ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
x✝ x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)

⊢ x < tan x
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
x✝ x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2

⊢ x < tan x
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
x✝ x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y

⊢ x < tan x
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
x✝ x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y

⊢ x < tan x
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
x✝ x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U

⊢ x < tan x
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
x✝ x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y => tan y - y) U

⊢ x < tan x
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
x✝ x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y => tan y - y) U
deriv_pos : ∀ y ∈ interior U, 0 < deriv (fun y' => tan y' - y') y

⊢ x < tan x
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
x✝ x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y => tan y - y) U
deriv_pos : ∀ y ∈ interior U, 0 < deriv (fun y' => tan y' - y') y
mono : StrictMonoOn (fun y => tan y - y) (Ico 0 (π / 2))

⊢ x < tan x
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
x✝ x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y => tan y - y) U
deriv_pos : ∀ y ∈ interior U, 0 < deriv (fun y' => tan y' - y') y
mono : StrictMonoOn (fun y => tan y - y) (Ico 0 (π / 2))
zero_in_U : 0 ∈ U

⊢ x < tan x
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
x✝ x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y
tan_cts_U : ContinuousOn tan U
tan_minus_id_cts : ContinuousOn (fun y => tan y - y) U
deriv_pos : ∀ y ∈ interior U, 0 < deriv (fun y' => tan y' - y') y
mono : StrictMonoOn (fun y => tan y - y) (Ico 0 (π / 2))
zero_in_U : 0 ∈ U
x_in_U : x ∈ U

⊢ x < tan x
-- Mathlib/Data/Nat/Fib/Basic.lean
n : ℕ

⊢ fib (2 * n + 2) = fib (n + 1) * (2 * fib n + fib (n + 1))
-- Mathlib/Data/Nat/Fib/Basic.lean
n : ℕ

⊢ fib n * (2 * fib (n + 1) - fib n) + (fib (n + 1) ^ 2 + fib n ^ 2) = fib (n + 1) * (2 * fib n + fib (n + 1))
-- Mathlib/Data/Nat/Fib/Basic.lean
n : ℕ
this : fib n ≤ 2 * fib (n + 1)

⊢ fib n * (2 * fib (n + 1) - fib n) + (fib (n + 1) ^ 2 + fib n ^ 2) = fib (n + 1) * (2 * fib n + fib (n + 1))
-- Mathlib/Data/Set/Function.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
π : α → Type u_5
s s₁ s₂ : Set α
t t₁ t₂ : Set β
p : Set γ
f f₁ f₂ f₃ : α → β
g g₁ g₂ : β → γ
f' f₁' f₂' : β → α
g' : γ → β
a : α
b : β

⊢ Surjective f ↔ SurjOn f univ univ
-- Mathlib/RingTheory/Localization/Basic.lean
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
x : R
y : ↥M

⊢ mk x y = mk' (Localization M) x y
-- Mathlib/CategoryTheory/Localization/SmallHom.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
W : MorphismProperty C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
D' : Type u₃
inst✝² : Category.{v₃, u₃} D'
L : C ⥤ D
inst✝¹ : L.IsLocalization W
X Y Z : C
inst✝ : HasSmallLocalizedHom W X Y

⊢ Small.{w, v₂} (L.obj X ⟶ L.obj Y)
-- Mathlib/LinearAlgebra/Span.lean
R : Type u_1
R₂ : Type u_2
K : Type u_3
M : Type u_4
M₂ : Type u_5
V : Type u_6
S : Type u_7
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module R M
x : M
p p' : Submodule R M
inst✝⁴ : Semiring R₂
σ₁₂ : R →+* R₂
inst✝³ : AddCommMonoid M₂
inst✝² : Module R₂ M₂
F : Type u_8
inst✝¹ : FunLike F M M₂
inst✝ : SemilinearMapClass F σ₁₂ M M₂
s t : Set M

⊢ p ⊔ span R s = span R (↑p ∪ s)
-- Mathlib/Topology/Instances/Rat.lean
x y : ℕ

⊢ dist ↑x ↑y = dist x y
-- Mathlib/Topology/Instances/Rat.lean
x y : ℕ

⊢ dist ↑↑x ↑↑y = dist ↑x ↑y
-- Mathlib/MeasureTheory/Constructions/Cylinders.lean
ι : Type u_1
α : ι → Type u_2
s : Finset ι
S₁ S₂ : Set ((i : { x // x ∈ s }) → α ↑i)

⊢ cylinder s S₁ ∩ cylinder s S₂ = cylinder s (S₁ ∩ S₂)
-- Mathlib/Order/Defs.lean
α : Type u
inst✝ : LinearOrder α
a b : α

⊢ compare a b = Ordering.eq ↔ a = b
-- Mathlib/Order/Defs.lean
α : Type u
inst✝ : LinearOrder α
a b : α

⊢ (if a < b then Ordering.lt else if a = b then Ordering.eq else Ordering.gt) = Ordering.eq ↔ a = b
-- Mathlib/Order/Defs.lean
α : Type u
inst✝ : LinearOrder α
a b : α
h✝ : a < b

⊢ False ↔ a = b
-- Mathlib/Order/Defs.lean
α : Type u
inst✝ : LinearOrder α
a b : α
h✝¹ : ¬a < b
h✝ : a = b

⊢ True ↔ a = b
-- Mathlib/Order/Defs.lean
α : Type u
inst✝ : LinearOrder α
a b : α
h✝¹ : ¬a < b
h✝ : ¬a = b

⊢ False ↔ a = b
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g
x : α
hx : x ∈ filter (fun a => a ^ n = 1) univ
m : ℕ
hm : g ^ m = x

⊢ (fun x => (g ^ (Fintype.card α / n.gcd (Fintype.card α))) ^ x) ↑(m / (Fintype.card α / n.gcd (Fintype.card α))) = x
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g
x : α
hx : x ∈ filter (fun a => a ^ n = 1) univ
m : ℕ
hm : g ^ m = x
hgmn : g ^ (m * n.gcd (Fintype.card α)) = 1

⊢ (fun x => (g ^ (Fintype.card α / n.gcd (Fintype.card α))) ^ x) ↑(m / (Fintype.card α / n.gcd (Fintype.card α))) = x
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g
x : α
hx : x ∈ filter (fun a => a ^ n = 1) univ
m : ℕ
hm : g ^ m = x
hgmn : g ^ (m * n.gcd (Fintype.card α)) = 1

⊢ (g ^ (Fintype.card α / n.gcd (Fintype.card α))) ^ ↑(m / (Fintype.card α / n.gcd (Fintype.card α))) = x
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g
x : α
hx : x ∈ filter (fun a => a ^ n = 1) univ
m : ℕ
hm : g ^ m = x
hgmn : g ^ (m * n.gcd (Fintype.card α)) = 1

⊢ Fintype.card α / n.gcd (Fintype.card α) ∣ m
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g
x : α
hx : x ∈ filter (fun a => a ^ n = 1) univ
m : ℕ
hm : g ^ m = x
hgmn : g ^ (m * n.gcd (Fintype.card α)) = 1

⊢ Fintype.card α / n.gcd (Fintype.card α) * n.gcd (Fintype.card α) ∣ m * n.gcd (Fintype.card α)
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g
x : α
hx : x ∈ filter (fun a => a ^ n = 1) univ
m : ℕ
hm : g ^ m = x
hgmn : g ^ (m * n.gcd (Fintype.card α)) = 1

⊢ orderOf g ∣ m * n.gcd (Fintype.card α)
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g

⊢ (↑(zpowers (g ^ (Fintype.card α / n.gcd (Fintype.card α))))).toFinset.card ≤ n
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g
m : ℕ
hm : Fintype.card α = n.gcd (Fintype.card α) * m

⊢ (↑(zpowers (g ^ (Fintype.card α / n.gcd (Fintype.card α))))).toFinset.card ≤ n
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g
m : ℕ
hm : Fintype.card α = n.gcd (Fintype.card α) * m
hm0 : 0 < m

⊢ (↑(zpowers (g ^ (Fintype.card α / n.gcd (Fintype.card α))))).toFinset.card ≤ n
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g
m : ℕ
hm : Fintype.card α = n.gcd (Fintype.card α) * m
hm0 : 0 < m

⊢ Fintype.card ↥(zpowers (g ^ (Fintype.card α / n.gcd (Fintype.card α)))) ≤ n
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g
m : ℕ
hm : Fintype.card α = n.gcd (Fintype.card α) * m
hm0 : 0 < m

⊢ Fintype.card α / (Fintype.card α).gcd (Fintype.card α / n.gcd (Fintype.card α)) ≤ n
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g
m : ℕ
hm : Fintype.card α = n.gcd (Fintype.card α) * m
hm0 : 0 < m

⊢ Fintype.card α / (n.gcd (Fintype.card α) * m).gcd (n.gcd (Fintype.card α) * m / n.gcd (Fintype.card α)) ≤ n
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝³ : Group α
inst✝² : DecidableEq α
inst✝¹ : Fintype α
inst✝ : IsCyclic α
n : ℕ
hn0 : 0 < n
g : α
hg : ∀ (x : α), x ∈ zpowers g
m : ℕ
hm : Fintype.card α = n.gcd (Fintype.card α) * m
hm0 : 0 < m

⊢ n.gcd (Fintype.card α) ≤ n
-- Mathlib/Algebra/Lie/IdealOperations.lean
R : Type u
L : Type v
M : Type w
M₂ : Type w₁
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : AddCommGroup M₂
inst✝² : Module R M₂
inst✝¹ : LieRingModule L M₂
inst✝ : LieModule R L M₂
N N' : LieSubmodule R L M
I J : LieIdeal R L
N₂ : LieSubmodule R L M₂
h₁ : I ≤ J
h₂ : N ≤ N'
m : M
h : m ∈ ⁅I, N⁆

⊢ m ∈ ⁅J, N'⁆
-- Mathlib/Algebra/Lie/IdealOperations.lean
R : Type u
L : Type v
M : Type w
M₂ : Type w₁
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : AddCommGroup M₂
inst✝² : Module R M₂
inst✝¹ : LieRingModule L M₂
inst✝ : LieModule R L M₂
N N' : LieSubmodule R L M
I J : LieIdeal R L
N₂ : LieSubmodule R L M₂
h₁ : I ≤ J
h₂ : N ≤ N'
m : M
h : ∀ (N_1 : LieSubmodule R L M), {m | ∃ x n, ⁅↑x, ↑n⁆ = m} ⊆ ↑N_1 → m ∈ N_1

⊢ m ∈ ⁅J, N'⁆
-- Mathlib/Algebra/Lie/IdealOperations.lean
R : Type u
L : Type v
M : Type w
M₂ : Type w₁
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : AddCommGroup M₂
inst✝² : Module R M₂
inst✝¹ : LieRingModule L M₂
inst✝ : LieModule R L M₂
N N' : LieSubmodule R L M
I J : LieIdeal R L
N₂ : LieSubmodule R L M₂
h₁ : I ≤ J
h₂ : N ≤ N'
m : M
h : ∀ (N_1 : LieSubmodule R L M), {m | ∃ x n, ⁅↑x, ↑n⁆ = m} ⊆ ↑N_1 → m ∈ N_1

⊢ ∀ (N : LieSubmodule R L M), {m | ∃ x n, ⁅↑x, ↑n⁆ = m} ⊆ ↑N → m ∈ N
-- Mathlib/FieldTheory/PurelyInseparable.lean
F : Type u
E : Type v
inst✝¹¹ : Field F
inst✝¹⁰ : Field E
inst✝⁹ : Algebra F E
K✝ : Type w
inst✝⁸ : Field K✝
inst✝⁷ : Algebra F K✝
inst✝⁶ : Algebra E K✝
inst✝⁵ : IsScalarTower F E K✝
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsSeparable F E

⊢ Module.rank F E * sepDegree E K = sepDegree F K
-- Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
l : Type u_1
m : Type u
n : Type u'
α : Type v
inst✝⁴ : DecidableEq m
inst✝³ : DecidableEq n
R : Type u_2
inst✝² : Semiring R
inst✝¹ : Finite m
inst✝ : Fintype n
A : Matrix m n R

⊢ Function.Surjective A.mulVec ↔ ∃ B, A * B = 1
-- Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
l : Type u_1
m : Type u
n : Type u'
α : Type v
inst✝⁴ : DecidableEq m
inst✝³ : DecidableEq n
R : Type u_2
inst✝² : Semiring R
inst✝¹ : Finite m
inst✝ : Fintype n
A : Matrix m n R
val✝ : Fintype m

⊢ Function.Surjective A.mulVec ↔ ∃ B, A * B = 1
-- Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
l : Type u_1
m : Type u
n : Type u'
α : Type v
inst✝⁴ : DecidableEq m
inst✝³ : DecidableEq n
R : Type u_2
inst✝² : Semiring R
inst✝¹ : Finite m
inst✝ : Fintype n
A : Matrix m n R
val✝ : Fintype m
h : Function.Surjective A.mulVec
cols : m → n → R
hcols : ∀ (x : m), A *ᵥ cols x = Pi.single x 1

⊢ ∃ B, A * B = 1
-- Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
l : Type u_1
m : Type u
n : Type u'
α : Type v
inst✝⁴ : DecidableEq m
inst✝³ : DecidableEq n
R : Type u_2
inst✝² : Semiring R
inst✝¹ : Finite m
inst✝ : Fintype n
A : Matrix m n R
val✝ : Fintype m
h : Function.Surjective A.mulVec
cols : m → n → R
hcols : ∀ (x : m), A *ᵥ cols x = Pi.single x 1
i j : m

⊢ (A * (of cols)ᵀ) i j = 1 i j
-- Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
l : Type u_1
m : Type u
n : Type u'
α : Type v
inst✝⁴ : DecidableEq m
inst✝³ : DecidableEq n
R : Type u_2
inst✝² : Semiring R
inst✝¹ : Finite m
inst✝ : Fintype n
A : Matrix m n R
val✝ : Fintype m
h : Function.Surjective A.mulVec
cols : m → n → R
hcols : ∀ (x : m), A *ᵥ cols x = Pi.single x 1
i j : m

⊢ (A * (of cols)ᵀ) i j = (A *ᵥ cols j) i
-- Mathlib/Topology/UniformSpace/Cauchy.lean
α : Type u
β : Type v
uniformSpace : UniformSpace α
u v : UniformSpace β
F : Filter β

⊢ F.NeBot ∧ F ×ˢ F ≤ 𝓤 β ↔ (F.NeBot ∧ F ×ˢ F ≤ 𝓤 β) ∧ F.NeBot ∧ F ×ˢ F ≤ 𝓤 β
-- Mathlib/CategoryTheory/Shift/Basic.lean
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddGroup A
inst✝ : HasShift C A
i j : A
h : i + j = 0

⊢ j + i = 0
-- Mathlib/CategoryTheory/Shift/Basic.lean
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddGroup A
inst✝ : HasShift C A
i j : A
h : i + j = 0
X : C

⊢ (shiftFunctor C i).map ((shiftFunctorCompIsoId C i j h).symm.hom.app X) ≫ (shiftFunctorCompIsoId C j i ⋯).hom.app ((shiftFunctor C i).obj X) = 𝟙 ((shiftFunctor C i).obj X)
-- Mathlib/CategoryTheory/Shift/Basic.lean
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddGroup A
inst✝ : HasShift C A
i j : A
h : i + j = 0
X : C
e_1✝ : ((shiftFunctor C i).obj ((𝟭 C).obj X) ⟶ (𝟭 C).obj ((shiftFunctor C i).obj X)) = ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor.obj ((𝟭 C).obj X) ⟶ (𝟭 C).obj ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor.obj X))
e_3✝ : (shiftFunctor C i).obj ((𝟭 C).obj X) = (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor.obj ((𝟭 C).obj X)
e_4✝¹ : (shiftFunctor C i).obj ((shiftFunctor C i ⋙ shiftFunctor C j).obj X) = (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor.obj (((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor ⋙ (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).inverse).obj X)
e_7✝ : (shiftFunctor C i ⋙ shiftFunctor C j).obj X = ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor ⋙ (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).inverse).obj X
e_6✝ : shiftFunctor C i ⋙ shiftFunctor C j = (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor ⋙ (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).inverse
e_4✝ : shiftFunctor C i ⋙ shiftFunctor C j = (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor ⋙ (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).inverse

⊢ (shiftFunctorCompIsoId C i j h).symm = (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).unitIso
-- Mathlib/CategoryTheory/Shift/Basic.lean
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddGroup A
inst✝ : HasShift C A
i j : A
h : i + j = 0
X : C
e_1✝ : ((shiftFunctor C i).obj ((𝟭 C).obj X) ⟶ (𝟭 C).obj ((shiftFunctor C i).obj X)) = ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor.obj ((𝟭 C).obj X) ⟶ (𝟭 C).obj ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor.obj X))
e_4✝ : (shiftFunctor C i).obj ((shiftFunctor C i ⋙ shiftFunctor C j).obj X) = (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor.obj (((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor ⋙ (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).inverse).obj X)
e_5✝¹ : (𝟭 C).obj ((shiftFunctor C i).obj X) = (𝟭 C).obj ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor.obj X)
e_5✝ : shiftFunctor C j ⋙ shiftFunctor C i = (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).inverse ⋙ (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor
e_3✝ : shiftFunctor C j ⋙ shiftFunctor C i = (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).inverse ⋙ (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).functor

⊢ shiftFunctorCompIsoId C j i ⋯ = (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h) (Discrete.eqToIso ⋯) ⋯).counitIso
-- Mathlib/Analysis/Calculus/BumpFunction/Normed.lean
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
x : E
n : ℕ∞
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsOpenPosMeasure

⊢ (μ (closedBall c f.rIn)).toReal ≤ ∫ (x : E), ↑f x ∂μ
-- Mathlib/Combinatorics/Additive/AP/Three/Defs.lean
F : Type u_1
α : Type u_2
β : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝³ : DecidableEq α
inst✝² : Monoid α
inst✝¹ : DecidableEq β
inst✝ : Monoid β
s t : Finset α

⊢ ThreeGPFree ∅
-- 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
s : (i : ι) → κ i → Set α

⊢ ⋂ i, ⋂ j, s i j = ∅ ↔ ∀ (a : α), ∃ i j, a ∉ s i j
-- Mathlib/Analysis/Analytic/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
h : Summable fun n => ‖p n‖ * ↑r ^ n

⊢ Summable fun n => ‖p n‖₊ * r ^ n
-- Mathlib/Analysis/Analytic/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
h : Summable fun n => ↑‖p n‖₊ * ↑r ^ n

⊢ Summable fun n => ‖p n‖₊ * r ^ n
-- Mathlib/LinearAlgebra/Matrix/Transvection.lean
n : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : DecidableEq n
inst✝² : DecidableEq p
inst✝¹ : CommRing R
i j : n
inst✝ : Fintype n
a b : n
hb : b ≠ j
c : R
M : Matrix n n R

⊢ (M * transvection i j c) a b = M a b
-- Mathlib/CategoryTheory/Types.lean
C : Type u
inst✝ : Category.{v, u} C
F G H : C ⥤ Type w
X Y Z : C
σ : F ⟶ G
τ : G ⟶ H
f : X ⟶ Y
g : Y ⟶ Z
a : F.obj X

⊢ F.map (f ≫ g) a = F.map g (F.map f a)
-- Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean
α : Type u_1
a b c d : α
inst✝⁴ : MulOneClass α
inst✝³ : Zero α
inst✝² : Preorder α
inst✝¹ : MulPosStrictMono α
inst✝ : MulPosReflectLT α
a0 : 0 < a

⊢ a < b * a ↔ 1 * a < b * a
-- 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

⊢ M ^ n = span R (↑M ^ n)
-- Mathlib/CategoryTheory/Sites/NonabelianCohomology/H1.lean
C : Type u
inst✝ : Category.{v, u} C
G : Cᵒᵖ ⥤ Grp
X : C
I : Type w'
U : I → C
γ : OneCocycle G U
i j : I
T : C
a : T ⟶ U i
b : T ⟶ U j

⊢ γ.ev i j a b = (γ.ev j i b a)⁻¹
-- 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
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid P
l✝ l₁ l₂ : List M
a : M
l : List M
m : M
h : ∀ x ∈ l, x = m

⊢ l.prod = m ^ l.length
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Field R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : CompleteSpace R
inst✝⁷ : TopologicalRing R
inst✝⁶ : ContinuousStar R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
inst✝ : UniqueNonUnitalContinuousFunctionalCalculus R A
a : A
ha : p a

⊢ cfcₙHom ha = cfcₙHom_of_cfcHom R ha
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Field R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : CompleteSpace R
inst✝⁷ : TopologicalRing R
inst✝⁶ : ContinuousStar R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
inst✝ : UniqueNonUnitalContinuousFunctionalCalculus R A
a : A
ha : p a
h_cpct' : CompactSpace ↑(σₙ R a)

⊢ cfcₙHom ha = cfcₙHom_of_cfcHom R ha
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Field R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : CompleteSpace R
inst✝⁷ : TopologicalRing R
inst✝⁶ : ContinuousStar R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
inst✝ : UniqueNonUnitalContinuousFunctionalCalculus R A
a : A
ha : p a
h_cpct' : CompactSpace ↑(σₙ R a)

⊢ ↑0 = 0