ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/Asymptotics/Asymptotics.lean
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
E' : Type u_6
F' : Type u_7
G' : Type u_8
E'' : Type u_9
F'' : Type u_10
G'' : Type u_11
E''' : Type u_12
R : Type u_13
R' : Type u_14
𝕜 : Type u_15
𝕜' : Type u_16
inst✝¹³ : Norm E
inst✝¹² : Norm F
inst✝¹¹ : Norm G
inst✝¹⁰ : SeminormedAddCommGroup E'
inst✝⁹ : SeminormedAddCommGroup F'
inst✝⁸ : SeminormedAddCommGroup G'
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedAddCommGroup F''
inst✝⁵ : NormedAddCommGroup G''
inst✝⁴ : SeminormedRing R
inst✝³ : SeminormedAddGroup E'''
inst✝² : SeminormedRing R'
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : NormedDivisionRing 𝕜'
c c' c₁ c₂ : ℝ
f : α → E
g : α → F
k : α → G
f' : α → E'
g' : α → F'
k' : α → G'
f'' : α → E''
g'' : α → F''
k'' : α → G''
l l' : Filter α
\u22a2 (fun x => -f' x) =o[l] g ↔ f' =o[l] g
-- Mathlib/Analysis/Asymptotics/Asymptotics.lean
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
E' : Type u_6
F' : Type u_7
G' : Type u_8
E'' : Type u_9
F'' : Type u_10
G'' : Type u_11
E''' : Type u_12
R : Type u_13
R' : Type u_14
𝕜 : Type u_15
𝕜' : Type u_16
inst✝¹³ : Norm E
inst✝¹² : Norm F
inst✝¹¹ : Norm G
inst✝¹⁰ : SeminormedAddCommGroup E'
inst✝⁹ : SeminormedAddCommGroup F'
inst✝⁸ : SeminormedAddCommGroup G'
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedAddCommGroup F''
inst✝⁵ : NormedAddCommGroup G''
inst✝⁴ : SeminormedRing R
inst✝³ : SeminormedAddGroup E'''
inst✝² : SeminormedRing R'
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : NormedDivisionRing 𝕜'
c c' c₁ c₂ : ℝ
f : α → E
g : α → F
k : α → G
f' : α → E'
g' : α → F'
k' : α → G'
f'' : α → E''
g'' : α → F''
k'' : α → G''
l l' : Filter α
\u22a2 (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l (fun x => -f' x) g) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f' g
-- Mathlib/Probability/ProbabilityMassFunction/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : Countable α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSingletonClass α
p : PMF α
μ✝ : Measure α
inst✝¹ : IsProbabilityMeasure μ✝
μ : Measure α
inst✝ : IsProbabilityMeasure μ
\u22a2 p.toMeasure = μ ↔ p = μ.toPMF
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝² : PseudoEMetricSpace α
γ : Type w
inst✝¹ : EMetricSpace γ
inst✝ : EMetricSpace β
f : γ → β
\u22a2 UniformEmbedding f ↔ (∀ ε > 0, ∃ δ > 0, ∀ {a b : γ}, edist a b < δ → edist (f a) (f b) < ε) ∧ ∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, edist (f a) (f b) < ε → edist a b < δ
-- Mathlib/Analysis/Complex/Hadamard.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ : ℂ → E
z✝ : ℂ
f : ℂ → E
ε : ℝ
hε : ε > 0
z : ℂ
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ re ⁻¹' {0, 1}
\u22a2 ‖F f ε z‖ ≤ 1
-- Mathlib/Analysis/Complex/Hadamard.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ : ℂ → E
z✝ : ℂ
f : ℂ → E
ε : ℝ
hε : ε > 0
z : ℂ
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ re ⁻¹' {0, 1}
\u22a2 (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re) * ‖f z‖ ≤ 1
-- Mathlib/Order/Filter/Prod.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
s : Set α
t : Set β
f✝ : Filter α
g✝ : Filter β
m : α × β → γ
f : Filter α
g : Filter β
\u22a2 map m (f ×ˢ g) = (map (fun a b => m (a, b)) f).seq g
-- Mathlib/Order/Filter/Prod.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
s : Set α
t : Set β
f✝ : Filter α
g✝ : Filter β
m : α × β → γ
f : Filter α
g : Filter β
\u22a2 ∀ (s : Set γ), (∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ×ˢ t₂ ⊆ m ⁻¹' s) ↔ ∃ t ∈ g, ∃ x ∈ f, ∀ x_1 ∈ x, ∀ y ∈ t, m (x_1, y) ∈ s
-- Mathlib/Data/Set/Card.lean
α : Type u_1
β : Type u_2
s✝ t : Set α
s : Finset α
\u22a2 (↑s).encard = ↑s.card
-- Mathlib/Data/Set/Card.lean
α : Type u_1
β : Type u_2
s✝ t : Set α
s : Finset α
\u22a2 ↑⋯.toFinset.card = ↑s.card
-- Mathlib/Analysis/Convex/Cone/Proper.lean
𝕜 : Type u_1
inst✝⁴ : OrderedSemiring 𝕜
E : Type u_2
inst✝³ : AddCommMonoid E
inst✝² : TopologicalSpace E
inst✝¹ : T1Space E
inst✝ : Module 𝕜 E
\u22a2 (↑↑0).Pointed
-- Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
h : Collinear k s
p₁ p₂ p₃ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃ : p₃ ∈ s
hp₁p₂ : p₁ ≠ p₂
\u22a2 p₃ ∈ affineSpan k {p₁, p₂}
-- Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₁ : P
h : ∃ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₁
p₂ p₃ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃ : p₃ ∈ s
hp₁p₂ : p₁ ≠ p₂
\u22a2 p₃ ∈ affineSpan k {p₁, p₂}
-- Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₁ : P
hp₁ : p₁ ∈ s
v : V
h : ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₁
r₂ : k
hp₂ : r₂ • v +ᵥ p₁ ∈ s
hp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁
r₃ : k
hp₃ : r₃ • v +ᵥ p₁ ∈ s
\u22a2 r₃ • v +ᵥ p₁ ∈ affineSpan k {p₁, r₂ • v +ᵥ p₁}
-- Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₁ : P
hp₁ : p₁ ∈ s
v : V
h : ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₁
r₂ : k
hp₂ : r₂ • v +ᵥ p₁ ∈ s
hp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁
r₃ : k
hp₃ : r₃ • v +ᵥ p₁ ∈ s
\u22a2 ∃ r, r • (r₂ • v +ᵥ p₁ -ᵥ p₁) = r₃ • v
-- Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₁ : P
hp₁ : p₁ ∈ s
v : V
h : ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₁
r₂ : k
hp₂ : r₂ • v +ᵥ p₁ ∈ s
hp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁
r₃ : k
hp₃ : r₃ • v +ᵥ p₁ ∈ s
h₂ : r₂ ≠ 0
\u22a2 (r₃ / r₂) • (r₂ • v +ᵥ p₁ -ᵥ p₁) = r₃ • v
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x y z : ℝ
n : ℕ
hx : 0 < x
hxy : x ≤ y
hz : z ≤ 0
this : 0 < y
\u22a2 y ^ z ≤ x ^ z
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x y z : ℝ
n : ℕ
hx : 0 < x
hxy : x ≤ y
hz : z ≤ 0
this : 0 < y
\u22a2 x ^ (-z) ≤ y ^ (-z)
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x y z : ℝ
n : ℕ
hx : 0 < x
hxy : x ≤ y
hz : z ≤ 0
this : 0 < y
\u22a2 0 ≤ y
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x y z : ℝ
n : ℕ
hx : 0 < x
hxy : x ≤ y
hz : z ≤ 0
this : 0 < y
\u22a2 0 ≤ x
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x y z : ℝ
n : ℕ
hx : 0 < x
hxy : x ≤ y
hz : z ≤ 0
this : 0 < y
\u22a2 0 < x ^ z
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x y z : ℝ
n : ℕ
hx : 0 < x
hxy : x ≤ y
hz : z ≤ 0
this : 0 < y
\u22a2 0 < y ^ z
-- Mathlib/Analysis/Calculus/FDeriv/Add.lean
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hg : DifferentiableOn 𝕜 g s
\u22a2 DifferentiableOn 𝕜 (fun y => f y - g y) s ↔ DifferentiableOn 𝕜 f s
-- Mathlib/Data/Finset/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s✝ t✝ u v : Finset α
a✝ b a : α
s t : Finset α
\u22a2 s.erase a ∩ t = (s ∩ t).erase a
-- Mathlib/Order/Filter/Cofinite.lean
ι : Type u_1
α : Type u_2
β : Type u_3
l : Filter α
\u22a2 Disjoint cofinite l ↔ ∃ s ∈ l, s.Finite
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 e.unit.app (e.inverse.obj Y) ≫ e.inverse.map (e.counit.app Y) = 𝟙 (e.inverse.obj Y)
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 e.unit.app (e.inverse.obj Y) ≫ (e.inverse.map (e.counitInv.app (e.functor.obj (e.inverse.obj Y))) ≫ e.inverse.map (e.functor.map (e.unitInv.app (e.inverse.obj Y)))) ≫ e.inverse.map (e.counit.app Y) = 𝟙 (e.inverse.obj Y)
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 e.unit.app (e.inverse.obj Y) ≫ (e.inverse.map (e.counitInv.app (e.functor.obj (e.inverse.obj Y))) ≫ e.unitIso.hom.app (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y))))) ≫ e.unitIso.inv.app (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y))))) ≫ e.inverse.map (e.functor.map (e.unitInv.app (e.inverse.obj Y)))) ≫ e.inverse.map (e.counit.app Y) = 𝟙 (e.inverse.obj Y)
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 e.unit.app (e.inverse.obj Y) ≫ (((e.unit.app (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj Y))) ≫ (e.functor ⋙ e.inverse).map (e.inverse.toPrefunctor.2 (e.counitInv.app (e.functor.obj (e.inverse.obj Y))))) ≫ e.unitIso.inv.app (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫ e.inverse.map (e.functor.map (e.unitInv.app (e.inverse.obj Y)))) ≫ e.inverse.map (e.counit.app Y) = 𝟙 (e.inverse.obj Y)
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 ((((e.unit.app ((𝟭 C).obj (e.inverse.obj Y)) ≫ (e.functor ⋙ e.inverse).map (e.unit.1 (e.inverse.obj Y))) ≫ (e.functor ⋙ e.inverse).map (e.inverse.toPrefunctor.2 (e.counitInv.app (e.functor.obj (e.inverse.obj Y))))) ≫ e.unitIso.inv.app (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫ e.inverse.map (e.functor.map (e.unitInv.app (e.inverse.obj Y)))) ≫ e.inverse.map (e.counit.app Y) = 𝟙 (e.inverse.obj Y)
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 e.unit.app ((𝟭 C).obj (e.inverse.obj Y)) ≫ (e.functor ⋙ e.inverse).map (e.unit.1 (e.inverse.obj Y)) ≫ (e.functor ⋙ e.inverse).map (e.inverse.toPrefunctor.2 (e.counitInv.app (e.functor.obj (e.inverse.obj Y)))) ≫ (((e.inverse.mapIso (e.counitIso.app (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))).hom ≫ (e.inverse.mapIso (e.counitIso.app (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))).inv ≫ e.unitInv.app (e.inverse.obj (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫ e.inverse.map (e.functor.map (e.unitInv.app (e.inverse.obj Y)))) ≫ e.inverse.map (e.counit.app Y) = 𝟙 (e.inverse.obj Y)
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 e.unit.app ((𝟭 C).obj (e.inverse.obj Y)) ≫ (e.functor ⋙ e.inverse).map (e.unit.1 (e.inverse.obj Y)) ≫ (((𝟙 (e.inverse.obj (e.functor.obj (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj Y))))) ≫ (e.inverse.mapIso (e.counitIso.app (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))).inv) ≫ e.unitInv.app (e.inverse.obj (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫ e.inverse.map (e.functor.map (e.unitInv.app (e.inverse.obj Y)))) ≫ e.inverse.map (e.counit.app Y) = 𝟙 (e.inverse.obj Y)
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 e.unit.app ((𝟭 C).obj (e.inverse.obj Y)) ≫ (e.functor ⋙ e.inverse).map (e.unit.1 (e.inverse.obj Y)) ≫ (((e.inverse.mapIso (e.counitIso.app (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))).inv ≫ e.unitInv.app (e.inverse.obj (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫ e.inverse.map (e.functor.map (e.unitInv.app (e.inverse.obj Y)))) ≫ e.inverse.map (e.counit.app Y) = 𝟙 (e.inverse.obj Y)
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 e.unit.app ((𝟭 C).obj (e.inverse.obj Y)) ≫ (((e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫ e.inverse.map ((e.inverse ⋙ e.functor).map (e.functor.toPrefunctor.2 (e.unit.1 (e.inverse.obj Y))))) ≫ e.unitInv.app (e.inverse.obj (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y)))))) ≫ e.inverse.map (e.functor.map (e.unitInv.app (e.inverse.obj Y)))) ≫ e.inverse.map (e.counit.app Y) = 𝟙 (e.inverse.obj Y)
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 e.unit.app ((𝟭 C).obj (e.inverse.obj Y)) ≫ e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫ ((e.unitInv.app (e.inverse.obj (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫ (𝟭 C).map (e.inverse.map (e.functor.toPrefunctor.2 (e.unit.1 (e.inverse.obj Y))))) ≫ e.inverse.map (e.functor.map (e.unitInv.app (e.inverse.obj Y)))) ≫ e.inverse.map (e.counit.app Y) = 𝟙 (e.inverse.obj Y)
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 e.unit.app ((𝟭 C).obj (e.inverse.obj Y)) ≫ e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫ e.unitInv.app (e.inverse.obj (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫ 𝟙 ((e.functor ⋙ e.inverse).obj ((𝟭 C).obj (e.inverse.obj Y))) ≫ e.inverse.map (e.counit.app Y) = 𝟙 (e.inverse.obj Y)
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 e.unit.app ((𝟭 C).obj (e.inverse.obj Y)) ≫ e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫ e.unitInv.app (e.inverse.obj (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫ e.inverse.map (e.counit.app Y) = 𝟙 (e.inverse.obj Y)
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 e.unit.app ((𝟭 C).obj (e.inverse.obj Y)) ≫ e.inverse.map (e.counitIso.inv.app (e.functor.toPrefunctor.1 ((𝟭 C).obj (e.inverse.obj Y)))) ≫ (e.functor ⋙ e.inverse).map (e.inverse.toPrefunctor.2 (e.counit.app Y)) ≫ e.unitInv.app (e.inverse.toPrefunctor.1 Y) = 𝟙 (e.inverse.obj Y)
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 e.unit.app ((𝟭 C).obj (e.inverse.obj Y)) ≫ 𝟙 (e.inverse.obj (e.functor.obj ((𝟭 C).obj (e.inverse.obj Y)))) ≫ e.unitInv.app (e.inverse.toPrefunctor.1 Y) = 𝟙 (e.inverse.obj Y)
-- Mathlib/CategoryTheory/Equivalence.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
\u22a2 𝟙 ((𝟭 C).obj ((𝟭 C).obj (e.inverse.obj Y))) = 𝟙 (e.inverse.obj Y)
-- Mathlib/RingTheory/PowerSeries/WellKnown.lean
A : Type u_1
A' : Type u_2
inst✝⁷ : Ring A
inst✝⁶ : Ring A'
inst✝⁵ : Algebra ℚ A
inst✝⁴ : Algebra ℚ A'
inst✝³ : Ring A
inst✝² : Ring A'
inst✝¹ : Algebra ℚ A
inst✝ : Algebra ℚ A'
n : ℕ
f : A →+* A'
n✝ : ℕ
\u22a2 (coeff A' n✝) ((map f) (cos A)) = (coeff A' n✝) (cos A')
-- 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✝ : CommGroupWithZero G₀
a✝ b✝ c✝ d a b c : G₀
\u22a2 a / c * b = a * b / c
-- Mathlib/CategoryTheory/GradedObject/Trifunctor.lean
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁵ : Category.{?u.45682, u_1} C₁
inst✝⁴ : Category.{?u.45686, u_2} C₂
inst✝³ : Category.{?u.45690, u_3} C₃
inst✝² : Category.{?u.45694, u_4} C₄
inst✝¹ : Category.{?u.45698, u_5} C₁₂
inst✝ : Category.{?u.45702, u_6} C₂₃
F F' : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄
e : F ≅ F'
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
i₁ : I₁
i₂ : I₂
i₃ : I₃
\u22a2 ((e.hom.app (X₁ i₁)).app (X₂ i₂)).app (X₃ i₃) ≫ ((e.inv.app (X₁ i₁)).app (X₂ i₂)).app (X₃ i₃) = 𝟙 (((F.obj (X₁ i₁)).obj (X₂ i₂)).obj (X₃ i₃))
-- Mathlib/CategoryTheory/GradedObject/Trifunctor.lean
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁵ : Category.{?u.45682, u_1} C₁
inst✝⁴ : Category.{?u.45686, u_2} C₂
inst✝³ : Category.{?u.45690, u_3} C₃
inst✝² : Category.{?u.45694, u_4} C₄
inst✝¹ : Category.{?u.45698, u_5} C₁₂
inst✝ : Category.{?u.45702, u_6} C₂₃
F F' : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄
e : F ≅ F'
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
i₁ : I₁
i₂ : I₂
i₃ : I₃
\u22a2 ((e.inv.app (X₁ i₁)).app (X₂ i₂)).app (X₃ i₃) ≫ ((e.hom.app (X₁ i₁)).app (X₂ i₂)).app (X₃ i₃) = 𝟙 (((F'.obj (X₁ i₁)).obj (X₂ i₂)).obj (X₃ i₃))
-- Mathlib/Algebra/Order/Sub/WithTop.lean
α : Type u_1
β : Type u_2
inst✝¹ : Sub α
inst✝ : Bot α
a b : WithTop α
\u22a2 a - b = ⊤ ↔ a = ⊤ ∧ b ≠ ⊤
-- Mathlib/Topology/Connected/PathConnected.lean
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
inst✝ : LocPathConnectedSpace X
U : Set X
U_op : IsOpen U
\u22a2 IsPathConnected U ↔ IsConnected U
-- Mathlib/Topology/Connected/PathConnected.lean
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
inst✝ : LocPathConnectedSpace X
U : Set X
U_op : IsOpen U
\u22a2 PathConnectedSpace ↑U ↔ ConnectedSpace ↑U
-- Mathlib/Topology/Connected/PathConnected.lean
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
inst✝ : LocPathConnectedSpace X
U : Set X
U_op : IsOpen U
this : LocPathConnectedSpace ↑U
\u22a2 PathConnectedSpace ↑U ↔ ConnectedSpace ↑U
-- Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
J : Type v
inst✝ : SmallCategory J
X✝ Y✝ Z : TopCat
X Y S : TopCat
f : X ⟶ S
g : Y ⟶ S
H₁ : Embedding ⇑f
H₂ : Embedding ⇑g
\u22a2 Embedding ⇑(limit.π (cospan f g) WalkingCospan.one)
-- Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
J : Type v
inst✝ : SmallCategory J
X✝ Y✝ Z : TopCat
X Y S : TopCat
f : X ⟶ S
g : Y ⟶ S
H₁ : Embedding ⇑f
H₂ : Embedding ⇑g
e_2✝ : ↑((cospan f g).obj WalkingCospan.one) = ↑S
\u22a2 ⇑(limit.π (cospan f g) WalkingCospan.one) = ⇑g ∘ ⇑(pullback.snd f g)
-- Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
J : Type v
inst✝ : SmallCategory J
X✝ Y✝ Z : TopCat
X Y S : TopCat
f : X ⟶ S
g : Y ⟶ S
H₁ : Embedding ⇑f
H₂ : Embedding ⇑g
e_2✝ : ↑((cospan f g).obj WalkingCospan.one) = ↑S
\u22a2 ⇑(limit.π (cospan f g) WalkingCospan.one) = ⇑(pullback.snd f g ≫ g)
-- Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
J : Type v
inst✝ : SmallCategory J
X✝ Y✝ Z : TopCat
X Y S : TopCat
f : X ⟶ S
g : Y ⟶ S
H₁ : Embedding ⇑f
H₂ : Embedding ⇑g
e_2✝ : ↑((cospan f g).obj WalkingCospan.one) = ↑S
\u22a2 ⇑(limit.π (cospan f g) WalkingCospan.right ≫ (cospan f g).map WalkingCospan.Hom.inr) = ⇑(pullback.snd f g ≫ g)
-- Mathlib/Combinatorics/SimpleGraph/Clique.lean
α : Type u_1
β : Type u_2
G H : SimpleGraph α
inst✝⁵ : Fintype α
inst✝⁴ : DecidableEq α
inst✝³ : DecidableRel G.Adj
n✝ : ℕ
a b c : α
s : Finset α
inst✝² : DecidableRel H.Adj
inst✝¹ : Fintype β
inst✝ : DecidableEq β
e : α ≃ β
n : ℕ
\u22a2 (SimpleGraph.map e.toEmbedding G).cliqueSet n = ⇑{ toFun := map e.toEmbedding, inj' := ⋯ } '' G.cliqueSet n
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α✝ : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α✝
α : Type u_3
β : Type u_4
t : TopologicalSpace α
inst✝⁵ : PolishSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
tβ : TopologicalSpace β
inst✝² : MeasurableSpace β
inst✝¹ : OpensMeasurableSpace β
f : α → β
inst✝ : SecondCountableTopology ↑(range f)
hf : Measurable f
\u22a2 ∃ t' ≤ t, Continuous f ∧ PolishSpace α
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α✝ : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α✝
α : Type u_3
β : Type u_4
t : TopologicalSpace α
inst✝⁵ : PolishSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
tβ : TopologicalSpace β
inst✝² : MeasurableSpace β
inst✝¹ : OpensMeasurableSpace β
f : α → β
inst✝ : SecondCountableTopology ↑(range f)
hf : Measurable f
b : Set (Set ↑(range f))
b_count : b.Countable
hb : IsTopologicalBasis b
\u22a2 ∃ t' ≤ t, Continuous f ∧ PolishSpace α
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α✝ : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α✝
α : Type u_3
β : Type u_4
t : TopologicalSpace α
inst✝⁵ : PolishSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
tβ : TopologicalSpace β
inst✝² : MeasurableSpace β
inst✝¹ : OpensMeasurableSpace β
f : α → β
inst✝ : SecondCountableTopology ↑(range f)
hf : Measurable f
b : Set (Set ↑(range f))
b_count : b.Countable
hb : IsTopologicalBasis b
this : Countable ↑b
\u22a2 ∃ t' ≤ t, Continuous f ∧ PolishSpace α
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α✝ : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α✝
α : Type u_3
β : Type u_4
t : TopologicalSpace α
inst✝⁵ : PolishSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
tβ : TopologicalSpace β
inst✝² : MeasurableSpace β
inst✝¹ : OpensMeasurableSpace β
f : α → β
inst✝ : SecondCountableTopology ↑(range f)
hf : Measurable f
b : Set (Set ↑(range f))
b_count : b.Countable
hb : IsTopologicalBasis b
this✝ : Countable ↑b
this : ∀ (s : ↑b), IsClopenable (rangeFactorization f ⁻¹' ↑s)
\u22a2 ∃ t' ≤ t, Continuous f ∧ PolishSpace α
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α✝ : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α✝
α : Type u_3
β : Type u_4
t : TopologicalSpace α
inst✝⁵ : PolishSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
tβ : TopologicalSpace β
inst✝² : MeasurableSpace β
inst✝¹ : OpensMeasurableSpace β
f : α → β
inst✝ : SecondCountableTopology ↑(range f)
hf : Measurable f
b : Set (Set ↑(range f))
b_count : b.Countable
hb : IsTopologicalBasis b
this : Countable ↑b
T : ↑b → TopologicalSpace α
Tt : ∀ (s : ↑b), T s ≤ t
Tpolish : ∀ (s : ↑b), PolishSpace α
h✝ : ∀ (s : ↑b), IsClosed (rangeFactorization f ⁻¹' ↑s)
Topen : ∀ (s : ↑b), IsOpen (rangeFactorization f ⁻¹' ↑s)
\u22a2 ∃ t' ≤ t, Continuous f ∧ PolishSpace α
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α✝ : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α✝
α : Type u_3
β : Type u_4
t : TopologicalSpace α
inst✝⁵ : PolishSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
tβ : TopologicalSpace β
inst✝² : MeasurableSpace β
inst✝¹ : OpensMeasurableSpace β
f : α → β
inst✝ : SecondCountableTopology ↑(range f)
hf : Measurable f
b : Set (Set ↑(range f))
b_count : b.Countable
hb : IsTopologicalBasis b
this : Countable ↑b
T : ↑b → TopologicalSpace α
Tt : ∀ (s : ↑b), T s ≤ t
Tpolish : ∀ (s : ↑b), PolishSpace α
h✝ : ∀ (s : ↑b), IsClosed (rangeFactorization f ⁻¹' ↑s)
Topen : ∀ (s : ↑b), IsOpen (rangeFactorization f ⁻¹' ↑s)
t' : TopologicalSpace α
t'T : ∀ (i : ↑b), t' ≤ T i
t't : t' ≤ t
t'_polish : PolishSpace α
\u22a2 ∃ t' ≤ t, Continuous f ∧ PolishSpace α
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α✝ : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α✝
α : Type u_3
β : Type u_4
t : TopologicalSpace α
inst✝⁵ : PolishSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
tβ : TopologicalSpace β
inst✝² : MeasurableSpace β
inst✝¹ : OpensMeasurableSpace β
f : α → β
inst✝ : SecondCountableTopology ↑(range f)
hf : Measurable f
b : Set (Set ↑(range f))
b_count : b.Countable
hb : IsTopologicalBasis b
this : Countable ↑b
T : ↑b → TopologicalSpace α
Tt : ∀ (s : ↑b), T s ≤ t
Tpolish : ∀ (s : ↑b), PolishSpace α
h✝ : ∀ (s : ↑b), IsClosed (rangeFactorization f ⁻¹' ↑s)
Topen : ∀ (s : ↑b), IsOpen (rangeFactorization f ⁻¹' ↑s)
t' : TopologicalSpace α
t'T : ∀ (i : ↑b), t' ≤ T i
t't : t' ≤ t
t'_polish : PolishSpace α
\u22a2 Continuous f