ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Subsingleton A
\u22a2 spectrum ℂ (star a * a) ⊆ Set.Icc 0 ↑‖star a * a‖
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h✝ : Nontrivial A
\u22a2 spectrum ℂ (star a * a) ⊆ Set.Icc 0 ↑‖star a * a‖
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β : Type u_2
δ : Type u_3
ι✝ : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s t : Set α
ι : Type u_5
x✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
Union_As_finite : μ (⋃ i, As i) ≠ ⊤
posmeas : Set ι := {i | 0 < μ (As i)}
posmeas_def : posmeas = {i | 0 < μ (As i)}
\u22a2 posmeas.Countable
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β : Type u_2
δ : Type u_3
ι✝ : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s t : Set α
ι : Type u_5
x✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
Union_As_finite : μ (⋃ i, As i) ≠ ⊤
posmeas : Set ι := {i | 0 < μ (As i)}
posmeas_def : posmeas = {i | 0 < μ (As i)}
as : ℕ → ℝ≥0∞
left✝ : StrictAnti as
as_mem : ∀ (n : ℕ), as n ∈ Ioo 0 1
as_lim : Tendsto as atTop (𝓝 0)
\u22a2 posmeas.Countable
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β : Type u_2
δ : Type u_3
ι✝ : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s t : Set α
ι : Type u_5
x✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
Union_As_finite : μ (⋃ i, As i) ≠ ⊤
posmeas : Set ι := {i | 0 < μ (As i)}
posmeas_def : posmeas = {i | 0 < μ (As i)}
as : ℕ → ℝ≥0∞
left✝ : StrictAnti as
as_mem : ∀ (n : ℕ), as n ∈ Ioo 0 1
as_lim : Tendsto as atTop (𝓝 0)
fairmeas : ℕ → Set ι := fun n => {i | as n ≤ μ (As i)}
\u22a2 posmeas.Countable
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β : Type u_2
δ : Type u_3
ι✝ : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s t : Set α
ι : Type u_5
x✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
Union_As_finite : μ (⋃ i, As i) ≠ ⊤
posmeas : Set ι := {i | 0 < μ (As i)}
posmeas_def : posmeas = {i | 0 < μ (As i)}
as : ℕ → ℝ≥0∞
left✝ : StrictAnti as
as_mem : ∀ (n : ℕ), as n ∈ Ioo 0 1
as_lim : Tendsto as atTop (𝓝 0)
fairmeas : ℕ → Set ι := fun n => {i | as n ≤ μ (As i)}
countable_union : posmeas = ⋃ n, fairmeas n
\u22a2 posmeas.Countable
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β : Type u_2
δ : Type u_3
ι✝ : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s t : Set α
ι : Type u_5
x✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
Union_As_finite : μ (⋃ i, As i) ≠ ⊤
posmeas : Set ι := {i | 0 < μ (As i)}
posmeas_def : posmeas = {i | 0 < μ (As i)}
as : ℕ → ℝ≥0∞
left✝ : StrictAnti as
as_mem : ∀ (n : ℕ), as n ∈ Ioo 0 1
as_lim : Tendsto as atTop (𝓝 0)
fairmeas : ℕ → Set ι := fun n => {i | as n ≤ μ (As i)}
countable_union : posmeas = ⋃ n, fairmeas n
\u22a2 (⋃ n, fairmeas n).Countable
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β : Type u_2
δ : Type u_3
ι✝ : Type u_4
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s t : Set α
ι : Type u_5
x✝ : MeasurableSpace α
μ : Measure α
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
As_disj : Pairwise (AEDisjoint μ on As)
Union_As_finite : μ (⋃ i, As i) ≠ ⊤
posmeas : Set ι := {i | 0 < μ (As i)}
posmeas_def : posmeas = {i | 0 < μ (As i)}
as : ℕ → ℝ≥0∞
left✝ : StrictAnti as
as_mem : ∀ (n : ℕ), as n ∈ Ioo 0 1
as_lim : Tendsto as atTop (𝓝 0)
fairmeas : ℕ → Set ι := fun n => {i | as n ≤ μ (As i)}
countable_union : posmeas = ⋃ n, fairmeas n
n : ℕ
\u22a2 (fairmeas n).Finite
-- Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
\u22a2 (o.oangle y (y - x)).tan * ‖y‖ = ‖x‖
-- Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
hs : (o.oangle y (y - x)).sign = 1
\u22a2 (o.oangle y (y - x)).tan * ‖y‖ = ‖x‖
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
φ₁ φ₂ : F ⟶ G
ho : Homotopy φ₁ φ₂
\u22a2 ofHom φ₁ = δ (-1) 0 (ofHomotopy ho) + ofHom φ₂
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
φ₁ φ₂ : F ⟶ G
z : { z // ofHom φ₁ = δ (-1) 0 z + ofHom φ₂ }
p : ℤ
\u22a2 φ₁.f p = (((dNext p) fun i j => if hij : i + -1 = j then (↑z).v i j hij else 0) + (prevD p) fun i j => if hij : i + -1 = j then (↑z).v i j hij else 0) + φ₂.f p
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
φ₁ φ₂ : F ⟶ G
z : { z // ofHom φ₁ = δ (-1) 0 z + ofHom φ₂ }
p : ℤ
eq : (ofHom φ₁).v p p ⋯ = (δ (-1) 0 ↑z + ofHom φ₂).v p p ⋯
\u22a2 φ₁.f p = (((dNext p) fun i j => if hij : i + -1 = j then (↑z).v i j hij else 0) + (prevD p) fun i j => if hij : i + -1 = j then (↑z).v i j hij else 0) + φ₂.f p
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
φ₁ φ₂ : F ⟶ G
z : { z // ofHom φ₁ = δ (-1) 0 z + ofHom φ₂ }
p : ℤ
eq : (ofHom φ₁).v p p ⋯ = (δ (-1) 0 ↑z + ofHom φ₂).v p p ⋯
h₁ : (ComplexShape.up ℤ).Rel (p - 1) p
h₂ : (ComplexShape.up ℤ).Rel p (p + 1)
\u22a2 φ₁.f p = (((dNext p) fun i j => if hij : i + -1 = j then (↑z).v i j hij else 0) + (prevD p) fun i j => if hij : i + -1 = j then (↑z).v i j hij else 0) + φ₂.f p
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
φ₁ φ₂ : F ⟶ G
z : { z // ofHom φ₁ = δ (-1) 0 z + ofHom φ₂ }
p : ℤ
h₁ : (ComplexShape.up ℤ).Rel (p - 1) p
h₂ : (ComplexShape.up ℤ).Rel p (p + 1)
eq : φ₁.f p = F.d p (p + 1) ≫ (↑z).v (p + 1) p ⋯ + (↑z).v p (p - 1) ⋯ ≫ G.d (p - 1) p + φ₂.f p
\u22a2 φ₁.f p = (((dNext p) fun i j => if hij : i + -1 = j then (↑z).v i j hij else 0) + (prevD p) fun i j => if hij : i + -1 = j then (↑z).v i j hij else 0) + φ₂.f p
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
φ₁ φ₂ : F ⟶ G
z : { z // ofHom φ₁ = δ (-1) 0 z + ofHom φ₂ }
p q : ℤ
hpq : p + -1 = q
\u22a2 (if hij : p + -1 = q then (↑z).v p q hij else 0) = (↑z).v p q hpq
-- Mathlib/GroupTheory/GroupAction/Blocks.lean
G : Type u_1
inst✝³ : Group G
X : Type u_2
inst✝² : MulAction G X
H : Type u_3
Y : Type u_4
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
\u22a2 IsBlock H (⇑j ⁻¹' B)
-- Mathlib/GroupTheory/GroupAction/Blocks.lean
G : Type u_1
inst✝³ : Group G
X : Type u_2
inst✝² : MulAction G X
H : Type u_3
Y : Type u_4
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
\u22a2 ∀ (g : H), g • ⇑j ⁻¹' B = ⇑j ⁻¹' B ∨ Disjoint (g • ⇑j ⁻¹' B) (⇑j ⁻¹' B)
-- Mathlib/GroupTheory/GroupAction/Blocks.lean
G : Type u_1
inst✝³ : Group G
X : Type u_2
inst✝² : MulAction G X
H : Type u_3
Y : Type u_4
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
g : H
\u22a2 g • ⇑j ⁻¹' B = ⇑j ⁻¹' B ∨ Disjoint (g • ⇑j ⁻¹' B) (⇑j ⁻¹' B)
-- Mathlib/GroupTheory/GroupAction/Blocks.lean
G : Type u_1
inst✝³ : Group G
X : Type u_2
inst✝² : MulAction G X
H : Type u_3
Y : Type u_4
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
g : H
\u22a2 ⇑j ⁻¹' (φ g • B) = ⇑j ⁻¹' B ∨ Disjoint (⇑j ⁻¹' (φ g • B)) (⇑j ⁻¹' B)
-- Mathlib/GroupTheory/GroupAction/Blocks.lean
G : Type u_1
inst✝³ : Group G
X : Type u_2
inst✝² : MulAction G X
H : Type u_3
Y : Type u_4
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
g : H
\u22a2 φ g • B = B → ⇑j ⁻¹' (φ g • B) = ⇑j ⁻¹' B
-- Mathlib/GroupTheory/GroupAction/Blocks.lean
G : Type u_1
inst✝³ : Group G
X : Type u_2
inst✝² : MulAction G X
H : Type u_3
Y : Type u_4
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
B : Set X
hB : IsBlock G B
g : H
\u22a2 Disjoint (φ g • B) B → Disjoint (⇑j ⁻¹' (φ g • B)) (⇑j ⁻¹' B)
-- Mathlib/Analysis/Convex/Jensen.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
β : Type u_4
ι : Type u_5
inst✝⁵ : LinearOrderedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : LinearOrderedAddCommGroup β
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
t : Finset ι
w : ι → 𝕜
p : ι → E
x y✝ z : E
h : ConvexOn 𝕜 s f
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : 0 < ∑ i ∈ t, w i
hp : ∀ i ∈ t, p i ∈ s
y : E := t.centerMass w p
\u22a2 ∃ i ∈ t, f y ≤ f (p i)
-- Mathlib/Analysis/Convex/Jensen.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
β : Type u_4
ι : Type u_5
inst✝⁵ : LinearOrderedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : LinearOrderedAddCommGroup β
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
t : Finset ι
w : ι → 𝕜
p : ι → E
x y✝ z : E
h : ConvexOn 𝕜 s f
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : 0 < ∑ i ∈ t, w i
hp : ∀ i ∈ t, p i ∈ s
y : E := t.centerMass w p
i : ι
hi : i ∈ filter (fun i => w i ≠ 0) t
hfi : w i • f y ≤ w i • (f ∘ p) i
\u22a2 ∃ i ∈ t, f y ≤ f (p i)
-- Mathlib/Analysis/Convex/Jensen.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
β : Type u_4
ι : Type u_5
inst✝⁵ : LinearOrderedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : LinearOrderedAddCommGroup β
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
t : Finset ι
w : ι → 𝕜
p : ι → E
x y✝ z : E
h : ConvexOn 𝕜 s f
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : 0 < ∑ i ∈ t, w i
hp : ∀ i ∈ t, p i ∈ s
y : E := t.centerMass w p
i : ι
hi : i ∈ t ∧ w i ≠ 0
hfi : w i • f y ≤ w i • (f ∘ p) i
\u22a2 ∃ i ∈ t, f y ≤ f (p i)
-- Mathlib/Algebra/Group/Subsemigroup/Membership.lean
ι : Sort u_1
M : Type u_2
A : Type u_3
B : Type u_4
inst✝ : Mul M
S : Set (Subsemigroup M)
s : Subsemigroup M
hs : s ∈ S
\u22a2 ∀ {x : M}, x ∈ s → x ∈ sSup S
-- Mathlib/Algebra/Group/Subsemigroup/Membership.lean
ι : Sort u_1
M : Type u_2
A : Type u_3
B : Type u_4
inst✝ : Mul M
S : Set (Subsemigroup M)
s : Subsemigroup M
hs : s ∈ S
this : s ≤ sSup S
\u22a2 ∀ {x : M}, x ∈ s → x ∈ sSup S
-- Mathlib/Analysis/Normed/Group/AddCircle.lean
p : ℝ
hp : Fact (0 < p)
m n : ℕ
\u22a2 ‖↑(↑m / ↑n * p)‖ = p * (↑(min (m % n) (n - m % n)) / ↑n)
-- Mathlib/Analysis/Normed/Group/AddCircle.lean
p : ℝ
hp : Fact (0 < p)
m n : ℕ
this : p⁻¹ * (↑m / ↑n * p) = ↑m / ↑n
\u22a2 ‖↑(↑m / ↑n * p)‖ = p * (↑(min (m % n) (n - m % n)) / ↑n)
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y✝ : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝ g✝ : X → E
s✝ t : Set X
μ ν : Measure X
l l' : Filter X
Y : Type u_5
inst✝ : MeasurableSpace Y
g : X → Y
f : Y → E
s : Set Y
hs : MeasurableSet s
hf : AEStronglyMeasurable f (Measure.map g μ)
hg : AEMeasurable g μ
\u22a2 ∫ (y : Y) in s, f y ∂Measure.map g μ = ∫ (x : X) in g ⁻¹' s, f (g x) ∂μ
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y✝ : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝ g✝ : X → E
s✝ t : Set X
μ ν : Measure X
l l' : Filter X
Y : Type u_5
inst✝ : MeasurableSpace Y
g : X → Y
f : Y → E
s : Set Y
hs : MeasurableSet s
hf : AEStronglyMeasurable f (Measure.map g μ)
hg : AEMeasurable g μ
\u22a2 Measure.map g (μ.restrict (g ⁻¹' s)) ≤ Measure.map g μ
-- Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
ψ : L ⟶ M
i j k : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
\u22a2 ShortComplex.homologyMap ((shortComplexFunctor C c i).map (φ ≫ ψ)) = ShortComplex.homologyMap ((shortComplexFunctor C c i).map φ) ≫ ShortComplex.homologyMap ((shortComplexFunctor C c i).map ψ)
-- Mathlib/GroupTheory/PGroup.lean
p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
this✝ : Fintype α
this : Fintype ↑(fixedPoints G α)
\u22a2 Nat.card α ≡ Nat.card ↑(fixedPoints G α) [MOD p]
-- Mathlib/GroupTheory/PGroup.lean
p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
this✝ : Fintype α
this : Fintype ↑(fixedPoints G α)
\u22a2 card α ≡ card ↑(fixedPoints G α) [MOD p]
-- Mathlib/GroupTheory/Perm/ClosureSwap.lean
G : Type u_1
α : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : DecidableEq α
x y : α
\u22a2 (fixedBy α (swap x y))ᶜ.Finite
-- Mathlib/Combinatorics/SetFamily/Shadow.lean
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
s✝ t : Finset α
a : α
k r : ℕ
s : Finset α
\u22a2 s ∈ ∂⁺ 𝒜ᶜˢ ↔ s ∈ (∂ 𝒜)ᶜˢ
-- Mathlib/Combinatorics/SetFamily/Shadow.lean
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
s✝ t : Finset α
a : α
k r : ℕ
s : Finset α
\u22a2 (∃ s_1, s_1ᶜ ∈ 𝒜 ∧ ∃ a ∉ s_1, insert a s_1 = s) ↔ ∃ s_1 ∈ 𝒜, ∃ a ∈ s_1, s_1.erase a = sᶜ
-- Mathlib/Combinatorics/SetFamily/Shadow.lean
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
s✝ t : Finset α
a : α
k r : ℕ
s : Finset α
\u22a2 (∃ b, ((Equiv.symm (Function.Involutive.toPerm compl ⋯)) b)ᶜ ∈ 𝒜 ∧ ∃ a ∉ (Equiv.symm (Function.Involutive.toPerm compl ⋯)) b, insert a ((Equiv.symm (Function.Involutive.toPerm compl ⋯)) b) = s) ↔ ∃ s_1 ∈ 𝒜, ∃ a ∈ s_1, s_1.erase a = sᶜ
-- Mathlib/CategoryTheory/Generator.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
G : Cᵒᵖ
\u22a2 IsCodetector (unop G) ↔ IsDetector G
-- Mathlib/GroupTheory/Coprod/Basic.lean
M : Type u_1
N : Type u_2
P : Type u_3
inst✝² : MulOneClass M
inst✝¹ : MulOneClass N
inst✝ : Monoid P
f : M →* P
g : N →* P
\u22a2 MonoidHom.mrange (lift f g) = MonoidHom.mrange f ⊔ MonoidHom.mrange g
-- Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
x y : α
r : ℝ
hr : 0 ≤ r
\u22a2 edist x y ≤ ENNReal.ofReal r ↔ dist x y ≤ r
-- Mathlib/Algebra/Order/Group/Unbundled/Basic.lean
α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
a b c : α
\u22a2 a * a⁻¹ < a * b ↔ 1 < a * b
-- Mathlib/Order/Interval/Finset/Basic.lean
ι : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : LocallyFiniteOrder α
a✝ b✝ a b c : α
h₁ : a ≤ b
h₂ : b ≤ c
\u22a2 Ioc a b ∪ Ioc b c = Ioc a c
-- Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
E : Type u_1
f : ℝ → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : ℝ → E
p : ℝ
hp : 0 < p
\u22a2 ∫ (x : ℝ) in Ioi 0, (p * x ^ (p - 1)) • g (x ^ p) = ∫ (y : ℝ) in Ioi 0, g y
-- Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
E : Type u_1
f : ℝ → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : ℝ → E
p : ℝ
hp : 0 < p
x✝ : ℝ
\u22a2 p = |p|
-- Mathlib/Algebra/Ring/Subsemiring/MulOpposite.lean
ι : Sort u_1
R : Type u_2
inst✝ : NonAssocSemiring R
s : Set R
\u22a2 (closure s).op = closure (MulOpposite.unop ⁻¹' s)
-- Mathlib/Algebra/Ring/Subsemiring/MulOpposite.lean
ι : Sort u_1
R : Type u_2
inst✝ : NonAssocSemiring R
s : Set R
\u22a2 sInf {a | s ⊆ MulOpposite.op ⁻¹' ↑a} = sInf {S | MulOpposite.unop ⁻¹' s ⊆ ↑S}
-- Mathlib/Algebra/Ring/Subsemiring/MulOpposite.lean
ι : Sort u_1
R : Type u_2
inst✝ : NonAssocSemiring R
s : Set R
a : Subsemiring Rᵐᵒᵖ
\u22a2 a ∈ {a | s ⊆ MulOpposite.op ⁻¹' ↑a} ↔ a ∈ {S | MulOpposite.unop ⁻¹' s ⊆ ↑S}
-- Mathlib/SetTheory/Cardinal/ToNat.lean
α : Type u
c d : Cardinal.{u}
\u22a2 toNat c = 1 ↔ c = 1
-- Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : SnakeInput C
\u22a2 pullback.lift S.L₁.f 0 ⋯ ≫ S.φ₁ = S.v₁₂.τ₁
-- Mathlib/Geometry/Euclidean/Inversion/Basic.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a✝ b c✝ d x✝ y z : P
r R✝ : ℝ
c : P
a R : ℝ
x : P
\u22a2 inversion c (a * R) x = (homothety c (a ^ 2)) (inversion c R x)
-- Mathlib/LinearAlgebra/Lagrange.lean
R : Type u_1
inst✝² : CommRing R
ι : Type u_2
s : Finset ι
v : ι → R
inst✝¹ : IsDomain R
G : Subgroup Rˣ
inst✝ : Fintype ↥G
\u22a2 nodal (↑G).toFinset Units.val = X ^ Fintype.card ↥G - 1
-- Mathlib/LinearAlgebra/Lagrange.lean
R : Type u_1
inst✝² : CommRing R
ι : Type u_2
s : Finset ι
v : ι → R
inst✝¹ : IsDomain R
G : Subgroup Rˣ
inst✝ : Fintype ↥G
h : degree 1 < (X ^ Fintype.card ↥G).degree
\u22a2 nodal (↑G).toFinset Units.val = X ^ Fintype.card ↥G - 1
-- Mathlib/LinearAlgebra/Lagrange.lean
R : Type u_1
inst✝² : CommRing R
ι : Type u_2
s : Finset ι
v : ι → R
inst✝¹ : IsDomain R
G : Subgroup Rˣ
inst✝ : Fintype ↥G
h : degree 1 < (X ^ Fintype.card ↥G).degree
\u22a2 Set.InjOn Units.val ↑(↑G).toFinset
-- Mathlib/LinearAlgebra/Lagrange.lean
R : Type u_1
inst✝² : CommRing R
ι : Type u_2
s : Finset ι
v : ι → R
inst✝¹ : IsDomain R
G : Subgroup Rˣ
inst✝ : Fintype ↥G
h : degree 1 < (X ^ Fintype.card ↥G).degree
\u22a2 (nodal (↑G).toFinset Units.val).degree ≤ ↑(↑G).toFinset.card
-- Mathlib/LinearAlgebra/Lagrange.lean
R : Type u_1
inst✝² : CommRing R
ι : Type u_2
s : Finset ι
v : ι → R
inst✝¹ : IsDomain R
G : Subgroup Rˣ
inst✝ : Fintype ↥G
h : degree 1 < (X ^ Fintype.card ↥G).degree
\u22a2 (nodal (↑G).toFinset Units.val).degree = (X ^ Fintype.card ↥G - 1).degree
-- Mathlib/LinearAlgebra/Lagrange.lean
R : Type u_1
inst✝² : CommRing R
ι : Type u_2
s : Finset ι
v : ι → R
inst✝¹ : IsDomain R
G : Subgroup Rˣ
inst✝ : Fintype ↥G
h : degree 1 < (X ^ Fintype.card ↥G).degree
\u22a2 (nodal (↑G).toFinset Units.val).leadingCoeff = (X ^ Fintype.card ↥G - 1).leadingCoeff
-- Mathlib/LinearAlgebra/Lagrange.lean
R : Type u_1
inst✝² : CommRing R
ι : Type u_2
s : Finset ι
v : ι → R
inst✝¹ : IsDomain R
G : Subgroup Rˣ
inst✝ : Fintype ↥G
h : degree 1 < (X ^ Fintype.card ↥G).degree
\u22a2 ∀ i ∈ (↑G).toFinset, eval (↑i) (nodal (↑G).toFinset Units.val) = eval (↑i) (X ^ Fintype.card ↥G - 1)
-- Mathlib/Algebra/MvPolynomial/Division.lean
σ : Type u_1
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
i j : σ →₀ ℕ
\u22a2 (monomial i) 1 ∣ (monomial j) 1 ↔ i ≤ j
-- Mathlib/Algebra/MvPolynomial/Division.lean
σ : Type u_1
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
i j : σ →₀ ℕ
\u22a2 (1 = 0 ∨ i ≤ j) ∧ 1 ∣ 1 ↔ i ≤ j
-- Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g : α → E
ι : Type u_3
s : Finset ι
f : ι → α → E
this : DecidableEq ι
\u22a2 (∀ i ∈ s, Memℒp (f i) p μ) → Memℒp (fun a => ∑ i ∈ s, f i a) p μ
-- Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g : α → E
ι : Type u_3
s : Finset ι
f : ι → α → E
this : DecidableEq ι
\u22a2 (∀ i ∈ ∅, Memℒp (f i) p μ) → Memℒp (fun a => ∑ i ∈ ∅, f i a) p μ
-- Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g : α → E
ι : Type u_3
s : Finset ι
f : ι → α → E
this : DecidableEq ι
\u22a2 ∀ ⦃a : ι⦄ {s : Finset ι}, a ∉ s → ((∀ i ∈ s, Memℒp (f i) p μ) → Memℒp (fun a => ∑ i ∈ s, f i a) p μ) → (∀ i ∈ insert a s, Memℒp (f i) p μ) → Memℒp (fun a_3 => ∑ i ∈ insert a s, f i a_3) p μ
-- Mathlib/Topology/Instances/ENNReal.lean
α✝ : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PseudoEMetricSpace α✝
α : Type u_4
inst✝ : PseudoMetricSpace α
s : Set α
\u22a2 diam (closure s) = diam s
-- Mathlib/Data/DFinsupp/NeLocus.lean
α : Type u_1
N : α → Type u_2
inst✝² : DecidableEq α
inst✝¹ : (a : α) → DecidableEq (N a)
inst✝ : (a : α) → Zero (N a)
f g : Π₀ (a : α), N a
a✝ : α
\u22a2 a✝ ∈ f.neLocus 0 ↔ a✝ ∈ f.support
-- Mathlib/GroupTheory/CoprodI.lean
ι : Type u_1
M : ι → Type u_2
inst✝³ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝² : Monoid N
inst✝¹ : (i : ι) → DecidableEq (M i)
inst✝ : DecidableEq ι
i : ι
p : Pair M i
\u22a2 rcons p = of p.head • p.tail
-- Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean
α : Type u
inst✝ : CanonicallyOrderedCommMonoid α
a c✝ d c : α
\u22a2 a ≤ a * c
-- Mathlib/RingTheory/HahnSeries/Addition.lean
Γ : Type u_1
R : Type u_2
inst✝¹ : PartialOrder Γ
inst✝ : AddGroup R
x : HahnSeries Γ R
\u22a2 (-x).orderTop = x.orderTop
-- Mathlib/CategoryTheory/WithTerminal.lean
C : Type u
inst✝ : Category.{v, u} C
a✝ b✝ : Cat
f✝ : a✝ ⟶ b✝
\u22a2 ({ obj := fun C => Cat.of (WithTerminal ↑C), map := fun {X Y} => map, map₂ := fun {a b} {f g} => map₂ }.map₂ (𝟙 f✝)).app = (𝟙 ({ obj := fun C => Cat.of (WithTerminal ↑C), map := fun {X Y} => map, map₂ := fun {a b} {f g} => map₂ }.map f✝)).app
-- Mathlib/CategoryTheory/WithTerminal.lean
C : Type u
inst✝ : Category.{v, u} C
a✝ b✝ : Cat
f✝ : a✝ ⟶ b✝
X : ↑({ obj := fun C => Cat.of (WithTerminal ↑C), map := fun {X Y} => map, map₂ := fun {a b} {f g} => map₂ }.obj a✝)
\u22a2 ({ obj := fun C => Cat.of (WithTerminal ↑C), map := fun {X Y} => map, map₂ := fun {a b} {f g} => map₂ }.map₂ (𝟙 f✝)).app X = (𝟙 ({ obj := fun C => Cat.of (WithTerminal ↑C), map := fun {X Y} => map, map₂ := fun {a b} {f g} => map₂ }.map f✝)).app X
-- Mathlib/CategoryTheory/WithTerminal.lean
C : Type u
inst✝ : Category.{v, u} C
a✝ b✝ : Cat
f✝ g✝ h✝ : a✝ ⟶ b✝
η✝ : f✝ ⟶ g✝
θ✝ : g✝ ⟶ h✝
\u22a2 ({ obj := fun C => Cat.of (WithTerminal ↑C), map := fun {X Y} => map, map₂ := fun {a b} {f g} => map₂ }.map₂ (η✝ ≫ θ✝)).app = ({ obj := fun C => Cat.of (WithTerminal ↑C), map := fun {X Y} => map, map₂ := fun {a b} {f g} => map₂ }.map₂ η✝ ≫ { obj := fun C => Cat.of (WithTerminal ↑C), map := fun {X Y} => map, map₂ := fun {a b} {f g} => map₂ }.map₂ θ✝).app
-- Mathlib/CategoryTheory/WithTerminal.lean
C : Type u
inst✝ : Category.{v, u} C
a✝ b✝ : Cat
f✝ g✝ h✝ : a✝ ⟶ b✝
η✝ : f✝ ⟶ g✝
θ✝ : g✝ ⟶ h✝
X : ↑({ obj := fun C => Cat.of (WithTerminal ↑C), map := fun {X Y} => map, map₂ := fun {a b} {f g} => map₂ }.obj a✝)
\u22a2 ({ obj := fun C => Cat.of (WithTerminal ↑C), map := fun {X Y} => map, map₂ := fun {a b} {f g} => map₂ }.map₂ (η✝ ≫ θ✝)).app X = ({ obj := fun C => Cat.of (WithTerminal ↑C), map := fun {X Y} => map, map₂ := fun {a b} {f g} => map₂ }.map₂ η✝ ≫ { obj := fun C => Cat.of (WithTerminal ↑C), map := fun {X Y} => map, map₂ := fun {a b} {f g} => map₂ }.map₂ θ✝).app X
-- Mathlib/Analysis/InnerProductSpace/PiL2.lean
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁹ : _root_.RCLike 𝕜
E : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : InnerProductSpace 𝕜 E
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : InnerProductSpace 𝕜 E'
F : Type u_6
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
F' : Type u_7
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : Fintype ι
b : OrthonormalBasis ι 𝕜 E
\u22a2 Orthonormal 𝕜 ⇑b