ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim : ∀ (a : ℝ), 1 < a → ∃ c, (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧ Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l

⊢ ∀ᶠ (b : ℕ) in atTop, d < u b / ↑b
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim : ∀ (a : ℝ), 1 < a → ∃ c, (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧ Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l

⊢ ∀ᶠ (b : ℕ) in atTop, u b / ↑b < d
-- Mathlib/RingTheory/WittVector/Truncated.lean
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
R : Type u_1
inst✝ : CommRing R
i : Fin n

⊢ coeff i (truncateFun n 0) = 0
-- Mathlib/Algebra/Order/Group/Abs.lean
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
a c d : α

⊢ |a - a| ≤ 0
-- Mathlib/Algebra/Module/Torsion.lean
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set R
a : R
h : torsionBySet R M s = ⊤
x : M

⊢ ∀ ⦃a : ↑s⦄, ↑a • x = 0
-- Mathlib/Algebra/Module/Torsion.lean
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set R
a : R
h : torsionBySet R M s = ⊤
x : M

⊢ x ∈ ⊤
-- Mathlib/Algebra/Group/Defs.lean
G : Type u_1
M : Type u_2
inst✝ : Monoid M
a✝ b c : M
m n : ℕ
a : M

⊢ a ^ 1 = a
-- Mathlib/Topology/Gluing.lean
D : GlueData
i : D.J
a : ↑(D.U i)
j : D.J
b : ↑(D.U j)
k : D.J
c : ↑(D.U k)
x : ↑(D.V (⟨i, a⟩.fst, ⟨j, b⟩.fst))
e₁ : (D.f ⟨i, a⟩.fst ⟨j, b⟩.fst) x = ⟨i, a⟩.snd
e₂ : (D.f ⟨j, b⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x) = ⟨j, b⟩.snd
y : ↑(D.V (⟨j, b⟩.fst, ⟨k, c⟩.fst))
e₃ : (D.f ⟨j, b⟩.fst ⟨k, c⟩.fst) y = ⟨j, b⟩.snd
e₄ : (D.f ⟨k, c⟩.fst ⟨j, b⟩.fst) ((D.t ⟨j, b⟩.fst ⟨k, c⟩.fst) y) = ⟨k, c⟩.snd

⊢ D.Rel ⟨i, a⟩ ⟨k, c⟩
-- Mathlib/Topology/Gluing.lean
D : GlueData
i : D.J
a : ↑(D.U i)
j : D.J
b : ↑(D.U j)
k : D.J
c : ↑(D.U k)
x : ↑(D.V (⟨i, a⟩.fst, ⟨j, b⟩.fst))
e₁ : (D.f ⟨i, a⟩.fst ⟨j, b⟩.fst) x = ⟨i, a⟩.snd
e₂ : (D.f ⟨j, b⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x) = ⟨j, b⟩.snd
y : ↑(D.V (⟨j, b⟩.fst, ⟨k, c⟩.fst))
e₃ : (D.f ⟨j, b⟩.fst ⟨k, c⟩.fst) y = ⟨j, b⟩.snd
e₄ : (D.f ⟨k, c⟩.fst ⟨j, b⟩.fst) ((D.t ⟨j, b⟩.fst ⟨k, c⟩.fst) y) = ⟨k, c⟩.snd
z : ↑(pullback (D.f j i) (D.f j k)) := (pullbackIsoProdSubtype (D.f j i) (D.f j k)).inv ⟨((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x, y), ⋯⟩

⊢ D.Rel ⟨i, a⟩ ⟨k, c⟩
-- Mathlib/Topology/Gluing.lean
D : GlueData
i : D.J
a : ↑(D.U i)
j : D.J
b : ↑(D.U j)
k : D.J
c : ↑(D.U k)
x : ↑(D.V (⟨i, a⟩.fst, ⟨j, b⟩.fst))
e₁ : (D.f ⟨i, a⟩.fst ⟨j, b⟩.fst) x = ⟨i, a⟩.snd
e₂ : (D.f ⟨j, b⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x) = ⟨j, b⟩.snd
y : ↑(D.V (⟨j, b⟩.fst, ⟨k, c⟩.fst))
e₃ : (D.f ⟨j, b⟩.fst ⟨k, c⟩.fst) y = ⟨j, b⟩.snd
e₄ : (D.f ⟨k, c⟩.fst ⟨j, b⟩.fst) ((D.t ⟨j, b⟩.fst ⟨k, c⟩.fst) y) = ⟨k, c⟩.snd
z : ↑(pullback (D.f j i) (D.f j k)) := (pullbackIsoProdSubtype (D.f j i) (D.f j k)).inv ⟨((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x, y), ⋯⟩
eq₁ : (D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z) = x

⊢ D.Rel ⟨i, a⟩ ⟨k, c⟩
-- Mathlib/Topology/Gluing.lean
D : GlueData
i : D.J
a : ↑(D.U i)
j : D.J
b : ↑(D.U j)
k : D.J
c : ↑(D.U k)
x : ↑(D.V (⟨i, a⟩.fst, ⟨j, b⟩.fst))
e₁ : (D.f ⟨i, a⟩.fst ⟨j, b⟩.fst) x = ⟨i, a⟩.snd
e₂ : (D.f ⟨j, b⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x) = ⟨j, b⟩.snd
y : ↑(D.V (⟨j, b⟩.fst, ⟨k, c⟩.fst))
e₃ : (D.f ⟨j, b⟩.fst ⟨k, c⟩.fst) y = ⟨j, b⟩.snd
e₄ : (D.f ⟨k, c⟩.fst ⟨j, b⟩.fst) ((D.t ⟨j, b⟩.fst ⟨k, c⟩.fst) y) = ⟨k, c⟩.snd
z : ↑(pullback (D.f j i) (D.f j k)) := (pullbackIsoProdSubtype (D.f j i) (D.f j k)).inv ⟨((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x, y), ⋯⟩
eq₁ : (D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z) = x
eq₂ : (pullback.snd (D.f j i) (D.f j k)) z = y

⊢ D.Rel ⟨i, a⟩ ⟨k, c⟩
-- Mathlib/Topology/Gluing.lean
D : GlueData
i : D.J
a : ↑(D.U i)
j : D.J
b : ↑(D.U j)
k : D.J
c : ↑(D.U k)
x : ↑(D.V (⟨i, a⟩.fst, ⟨j, b⟩.fst))
e₁ : (D.f ⟨i, a⟩.fst ⟨j, b⟩.fst) x = ⟨i, a⟩.snd
e₂ : (D.f ⟨j, b⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x) = ⟨j, b⟩.snd
y : ↑(D.V (⟨j, b⟩.fst, ⟨k, c⟩.fst))
e₃ : (D.f ⟨j, b⟩.fst ⟨k, c⟩.fst) y = ⟨j, b⟩.snd
e₄ : (D.f ⟨k, c⟩.fst ⟨j, b⟩.fst) ((D.t ⟨j, b⟩.fst ⟨k, c⟩.fst) y) = ⟨k, c⟩.snd
z : ↑(pullback (D.f j i) (D.f j k))
eq₁ : (D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z) = x
eq₂ : (pullback.snd (D.f j i) (D.f j k)) z = y

⊢ ∃ x, (D.f ⟨i, a⟩.fst ⟨k, c⟩.fst) x = ⟨i, a⟩.snd ∧ (D.f ⟨k, c⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨k, c⟩.fst) x) = ⟨k, c⟩.snd
-- Mathlib/Topology/Gluing.lean
D : GlueData
i : D.J
a : ↑(D.U i)
j : D.J
b : ↑(D.U j)
k : D.J
c : ↑(D.U k)
x : ↑(D.V (⟨i, a⟩.fst, ⟨j, b⟩.fst))
e₁ : (D.f ⟨i, a⟩.fst ⟨j, b⟩.fst) x = ⟨i, a⟩.snd
e₂ : (D.f ⟨j, b⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x) = ⟨j, b⟩.snd
y : ↑(D.V (⟨j, b⟩.fst, ⟨k, c⟩.fst))
e₃ : (D.f ⟨j, b⟩.fst ⟨k, c⟩.fst) y = ⟨j, b⟩.snd
e₄ : (D.f ⟨k, c⟩.fst ⟨j, b⟩.fst) ((D.t ⟨j, b⟩.fst ⟨k, c⟩.fst) y) = ⟨k, c⟩.snd
z : ↑(pullback (D.f j i) (D.f j k))
eq₁ : (D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z) = x
eq₂ : (pullback.snd (D.f j i) (D.f j k)) z = y

⊢ (D.f ⟨i, a⟩.fst ⟨k, c⟩.fst) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z)) = ⟨i, a⟩.snd ∧ (D.f ⟨k, c⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨k, c⟩.fst) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z))) = ⟨k, c⟩.snd
-- Mathlib/Topology/Gluing.lean
D : GlueData
i j k : D.J
z : ↑(pullback (D.f j i) (D.f j k))
e₂ : (D.f j i) ((D.t i j) ((D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z))) = (D.f j k) ((pullback.snd (D.f j i) (D.f j k)) z)

⊢ (D.f i k) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z)) = (D.f i j) ((D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z)) ∧ (D.f k i) ((D.t i k) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z))) = (D.f k j) ((D.t j k) ((pullback.snd (D.f j i) (D.f j k)) z))
-- Mathlib/Topology/Gluing.lean
D : GlueData
i j k : D.J
z : ↑(pullback (D.f j i) (D.f j k))
e₂ : (D.f j i) ((D.t i j) ((D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z))) = (D.f j k) ((pullback.snd (D.f j i) (D.f j k)) z)
h₁ : D.t' j i k ≫ pullback.fst (D.f i k) (D.f i j) ≫ D.f i k = pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j

⊢ (D.f i k) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z)) = (D.f i j) ((D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z)) ∧ (D.f k i) ((D.t i k) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z))) = (D.f k j) ((D.t j k) ((pullback.snd (D.f j i) (D.f j k)) z))
-- Mathlib/Topology/Gluing.lean
D : GlueData
i j k : D.J
z : ↑(pullback (D.f j i) (D.f j k))
e₂ : (D.f j i) ((D.t i j) ((D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z))) = (D.f j k) ((pullback.snd (D.f j i) (D.f j k)) z)
h₁ : D.t' j i k ≫ pullback.fst (D.f i k) (D.f i j) ≫ D.f i k = pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j
h₂ : D.t' j i k ≫ pullback.fst (D.f i k) (D.f i j) ≫ D.t i k ≫ D.f k i = pullback.snd (D.f j i) (D.f j k) ≫ D.t j k ≫ D.f k j

⊢ (D.f i k) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z)) = (D.f i j) ((D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z)) ∧ (D.f k i) ((D.t i k) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z))) = (D.f k j) ((D.t j k) ((pullback.snd (D.f j i) (D.f j k)) z))
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s₁ : AffineSubspace k P
v₁₂ v₂₃ : V

⊢ map (↑(constVAdd k P v₂₃)) (map (↑(constVAdd k P v₁₂)) s₁) = map (↑(constVAdd k P (v₂₃ + v₁₂))) s₁
-- Mathlib/Logic/Function/Basic.lean
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
g : α → γ
e' : β → γ
b : β
hb : ¬∃ a, f a = b

⊢ extend f g e' b = e' b
-- Mathlib/Algebra/BigOperators/Intervals.lean
α✝ : Type u_1
M : Type u_2
inst✝¹ : CommMonoid M
s₂ s₁ s : Finset α✝
a : α✝
g f✝ : α✝ → M
α : Type u_3
inst✝ : CommGroup α
f : ℕ → α
n m : ℕ
hnm : n ≤ m

⊢ (∏ k ∈ range m, f k) / ∏ k ∈ range n, f k = ∏ k ∈ filter (fun k => n ≤ k) (range m), f k
-- Mathlib/Algebra/BigOperators/Intervals.lean
α✝ : Type u_1
M : Type u_2
inst✝¹ : CommMonoid M
s₂ s₁ s : Finset α✝
a : α✝
g f✝ : α✝ → M
α : Type u_3
inst✝ : CommGroup α
f : ℕ → α
n m : ℕ
hnm : n ≤ m

⊢ ∏ k ∈ Ico n m, f k = ∏ k ∈ filter (fun k => n ≤ k) (range m), f k
-- Mathlib/Algebra/BigOperators/Intervals.lean
α✝ : Type u_1
M : Type u_2
inst✝¹ : CommMonoid M
s₂ s₁ s : Finset α✝
a : α✝
g f✝ : α✝ → M
α : Type u_3
inst✝ : CommGroup α
f : ℕ → α
n m : ℕ
hnm : n ≤ m

⊢ Ico n m = filter (fun k => n ≤ k) (range m)
-- Mathlib/Algebra/BigOperators/Intervals.lean
α✝ : Type u_1
M : Type u_2
inst✝¹ : CommMonoid M
s₂ s₁ s : Finset α✝
a : α✝
g f✝ : α✝ → M
α : Type u_3
inst✝ : CommGroup α
f : ℕ → α
n m : ℕ
hnm : n ≤ m

⊢ ∀ (a : ℕ), a ∈ Ico n m ↔ a ∈ filter (fun k => n ≤ k) (range m)
-- Mathlib/Algebra/BigOperators/Intervals.lean
α✝ : Type u_1
M : Type u_2
inst✝¹ : CommMonoid M
s₂ s₁ s : Finset α✝
a : α✝
g f✝ : α✝ → M
α : Type u_3
inst✝ : CommGroup α
f : ℕ → α
n m : ℕ
hnm : n ≤ m

⊢ ∀ (a : ℕ), n ≤ a ∧ a < m ↔ a < m ∧ n ≤ a
-- Mathlib/LinearAlgebra/Dual.lean
R : Type uR
M : Type uM
K : Type uK
V : Type uV
ι : Type uι
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : _root_.Finite ι

⊢ range b.toDual = ⊤
-- Mathlib/LinearAlgebra/Dual.lean
R : Type uR
M : Type uM
K : Type uK
V : Type uV
ι : Type uι
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : _root_.Finite ι
f : Dual R M

⊢ f ∈ range b.toDual
-- Mathlib/LinearAlgebra/Dual.lean
R : Type uR
M : Type uM
K : Type uK
V : Type uV
ι : Type uι
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : _root_.Finite ι
f : Dual R M
lin_comb : ι →₀ R := Finsupp.equivFunOnFinite.symm fun i => f (b i)

⊢ f ∈ range b.toDual
-- Mathlib/LinearAlgebra/Dual.lean
R : Type uR
M : Type uM
K : Type uK
V : Type uV
ι : Type uι
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : _root_.Finite ι
f : Dual R M
lin_comb : ι →₀ R := Finsupp.equivFunOnFinite.symm fun i => f (b i)
i : ι

⊢ (b.toDual ((Finsupp.total ι M R ⇑b) lin_comb)) (b i) = f (b i)
-- Mathlib/LinearAlgebra/Dual.lean
R : Type uR
M : Type uM
K : Type uK
V : Type uV
ι : Type uι
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : _root_.Finite ι
f : Dual R M
lin_comb : ι →₀ R := Finsupp.equivFunOnFinite.symm fun i => f (b i)
i : ι

⊢ lin_comb i = f (b i)
-- Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
inst✝ : S.HasLeftHomology

⊢ h.cyclesIso.inv ≫ S.iCycles = h.i
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/Zero.lean
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
E : Type u₃
inst✝⁵ : Category.{v₃, u₃} E
inst✝⁴ : HasZeroObject C
inst✝³ : HasZeroObject D
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroMorphisms D
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms

⊢ 0 ≫ 0 = 𝟙 (F.obj 0)
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/Zero.lean
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
E : Type u₃
inst✝⁵ : Category.{v₃, u₃} E
inst✝⁴ : HasZeroObject C
inst✝³ : HasZeroObject D
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroMorphisms D
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms

⊢ 0 ≫ 0 = 𝟙 0
-- Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
n : ℕ
h : fract v ≠ 0

⊢ (of v).s.get? 0 = some { a := 1, b := ↑⌊(fract v)⁻¹⌋ }
-- Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
n : ℕ
h : fract v ≠ 0

⊢ (match some (IntFractPair.of (IntFractPair.of v).fr⁻¹), some ∘ fun p => { a := 1, b := ↑p.b } with | none, x => none | some a, f => f a) = some { a := 1, b := ↑⌊(fract v)⁻¹⌋ }
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier

⊢ ↑(index (↑K₁) U) / ↑(index (↑K₀) U) ≤ ↑(index (↑K₂) U) / ↑(index (↑K₀) U)
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier

⊢ ↑(index (↑K₁) U) ≤ ↑(index (↑K₂) U)
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier

⊢ 0 < ↑(index (↑K₀) U)
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
ι : Type (max u_2 u_3)
f : ι → Ordinal.{max u_2 u_3}
c : Cardinal.{max u_2 u_3}
hc : c.IsRegular
hι : #ι < c

⊢ #ι < c.ord.cof
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : Nontrivial R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : TopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module R A
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
a : A
ha : p a
inst✝ : UniqueNonUnitalContinuousFunctionalCalculus R A
φ : C(↑(σₙ R a), R)₀ →⋆ₙₐ[R] A
hφ₁ : Continuous ⇑φ
hφ₂ : φ { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ } = a

⊢ (cfcₙHom ha) { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ } = φ { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ }
-- Mathlib/GroupTheory/GroupAction/Blocks.lean
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B : Set X

⊢ IsBlock G B ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
-- Mathlib/GroupTheory/GroupAction/Blocks.lean
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B : Set X

⊢ (∀ (g : G), g • B ∩ B ≠ ∅ → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
-- Mathlib/GroupTheory/GroupAction/Blocks.lean
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B : Set X

⊢ (∀ (g : G) (x : X), g⁻¹ • x ∈ B → x ∈ B → g • B = B) ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B
-- Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M

⊢ ⨆ i, LinearMap.range (ι Q) ^ i = ⊤
-- Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M

⊢ iSup (evenOdd Q) = ⨆ i, LinearMap.range (ι Q) ^ i
-- Mathlib/Algebra/GroupWithZero/Basic.lean
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝ : CancelMonoidWithZero M₀
a b c : M₀
ha : a ≠ 0

⊢ a = a * b ↔ b = 1
-- Mathlib/NumberTheory/Padics/PadicVal.lean
p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hS : S.Nonempty
hF : ∀ i ∈ S, padicValRat p (F j) < padicValRat p (F i)
hn1 : ∀ (i : ℕ), 0 < F i

⊢ padicValRat p (F j) < padicValRat p (∑ i ∈ S, F i)
-- Mathlib/NumberTheory/Padics/PadicVal.lean
p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
k : ℕ
hF : ∀ i ∈ {k}, padicValRat p (F j) < padicValRat p (F i)

⊢ padicValRat p (F j) < padicValRat p (∑ i ∈ {k}, F i)
-- Mathlib/NumberTheory/Padics/PadicVal.lean
p✝ : ℕ
hp✝ : Fact (Nat.Prime p✝)
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
s : ℕ
S' : Finset ℕ
Hnot : s ∉ S'
Hne : S'.Nonempty
Hind : (∀ i ∈ S', padicValRat p (F j) < padicValRat p (F i)) → padicValRat p (F j) < padicValRat p (∑ i ∈ S', F i)
hF : ∀ i ∈ Finset.cons s S' Hnot, padicValRat p (F j) < padicValRat p (F i)

⊢ padicValRat p (F j) < padicValRat p (∑ i ∈ Finset.cons s S' Hnot, F i)
-- Mathlib/Data/List/TFAE.lean
α : Type u_1
l : List (α → Prop)
H : ∀ (a : α), (map (fun p => p a) l).TFAE

⊢ (map (fun p => ∃ a, p a) l).TFAE
-- Mathlib/Data/List/TFAE.lean
α : Type u_1
l : List (α → Prop)
H : ∀ (a : α), (map (fun p => p a) l).TFAE

⊢ ∀ (j : α → Prop), j ∈ l → ∀ (j_1 : α → Prop), j_1 ∈ l → ((∃ a, j a) ↔ ∃ a, j_1 a)
-- Mathlib/Data/List/TFAE.lean
α : Type u_1
l : List (α → Prop)
H : ∀ (a : α), (map (fun p => p a) l).TFAE
p₁ : α → Prop
hp₁ : p₁ ∈ l
p₂ : α → Prop
hp₂ : p₂ ∈ l

⊢ (∃ a, p₁ a) ↔ ∃ a, p₂ a
-- Mathlib/Data/Finset/Piecewise.lean
ι : Type u_1
π : ι → Sort u_2
s : Finset ι
f g : (i : ι) → π i
inst✝² : (j : ι) → Decidable (j ∈ s)
inst✝¹ : DecidableEq ι
j : ι
inst✝ : (i : ι) → Decidable (i ∈ insert j s)

⊢ (insert j s).piecewise f g = update (s.piecewise f g) j (f j)
-- Mathlib/Data/Finset/Piecewise.lean
ι : Type u_1
π : ι → Sort u_2
s : Finset ι
f g : (i : ι) → π i
inst✝² : (j : ι) → Decidable (j ∈ s)
inst✝¹ : DecidableEq ι
j : ι
inst✝ : (i : ι) → Decidable (i ∈ insert j s)

⊢ (↑(insert j s)).piecewise f g = (insert j ↑s).piecewise f g
-- Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
t : TransvectionStruct ι ℝ

⊢ MeasurePreserving (⇑(toLin' t.toMatrix)) volume volume
-- Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
t : TransvectionStruct ι ℝ
ht : Measurable ⇑(toLin' t.toMatrix)

⊢ MeasurePreserving (⇑(toLin' t.toMatrix)) volume volume
-- Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
t : TransvectionStruct ι ℝ
ht : Measurable ⇑(toLin' t.toMatrix)

⊢ Measure.map (⇑(toLin' t.toMatrix)) volume = volume
-- Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
t : TransvectionStruct ι ℝ
ht : Measurable ⇑(toLin' t.toMatrix)
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)

⊢ (Measure.map (⇑(toLin' t.toMatrix)) volume) (univ.pi s) = ∏ i : ι, volume (s i)
-- Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
t : TransvectionStruct ι ℝ
ht : Measurable ⇑(toLin' t.toMatrix)
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
h2s : MeasurableSet (univ.pi s)

⊢ (Measure.map (⇑(toLin' t.toMatrix)) volume) (univ.pi s) = ∏ i : ι, volume (s i)
-- Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
t : TransvectionStruct ι ℝ
ht : Measurable ⇑(toLin' t.toMatrix)
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
h2s : MeasurableSet (univ.pi s)

⊢ ∫⁻ (a : ι → ℝ), (univ.pi s).indicator 1 a ∂Measure.map (⇑(toLin' t.toMatrix)) volume = ∫⁻ (a : ι → ℝ), (univ.pi s).indicator 1 a ∂Measure.pi fun i => volume
-- Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
t : TransvectionStruct ι ℝ
ht : Measurable ⇑(toLin' t.toMatrix)
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
h2s : MeasurableSet (univ.pi s)

⊢ (∫⁻ (a : ι → ℝ), (univ.pi s).indicator 1 ((toLin' t.toMatrix) a) ∂Measure.pi fun x => volume) = ∫⁻ (a : ι → ℝ), (univ.pi s).indicator 1 a ∂Measure.pi fun i => volume
-- Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
t : TransvectionStruct ι ℝ
ht : Measurable ⇑(toLin' t.toMatrix)
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
h2s : MeasurableSet (univ.pi s)

⊢ (∫⋯∫⁻_{t.i}, fun a => (univ.pi s).indicator 1 ((toLin' t.toMatrix) a) ∂fun x => volume) = ∫⋯∫⁻_{t.i}, (univ.pi s).indicator 1 ∂fun x => volume
-- Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
t : TransvectionStruct ι ℝ
ht : Measurable ⇑(toLin' t.toMatrix)
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
h2s : MeasurableSet (univ.pi s)

⊢ (fun x => ∫⁻ (xᵢ : ℝ), (univ.pi s).indicator 1 ((toLin' t.toMatrix) (Function.update x t.i xᵢ))) = fun x => ∫⁻ (xᵢ : ℝ), (univ.pi s).indicator 1 (Function.update x t.i xᵢ)
-- Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
t : TransvectionStruct ι ℝ
ht : Measurable ⇑(toLin' t.toMatrix)
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
h2s : MeasurableSet (univ.pi s)
x : ι → ℝ

⊢ ∫⁻ (xᵢ : ℝ), (univ.pi s).indicator 1 ((toLin' t.toMatrix) (Function.update x t.i xᵢ)) = ∫⁻ (xᵢ : ℝ), (univ.pi s).indicator 1 (Function.update x t.i xᵢ)
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ✝ : Kernel α (γ × β)
ν : Kernel α γ
κ : Kernel α (γ × β)
inst✝ : IsFiniteKernel κ
n : ℕ
a : α

⊢ ∀ᵐ (x : γ) ∂κ.fst a, κ.densityProcess κ.fst n a x univ = 1
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ✝ : Kernel α (γ × β)
ν : Kernel α γ
κ : Kernel α (γ × β)
inst✝ : IsFiniteKernel κ
n : ℕ
a : α

⊢ (κ.fst a) {a_1 | ¬κ.densityProcess κ.fst n a a_1 univ = 1} = 0
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ✝ : Kernel α (γ × β)
ν : Kernel α γ
κ : Kernel α (γ × β)
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
this : {x | ¬κ.densityProcess κ.fst n a x univ = 1} ⊆ {x | (κ.fst a) (countablePartitionSet n x) = 0}

⊢ (κ.fst a) {a_1 | ¬κ.densityProcess κ.fst n a a_1 univ = 1} = 0
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ✝ : Kernel α (γ × β)
ν : Kernel α γ
κ : Kernel α (γ × β)
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
this : {x | ¬κ.densityProcess κ.fst n a x univ = 1} ⊆ {x | (κ.fst a) (countablePartitionSet n x) = 0}

⊢ (κ.fst a) {x | (κ.fst a) (countablePartitionSet n x) = 0} = 0
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ✝ : Kernel α (γ × β)
ν : Kernel α γ
κ : Kernel α (γ × β)
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
this✝ : {x | ¬κ.densityProcess κ.fst n a x univ = 1} ⊆ {x | (κ.fst a) (countablePartitionSet n x) = 0}
this : {x | (κ.fst a) (countablePartitionSet n x) = 0} ⊆ ⋃ u ∈ countablePartition γ n, ⋃ (_ : (κ.fst a) u = 0), u

⊢ (κ.fst a) {x | (κ.fst a) (countablePartitionSet n x) = 0} = 0
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ✝ : Kernel α (γ × β)
ν : Kernel α γ
κ : Kernel α (γ × β)
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
this✝ : {x | ¬κ.densityProcess κ.fst n a x univ = 1} ⊆ {x | (κ.fst a) (countablePartitionSet n x) = 0}
this : {x | (κ.fst a) (countablePartitionSet n x) = 0} ⊆ ⋃ u ∈ countablePartition γ n, ⋃ (_ : (κ.fst a) u = 0), u

⊢ (κ.fst a) (⋃ u ∈ countablePartition γ n, ⋃ (_ : (κ.fst a) u = 0), u) = 0
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ✝ : Kernel α (γ × β)
ν : Kernel α γ
κ : Kernel α (γ × β)
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
this✝ : {x | ¬κ.densityProcess κ.fst n a x univ = 1} ⊆ {x | (κ.fst a) (countablePartitionSet n x) = 0}
this : {x | (κ.fst a) (countablePartitionSet n x) = 0} ⊆ ⋃ u ∈ countablePartition γ n, ⋃ (_ : (κ.fst a) u = 0), u

⊢ ∑' (p : ↑(countablePartition γ n)), (κ.fst a) (⋃ (_ : (κ.fst a) ↑p = 0), ↑p) = 0