ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Data/Complex/Exponential.lean
x y : ℂ

⊢ sinh x ^ 2 = cosh x ^ 2 - (cosh x ^ 2 - sinh x ^ 2)
-- Mathlib/Analysis/RCLike/Basic.lean
K : Type u_1
E : Type u_2
inst✝ : RCLike K
z w : K

⊢ 0 < z ↔ 0 < re z ∧ im z = 0
-- Mathlib/Data/List/Sort.lean
α : Type u
r : α → α → Prop
a : α
l✝ l : List α
h : Sorted r l
k : ℕ
x y : α
hx : x ∈ take k l
hy : y ∈ drop k l

⊢ r x y
-- Mathlib/Data/List/Sort.lean
α : Type u
r : α → α → Prop
a : α
l✝ l : List α
h : Sorted r l
k : ℕ
x : α
hx : x ∈ take k l
iy : ℕ
hiy : iy < (drop k l).length
hy : (drop k l)[iy] ∈ drop k l

⊢ r x (drop k l)[iy]
-- Mathlib/Data/List/Sort.lean
α : Type u
r : α → α → Prop
a : α
l✝ l : List α
h : Sorted r l
k iy : ℕ
hiy : iy < (drop k l).length
hy : (drop k l)[iy] ∈ drop k l
ix : ℕ
hix : ix < (take k l).length
hx : (take k l)[ix] ∈ take k l

⊢ r (take k l)[ix] (drop k l)[iy]
-- Mathlib/Data/List/Sort.lean
α : Type u
r : α → α → Prop
a : α
l✝ l : List α
h : Sorted r l
k iy : ℕ
hiy : iy < (drop k l).length
hy : (drop k l)[iy] ∈ drop k l
ix : ℕ
hix : ix < (take k l).length
hx : (take k l)[ix] ∈ take k l

⊢ r l[ix] l[k + iy]
-- Mathlib/Data/List/Sort.lean
α : Type u
r : α → α → Prop
a : α
l✝ l : List α
h : Sorted r l
k iy : ℕ
hiy : iy < (drop k l).length
hy : (drop k l)[iy] ∈ drop k l
ix : ℕ
hix✝ : ix < (take k l).length
hix : ix < min k l.length
hx : (take k l)[ix] ∈ take k l

⊢ r l[ix] l[k + iy]
-- Mathlib/Data/Int/ConditionallyCompleteOrder.lean
s : Set ℤ
h1 : s.Nonempty
h2 : BddAbove s

⊢ sSup s ∈ s
-- Mathlib/Data/Int/ConditionallyCompleteOrder.lean
s : Set ℤ
h1 : s.Nonempty
h2 : BddAbove s

⊢ sSup s = ↑((Classical.choose h2).greatestOfBdd ⋯ h1)
-- Mathlib/Data/Set/Lattice.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : Set α

⊢ kernImage f sᶜ = (f '' s)ᶜ
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
W : Affine R
f : R →+* S
n : Fin 2

⊢ (CoordinateRing.basis W) n = (AdjoinRoot.powerBasis' ⋯).gen ^ ↑n
-- Mathlib/Algebra/Polynomial/Splits.lean
R : Type u_1
F : Type u
K : Type v
L : Type w
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Field F
i : K →+* L
inst✝¹ : Algebra R K
inst✝ : Algebra R L
p : R[X]
h : Splits (algebraMap R K) p
f : K →ₐ[R] L

⊢ ⇑f '' p.rootSet K = p.rootSet L
-- Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
v : E
inst✝ : SecondCountableTopology E
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E × E → 𝕜 → F := fun p t => f (p.1 + t • p.2)

⊢ StronglyMeasurable fun p => lineDeriv 𝕜 f p.1 p.2
-- Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
v : E
inst✝ : SecondCountableTopology E
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E × E → 𝕜 → F := fun p t => f (p.1 + t • p.2)
this : Continuous (Function.uncurry g)

⊢ StronglyMeasurable fun p => lineDeriv 𝕜 f p.1 p.2
-- Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
a : UnitAddCircle
s : ℂ

⊢ sinZeta (-a) s = -sinZeta a s
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
x✝ y x : ℝ

⊢ arcsin x = 0 ↔ x = 0
-- Mathlib/Data/Real/Pointwise.lean
ι : Sort u_1
α : Type u_2
inst✝ : LinearOrderedField α
r : ℝ
ha : r ≤ 0
f : ι → ℝ

⊢ (⨅ i, f i) * r = ⨆ i, f i * r
-- Mathlib/Algebra/Group/Basic.lean
α : Type u_1
β : Type u_2
G : Type u_3
M : Type u_4
inst✝ : CommGroup G
a✝ b✝ c✝ d a b c : G
h : a = b * c

⊢ a / b = c
-- Mathlib/Algebra/Order/Ring/Rat.lean
a b c p q : ℚ
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0

⊢ 0 ≤ n₁ /. ↑d₁ → 0 ≤ n₂ /. ↑d₂ → 0 ≤ n₁ /. ↑d₁ * (n₂ /. ↑d₂)
-- Mathlib/Algebra/Order/Ring/Rat.lean
a b c p q : ℚ
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
d₁0 : 0 < ↑d₁

⊢ 0 ≤ n₁ /. ↑d₁ → 0 ≤ n₂ /. ↑d₂ → 0 ≤ n₁ /. ↑d₁ * (n₂ /. ↑d₂)
-- Mathlib/Algebra/Order/Ring/Rat.lean
a b c p q : ℚ
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
d₁0 : 0 < ↑d₁
d₂0 : 0 < ↑d₂

⊢ 0 ≤ n₁ /. ↑d₁ → 0 ≤ n₂ /. ↑d₂ → 0 ≤ n₁ /. ↑d₁ * (n₂ /. ↑d₂)
-- Mathlib/Algebra/Order/Ring/Rat.lean
a b c p q : ℚ
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
d₁0 : 0 < ↑d₁
d₂0 : 0 < ↑d₂

⊢ 0 ≤ n₁ → 0 ≤ n₂ → 0 ≤ n₁ * n₂
-- Mathlib/Data/Finsupp/Defs.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝ : Zero M
a a' : α
b : M

⊢ single a b = 0 ↔ b = 0
-- Mathlib/Analysis/SpecialFunctions/Log/Base.lean
b x y : ℝ
hx : x ≠ 0
hy : y ≠ 0

⊢ logb b (x * y) = logb b x + logb b y
-- Mathlib/Data/Nat/Factorization/Basic.lean
a✝ b✝ m n p a b : ℕ
ha_pos : a ≠ 0
hb_pos : b ≠ 0
dfac : ℕ →₀ ℕ := a.factorization ⊓ b.factorization
d : ℕ := dfac.prod fun x x_1 => x ^ x_1

⊢ (a.gcd b).factorization = a.factorization ⊓ b.factorization
-- Mathlib/Data/Nat/Factorization/Basic.lean
a✝ b✝ m n p a b : ℕ
ha_pos : a ≠ 0
hb_pos : b ≠ 0
dfac : ℕ →₀ ℕ := a.factorization ⊓ b.factorization
d : ℕ := dfac.prod fun x x_1 => x ^ x_1
dfac_prime : ∀ p ∈ dfac.support, Prime p

⊢ (a.gcd b).factorization = a.factorization ⊓ b.factorization
-- Mathlib/Data/Nat/Factorization/Basic.lean
a✝ b✝ m n p a b : ℕ
ha_pos : a ≠ 0
hb_pos : b ≠ 0
dfac : ℕ →₀ ℕ := a.factorization ⊓ b.factorization
d : ℕ := dfac.prod fun x x_1 => x ^ x_1
dfac_prime : ∀ p ∈ dfac.support, Prime p
h1 : d.factorization = dfac

⊢ (a.gcd b).factorization = a.factorization ⊓ b.factorization
-- Mathlib/Data/Nat/Factorization/Basic.lean
a✝ b✝ m n p a b : ℕ
ha_pos : a ≠ 0
hb_pos : b ≠ 0
dfac : ℕ →₀ ℕ := a.factorization ⊓ b.factorization
d : ℕ := dfac.prod fun x x_1 => x ^ x_1
dfac_prime : ∀ p ∈ dfac.support, Prime p
h1 : d.factorization = dfac
hd_pos : d ≠ 0

⊢ (a.gcd b).factorization = a.factorization ⊓ b.factorization
-- Mathlib/Data/Nat/Factorization/Basic.lean
a✝ b✝ m n p a b : ℕ
ha_pos : a ≠ 0
hb_pos : b ≠ 0
dfac : ℕ →₀ ℕ := a.factorization ⊓ b.factorization
d : ℕ := dfac.prod fun x x_1 => x ^ x_1
dfac_prime : ∀ p ∈ dfac.support, Prime p
h1 : d.factorization = dfac
hd_pos : d ≠ 0

⊢ d = a.gcd b
-- Mathlib/Data/Nat/Factorization/Basic.lean
a✝ b✝ m n p a b : ℕ
ha_pos : a ≠ 0
hb_pos : b ≠ 0
dfac : ℕ →₀ ℕ := a.factorization ⊓ b.factorization
d : ℕ := dfac.prod fun x x_1 => x ^ x_1
dfac_prime : ∀ p ∈ dfac.support, Prime p
h1 : d.factorization = dfac
hd_pos : d ≠ 0

⊢ d ∣ a
-- Mathlib/Data/Nat/Factorization/Basic.lean
a✝ b✝ m n p a b : ℕ
ha_pos : a ≠ 0
hb_pos : b ≠ 0
dfac : ℕ →₀ ℕ := a.factorization ⊓ b.factorization
d : ℕ := dfac.prod fun x x_1 => x ^ x_1
dfac_prime : ∀ p ∈ dfac.support, Prime p
h1 : d.factorization = dfac
hd_pos : d ≠ 0

⊢ d ∣ b
-- Mathlib/Data/Nat/Factorization/Basic.lean
a✝ b✝ m n p a b : ℕ
ha_pos : a ≠ 0
hb_pos : b ≠ 0
dfac : ℕ →₀ ℕ := a.factorization ⊓ b.factorization
d : ℕ := dfac.prod fun x x_1 => x ^ x_1
dfac_prime : ∀ p ∈ dfac.support, Prime p
h1 : d.factorization = dfac
hd_pos : d ≠ 0

⊢ ∀ (e : ℕ), e ∣ a → e ∣ b → e ∣ d
-- Mathlib/NumberTheory/Padics/PadicNorm.lean
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hr : ¬r = 0

⊢ padicNorm p (q / r) * padicNorm p r = padicNorm p q
-- Mathlib/Data/Rat/Cast/Order.lean
F : Type u_1
ι : Type u_2
α : Type u_3
β : Type u_4
p q : ℚ
K : Type u_5
inst✝ : LinearOrderedField K

⊢ 0 ≤ ↑q ↔ 0 ≤ q
-- Mathlib/LinearAlgebra/Dual.lean
K : Type u
V : Type v
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
W✝ : Subspace K V
inst✝ : FiniteDimensional K V
W : Subspace K (Module.Dual K V)

⊢ finrank K ↥W + finrank K ↥(dualCoannihilator W) = finrank K V
-- Mathlib/LinearAlgebra/Dual.lean
K : Type u
V : Type v
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
W✝ : Subspace K V
inst✝ : FiniteDimensional K V
W : Subspace K (Module.Dual K V)

⊢ finrank K ↥W + finrank K ↥(dualAnnihilator W) = finrank K V
-- Mathlib/LinearAlgebra/Dual.lean
K : Type u
V : Type v
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
W✝ : Subspace K V
inst✝ : FiniteDimensional K V
W : Subspace K (Module.Dual K V)
equiv : (Module.Dual K V ⧸ W) ≃ₗ[K] ↥(dualAnnihilator W) := W.quotEquivAnnihilator

⊢ finrank K ↥W + finrank K ↥(dualAnnihilator W) = finrank K V
-- Mathlib/LinearAlgebra/Dual.lean
K : Type u
V : Type v
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
W✝ : Subspace K V
inst✝ : FiniteDimensional K V
W : Subspace K (Module.Dual K V)
equiv : (Module.Dual K V ⧸ W) ≃ₗ[K] ↥(dualAnnihilator W) := W.quotEquivAnnihilator
eq : finrank K (Module.Dual K V ⧸ W) = finrank K ↥(dualAnnihilator W)

⊢ finrank K ↥W + finrank K ↥(dualAnnihilator W) = finrank K V
-- Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
I J : HasZeroMorphisms C

⊢ I = J
-- Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
I J : HasZeroMorphisms C

⊢ ∀ (X Y : C), Zero.zero = Zero.zero
-- Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
I J : HasZeroMorphisms C
X Y : C

⊢ Zero.zero = Zero.zero
-- Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
I J : HasZeroMorphisms C
X Y : C
this : Zero.zero ≫ Zero.zero = Zero.zero

⊢ Zero.zero = Zero.zero
-- Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
I J : HasZeroMorphisms C
X Y : C
this : Zero.zero ≫ Zero.zero = Zero.zero
that : Zero.zero ≫ Zero.zero = Zero.zero

⊢ Zero.zero = Zero.zero
-- Mathlib/Data/Num/Lemmas.lean
α : Type u_1
m n : PosNum
h : m = n

⊢ ↑m < ↑n ↔ Ordering.eq = Ordering.lt
-- Mathlib/Data/Num/Lemmas.lean
α : Type u_1
m n : PosNum
h : Ordering.casesOn Ordering.gt (↑m < ↑n) (m = n) (↑n < ↑m)

⊢ ↑m < ↑n ↔ Ordering.gt = Ordering.lt
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hn : ∀ (n : ℕ), 0 < n → (filter (fun a => a ^ n = 1) univ).card ≤ n
d : ℕ
IH : ∀ m < d, m ∣ Fintype.card α → 0 < (filter (fun a => orderOf a = m) univ).card → (filter (fun a => orderOf a = m) univ).card = φ m
hd : d ∣ Fintype.card α
hpos : 0 < (filter (fun a => orderOf a = d) univ).card

⊢ (filter (fun a => orderOf a = d) univ).card = φ d
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hn : ∀ (n : ℕ), 0 < n → (filter (fun a => a ^ n = 1) univ).card ≤ n
IH : ∀ m < 0, m ∣ Fintype.card α → 0 < (filter (fun a => orderOf a = m) univ).card → (filter (fun a => orderOf a = m) univ).card = φ m
hd : 0 ∣ Fintype.card α
hpos : 0 < (filter (fun a => orderOf a = 0) univ).card

⊢ (filter (fun a => orderOf a = 0) univ).card = φ 0
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hn : ∀ (n : ℕ), 0 < n → (filter (fun a => a ^ n = 1) univ).card ≤ n
d : ℕ
IH : ∀ m < d, m ∣ Fintype.card α → 0 < (filter (fun a => orderOf a = m) univ).card → (filter (fun a => orderOf a = m) univ).card = φ m
hd : d ∣ Fintype.card α
hpos : 0 < (filter (fun a => orderOf a = d) univ).card
hd0 : d ≠ 0

⊢ (filter (fun a => orderOf a = d) univ).card = φ d
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a✝ : α
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hn : ∀ (n : ℕ), 0 < n → (filter (fun a => a ^ n = 1) univ).card ≤ n
d : ℕ
IH : ∀ m < d, m ∣ Fintype.card α → 0 < (filter (fun a => orderOf a = m) univ).card → (filter (fun a => orderOf a = m) univ).card = φ m
hd : d ∣ Fintype.card α
hpos : 0 < (filter (fun a => orderOf a = d) univ).card
hd0 : d ≠ 0
a : α
ha' : a ∈ filter (fun a => orderOf a = d) univ

⊢ (filter (fun a => orderOf a = d) univ).card = φ d
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a✝ : α
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hn : ∀ (n : ℕ), 0 < n → (filter (fun a => a ^ n = 1) univ).card ≤ n
d : ℕ
IH : ∀ m < d, m ∣ Fintype.card α → 0 < (filter (fun a => orderOf a = m) univ).card → (filter (fun a => orderOf a = m) univ).card = φ m
hd : d ∣ Fintype.card α
hpos : 0 < (filter (fun a => orderOf a = d) univ).card
hd0 : d ≠ 0
a : α
ha' : a ∈ filter (fun a => orderOf a = d) univ
ha : orderOf a = d

⊢ (filter (fun a => orderOf a = d) univ).card = φ d
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a✝ : α
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hn : ∀ (n : ℕ), 0 < n → (filter (fun a => a ^ n = 1) univ).card ≤ n
d : ℕ
IH : ∀ m < d, m ∣ Fintype.card α → 0 < (filter (fun a => orderOf a = m) univ).card → (filter (fun a => orderOf a = m) univ).card = φ m
hd : d ∣ Fintype.card α
hpos : 0 < (filter (fun a => orderOf a = d) univ).card
hd0 : d ≠ 0
a : α
ha' : a ∈ filter (fun a => orderOf a = d) univ
ha : orderOf a = d
h1 : ∑ m ∈ d.properDivisors, (filter (fun a => orderOf a = m) univ).card = ∑ m ∈ d.properDivisors, φ m

⊢ (filter (fun a => orderOf a = d) univ).card = φ d
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a✝ : α
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hn : ∀ (n : ℕ), 0 < n → (filter (fun a => a ^ n = 1) univ).card ≤ n
d : ℕ
IH : ∀ m < d, m ∣ Fintype.card α → 0 < (filter (fun a => orderOf a = m) univ).card → (filter (fun a => orderOf a = m) univ).card = φ m
hd : d ∣ Fintype.card α
hpos : 0 < (filter (fun a => orderOf a = d) univ).card
hd0 : d ≠ 0
a : α
ha' : a ∈ filter (fun a => orderOf a = d) univ
ha : orderOf a = d
h1 : ∑ m ∈ d.properDivisors, (filter (fun a => orderOf a = m) univ).card = ∑ m ∈ d.properDivisors, φ m
h2 : ∑ m ∈ d.divisors, (filter (fun a => orderOf a = m) univ).card = ∑ m ∈ d.divisors, φ m

⊢ (filter (fun a => orderOf a = d) univ).card = φ d
-- Mathlib/Data/Nat/GCD/Basic.lean
m n k : ℕ

⊢ (m * n).lcm (m * k) = m * n.lcm k
-- Mathlib/Data/Nat/GCD/Basic.lean
m n k : ℕ

⊢ (m * n).lcm (m * k) ∣ m * n.lcm k
-- Mathlib/Data/Nat/GCD/Basic.lean
m n k : ℕ

⊢ m * n.lcm k ∣ (m * n).lcm (m * k)
-- Mathlib/RingTheory/PolynomialAlgebra.lean
R : Type u_1
A : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
m : Matrix n n R[X]
i j : n
k : ℕ

⊢ k ∈ (m i j).support → k ∈ (matPolyEquiv m).support
-- Mathlib/RingTheory/PolynomialAlgebra.lean
R : Type u_1
A : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
m : Matrix n n R[X]
i j : n
k : ℕ

⊢ k ∉ (matPolyEquiv m).support → k ∉ (m i j).support
-- Mathlib/RingTheory/PolynomialAlgebra.lean
R : Type u_1
A : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
m : Matrix n n R[X]
i j : n
k : ℕ

⊢ (matPolyEquiv m).coeff k = 0 → (m i j).coeff k = 0
-- Mathlib/RingTheory/PolynomialAlgebra.lean
R : Type u_1
A : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
m : Matrix n n R[X]
i j : n
k : ℕ
hk : (matPolyEquiv m).coeff k = 0

⊢ (m i j).coeff k = 0
-- Mathlib/RingTheory/PolynomialAlgebra.lean
R : Type u_1
A : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
m : Matrix n n R[X]
i j : n
k : ℕ
hk : (matPolyEquiv m).coeff k = 0

⊢ 0 i j = 0
-- Mathlib/Topology/Algebra/InfiniteSum/Ring.lean
ι : Type u_1
κ : Type u_2
R : Type u_3
α : Type u_4
inst✝² : DivisionSemiring α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSemiring α
f g : ι → α
a a₁ a₂ : α
h : HasSum f a
b : α

⊢ HasSum (fun i => f i / b) (a / b)
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f✝ f : ℝ → ℝ
hf : GrowsPolynomially (-f)

⊢ GrowsPolynomially f
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f✝ f : ℝ → ℝ
hf : GrowsPolynomially (-f)

⊢ GrowsPolynomially (- -f)
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean
α : Type u_1
𝕜 : Type u_2
E : Type u_3
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ

⊢ μ.restrict s[f|m] =ᶠ[ae (μ.restrict s)] μ[f|m]
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean
α : Type u_1
𝕜 : Type u_2
E : Type u_3
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)

⊢ μ.restrict s[f|m] =ᶠ[ae (μ.restrict s)] μ[f|m]
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean
α : Type u_1
𝕜 : Type u_2
E : Type u_3
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)

⊢ s.indicator (μ.restrict s[f|m]) =ᶠ[ae μ] s.indicator (μ[f|m])
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean
α : Type u_1
𝕜 : Type u_2
E : Type u_3
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)

⊢ s.indicator (μ.restrict s[f|m]) =ᶠ[ae μ] μ[s.indicator f|m]
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean
α : Type u_1
𝕜 : Type u_2
E : Type u_3
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)

⊢ ∀ (s_1 : Set α), MeasurableSet s_1 → μ s_1 < ⊤ → IntegrableOn (s.indicator (μ.restrict s[f|m])) s_1 μ
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean
α : Type u_1
𝕜 : Type u_2
E : Type u_3
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)

⊢ ∀ (s_1 : Set α), MeasurableSet s_1 → μ s_1 < ⊤ → ∫ (x : α) in s_1, s.indicator (μ.restrict s[f|m]) x ∂μ = ∫ (x : α) in s_1, s.indicator f x ∂μ
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean
α : Type u_1
𝕜 : Type u_2
E : Type u_3
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)

⊢ AEStronglyMeasurable' m (s.indicator (μ.restrict s[f|m])) μ
-- Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
f✝ g✝ : ℝ → ℝ
a✝ b✝ : ℝ
μ : Measure ℝ
f g : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hgc : ContinuousOn g (Icc a b)
hle : ∀ x ∈ Ioc a b, f x ≤ g x
hlt : ∃ c ∈ Icc a b, f c < g c

⊢ ∫ (x : ℝ) in a..b, f x < ∫ (x : ℝ) in a..b, g x
-- Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
f✝ g✝ : ℝ → ℝ
a✝ b✝ : ℝ
μ : Measure ℝ
f g : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hgc : ContinuousOn g (Icc a b)
hle : ∀ x ∈ Ioc a b, f x ≤ g x
hlt : ∃ c ∈ Icc a b, f c < g c

⊢ f ≤ᶠ[ae (volume.restrict (Ioc a b))] g
-- Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
f✝ g✝ : ℝ → ℝ
a✝ b✝ : ℝ
μ : Measure ℝ
f g : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hgc : ContinuousOn g (Icc a b)
hle : ∀ x ∈ Ioc a b, f x ≤ g x
hlt : ∃ c ∈ Icc a b, f c < g c

⊢ (volume.restrict (Ioc a b)) {x | f x < g x} ≠ 0
-- Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
f✝ g✝ : ℝ → ℝ
a✝ b✝ : ℝ
μ : Measure ℝ
f g : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hgc : ContinuousOn g (Icc a b)
hle : ∀ x ∈ Ioc a b, f x ≤ g x
hlt : (volume.restrict (Ioc a b)) {x | f x < g x} = 0

⊢ ∀ c ∈ Icc a b, g c ≤ f c
-- Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
f✝ g✝ : ℝ → ℝ
a✝ b✝ : ℝ
μ : Measure ℝ
f g : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hgc : ContinuousOn g (Icc a b)
hle : ∀ x ∈ Ioc a b, f x ≤ g x
hlt : (volume.restrict (Ioc a b)) {x | f x < g x} = 0
h_eq : f =ᶠ[ae (volume.restrict (Ioc a b))] g

⊢ ∀ c ∈ Icc a b, g c ≤ f c
-- Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
f✝ g✝ : ℝ → ℝ
a✝ b✝ : ℝ
μ : Measure ℝ
f g : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hgc : ContinuousOn g (Icc a b)
hle : ∀ x ∈ Ioc a b, f x ≤ g x
hlt : (volume.restrict (Ioc a b)) {x | f x < g x} = 0
h_eq : f =ᶠ[ae (volume.restrict (Icc a b))] g

⊢ ∀ c ∈ Icc a b, g c ≤ f c
-- Mathlib/Analysis/Convex/Star.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : SMul 𝕜 E
inst✝ : SMul 𝕜 F
x : E
s t : Set E
S : Set (Set E)
hS : ∀ s ∈ S, StarConvex 𝕜 x s

⊢ StarConvex 𝕜 x (⋃₀ S)
-- Mathlib/Analysis/Convex/Star.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : SMul 𝕜 E
inst✝ : SMul 𝕜 F
x : E
s t : Set E
S : Set (Set E)
hS : ∀ s ∈ S, StarConvex 𝕜 x s

⊢ StarConvex 𝕜 x (⋃ i, ↑i)
-- Mathlib/Data/Set/Pointwise/Interval.lean
α : Type u_1
inst✝ : OrderedAddCommGroup α
a b c : α

⊢ (fun x => x - a) ⁻¹' Ioo b c = Ioo (b + a) (c + a)
-- Mathlib/Topology/Basic.lean
X : Type u
Y : Type v
ι : Sort w
α : Type u_1
β : Type u_2
x : X
s s₁ s₂ t : Set X
p p₁ p₂ : X → Prop
inst✝ : TopologicalSpace X
f : Filter X

⊢ ClusterPt x f ↔ ∃ u, ↑u ≤ f ∧ ↑u ≤ 𝓝 x
-- Mathlib/Data/Int/GCD.lean
s : ℕ

⊢ (xgcd 0 s).2 = 1
-- Mathlib/Analysis/Calculus/Deriv/Add.lean
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s t : Set 𝕜
L : Filter 𝕜
hf : HasStrictDerivAt f f' x
hg : HasStrictDerivAt g g' x

⊢ HasStrictDerivAt (fun x => f x - g x) (f' - g') x
-- Mathlib/Algebra/Polynomial/Basic.lean
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
p q : R[X]
c : R
h : c ≠ 0

⊢ (C c * X).support = {1}
-- Mathlib/Algebra/MvPolynomial/Degrees.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n✝ m : σ
s : σ →₀ ℕ
inst✝ : CommSemiring R
p q : MvPolynomial σ R
n : σ

⊢ degreeOf n 0 = 0
-- Mathlib/Topology/Connected/PathConnected.lean
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y z : X
ι : Type u_3
F : Set X
x : X

⊢ pathComponentIn x univ = pathComponent x