ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Data/Nat/Bitwise.lean
f✝ f : Bool → Bool → Bool
h : autoParam (f false false = false) _auto✝
a : Bool
m : ℕ
b : Bool
n : ℕ
h2 : ∀ (x : ℕ), (x + x + 1) % 2 = 1
h4 : ∀ (x : ℕ), (x + x + 1) / 2 = x

⊢ (if (if a = true then 2 * m + 1 else 2 * m) = 0 then if f false true = true then if b = true then 2 * n + 1 else 2 * n else 0 else if (if b = true then 2 * n + 1 else 2 * n) = 0 then if f true false = true then if a = true then 2 * m + 1 else 2 * m else 0 else if f (decide ((if a = true then 2 * m + 1 else 2 * m) % 2 = 1)) (decide ((if b = true then 2 * n + 1 else 2 * n) % 2 = 1)) = true then bitwise f ((if a = true then 2 * m + 1 else 2 * m) / 2) ((if b = true then 2 * n + 1 else 2 * n) / 2) + bitwise f ((if a = true then 2 * m + 1 else 2 * m) / 2) ((if b = true then 2 * n + 1 else 2 * n) / 2) + 1 else bitwise f ((if a = true then 2 * m + 1 else 2 * m) / 2) ((if b = true then 2 * n + 1 else 2 * n) / 2) + bitwise f ((if a = true then 2 * m + 1 else 2 * m) / 2) ((if b = true then 2 * n + 1 else 2 * n) / 2)) = if f a b = true then 2 * bitwise f m n + 1 else 2 * bitwise f m n
-- Mathlib/Topology/IsLocalHomeomorph.lean
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
g : Y → Z
f : X → Y
s : Set X
t : Set Y
h : ∀ x ∈ s, ∃ e, x ∈ e.source ∧ Set.EqOn f (↑e) e.source
x : X
hx✝ : x ∈ s
e : PartialHomeomorph X Y
hx : x ∈ e.source
he : Set.EqOn f (↑e) e.source

⊢ ∃ e, x ∈ e.source ∧ f = ↑e
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
inst✝² : OrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfa : Antitone f
k : ℕ

⊢ ∑ i ∈ Finset.range (2 * k), (-1) ^ i * f i ≤ l
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
inst✝² : OrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfa : Antitone f
k : ℕ
hm : Monotone fun n => ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i

⊢ ∑ i ∈ Finset.range (2 * k), (-1) ^ i * f i ≤ l
-- Mathlib/Algebra/Order/Ring/Rat.lean
a✝ b✝ c✝ p q : ℚ
a b c d : ℤ
b0 : 0 < b
d0 : 0 < d

⊢ a /. b ≤ c /. d ↔ a * d ≤ c * b
-- Mathlib/Algebra/Order/Ring/Rat.lean
a✝ b✝ c✝ p q : ℚ
a b c d : ℤ
b0 : 0 < b
d0 : 0 < d

⊢ 0 ≤ c /. d - a /. b ↔ 0 ≤ c * b - a * d
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean
x : ℝ
h : cos x = 0

⊢ deriv tan x = 1 / cos x ^ 2
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean
x : ℝ
h : cos x = 0
this : ¬DifferentiableAt ℝ tan x

⊢ deriv tan x = 1 / cos x ^ 2
-- Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : Measurable f

⊢ MeasurableSet {a | IsRatStieltjesPoint f a}
-- Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : Measurable f
h1 : MeasurableSet {a | Monotone (f a)}

⊢ MeasurableSet {a | IsRatStieltjesPoint f a}
-- Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : Measurable f
h1 : MeasurableSet {a | Monotone (f a)}
h2 : MeasurableSet {a | Tendsto (f a) atTop (𝓝 1)}

⊢ MeasurableSet {a | IsRatStieltjesPoint f a}
-- Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : Measurable f
h1 : MeasurableSet {a | Monotone (f a)}
h2 : MeasurableSet {a | Tendsto (f a) atTop (𝓝 1)}
h3 : MeasurableSet {a | Tendsto (f a) atBot (𝓝 0)}

⊢ MeasurableSet {a | IsRatStieltjesPoint f a}
-- Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : Measurable f
h1 : MeasurableSet {a | Monotone (f a)}
h2 : MeasurableSet {a | Tendsto (f a) atTop (𝓝 1)}
h3 : MeasurableSet {a | Tendsto (f a) atBot (𝓝 0)}
h4 : MeasurableSet {a | ∀ (t : ℚ), ⨅ r, f a ↑r = f a t}

⊢ MeasurableSet {a | IsRatStieltjesPoint f a}
-- Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : Measurable f
h1 : MeasurableSet {a | Monotone (f a)}
h2 : MeasurableSet {a | Tendsto (f a) atTop (𝓝 1)}
h3 : MeasurableSet {a | Tendsto (f a) atBot (𝓝 0)}
h4 : MeasurableSet {a | ∀ (t : ℚ), ⨅ r, f a ↑r = f a t}

⊢ {a | IsRatStieltjesPoint f a} = {a | Monotone (f a)} ∩ {a | Tendsto (f a) atTop (𝓝 1)} ∩ {a | Tendsto (f a) atBot (𝓝 0)} ∩ {a | ∀ (t : ℚ), ⨅ r, f a ↑r = f a t}
-- Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : Measurable f
h1 : MeasurableSet {a | Monotone (f a)}
h2 : MeasurableSet {a | Tendsto (f a) atTop (𝓝 1)}
h3 : MeasurableSet {a | Tendsto (f a) atBot (𝓝 0)}
h4 : MeasurableSet {a | ∀ (t : ℚ), ⨅ r, f a ↑r = f a t}
a : α

⊢ a ∈ {a | IsRatStieltjesPoint f a} ↔ a ∈ {a | Monotone (f a)} ∩ {a | Tendsto (f a) atTop (𝓝 1)} ∩ {a | Tendsto (f a) atBot (𝓝 0)} ∩ {a | ∀ (t : ℚ), ⨅ r, f a ↑r = f a t}
-- Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : Measurable f
h1 : MeasurableSet {a | Monotone (f a)}
h2 : MeasurableSet {a | Tendsto (f a) atTop (𝓝 1)}
h3 : MeasurableSet {a | Tendsto (f a) atBot (𝓝 0)}
h4 : MeasurableSet {a | ∀ (t : ℚ), ⨅ r, f a ↑r = f a t}
a : α

⊢ IsRatStieltjesPoint f a ↔ ((Monotone (f a) ∧ Tendsto (f a) atTop (𝓝 1)) ∧ Tendsto (f a) atBot (𝓝 0)) ∧ ∀ (t : ℚ), ⨅ r, f a ↑r = f a t
-- Mathlib/Data/Num/Lemmas.lean
α : Type u_1
inst✝ : Semiring α
m n : PosNum

⊢ ↑(m * n) = ↑m * ↑n
-- Mathlib/GroupTheory/Nilpotent.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : H.Normal
x✝ : G

⊢ x✝ ∈ upperCentralSeries G 1 ↔ x✝ ∈ center G
-- Mathlib/GroupTheory/Nilpotent.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : H.Normal
x✝ : G

⊢ (∀ (y : G), x✝ * y * x✝⁻¹ * y⁻¹ = 1) ↔ ∀ (g : G), g * x✝ = x✝ * g
-- Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean
R : Type u
S : Type v
a b : R
n✝ m : ℕ
inst✝ : Semiring R
p✝ q r p : R[X]
hp : p ≠ 0
n : ℕ

⊢ (p * X ^ n).natTrailingDegree = p.natTrailingDegree + n
-- Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean
R : Type u
S : Type v
a b : R
n✝ m : ℕ
inst✝ : Semiring R
p✝ q r p : R[X]
hp : p ≠ 0
n : ℕ

⊢ (p * X ^ n).natTrailingDegree ≤ p.natTrailingDegree + n
-- Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean
R : Type u
S : Type v
a b : R
n✝ m : ℕ
inst✝ : Semiring R
p✝ q r p : R[X]
hp : p ≠ 0
n : ℕ

⊢ p.natTrailingDegree + n ≤ (p * X ^ n).natTrailingDegree
-- Mathlib/Combinatorics/SimpleGraph/Matching.lean
V : Type u_1
G G' : SimpleGraph V
M M' : G.Subgraph
v w : V
h : M.IsMatching
hv : v ∈ M.verts
hvw : M.Adj v w

⊢ h.toEdge ⟨v, hv⟩ = ⟨s(v, w), hvw⟩
-- Mathlib/Combinatorics/SimpleGraph/Matching.lean
V : Type u_1
G G' : SimpleGraph V
M M' : G.Subgraph
v w : V
h : M.IsMatching
hv : v ∈ M.verts
hvw : M.Adj v w

⊢ s(v, Exists.choose ⋯) = s(v, w)
-- Mathlib/Combinatorics/SimpleGraph/Matching.lean
V : Type u_1
G G' : SimpleGraph V
M M' : G.Subgraph
v w : V
h : M.IsMatching
hv : v ∈ M.verts
hvw : M.Adj v w

⊢ Exists.choose ⋯ = w
-- Mathlib/Algebra/Periodic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
f g : α → β
c c₁ c₂ x : α
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : Archimedean α
h : Periodic f c
hc : c ≠ 0
a : α

⊢ f '' uIcc a (a + c) = range f
-- Mathlib/Analysis/InnerProductSpace/Adjoint.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : FiniteDimensional 𝕜 G
A : E →ₗ[𝕜] F
x : E
y : F

⊢ (sesqFormOfInner (A x)) y = (sesqFormOfInner x) ((adjoint A) y)
-- Mathlib/Algebra/Polynomial/Reverse.lean
R : Type u_1
inst✝ : Semiring R
f✝ f : R[X]

⊢ f.reverse.natTrailingDegree = 0
-- Mathlib/Algebra/Polynomial/Reverse.lean
R : Type u_1
inst✝ : Semiring R
f✝ f : R[X]

⊢ f = 0 ∨ f ≠ 0
-- Mathlib/NumberTheory/VonMangoldt.lean
n a b : ℕ
ha' : 1 < a
hb' : 1 < b
hab : a.Coprime b
ha : ∑ i ∈ a.divisors, Λ i = Real.log ↑a
hb : ∑ i ∈ b.divisors, Λ i = Real.log ↑b

⊢ ∑ i ∈ (a * b).divisors, Λ i = Real.log ↑(a * b)
-- Mathlib/NumberTheory/VonMangoldt.lean
n a b : ℕ
ha' : 1 < a
hb' : 1 < b
hab : a.Coprime b
ha : ∑ a ∈ filter IsPrimePow a.divisors, Real.log ↑a.minFac = Real.log ↑a
hb : ∑ a ∈ filter IsPrimePow b.divisors, Real.log ↑a.minFac = Real.log ↑b

⊢ ∑ a ∈ filter IsPrimePow (a * b).divisors, Real.log ↑a.minFac = Real.log ↑(a * b)
-- Mathlib/Algebra/CharZero/Lemmas.lean
R✝ : Type u_1
inst✝³ : AddMonoidWithOne R✝
inst✝² : CharZero R✝
R : Type u_2
inst✝¹ : Semiring R
inst✝ : CharZero R
q n : ℕ
hn : n ≠ 0

⊢ ↑q ^ n = 1 ↔ q = 1
-- Mathlib/Algebra/CharZero/Lemmas.lean
R✝ : Type u_1
inst✝³ : AddMonoidWithOne R✝
inst✝² : CharZero R✝
R : Type u_2
inst✝¹ : Semiring R
inst✝ : CharZero R
q n : ℕ
hn : n ≠ 0

⊢ q ^ n = 1 ↔ q = 1
-- Mathlib/Data/Set/Prod.lean
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
s s₁ s₂ : Set ι
t✝ t₁ t₂ : (i : ι) → Set (α i)
i : ι
inst✝ : DecidableEq ι
hi : i ∉ s
f : (j : ι) → α j
a : α i
t : (j : ι) → α j → Set (β j)
j : ι
hj : j ∈ s

⊢ t j (update f i a j) = t j (f j)
-- Mathlib/Data/Set/Prod.lean
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
s s₁ s₂ : Set ι
t✝ t₁ t₂ : (i : ι) → Set (α i)
i : ι
inst✝ : DecidableEq ι
hi : i ∉ s
f : (j : ι) → α j
a : α i
t : (j : ι) → α j → Set (β j)
j : ι
hj : j ∈ s

⊢ j ≠ i
-- Mathlib/FieldTheory/SeparableClosure.lean
F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
K : Type w
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : Algebra E K
inst✝ : IsScalarTower F E K

⊢ finInsepDegree F ↥⊤ = finInsepDegree F K
-- 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✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid M
inst✝⁵ : AddCommMonoid M'
inst✝⁴ : AddCommMonoid M''
inst✝³ : Module R M
inst✝² : Module R M'
inst✝¹ : Module R M''
a b : R
x y : M
inst✝ : Fintype ι

⊢ ¬LinearIndependent R v ↔ ∃ g, ∑ i : ι, g i • v i = 0 ∧ ∃ i, g i ≠ 0
-- Mathlib/LinearAlgebra/Dimension/Basic.lean
R✝ : Type u
R'✝ : Type u'
M M₁ : Type v
M' : Type v'
inst✝¹⁵ : Ring R✝
inst✝¹⁴ : Ring R'✝
inst✝¹³ : AddCommGroup M
inst✝¹² : AddCommGroup M'
inst✝¹¹ : AddCommGroup M₁
inst✝¹⁰ : Module R✝ M
inst✝⁹ : Module R✝ M'
inst✝⁸ : Module R✝ M₁
inst✝⁷ : Module R'✝ M'
inst✝⁶ : Module R'✝ M₁
R : Type w
S : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Ring S
inst✝³ : Algebra R S
R' : Type w'
S' : Type v'
inst✝² : CommRing R'
inst✝¹ : Ring S'
inst✝ : Algebra R' S'
i : R ≃+* R'
j : S ≃+* S'
hc : (algebraMap R' S').comp i.toRingHom = j.toRingHom.comp (algebraMap R S)

⊢ lift.{v', v} (Module.rank R S) = lift.{v, v'} (Module.rank R' S')
-- Mathlib/LinearAlgebra/Dimension/Basic.lean
R✝ : Type u
R'✝ : Type u'
M M₁ : Type v
M' : Type v'
inst✝¹⁵ : Ring R✝
inst✝¹⁴ : Ring R'✝
inst✝¹³ : AddCommGroup M
inst✝¹² : AddCommGroup M'
inst✝¹¹ : AddCommGroup M₁
inst✝¹⁰ : Module R✝ M
inst✝⁹ : Module R✝ M'
inst✝⁸ : Module R✝ M₁
inst✝⁷ : Module R'✝ M'
inst✝⁶ : Module R'✝ M₁
R : Type w
S : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Ring S
inst✝³ : Algebra R S
R' : Type w'
S' : Type v'
inst✝² : CommRing R'
inst✝¹ : Ring S'
inst✝ : Algebra R' S'
i : R ≃+* R'
j : S ≃+* S'
hc : (algebraMap R' S').comp i.toRingHom = j.toRingHom.comp (algebraMap R S)
r : R
x✝ : S

⊢ ↑j (r • x✝) = ↑i r • ↑j x✝
-- Mathlib/LinearAlgebra/Dimension/Basic.lean
R✝ : Type u
R'✝ : Type u'
M M₁ : Type v
M' : Type v'
inst✝¹⁵ : Ring R✝
inst✝¹⁴ : Ring R'✝
inst✝¹³ : AddCommGroup M
inst✝¹² : AddCommGroup M'
inst✝¹¹ : AddCommGroup M₁
inst✝¹⁰ : Module R✝ M
inst✝⁹ : Module R✝ M'
inst✝⁸ : Module R✝ M₁
inst✝⁷ : Module R'✝ M'
inst✝⁶ : Module R'✝ M₁
R : Type w
S : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Ring S
inst✝³ : Algebra R S
R' : Type w'
S' : Type v'
inst✝² : CommRing R'
inst✝¹ : Ring S'
inst✝ : Algebra R' S'
i : R ≃+* R'
j : S ≃+* S'
hc : (algebraMap R' S').comp i.toRingHom = j.toRingHom.comp (algebraMap R S)
r : R
x✝ : S
this : ((algebraMap R' S').comp i.toRingHom) r = (j.toRingHom.comp (algebraMap R S)) r

⊢ ↑j (r • x✝) = ↑i r • ↑j x✝
-- Mathlib/LinearAlgebra/Dimension/Basic.lean
R✝ : Type u
R'✝ : Type u'
M M₁ : Type v
M' : Type v'
inst✝¹⁵ : Ring R✝
inst✝¹⁴ : Ring R'✝
inst✝¹³ : AddCommGroup M
inst✝¹² : AddCommGroup M'
inst✝¹¹ : AddCommGroup M₁
inst✝¹⁰ : Module R✝ M
inst✝⁹ : Module R✝ M'
inst✝⁸ : Module R✝ M₁
inst✝⁷ : Module R'✝ M'
inst✝⁶ : Module R'✝ M₁
R : Type w
S : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Ring S
inst✝³ : Algebra R S
R' : Type w'
S' : Type v'
inst✝² : CommRing R'
inst✝¹ : Ring S'
inst✝ : Algebra R' S'
i : R ≃+* R'
j : S ≃+* S'
hc : (algebraMap R' S').comp i.toRingHom = j.toRingHom.comp (algebraMap R S)
r : R
x✝ : S
this : (algebraMap R' S') (i r) = j ((algebraMap R S) r)

⊢ ↑j (r • x✝) = ↑i r • ↑j x✝
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Field R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : CompleteSpace R
inst✝⁶ : TopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
a : A
ha : p a

⊢ ClosedEmbedding ⇑(cfcₙHom_of_cfcHom R ha)
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Field R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : CompleteSpace R
inst✝⁶ : TopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
a : A
ha : p a
f : C(↑(spectrum R a), ↑(σₙ R a)) := { toFun := inclusion ⋯, continuous_toFun := ⋯ }

⊢ ClosedEmbedding ⇑(cfcₙHom_of_cfcHom R ha)
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Field R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : CompleteSpace R
inst✝⁶ : TopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
a : A
ha : p a
f : C(↑(spectrum R a), ↑(σₙ R a)) := { toFun := inclusion ⋯, continuous_toFun := ⋯ }
h_cpct' : CompactSpace ↑(σₙ R a)

⊢ ClosedEmbedding ⇑(cfcₙHom_of_cfcHom R ha)
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Field R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : CompleteSpace R
inst✝⁶ : TopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
a : A
ha : p a
f : C(↑(spectrum R a), ↑(σₙ R a)) := { toFun := inclusion ⋯, continuous_toFun := ⋯ }
h_cpct' : CompactSpace ↑(σₙ R a)

⊢ Filter.comap (fun x => (↑((↑(compStarAlgHom' R R { toFun := inclusion ⋯, continuous_toFun := ⋯ })).comp toContinuousMapHom).toNonUnitalAlgHom x.1, ↑((↑(compStarAlgHom' R R { toFun := inclusion ⋯, continuous_toFun := ⋯ })).comp toContinuousMapHom).toNonUnitalAlgHom x.2)) (𝓤 C(↑(spectrum R a), R)) = 𝓤 C(↑(σₙ R a), R)₀
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Field R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : CompleteSpace R
inst✝⁶ : TopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
a : A
ha : p a
f : C(↑(spectrum R a), ↑(σₙ R a)) := { toFun := inclusion ⋯, continuous_toFun := ⋯ }
h_cpct' : CompactSpace ↑(σₙ R a)
this : inferInstanceAs (UniformSpace C(↑(σₙ R a), R)) = UniformSpace.comap (fun x => x.comp f) inferInstance ⊓ UniformSpace.comap (fun x => x.comp 0) inferInstance

⊢ Filter.comap (fun x => (↑((↑(compStarAlgHom' R R { toFun := inclusion ⋯, continuous_toFun := ⋯ })).comp toContinuousMapHom).toNonUnitalAlgHom x.1, ↑((↑(compStarAlgHom' R R { toFun := inclusion ⋯, continuous_toFun := ⋯ })).comp toContinuousMapHom).toNonUnitalAlgHom x.2)) (𝓤 C(↑(spectrum R a), R)) = 𝓤 C(↑(σₙ R a), R)₀
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Field R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : CompleteSpace R
inst✝⁶ : TopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
a : A
ha : p a
f : C(↑(spectrum R a), ↑(σₙ R a)) := { toFun := inclusion ⋯, continuous_toFun := ⋯ }
h_cpct' : CompactSpace ↑(σₙ R a)
this : inferInstanceAs (UniformSpace C(↑(σₙ R a), R)) = UniformSpace.comap (fun x => x.comp f) inferInstance ⊓ UniformSpace.comap (fun x => x.comp 0) inferInstance

⊢ Filter.comap (fun x => (↑((↑(compStarAlgHom' R R { toFun := inclusion ⋯, continuous_toFun := ⋯ })).comp toContinuousMapHom).toNonUnitalAlgHom x.1, ↑((↑(compStarAlgHom' R R { toFun := inclusion ⋯, continuous_toFun := ⋯ })).comp toContinuousMapHom).toNonUnitalAlgHom x.2)) (𝓤 C(↑(spectrum R a), R)) = Filter.comap ((Prod.map (fun x => x.comp f) fun x => x.comp f) ∘ Prod.map ContinuousMapZero.toContinuousMap ContinuousMapZero.toContinuousMap) (𝓤 C(↑(spectrum R a), R)) ⊓ Filter.comap ((Prod.map (fun x => x.comp 0) fun x => x.comp 0) ∘ Prod.map ContinuousMapZero.toContinuousMap ContinuousMapZero.toContinuousMap) (𝓤 C(Unit, R))
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Field R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : CompleteSpace R
inst✝⁶ : TopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
a : A
ha : p a
f : C(↑(spectrum R a), ↑(σₙ R a)) := { toFun := inclusion ⋯, continuous_toFun := ⋯ }
h_cpct' : CompactSpace ↑(σₙ R a)
this : inferInstanceAs (UniformSpace C(↑(σₙ R a), R)) = UniformSpace.comap (fun x => x.comp f) inferInstance ⊓ UniformSpace.comap (fun x => x.comp 0) inferInstance

⊢ Filter.comap ((Prod.map (fun x => x.comp 0) fun x => x.comp 0) ∘ Prod.map ContinuousMapZero.toContinuousMap ContinuousMapZero.toContinuousMap) (𝓤 C(Unit, R)) = ⊤
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Field R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : CompleteSpace R
inst✝⁶ : TopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
a : A
ha : p a
f : C(↑(spectrum R a), ↑(σₙ R a)) := { toFun := inclusion ⋯, continuous_toFun := ⋯ }
h_cpct' : CompactSpace ↑(σₙ R a)
this✝ : inferInstanceAs (UniformSpace C(↑(σₙ R a), R)) = UniformSpace.comap (fun x => x.comp f) inferInstance ⊓ UniformSpace.comap (fun x => x.comp 0) inferInstance
this : ∀ U ∈ 𝓤 C(Unit, R), (0, 0) ∈ U

⊢ Filter.comap ((Prod.map (fun x => x.comp 0) fun x => x.comp 0) ∘ Prod.map ContinuousMapZero.toContinuousMap ContinuousMapZero.toContinuousMap) (𝓤 C(Unit, R)) = ⊤
-- Mathlib/Topology/Order/ScottTopology.lean
α : Type u_1
β : Type u_2
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α

⊢ scottHausdorff α ≤ inst✝¹
-- Mathlib/Topology/Order/ScottTopology.lean
α : Type u_1
β : Type u_2
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α

⊢ scottHausdorff α ≤ upperSet α ⊔ scottHausdorff α
-- Mathlib/Data/List/Duplicate.lean
α : Type u_1
l : List α
x : α

⊢ l.Nodup ↔ ∀ (x : α), ¬x ∈+ l
-- Mathlib/CategoryTheory/Preadditive/Biproducts.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
J : Type
inst✝¹ : Fintype J
f : J → C
inst✝ : HasBiproduct f
T U : C
g : (j : J) → T ⟶ f j
h : (j : J) → f j ⟶ U

⊢ lift g ≫ desc h = ∑ j : J, g j ≫ h j
-- Mathlib/Analysis/Convex/Cone/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
S✝ S : ConvexCone 𝕜 E

⊢ S.Blunt ↔ ¬S.Pointed
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean
x y : ℂ
h : ((∀ (k : ℤ), x ≠ (2 * ↑k + 1) * ↑π / 2) ∧ ∀ (l : ℤ), y * I ≠ (2 * ↑l + 1) * ↑π / 2) ∨ (∃ k, x = (2 * ↑k + 1) * ↑π / 2) ∧ ∃ l, y * I = (2 * ↑l + 1) * ↑π / 2

⊢ tan (x + y * I) = (tan x + tanh y * I) / (1 - tan x * tanh y * I)
-- Mathlib/MeasureTheory/Group/Arithmetic.lean
α : Type u_1
G : Type u_2
inst✝² : DivInvMonoid G
inst✝¹ : MeasurableSpace G
inst✝ : MeasurableMul G
g : G

⊢ Measurable fun h => h / g
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E✝ : Type u_1
inst✝¹² : NormedAddCommGroup E✝
inst✝¹¹ : NormedSpace ℝ E✝
inst✝¹⁰ : MeasurableSpace E✝
inst✝⁹ : BorelSpace E✝
inst✝⁸ : FiniteDimensional ℝ E✝
μ✝ : Measure E✝
inst✝⁷ : μ✝.IsAddHaarMeasure
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : CompleteSpace F
s : Set E✝
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
x : E
r : ℝ
this : ball 0 r = (fun x_1 => x + x_1) ⁻¹' ball x r

⊢ μ (ball x r) = μ (ball 0 r)
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
a : ℕ × ℕ
ha : a ∈ antidiagonal k

⊢ ∑ A ∈ powersetCard a.1 univ, ∑ j : σ, MvPolynomial.NewtonIdentities.weight σ R k (A, j) = (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
a : ℕ × ℕ
ha : a ∈ antidiagonal k

⊢ ∑ x ∈ powersetCard a.1 univ, ∑ x_1 : σ, ((-1) ^ x.card * ∏ a ∈ x, X a) * X x_1 ^ (k - x.card) = ∑ i : σ, (∑ i ∈ powersetCard a.1 univ, (-1) ^ a.1 * ∏ a ∈ i, X a) * X i ^ a.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
a : ℕ × ℕ
ha : a ∈ antidiagonal k

⊢ ∑ y : σ, ∑ x ∈ powersetCard a.1 univ, ((-1) ^ x.card * ∏ a ∈ x, X a) * X y ^ (k - x.card) = ∑ i : σ, (∑ i ∈ powersetCard a.1 univ, (-1) ^ a.1 * ∏ a ∈ i, X a) * X i ^ a.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
a : ℕ × ℕ
ha : a ∈ antidiagonal k
x : σ
x✝ : x ∈ univ

⊢ ∑ x_1 ∈ powersetCard a.1 univ, ((-1) ^ x_1.card * ∏ a ∈ x_1, X a) * X x ^ (k - x_1.card) = (∑ i ∈ powersetCard a.1 univ, (-1) ^ a.1 * ∏ a ∈ i, X a) * X x ^ a.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
a : ℕ × ℕ
ha : a ∈ antidiagonal k
x : σ
x✝ : x ∈ univ

⊢ ∑ x_1 ∈ powersetCard a.1 univ, ((-1) ^ x_1.card * ∏ a ∈ x_1, X a) * X x ^ (k - x_1.card) = ∑ i ∈ powersetCard a.1 univ, ((-1) ^ a.1 * ∏ a ∈ i, X a) * X x ^ a.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
a : ℕ × ℕ
ha : a ∈ antidiagonal k
x : σ
x✝ : x ∈ univ
s : Finset σ
hs : s ∈ powersetCard a.1 univ

⊢ ((-1) ^ s.card * ∏ a ∈ s, X a) * X x ^ (k - s.card) = ((-1) ^ a.1 * ∏ a ∈ s, X a) * X x ^ a.2