ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Probability/Moments.lean
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t✝ : ℝ
X Y : Ω → ℝ
h_indep : IndepFun X Y μ
s t : ℝ
h_meas : ∀ (t : ℝ), Measurable fun x => rexp (t * x)

⊢ IndepFun (fun ω => rexp (s * X ω)) (fun ω => rexp (t * Y ω)) μ
-- Mathlib/Probability/Moments.lean
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t✝ : ℝ
X Y : Ω → ℝ
h_indep : IndepFun X Y μ
s t : ℝ
h_meas : ∀ (t : ℝ), Measurable fun x => rexp (t * x)

⊢ IndepFun ((fun x => rexp (s * x)) ∘ X) ((fun x => rexp (t * x)) ∘ Y) μ
-- Mathlib/Data/Set/Card.lean
α : Type u_1
β : Type u_2
s t : Set α

⊢ s.encard = ⊤ ↔ s.Infinite
-- Mathlib/RingTheory/Binomial.lean
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R

⊢ choose r 1 = r
-- Mathlib/Computability/Encoding.lean
n : ℕ

⊢ decodeNat (encodeNat n) = n
-- Mathlib/Computability/Encoding.lean
n : ℕ

⊢ decodeNat (encodeNat n) = ↑↑n
-- Mathlib/Combinatorics/Schnirelmann.lean
A : Set ℕ
inst✝¹ : DecidablePred fun x => x ∈ A
B : Set ℕ
inst✝ : DecidablePred fun x => x ∈ B
h : A ⊆ B
x✝ : { n // 0 < n }

⊢ filter (fun x => x ∈ A) (Ioc 0 ↑x✝) ⊆ filter (fun x => x ∈ B) (Ioc 0 ↑x✝)
-- Mathlib/Algebra/Algebra/Operations.lean
ι : Sort uι
R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : Semiring A
inst✝ : Algebra R A
S T : Set A
M N P Q : Submodule R A
m n✝ : A
n : ℕ

⊢ (Nat.castAddMonoidHom A) n ∈ toAddSubmonoid 1
-- Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
f f' : PartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜 E'
inst✝¹ : TopologicalSpace H'
inst✝ : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
s t : Set M

⊢ ↑(f.extend I) '' (f.source ∩ f'.source) = ((f.extend I).symm ≫ f'.extend I).source
-- Mathlib/Logic/Function/Iterate.lean
α : Type u
β : Type v
f : α → α
m n : ℕ
x : α

⊢ f^[m + n] x = f^[m] (f^[n] x)
-- Mathlib/Logic/Function/Iterate.lean
α : Type u
β : Type v
f : α → α
m n : ℕ
x : α

⊢ (f^[m] ∘ f^[n]) x = f^[m] (f^[n] x)
-- Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean
Ω : Type u_1
inst✝⁶ : TopologicalSpace Ω
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝³ : L.IsCountablyGenerated
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ι → Ω →ᵇ ℝ≥0
fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => (fs i) ω) L (𝓝 (E.indicator (fun x => 1) ω))

⊢ Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂μ) L (𝓝 (μ E))
-- Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean
Ω : Type u_1
inst✝⁶ : TopologicalSpace Ω
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝³ : L.IsCountablyGenerated
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ι → Ω →ᵇ ℝ≥0
fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => (fs i) ω) L (𝓝 (E.indicator (fun x => 1) ω))

⊢ μ E = ∫⁻ (ω : Ω), ↑(E.indicator (fun x => 1) ω) ∂μ
-- Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean
Ω : Type u_1
inst✝⁶ : TopologicalSpace Ω
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝³ : L.IsCountablyGenerated
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ι → Ω →ᵇ ℝ≥0
fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => (fs i) ω) L (𝓝 (E.indicator (fun x => 1) ω))
aux : ∀ (ω : Ω), E.indicator (fun x => 1) ω = ↑(E.indicator (fun x => 1) ω)

⊢ μ E = ∫⁻ (ω : Ω), ↑(E.indicator (fun x => 1) ω) ∂μ
-- Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean
Ω : Type u_1
inst✝⁶ : TopologicalSpace Ω
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝³ : L.IsCountablyGenerated
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
c : ℝ≥0
E : Set Ω
E_mble : MeasurableSet E
fs : ι → Ω →ᵇ ℝ≥0
fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c
fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => (fs i) ω) L (𝓝 (E.indicator (fun x => 1) ω))
aux : ∀ (ω : Ω), E.indicator (fun x => 1) ω = ↑(E.indicator (fun x => 1) ω)

⊢ μ E = ∫⁻ (x : Ω) in E, 1 ∂μ
-- Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s t : Set (PrimeSpectrum R)
ht : IsClosed t
h : vanishingIdeal t ≤ vanishingIdeal s

⊢ s ⊆ t
-- Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s t : Set (PrimeSpectrum R)
ht : IsClosed t
h : vanishingIdeal t ≤ vanishingIdeal s

⊢ closure s ⊆ closure t
-- Mathlib/Computability/Primrec.lean
α✝ : Type u_1
σ : Type u_2
inst✝⁴ : Primcodable α✝
inst✝³ : Primcodable σ
α : Type u_3
β : Type u_4
γ : Type u_5
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable γ
f : α → β
g : α → γ
hf : Primrec f
hg : Primrec g
n : ℕ

⊢ (Nat.casesOn (encode (decode n)) 0 fun n => (Nat.pair (encode (Option.map f (decode n))).pred (encode (Option.map g (decode n))).pred).succ) = encode (Option.map (fun a => (f a, g a)) (decode n))
-- Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
α : Type u_1
r c : ℝ
l : Filter α
f g : α → ℝ
h : IsBigOWith c l f g
hc : 0 ≤ c
hr : 0 ≤ r
hg : 0 ≤ᶠ[l] g

⊢ ∀ᶠ (x : α) in l, ‖f x ^ r‖ ≤ c ^ r * ‖g x ^ r‖
-- Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
α : Type u_1
r c : ℝ
l : Filter α
f g : α → ℝ
h : IsBigOWith c l f g
hc : 0 ≤ c
hr : 0 ≤ r
hg : 0 ≤ᶠ[l] g
x : α
hgx : 0 x ≤ g x
hx : ‖f x‖ ≤ c * ‖g x‖

⊢ ‖f x ^ r‖ ≤ c ^ r * ‖g x ^ r‖
-- Mathlib/MeasureTheory/OuterMeasure/Induced.lean
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
h : m.trim s = 0

⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = 0
-- Mathlib/MeasureTheory/OuterMeasure/Induced.lean
α : Type u_1
inst✝ : MeasurableSpace α
m✝ m : OuterMeasure α
s : Set α
h : m.trim s = 0
t : Set α
hst : s ⊆ t
ht : MeasurableSet t
hm : m t = m.trim s

⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = 0
-- Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean
f : ℝ → ℝ
x✝ : ℝ
n✝ : ℕ
x : ℝ
n : ℕ

⊢ logGammaSeq (x + 1) n = logGammaSeq x (n + 1) + log x - (x + 1) * (log (↑n + 1) - log ↑n)
-- Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean
f : ℝ → ℝ
x✝ : ℝ
n✝ : ℕ
x : ℝ
n : ℕ

⊢ (x + 1) * log ↑n + log ↑n ! - ∑ m ∈ Finset.range (n + 1), log (x + 1 + ↑m) = x * log ↑(n + 1) + log ↑((n + 1) * n !) - ∑ m ∈ Finset.range (n + 1 + 1), log (x + ↑m) + log x - (x + 1) * (log (↑n + 1) - log ↑n)
-- Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean
f : ℝ → ℝ
x✝ : ℝ
n✝ : ℕ
x : ℝ
n : ℕ

⊢ (x + 1) * log ↑n + log ↑n ! - ∑ m ∈ Finset.range (n + 1), log (x + 1 + ↑m) = x * log ↑(n + 1) + log ↑((n + 1) * n !) - (∑ k ∈ Finset.range (n + 1), log (x + ↑(k + 1)) + log x) + log x - (x + 1) * (log (↑n + 1) - log ↑n)
-- Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean
f : ℝ → ℝ
x✝ : ℝ
n✝ : ℕ
x : ℝ
n : ℕ

⊢ (x + 1) * log ↑n + log ↑n ! - ∑ m ∈ Finset.range (n + 1), log (x + 1 + ↑m) = x * log ↑(n + 1) + (log ↑(n + 1) + log ↑n !) - (∑ k ∈ Finset.range (n + 1), log (x + ↑(k + 1)) + log x) + log x - (x + 1) * (log (↑n + 1) - log ↑n)
-- Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean
f : ℝ → ℝ
x✝ : ℝ
n✝ : ℕ
x : ℝ
n : ℕ

⊢ ↑(n + 1) ≠ 0
-- Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean
f : ℝ → ℝ
x✝ : ℝ
n✝ : ℕ
x : ℝ
n : ℕ

⊢ ↑n ! ≠ 0
-- Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean
f : ℝ → ℝ
x✝ : ℝ
n✝ : ℕ
x : ℝ
n : ℕ
this : ∑ m ∈ Finset.range (n + 1), log (x + 1 + ↑m) = ∑ k ∈ Finset.range (n + 1), log (x + ↑(k + 1))

⊢ (x + 1) * log ↑n + log ↑n ! - ∑ m ∈ Finset.range (n + 1), log (x + 1 + ↑m) = x * log ↑(n + 1) + (log ↑(n + 1) + log ↑n !) - (∑ k ∈ Finset.range (n + 1), log (x + ↑(k + 1)) + log x) + log x - (x + 1) * (log (↑n + 1) - log ↑n)
-- Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean
f : ℝ → ℝ
x✝ : ℝ
n✝ : ℕ
x : ℝ
n : ℕ
this : ∑ m ∈ Finset.range (n + 1), log (x + 1 + ↑m) = ∑ k ∈ Finset.range (n + 1), log (x + ↑(k + 1))

⊢ (x + 1) * log ↑n + log ↑n ! - ∑ m ∈ Finset.range (n + 1), log (x + 1 + ↑m) = x * log (↑n + 1) + (log (↑n + 1) + log ↑n !) - (∑ m ∈ Finset.range (n + 1), log (x + 1 + ↑m) + log x) + log x - (x + 1) * (log (↑n + 1) - log ↑n)
-- Mathlib/Data/Nat/Choose/Multinomial.lean
α : Type u_1
s : Finset α
f✝ : α → ℕ
a✝ b : α
n : ℕ
a : α
f : α → ℕ

⊢ multinomial {a} f = 1
-- Mathlib/Data/Nat/Choose/Multinomial.lean
α : Type u_1
s : Finset α
f✝ : α → ℕ
a✝ b : α
n : ℕ
a : α
f : α → ℕ

⊢ (f a + ∑ i ∈ ∅, f i).choose (f a) * multinomial ∅ f = 1
-- Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
R : Type u
inst✝ : CommRing R
W : Projective R
P✝ : Fin 3 → R
u✝ u : Rˣ
P : Fin 3 → R
h : W.Equation P

⊢ W.Equation (u • P)
-- Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
R : Type u
inst✝ : CommRing R
W : Projective R
P✝ : Fin 3 → R
u✝ u : Rˣ
P : Fin 3 → R
h : P 1 ^ 2 * P 2 + W.a₁ * P 0 * P 1 * P 2 + W.a₃ * P 1 * P 2 ^ 2 = P 0 ^ 3 + W.a₂ * P 0 ^ 2 * P 2 + W.a₄ * P 0 * P 2 ^ 2 + W.a₆ * P 2 ^ 3

⊢ (u • P) 1 ^ 2 * (u • P) 2 + W.a₁ * (u • P) 0 * (u • P) 1 * (u • P) 2 + W.a₃ * (u • P) 1 * (u • P) 2 ^ 2 = (u • P) 0 ^ 3 + W.a₂ * (u • P) 0 ^ 2 * (u • P) 2 + W.a₄ * (u • P) 0 * (u • P) 2 ^ 2 + W.a₆ * (u • P) 2 ^ 3
-- Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
R : Type u
inst✝ : CommRing R
W : Projective R
P : Fin 3 → R
u : Rˣ
this : ∀ (u : Rˣ) {P : Fin 3 → R}, W.Equation P → W.Equation (u • P)
h : W.Equation (u • P)

⊢ P = u⁻¹ • u • P
-- Mathlib/NumberTheory/KummerDedekind.lean
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)

⊢ multiplicity J (Ideal.map (algebraMap R S) I) = multiplicity (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩)) (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α
μ ν : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
h : ∀ (a b : ℚ), μ (Ioo ↑a ↑b) = ν (Ioo ↑a ↑b)

⊢ ∀ s ∈ ⋃ a, ⋃ b, ⋃ (_ : a < b), {Ioo ↑a ↑b}, μ s = ν s
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α
μ ν : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
h : ∀ (a b : ℚ), μ (Ioo ↑a ↑b) = ν (Ioo ↑a ↑b)

⊢ ∀ (s : Set ℝ), (∃ i i_1, ∃ (_ : i < i_1), s = Ioo ↑i ↑i_1) → μ s = ν s
-- Mathlib/Probability/Distributions/Exponential.lean
r x : ℝ

⊢ HasDerivAt (fun a => -rexp (-(r * a))) (r * rexp (-(r * x))) x
-- Mathlib/Probability/Distributions/Exponential.lean
r x : ℝ

⊢ (fun a => -rexp (-(r * a))) = fun y => -1 * rexp (-r * id y)
-- Mathlib/Probability/Distributions/Exponential.lean
r x : ℝ

⊢ r * rexp (-(r * x)) = -1 * (rexp (-r * id x) * (-r * 1))
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
α : Type u_1
inst✝ : TopologicalSpace α
f : α → Angle
s : Set α
hf : ContinuousOn f s
hs : ∀ z ∈ s, f z ≠ 0 ∧ f z ≠ ↑π

⊢ ContinuousOn (sign ∘ f) s
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
α : Type u_1
inst✝ : TopologicalSpace α
f : α → Angle
s : Set α
hf : ContinuousOn f s
hs : ∀ z ∈ s, f z ≠ 0 ∧ f z ≠ ↑π
θ : Angle
hθ : θ ∈ f '' s

⊢ ContinuousAt sign θ
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
α : Type u_1
inst✝ : TopologicalSpace α
f : α → Angle
s : Set α
hf : ContinuousOn f s
hs : ∀ z ∈ s, f z ≠ 0 ∧ f z ≠ ↑π
z : α
hz : z ∈ s

⊢ ContinuousAt sign (f z)
-- Mathlib/MeasureTheory/Measure/Restrict.lean
R : Type u_1
α : Type u_2
β : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure β
s : Set α

⊢ (Measure.comap f μ) s = (Measure.comap f μ) (f ⁻¹' (f '' s))
-- Mathlib/MeasureTheory/Measure/Restrict.lean
R : Type u_1
α : Type u_2
β : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure β
s : Set α

⊢ (Measure.map f (Measure.comap f μ)) (f '' s) = μ (f '' s)
-- .lake/packages/batteries/Batteries/Data/List/Pairwise.lean
α : Type u_1
R : α → α → Prop
L : List (List α)

⊢ Pairwise R L.join ↔ (∀ (l : List α), l ∈ L → Pairwise R l) ∧ Pairwise (fun l₁ l₂ => ∀ (x : α), x ∈ l₁ → ∀ (y : α), y ∈ l₂ → R x y) L
-- Mathlib/MeasureTheory/Integral/Bochner.lean
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m✝ : MeasurableSpace α
μ✝ : Measure α
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
x : F

⊢ (weightedSMul (c • μ) s) x = (c.toReal • ⇑(weightedSMul μ s)) x
-- Mathlib/MeasureTheory/Integral/Bochner.lean
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m✝ : MeasurableSpace α
μ✝ : Measure α
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
x : F

⊢ ((c • μ) s).toReal • x = c.toReal • (μ s).toReal • x
-- Mathlib/MeasureTheory/Integral/Bochner.lean
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m✝ : MeasurableSpace α
μ✝ : Measure α
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
x : F

⊢ ((c • ⇑μ) s).toReal • x = c.toReal • (μ s).toReal • x
-- Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean
J : Type w
C✝ : Type u
inst✝² : Category.{v, u} C✝
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → B ⟶ objs j
inst✝ : HasWidePushout B objs arrows
X : C
f : B ⟶ X
fs : (j : J) → objs j ⟶ X
w : ∀ (j : J), arrows j ≫ fs j = f
g1 g2 : widePushout B objs arrows ⟶ X
h1 : ∀ (j : J), ι arrows j ≫ g1 = ι arrows j ≫ g2
h2 : head arrows ≫ g1 = head arrows ≫ g2

⊢ g1 = g2
-- Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean
J : Type w
C✝ : Type u
inst✝² : Category.{v, u} C✝
C : Type u
inst✝¹ : Category.{v, u} C
B : C
objs : J → C
arrows : (j : J) → B ⟶ objs j
inst✝ : HasWidePushout B objs arrows
X : C
f : B ⟶ X
fs : (j : J) → objs j ⟶ X
w : ∀ (j : J), arrows j ≫ fs j = f
g1 g2 : widePushout B objs arrows ⟶ X
h1 : ∀ (j : J), ι arrows j ≫ g1 = ι arrows j ≫ g2
h2 : head arrows ≫ g1 = head arrows ≫ g2

⊢ ∀ (j : WidePushoutShape J), colimit.ι (WidePushoutShape.wideSpan B objs arrows) j ≫ g1 = colimit.ι (WidePushoutShape.wideSpan B objs arrows) j ≫ g2
-- Mathlib/Probability/Process/Stopping.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
hτ : IsStoppingTime f τ
i j : ι

⊢ MeasurableSet ({ω | τ ω ≤ i} ∩ {ω | τ ω ≤ j})
-- Mathlib/Probability/Process/Stopping.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
hτ : IsStoppingTime f τ
i j : ι
this : {ω | τ ω ≤ i} ∩ {ω | τ ω ≤ j} = {ω | τ ω ≤ min i j}

⊢ MeasurableSet ({ω | τ ω ≤ i} ∩ {ω | τ ω ≤ j})
-- Mathlib/Probability/Process/Stopping.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
hτ : IsStoppingTime f τ
i j : ι
this : {ω | τ ω ≤ i} ∩ {ω | τ ω ≤ j} = {ω | τ ω ≤ min i j}

⊢ MeasurableSet {ω | τ ω ≤ min i j}
-- Mathlib/Computability/Primrec.lean
p : ℕ

⊢ Nat.rec 1 (fun y IH => IH * (unpair p).1) (unpair p).2 = (unpair p).1 ^ (unpair p).2
-- Mathlib/Order/Filter/Basic.lean
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
l f : Filter α
g : α → Filter β
s✝ s : Set β

⊢ (s ∈ f.bind fun x => g x ⊓ 𝓟 s✝) ↔ s ∈ f.bind g ⊓ 𝓟 s✝
-- Mathlib/Data/Sym/Card.lean
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
s : Finset α

⊢ (image Sym2.mk s.diag).card = s.card
-- Mathlib/Data/Sym/Card.lean
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
s : Finset α

⊢ Set.InjOn Sym2.mk ↑s.diag
-- Mathlib/Data/Sym/Card.lean
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
s : Finset α
x₀ x₁ : α
hx : (x₀, x₁) ∈ ↑s.diag
x₂✝ : α × α
a✝ : x₂✝ ∈ ↑s.diag
h : s(x₀, x₁) = Sym2.mk x₂✝

⊢ (x₀, x₁) = x₂✝
-- Mathlib/Data/Sym/Card.lean
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
s : Finset α
x₀ x₁ : α
hx a✝ : (x₀, x₁) ∈ ↑s.diag
h : s(x₀, x₁) = s(x₀, x₁)

⊢ (x₀, x₁) = (x₀, x₁)
-- Mathlib/Data/Sym/Card.lean
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
s : Finset α
x₀ x₁ : α
hx : (x₀, x₁) ∈ ↑s.diag
a✝ : (x₁, x₀) ∈ ↑s.diag
h : s(x₀, x₁) = s(x₁, x₀)

⊢ (x₀, x₁) = (x₁, x₀)
-- Mathlib/Analysis/Normed/Order/Lattice.lean
α : Type u_1
inst✝ : NormedLatticeAddCommGroup α
x y : α

⊢ ‖x ⊓ y‖ ≤ ‖x‖ + ‖y‖
-- Mathlib/Analysis/Normed/Order/Lattice.lean
α : Type u_1
inst✝ : NormedLatticeAddCommGroup α
x y : α
h : ‖x ⊓ y - 0 ⊓ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖

⊢ ‖x ⊓ y‖ ≤ ‖x‖ + ‖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✝ : CompleteLattice α
f✝ g s✝ t : ι✝ → α
a b : α
ι : Type u_9
s : Set ι
f : ι → α
i : ι
hi : i ∈ s

⊢ ⨅ i ∈ s, f i ≤ f i
-- Mathlib/RingTheory/Ideal/Norm.lean
S : Type u_1
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Nontrivial S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
x : S
hx : Prime ((Algebra.norm ℤ) x)
y : ℤ

⊢ (Algebra.norm ℤ) x ∣ y ↔ x ∣ ↑y
-- Mathlib/RingTheory/Ideal/Norm.lean
S : Type u_1
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain S
inst✝³ : Nontrivial S
inst✝² : IsDedekindDomain S
inst✝¹ : Module.Free ℤ S
inst✝ : Module.Finite ℤ S
x : S
hx : Prime ((Algebra.norm ℤ) x)
y : ℤ

⊢ Nat.Prime (absNorm (span {x}))
-- Mathlib/Algebra/Order/Ring/Canonical.lean
α : Type u
β : Type u_1
inst✝³ : CanonicallyOrderedCommSemiring α
a b c : α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : IsTotal α fun x x_1 => x ≤ x_1
h : AddLECancellable (b * c)

⊢ (a - b) * c = a * c - b * c
-- Mathlib/Algebra/Order/Ring/Canonical.lean
α : Type u
β : Type u_1
inst✝³ : CanonicallyOrderedCommSemiring α
a b c : α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : IsTotal α fun x x_1 => x ≤ x_1
h : AddLECancellable (c * b)

⊢ c * (a - b) = c * a - c * b
-- Mathlib/Algebra/BigOperators/Group/List.lean
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
G : Type u_7
L : List (List α)

⊢ L.join.length = (map length L).sum
-- Mathlib/Order/RelClasses.lean
α : Type u
β : Type v
r✝ : α → α → Prop
s : β → β → Prop
r : α → α → Prop
inst✝ : IsTrichotomous α r
a b : α

⊢ Function.swap r a b ∨ a = b ∨ Function.swap r b a
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β

⊢ ∀ (l : List α), ContinuousAt length l
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β

⊢ ∀ (l : List α), Tendsto length (𝓝 l) (pure l.length)
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β

⊢ Tendsto length (pure []) (pure [].length)
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β

⊢ ∀ (l : List α) (a : α), Tendsto length (𝓝 l) (pure l.length) → Tendsto (fun p => (p.1 :: p.2).length) (𝓝 a ×ˢ 𝓝 l) (pure (a :: l).length)
-- Mathlib/Algebra/Order/Field/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedField α
a✝ b c d : α
n : ℤ
a : α

⊢ a / 2 - a = -(a / 2)
-- Mathlib/Algebra/Order/Field/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedField α
a✝ b c d : α
n : ℤ
a : α

⊢ a / 2 - (a / 2 + a / 2) = -(a / 2)
-- Mathlib/MeasureTheory/Constructions/Projective.lean
ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : { x // x ∈ J }) → α ↑j)
μ ν : Measure ((i : ι) → α i)
hμ : IsProjectiveLimit μ P
hν : IsProjectiveLimit ν P

⊢ μ univ = ν univ
-- Mathlib/Analysis/NormedSpace/Exponential.lean
𝕂 : Type u_1
𝔸 : Type u_2
inst✝³ : RCLike 𝕂
inst✝² : NormedDivisionRing 𝔸
inst✝¹ : NormedAlgebra 𝕂 𝔸
inst✝ : CompleteSpace 𝔸
z : ℤ
x : 𝔸

⊢ exp 𝕂 (z • x) = exp 𝕂 x ^ z
-- Mathlib/Analysis/NormedSpace/Exponential.lean
𝕂 : Type u_1
𝔸 : Type u_2
inst✝³ : RCLike 𝕂
inst✝² : NormedDivisionRing 𝔸
inst✝¹ : NormedAlgebra 𝕂 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
n : ℕ

⊢ exp 𝕂 (↑n • x) = exp 𝕂 x ^ ↑n
-- Mathlib/Analysis/NormedSpace/Exponential.lean
𝕂 : Type u_1
𝔸 : Type u_2
inst✝³ : RCLike 𝕂
inst✝² : NormedDivisionRing 𝔸
inst✝¹ : NormedAlgebra 𝕂 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
n : ℕ

⊢ exp 𝕂 (-↑n • x) = exp 𝕂 x ^ (-↑n)
-- Mathlib/FieldTheory/SplittingField/Construction.lean
F : Type u
K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field F
f : K[X]

⊢ f.removeFactor.natDegree = f.natDegree - 1
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
w x✝ y z p : ℝ
x : ℝ≥0
hx_pos : 0 < x

⊢ 0 < x ^ p
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
w x✝ y z p : ℝ
x : ℝ≥0
hx_pos : 0 < x
rpow_pos_of_nonneg : ∀ {p : ℝ}, 0 < p → 0 < x ^ p

⊢ 0 < x ^ p
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
w x✝ y z p : ℝ
x : ℝ≥0
hx_pos : 0 < x
rpow_pos_of_nonneg : ∀ {p : ℝ}, 0 < p → 0 < x ^ p
hp_pos : 0 < p

⊢ 0 < x ^ p
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
w x✝ y z : ℝ
x : ℝ≥0
hx_pos : 0 < x
rpow_pos_of_nonneg : ∀ {p : ℝ}, 0 < p → 0 < x ^ p

⊢ 0 < x ^ 0
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
w x✝ y z p : ℝ
x : ℝ≥0
hx_pos : 0 < x
rpow_pos_of_nonneg : ∀ {p : ℝ}, 0 < p → 0 < x ^ p
hp_neg : p < 0

⊢ 0 < x ^ p
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
θ ψ : Angle

⊢ θ.cos = ψ.cos ↔ θ = ψ ∨ θ = -ψ
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
θ : Angle
x✝ : ℝ

⊢ θ.cos = (↑x✝).cos ↔ θ = ↑x✝ ∨ θ = -↑x✝
-- Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : G.Adj v w
e : Sym2 V

⊢ e ∈ (G.subgraphOfAdj hvw).edgeSet ↔ e ∈ {s(v, w)}
-- Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
v w : V
hvw : G.Adj v w
e : Sym2 V

⊢ ∀ (x y : V), s(x, y) ∈ (G.subgraphOfAdj hvw).edgeSet ↔ s(x, y) ∈ {s(v, w)}
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
a x z : ℂ

⊢ arg 1 = 0
-- Mathlib/Topology/MetricSpace/Infsep.lean
α : Type u_1
β : Type u_2
inst✝ : PseudoMetricSpace α
x y z : α
s t : Set α
hsf : s.Finite
hs : s.Nontrivial

⊢ s.infsep = ⋯.toFinset.inf' ⋯ (uncurry dist)
-- Mathlib/Algebra/Polynomial/Eval.lean
R : Type u
S : Type v
T : Type w
ι : Type y
a b : R
m n : ℕ
inst✝ : Ring R
p q r : R[X]

⊢ (X - C a).IsRoot b ↔ a = b
-- Mathlib/NumberTheory/SmoothNumbers.lean
N k n : ℕ
hn : n ∈ N.smoothNumbersUpTo k

⊢ n ∈ Finset.image (fun x => match x with | (s, m) => m ^ 2 * s.prod id) (k.primesBelow.powerset ×ˢ (Finset.range N.sqrt.succ).erase 0)
-- Mathlib/NumberTheory/SmoothNumbers.lean
N k n : ℕ
hn : n ∈ N.smoothNumbersUpTo k
hn₁ : n ≤ N
hn₂ : n ∈ k.smoothNumbers

⊢ n ∈ Finset.image (fun x => match x with | (s, m) => m ^ 2 * s.prod id) (k.primesBelow.powerset ×ˢ (Finset.range N.sqrt.succ).erase 0)
-- Mathlib/NumberTheory/SmoothNumbers.lean
N k n : ℕ
hn : n ∈ N.smoothNumbersUpTo k
hn₁ : n ≤ N
hn₂ : n ∈ k.smoothNumbers
s : Finset ℕ
hs : s ∈ k.primesBelow.powerset
m : ℕ
hm : n = m ^ 2 * s.prod id

⊢ n ∈ Finset.image (fun x => match x with | (s, m) => m ^ 2 * s.prod id) (k.primesBelow.powerset ×ˢ (Finset.range N.sqrt.succ).erase 0)
-- Mathlib/NumberTheory/SmoothNumbers.lean
N k n : ℕ
hn : n ∈ N.smoothNumbersUpTo k
hn₁ : n ≤ N
hn₂ : n ∈ k.smoothNumbers
s : Finset ℕ
hs : s ∈ k.primesBelow.powerset
m : ℕ
hm : n = m ^ 2 * s.prod id

⊢ ∃ a b, (a ⊆ k.primesBelow ∧ b ≠ 0 ∧ b < N.sqrt.succ) ∧ b ^ 2 * ∏ x ∈ a, x = n
-- Mathlib/NumberTheory/SmoothNumbers.lean
N k n : ℕ
hn : n ∈ N.smoothNumbersUpTo k
hn₁ : n ≤ N
hn₂ : n ∈ k.smoothNumbers
s : Finset ℕ
hs : s ∈ k.primesBelow.powerset
m : ℕ
hm : n = m ^ 2 * s.prod id

⊢ m ≠ 0