
hg₁ : g₁ ∈ s

⊢ (fun h => g₁ * h) ⁻¹' ↑K₀ ⊆ ⋃ g ∈ t * s, (fun h => g * h) ⁻¹' V
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : (interior V).Nonempty
s : Finset G
h1s : ↑K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' ↑K₀
h2s : s.card = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' V
h2t : t.card = index (↑K₀) V
g₁ : G
hg₁ : g₁ ∈ s

⊢ ∀ (a : G), g₁ * a ∈ ↑K₀ → a ∈ ⋃ g ∈ t * s, (fun h => g * h) ⁻¹' V
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : (interior V).Nonempty
s : Finset G
h1s : ↑K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' ↑K₀
h2s : s.card = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' V
h2t : t.card = index (↑K₀) V
g₁ : G
hg₁ : g₁ ∈ s
g₂ : G
hg₂ : g₁ * g₂ ∈ ↑K₀
g₃ : G
hg₃ : g₃ ∈ t
h2V : g₁ * g₂ ∈ (fun h => (fun h => g₃ * h) ⁻¹' V) hg₃

⊢ g₂ ∈ ⋃ g ∈ t * s, (fun h => g * h) ⁻¹' V
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : (interior V).Nonempty
s : Finset G
h1s : ↑K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' ↑K₀
h2s : s.card = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' V
h2t : t.card = index (↑K₀) V
g₁ : G
hg₁ : g₁ ∈ s
g₂ : G
hg₂ : g₁ * g₂ ∈ ↑K₀
g₃ : G
hg₃ : g₃ ∈ t
h2V : g₃ * g₁ * g₂ ∈ V

⊢ g₂ ∈ ⋃ g ∈ t * s, (fun h => g * h) ⁻¹' V
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f : ℝ → ℝ
b : ℝ
hb : b ∈ Set.Ioo 0 1

⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, (fun x => x) u ∈ Set.Icc (c₁ * (fun x => x) x) (c₂ * (fun x => x) x)
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f : ℝ → ℝ
b : ℝ
hb : b ∈ Set.Ioo 0 1

⊢ ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, (fun x => x) u ∈ Set.Icc (b * (fun x => x) x) (c₂ * (fun x => x) x)
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f : ℝ → ℝ
b : ℝ
hb : b ∈ Set.Ioo 0 1

⊢ ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, (fun x => x) u ∈ Set.Icc (b * (fun x => x) x) (1 * (fun x => x) x)
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f : ℝ → ℝ
b : ℝ
hb : b ∈ Set.Ioo 0 1
x u : ℝ
hu : u ∈ Set.Icc (b * x) x

⊢ u ∈ Set.Icc (b * x) (1 * x)
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f : ℝ → ℝ
b : ℝ
hb : b ∈ Set.Ioo 0 1
x u : ℝ
hu : u ∈ Set.Icc (b * x) x

⊢ b * x ≤ u ∧ u ≤ x
-- Mathlib/Algebra/Group/Submonoid/Pointwise.lean
α : Type u_1
G : Type u_2
M : Type u_3
R : Type u_4
A : Type u_5
inst✝² : Monoid M
inst✝¹ : AddMonoid A
s✝ t u : Set M
inst✝ : Group G
s : Set G

⊢ closure s⁻¹ = (closure s)⁻¹
-- Mathlib/Algebra/Group/Submonoid/Pointwise.lean
α : Type u_1
G : Type u_2
M : Type u_3
R : Type u_4
A : Type u_5
inst✝² : Monoid M
inst✝¹ : AddMonoid A
s✝ t u : Set M
inst✝ : Group G
s : Set G

⊢ closure s⁻¹ ≤ (closure s)⁻¹
-- Mathlib/Algebra/Group/Submonoid/Pointwise.lean
α : Type u_1
G : Type u_2
M : Type u_3
R : Type u_4
A : Type u_5
inst✝² : Monoid M
inst✝¹ : AddMonoid A
s✝ t u : Set M
inst✝ : Group G
s : Set G

⊢ (closure s)⁻¹ ≤ closure s⁻¹
-- Mathlib/Order/Heyting/Hom.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
inst✝⁴ : FunLike F α β
inst✝³ : CoheytingAlgebra α
inst✝² : CoheytingAlgebra β
inst✝¹ : CoheytingAlgebra γ
inst✝ : CoheytingAlgebra δ
f f₁ f₂ : CoheytingHom α β
g g₁ g₂ : CoheytingHom β γ
hg : Injective ⇑g
h : g.comp f₁ = g.comp f₂
a : α

⊢ g (f₁ a) = g (f₂ a)
-- Mathlib/MeasureTheory/Function/UniformIntegrable.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑M
δ : ℝ
hδpos : 0 < δ
hδ : ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε

⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
-- Mathlib/MeasureTheory/Function/UniformIntegrable.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑M
δ : ℝ
hδpos : 0 < δ
hδ : ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), μ {x | C ≤ ‖f i x‖₊} ≤ ENNReal.ofReal δ

⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
-- Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : ContinuousMultilinearMap 𝕜 E₁ G
k : ℕ
x : (i : ι) → E₁ i

⊢ ‖f.iteratedFDeriv k x‖ ≤ ↑((Fintype.card ι).descFactorial k) * ‖f‖ * ‖x‖ ^ (Fintype.card ι - k)
-- Mathlib/FieldTheory/SeparableDegree.lean
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
f : F[X]
x : F
n : ℕ

⊢ ((X - C x) ^ n).natSepDegree = if n = 0 then 0 else 1
-- Mathlib/Combinatorics/Colex.lean
α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : PartialOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a b : α
inst✝ : DecidableEq α
ha : a ∉ s
hb : b ∉ s

⊢ { ofColex := insert a s } ≤ { ofColex := insert b s } ↔ a ≤ b
-- Mathlib/Algebra/Group/Subgroup/ZPowers.lean
G : Type u_1
inst✝² : Group G
A : Type u_2
inst✝¹ : AddGroup A
N : Type u_3
inst✝ : Group N
s : Set G
g✝ g : G

⊢ zpowers g = ⊥ ↔ g = 1
-- Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean
R : Type u_1
inst✝ : CommRing R
c₁ c₂ : R

⊢ (((ofQuaternion.toLinearMap ∘ₗ toQuaternion.toLinearMap) ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inl R R R) 1 = (((AlgHom.id R (CliffordAlgebra (Q c₁ c₂))).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inl R R R) 1
-- Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean
R : Type u_1
inst✝ : CommRing R
c₁ c₂ : R

⊢ (((ofQuaternion.toLinearMap ∘ₗ toQuaternion.toLinearMap) ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inr R R R) 1 = (((AlgHom.id R (CliffordAlgebra (Q c₁ c₂))).toLinearMap ∘ₗ ι (Q c₁ c₂)) ∘ₗ LinearMap.inr R R R) 1
-- Mathlib/Topology/UniformSpace/Equicontinuity.lean
ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
inst✝ : Finite ι
F : ι → X → α

⊢ Equicontinuous F ↔ ∀ (i : ι), Continuous (F i)
-- Mathlib/RingTheory/AlgebraicIndependent.lean
ι : Type u_1
ι' : Type u_2
R : Type u_3
K : Type u_4
A : Type u_5
A' : Type u_6
A'' : Type u_7
V : Type u
V' : Type u_8
x : ι → A
inst✝⁷ : CommRing R
inst✝⁶ : CommRing A
inst✝⁵ : CommRing A'
inst✝⁴ : CommRing A''
inst✝³ : Algebra R A
inst✝² : Algebra R A'
inst✝¹ : Algebra R A''
a b : R
inst✝ : IsEmpty ι
this : aeval x = (ofId R A).comp ↑(isEmptyAlgEquiv R ι)

⊢ AlgebraicIndependent R x ↔ Injective ⇑(algebraMap R A)
-- Mathlib/RingTheory/AlgebraicIndependent.lean
ι : Type u_1
ι' : Type u_2
R : Type u_3
K : Type u_4
A : Type u_5
A' : Type u_6
A'' : Type u_7
V : Type u
V' : Type u_8
x : ι → A
inst✝⁷ : CommRing R
inst✝⁶ : CommRing A
inst✝⁵ : CommRing A'
inst✝⁴ : CommRing A''
inst✝³ : Algebra R A
inst✝² : Algebra R A'
inst✝¹ : Algebra R A''
a b : R
inst✝ : IsEmpty ι
this : aeval x = (ofId R A).comp ↑(isEmptyAlgEquiv R ι)

⊢ Injective ⇑((ofId R A).comp ↑(isEmptyAlgEquiv R ι)) ↔ Injective (⇑(algebraMap R A) ∘ ⇑(isEmptyAlgEquiv R ι))
-- Mathlib/Data/Nat/Pairing.lean
m n : ℕ

⊢ m + n ≤ pair m n
-- Mathlib/Data/Nat/Pairing.lean
m n : ℕ

⊢ m + n ≤ if m < n then n * n + m else m * m + (m + n)
-- Mathlib/Algebra/Order/Field/Power.lean
α : Type u_1
inst✝ : LinearOrderedField α
a✝ b c d : α
n p : ℤ
hp : Even p
a : α

⊢ |a| ^ p = a ^ p
-- .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean
α : Type u_1
c₀ : RBColor
n₀ : Nat
c : RBColor
n : Nat
c' : RBColor
t : RBNode α
path : Path α
hp : Path.Balanced c₀ n₀ path c n
ht : DelProp c' t n
hc : c = black → c' ≠ red

⊢ ∃ n, (path.del t c').Balanced black n
-- Mathlib/Analysis/Complex/Hadamard.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : ε > 0
z : ℂ
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hz : z ∈ verticalClosedStrip 0 1

⊢ ‖f z‖ ≤ ‖(↑ε + ↑(sSupNormIm f 0)) ^ (1 - z) * (↑ε + ↑(sSupNormIm f 1)) ^ z‖
-- Mathlib/Analysis/Complex/Hadamard.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : ε > 0
z : ℂ
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hz : z ∈ verticalClosedStrip 0 1

⊢ ‖f z‖ ≤ (ε + sSupNormIm f 0) ^ (1 - z.re) * (ε + sSupNormIm f 1) ^ z.re
-- Mathlib/Analysis/Complex/Hadamard.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : ε > 0
z : ℂ
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hz : z ∈ verticalClosedStrip 0 1

⊢ ‖f z‖ * ((ε + sSupNormIm f 0) ^ (1 - z.re)) ^ (-1) * ((ε + sSupNormIm f 1) ^ z.re) ^ (-1) ≤ 1
-- Mathlib/Analysis/Complex/Hadamard.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : ε > 0
z : ℂ
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hz : z ∈ verticalClosedStrip 0 1

⊢ 0 < (ε + sSupNormIm f 1) ^ z.re
-- Mathlib/Analysis/Complex/Hadamard.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : ε > 0
z : ℂ
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hz : z ∈ verticalClosedStrip 0 1

⊢ 0 < (ε + sSupNormIm f 0) ^ (1 - z.re)
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k

⊢ psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ a ∈ filter (fun a => a.1 ∈ Set.Ioo 0 k) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k

⊢ psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => a.1 < k) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2

⊢ psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)

⊢ psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides : (fun x => x - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2) (↑k * esymm σ R k) = (fun x => x - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2) ((-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2))

⊢ psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides : ↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2

⊢ psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ : ↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides : (fun x => (-1) ^ (k + 1) * x) (↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2) = (fun x => (-1) ^ (k + 1) * x) ((-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2)

⊢ psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ : ↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides : (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2

⊢ psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ : ↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides : (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
this : filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) = {(0, k)}

⊢ psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ : ↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides : (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (0, k).1 * esymm σ R (0, k).1 * psum σ R (0, k).2
this : filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) = {(0, k)}

⊢ psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ : ↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
this : filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) = {(0, k)}
sub_both_sides : (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 = psum σ R k

⊢ psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/GradedAlgebra/Basic.lean
ι : Type u_1
R : Type u_2
A : Type u_3
σ : Type u_4
inst✝⁷ : DecidableEq ι
inst✝⁶ : AddMonoid ι
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
a : ⨁ (i : ι), ↥(𝒜 i)
i : ι

⊢ (proj 𝒜 i) ((decompose 𝒜).symm a) = (decompose 𝒜).symm ((of (fun i => ↥(𝒜 i)) i) (a i))
-- Mathlib/Order/Interval/Set/Basic.lean
α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
a₁ a₂ : α
b₁ b₂ : β
x : α
y : β

⊢ (x, y) ∈ Icc a₁ a₂ ×ˢ Icc b₁ b₂ ↔ (x, y) ∈ Icc (a₁, b₁) (a₂, b₂)
-- Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
E : Type u_1
f : ℝ → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : ℝ → E
a b : ℝ
hb : 0 < b

⊢ ∫ (x : ℝ) in Ioi a, g (x * b) = b⁻¹ • ∫ (x : ℝ) in Ioi (a * b), g x
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ

⊢ μ.toFiniteAux = 0 ↔ μ = 0
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
h : μ.toFiniteAux = 0

⊢ μ = 0
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
h : μ.toFiniteAux = 0
s : Set α
hs : MeasurableSet s

⊢ μ s = 0 s
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
h : ∀ (s : Set α), MeasurableSet s → μ.toFiniteAux s = 0 s
s : Set α
hs : MeasurableSet s

⊢ μ s = 0 s
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set α
hs : MeasurableSet s
h : μ.toFiniteAux s = 0 s

⊢ μ s = 0 s
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set α
hs : MeasurableSet s
h : ∀ (i : ℕ), 2 ^ (i + 1) * (sFiniteSeq μ i) Set.univ = ⊤ ∨ (sFiniteSeq μ i) s = 0

⊢ μ s = 0 s
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set α
hs : MeasurableSet s
h : ∀ (i : ℕ), 2 ^ (i + 1) * (sFiniteSeq μ i) Set.univ = ⊤ ∨ (sFiniteSeq μ i) s = 0

⊢ ∑' (i : ℕ), (sFiniteSeq μ i) s = 0 s
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set α
hs : MeasurableSet s
h : ∀ (i : ℕ), 2 ^ (i + 1) * (sFiniteSeq μ i) Set.univ = ⊤ ∨ (sFiniteSeq μ i) s = 0

⊢ ∀ (i : ℕ), (sFiniteSeq μ i) s = 0
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set α
hs : MeasurableSet s
n : ℕ
h : 2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ = ⊤ ∨ (sFiniteSeq μ n) s = 0

⊢ (sFiniteSeq μ n) s = 0
-- Mathlib/Algebra/GradedMonoid.lean
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝³ : SetLike S R
inst✝² : Monoid R
inst✝¹ : AddMonoid ι
A : ι → S
inst✝ : GradedMonoid A
n : ℕ
r : R
i : ι
h : r ∈ A i

⊢ r ^ n ∈ A (n • i)
-- Mathlib/Dynamics/Ergodic/Conservative.lean
ι : Type u_1
α : Type u_2
inst✝ : MeasurableSpace α
f : α → α
s : Set α
μ : Measure α
hf : Conservative f μ
hsm : NullMeasurableSet s μ
hs₀ : μ s ≠ 0

⊢ ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s