ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/CategoryTheory/Functor/FullyFaithful.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F F' : C ⥤ D
G : D ⥤ E
obj : C → D
map : {X Y : C} → (X ⟶ Y) → (obj X ⟶ obj Y)
F_obj : C → E
map✝¹ : {X Y : C} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)
map_id✝¹ : ∀ (X : C), { obj := F_obj, map := map✝¹ }.map (𝟙 X) = 𝟙 ({ obj := F_obj, map := map✝¹ }.obj X)
map_comp✝¹ : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), { obj := F_obj, map := map✝¹ }.map (f ≫ g) = { obj := F_obj, map := map✝¹ }.map f ≫ { obj := F_obj, map := map✝¹ }.map g
inst✝¹ : { obj := F_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.Faithful
G_obj : D → E
map✝ : {X Y : D} → (X ⟶ Y) → (G_obj X ⟶ G_obj Y)
map_id✝ : ∀ (X : D), { obj := G_obj, map := map✝ }.map (𝟙 X) = 𝟙 ({ obj := G_obj, map := map✝ }.obj X)
map_comp✝ : ∀ {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z), { obj := G_obj, map := map✝ }.map (f ≫ g) = { obj := G_obj, map := map✝ }.map f ≫ { obj := G_obj, map := map✝ }.map g
inst✝ : { obj := G_obj, map := map✝, map_id := map_id✝, map_comp := map_comp✝ }.Faithful
h_obj : ∀ (X : C), { obj := G_obj, map := map✝, map_id := map_id✝, map_comp := map_comp✝ }.obj (obj X) = { obj := F_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.obj X
h_map : ∀ {X Y : C} {f : X ⟶ Y}, HEq ({ obj := G_obj, map := map✝, map_id := map_id✝, map_comp := map_comp✝ }.map (map f)) ({ obj := F_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.map f)
this : F_obj = G_obj ∘ obj
\u22a2 { obj := fun X => { obj := G_obj, map := map✝, map_id := map_id✝, map_comp := map_comp✝ }.obj ({ obj := obj, map := map, map_id := ⋯, map_comp := ⋯ }.obj X), map := fun {X Y} f => { obj := G_obj, map := map✝, map_id := map_id✝, map_comp := map_comp✝ }.map ({ obj := obj, map := map, map_id := ⋯, map_comp := ⋯ }.map f), map_id := ⋯, map_comp := ⋯ } = { obj := F_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }
-- Mathlib/CategoryTheory/Functor/FullyFaithful.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F F' : C ⥤ D
G : D ⥤ E
obj : C → D
map : {X Y : C} → (X ⟶ Y) → (obj X ⟶ obj Y)
G_obj : D → E
map✝¹ : {X Y : D} → (X ⟶ Y) → (G_obj X ⟶ G_obj Y)
map_id✝¹ : ∀ (X : D), { obj := G_obj, map := map✝¹ }.map (𝟙 X) = 𝟙 ({ obj := G_obj, map := map✝¹ }.obj X)
map_comp✝¹ : ∀ {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z), { obj := G_obj, map := map✝¹ }.map (f ≫ g) = { obj := G_obj, map := map✝¹ }.map f ≫ { obj := G_obj, map := map✝¹ }.map g
inst✝¹ : { obj := G_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.Faithful
map✝ : {X Y : C} → (X ⟶ Y) → ((G_obj ∘ obj) X ⟶ (G_obj ∘ obj) Y)
map_id✝ : ∀ (X : C), { obj := G_obj ∘ obj, map := map✝ }.map (𝟙 X) = 𝟙 ({ obj := G_obj ∘ obj, map := map✝ }.obj X)
map_comp✝ : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), { obj := G_obj ∘ obj, map := map✝ }.map (f ≫ g) = { obj := G_obj ∘ obj, map := map✝ }.map f ≫ { obj := G_obj ∘ obj, map := map✝ }.map g
inst✝ : { obj := G_obj ∘ obj, map := map✝, map_id := map_id✝, map_comp := map_comp✝ }.Faithful
h_obj : ∀ (X : C), { obj := G_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.obj (obj X) = { obj := G_obj ∘ obj, map := map✝, map_id := map_id✝, map_comp := map_comp✝ }.obj X
h_map : ∀ {X Y : C} {f : X ⟶ Y}, HEq ({ obj := G_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.map (map f)) ({ obj := G_obj ∘ obj, map := map✝, map_id := map_id✝, map_comp := map_comp✝ }.map f)
\u22a2 (fun {X Y} f => { obj := G_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.map ({ obj := obj, map := map, map_id := ⋯, map_comp := ⋯ }.map f)) = map✝
-- Mathlib/CategoryTheory/Functor/FullyFaithful.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F F' : C ⥤ D
G : D ⥤ E
obj : C → D
map : {X Y : C} → (X ⟶ Y) → (obj X ⟶ obj Y)
G_obj : D → E
map✝¹ : {X Y : D} → (X ⟶ Y) → (G_obj X ⟶ G_obj Y)
map_id✝¹ : ∀ (X : D), { obj := G_obj, map := map✝¹ }.map (𝟙 X) = 𝟙 ({ obj := G_obj, map := map✝¹ }.obj X)
map_comp✝¹ : ∀ {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z), { obj := G_obj, map := map✝¹ }.map (f ≫ g) = { obj := G_obj, map := map✝¹ }.map f ≫ { obj := G_obj, map := map✝¹ }.map g
inst✝¹ : { obj := G_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.Faithful
map✝ : {X Y : C} → (X ⟶ Y) → ((G_obj ∘ obj) X ⟶ (G_obj ∘ obj) Y)
map_id✝ : ∀ (X : C), { obj := G_obj ∘ obj, map := map✝ }.map (𝟙 X) = 𝟙 ({ obj := G_obj ∘ obj, map := map✝ }.obj X)
map_comp✝ : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), { obj := G_obj ∘ obj, map := map✝ }.map (f ≫ g) = { obj := G_obj ∘ obj, map := map✝ }.map f ≫ { obj := G_obj ∘ obj, map := map✝ }.map g
inst✝ : { obj := G_obj ∘ obj, map := map✝, map_id := map_id✝, map_comp := map_comp✝ }.Faithful
h_obj : ∀ (X : C), { obj := G_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.obj (obj X) = { obj := G_obj ∘ obj, map := map✝, map_id := map_id✝, map_comp := map_comp✝ }.obj X
h_map✝ : ∀ {X Y : C} {f : X ⟶ Y}, HEq ({ obj := G_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.map (map f)) ({ obj := G_obj ∘ obj, map := map✝, map_id := map_id✝, map_comp := map_comp✝ }.map f)
h_map : ∀ {X Y : C} {f : X ⟶ Y}, map✝¹ (map f) = map✝ f
\u22a2 (fun {X Y} f => { obj := G_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.map ({ obj := obj, map := map, map_id := ⋯, map_comp := ⋯ }.map f)) = map✝
-- Mathlib/Topology/Bornology/Absorbs.lean
M : Type u_1
α : Type u_2
inst✝¹ : Bornology M
inst✝ : SMul M α
s t : Set α
hs : Absorbent M s
ht : t.Finite
\u22a2 Absorbs M s t
-- Mathlib/Topology/Bornology/Absorbs.lean
M : Type u_1
α : Type u_2
inst✝¹ : Bornology M
inst✝ : SMul M α
s t : Set α
hs : Absorbent M s
ht : t.Finite
\u22a2 Absorbs M s (⋃ x ∈ t, {x})
-- Mathlib/Algebra/Polynomial/Div.lean
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : CommRing R
p q : R[X]
h : (X - C a) * (p /ₘ (X - C a)) = p
\u22a2 eval a p = 0
-- Mathlib/Algebra/Polynomial/Div.lean
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : CommRing R
p q : R[X]
h : eval a p = 0
\u22a2 (X - C a) * (p /ₘ (X - C a)) = p
-- Mathlib/Topology/UniformSpace/Ascoli.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
α : Type u_4
β : Type u_5
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
F : ι → X → α
G : ι → β → α
𝔖 : Set (Set X)
𝔖_compact : ∀ K ∈ 𝔖, IsCompact K
F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K
ℱ : Filter ι
f : X → α
\u22a2 Tendsto (⇑(UniformOnFun.ofFun 𝔖) ∘ F) ℱ (𝓝 ((UniformOnFun.ofFun 𝔖) f)) ↔ Tendsto ((⋃₀ 𝔖).restrict ∘ F) ℱ (𝓝 ((⋃₀ 𝔖).restrict f))
-- Mathlib/Topology/UniformSpace/Ascoli.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
α : Type u_4
β : Type u_5
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
F : ι → X → α
G : ι → β → α
𝔖 : Set (Set X)
𝔖_compact : ∀ K ∈ 𝔖, IsCompact K
F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K
ℱ : Filter ι
f : X → α
\u22a2 Tendsto (⇑(UniformOnFun.ofFun 𝔖) ∘ F) ℱ (𝓝 ((UniformOnFun.ofFun 𝔖) f)) ↔ Tendsto F ℱ (𝓝 f)
-- Mathlib/Topology/UniformSpace/Ascoli.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
α : Type u_4
β : Type u_5
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
F : ι → X → α
G : ι → β → α
𝔖 : Set (Set X)
𝔖_compact : ∀ K ∈ 𝔖, IsCompact K
F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K
ℱ : Filter ι
f : X → α
\u22a2 (∀ i ∈ 𝔖, Tendsto ((⇑UniformFun.ofFun ∘ i.restrict ∘ ⇑(UniformOnFun.toFun 𝔖)) ∘ ⇑(UniformOnFun.ofFun 𝔖) ∘ F) ℱ (𝓝 ((⇑UniformFun.ofFun ∘ i.restrict ∘ ⇑(UniformOnFun.toFun 𝔖)) ((UniformOnFun.ofFun 𝔖) f)))) ↔ ∀ i ∈ 𝔖, Tendsto (i.restrict ∘ F) ℱ (𝓝 (i.restrict f))
-- Mathlib/Topology/UniformSpace/Ascoli.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
α : Type u_4
β : Type u_5
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
F : ι → X → α
G : ι → β → α
𝔖 : Set (Set X)
𝔖_compact : ∀ K ∈ 𝔖, IsCompact K
F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K
ℱ : Filter ι
f : X → α
K : Set X
hK : K ∈ 𝔖
\u22a2 Tendsto ((⇑UniformFun.ofFun ∘ K.restrict ∘ ⇑(UniformOnFun.toFun 𝔖)) ∘ ⇑(UniformOnFun.ofFun 𝔖) ∘ F) ℱ (𝓝 ((⇑UniformFun.ofFun ∘ K.restrict ∘ ⇑(UniformOnFun.toFun 𝔖)) ((UniformOnFun.ofFun 𝔖) f))) ↔ Tendsto (K.restrict ∘ F) ℱ (𝓝 (K.restrict f))
-- Mathlib/Topology/UniformSpace/Ascoli.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
α : Type u_4
β : Type u_5
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
F : ι → X → α
G : ι → β → α
𝔖 : Set (Set X)
𝔖_compact : ∀ K ∈ 𝔖, IsCompact K
F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K
ℱ : Filter ι
f : X → α
K : Set X
hK : K ∈ 𝔖
this : CompactSpace ↑K
\u22a2 Tendsto ((⇑UniformFun.ofFun ∘ K.restrict ∘ ⇑(UniformOnFun.toFun 𝔖)) ∘ ⇑(UniformOnFun.ofFun 𝔖) ∘ F) ℱ (𝓝 ((⇑UniformFun.ofFun ∘ K.restrict ∘ ⇑(UniformOnFun.toFun 𝔖)) ((UniformOnFun.ofFun 𝔖) f))) ↔ Tendsto (K.restrict ∘ F) ℱ (𝓝 (K.restrict f))
-- Mathlib/Topology/UniformSpace/Ascoli.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
α : Type u_4
β : Type u_5
inst✝² : TopologicalSpace X
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
F : ι → X → α
G : ι → β → α
𝔖 : Set (Set X)
𝔖_compact : ∀ K ∈ 𝔖, IsCompact K
F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K
ℱ : Filter ι
f : X → α
K : Set X
hK : K ∈ 𝔖
this : CompactSpace ↑K
\u22a2 Tendsto ((⇑UniformFun.ofFun ∘ K.restrict ∘ ⇑(UniformOnFun.toFun 𝔖)) ∘ ⇑(UniformOnFun.ofFun 𝔖) ∘ F) ℱ (𝓝 ((⇑UniformFun.ofFun ∘ K.restrict ∘ ⇑(UniformOnFun.toFun 𝔖)) ((UniformOnFun.ofFun 𝔖) f))) ↔ Tendsto (⇑UniformFun.ofFun ∘ K.restrict ∘ F) ℱ (𝓝 (UniformFun.ofFun (K.restrict f)))
-- Mathlib/Data/Rat/Cast/Defs.lean
F : Type u_1
ι : Type u_2
α : Type u_3
β : Type u_4
inst✝ : DivisionRing α
p q : ℚ
n : ℕ
\u22a2 ↑↑n = ↑n
-- Mathlib/FieldTheory/Adjoin.lean
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
pb : PowerBasis K L
\u22a2 pb.equivAdjoinSimple.symm pb.gen = AdjoinSimple.gen K pb.gen
-- Mathlib/MeasureTheory/Integral/Bochner.lean
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
inst✝² : SMulCommClass ℝ 𝕜 E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
f g : ↥(Lp E 1 μ)
\u22a2 integralCLM (f - g) = integralCLM f - integralCLM g
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : IsIntegralClosure B A L
inst✝⁴ : IsFractionRing B L
inst✝³ : FiniteDimensional K L
inst✝² : Algebra.IsSeparable K L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I J : FractionalIdeal B⁰ L
hI : I ≠ 0
hJ : J ≠ 0
\u22a2 dual A K (dual A K I) = I
-- Mathlib/Algebra/Group/Semiconj/Basic.lean
G : Type u_1
inst✝ : Group G
a x y : G
h : SemiconjBy a x y
n : ℕ
\u22a2 SemiconjBy a (x ^ ↑n) (y ^ ↑n)
-- Mathlib/Algebra/Group/Semiconj/Basic.lean
G : Type u_1
inst✝ : Group G
a x y : G
h : SemiconjBy a x y
n : ℕ
\u22a2 SemiconjBy a (x ^ Int.negSucc n) (y ^ Int.negSucc n)
-- Mathlib/Algebra/Group/Semiconj/Basic.lean
G : Type u_1
inst✝ : Group G
a x y : G
h : SemiconjBy a x y
n : ℕ
\u22a2 SemiconjBy a (x ^ (n + 1)) (y ^ (n + 1))
-- Mathlib/Algebra/Lie/Weights/RootSystem.lean
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
α β : Weight K (↥H) L
hα : α.IsNonZero
\u22a2 β (coroot α) = ↑(↑(chainBotCoeff (⇑α) β) - ↑(chainTopCoeff (⇑α) β))
-- Mathlib/Algebra/Lie/Weights/RootSystem.lean
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
α β : Weight K (↥H) L
hα : α.IsNonZero
\u22a2 ↑(↑(chainTopCoeff (⇑α) β + chainBotCoeff (⇑α) β) - 2 * ↑(chainTopCoeff (⇑α) β)) = ↑(↑(chainBotCoeff (⇑α) β) - ↑(chainTopCoeff (⇑α) β))
-- Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
k n : ℕ
f : ContinuousMultilinearMap 𝕜 (fun i => G) G'
s : Finset (Fin n)
hk : s.card = k
z : G
\u22a2 ‖f.restr s hk z‖ ≤ ‖f‖ * ‖z‖ ^ (n - k)
-- Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
k n : ℕ
f : ContinuousMultilinearMap 𝕜 (fun i => G) G'
s : Finset (Fin n)
hk : s.card = k
z : G
\u22a2 0 ≤ ‖f‖ * ‖z‖ ^ (n - k)
-- Mathlib/Analysis/InnerProductSpace/OfNorm.lean
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
y z : E
\u22a2 ‖I • y - 2 • z‖ * ‖I • y - 2 • z‖ = 2 * (‖I • y - z‖ * ‖I • y - z‖ + ‖z‖ * ‖z‖) - ‖I • y‖ * ‖I • y‖
-- Mathlib/Analysis/InnerProductSpace/OfNorm.lean
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
y z : E
\u22a2 ‖I • y‖ * ‖I • y‖ + ‖I • y - 2 • z‖ * ‖I • y - 2 • z‖ = 2 * (‖I • y - z‖ * ‖I • y - z‖ + ‖z‖ * ‖z‖)
-- Mathlib/Analysis/InnerProductSpace/OfNorm.lean
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
y z : E
h₀ : ‖I • y - z + z‖ * ‖I • y - z + z‖ + ‖I • y - z - z‖ * ‖I • y - z - z‖ = 2 * (‖I • y - z‖ * ‖I • y - z‖ + ‖z‖ * ‖z‖)
\u22a2 ‖I • y‖ * ‖I • y‖ + ‖I • y - 2 • z‖ * ‖I • y - 2 • z‖ = 2 * (‖I • y - z‖ * ‖I • y - z‖ + ‖z‖ * ‖z‖)
-- Mathlib/NumberTheory/LSeries/Deriv.lean
f : ℕ → ℂ
n : ℕ
s : ℂ
\u22a2 HasDerivAt (fun z => term f z n) (-term (logMul f) s n) s
-- Mathlib/NumberTheory/LSeries/Deriv.lean
f : ℕ → ℂ
s : ℂ
\u22a2 HasDerivAt (fun z => term f z 0) (-term (logMul f) s 0) s
-- Mathlib/NumberTheory/LSeries/Deriv.lean
f : ℕ → ℂ
n : ℕ
s : ℂ
hn : n ≠ 0
\u22a2 HasDerivAt (fun z => term f z n) (-term (logMul f) s n) s
-- Mathlib/NumberTheory/LSeries/Deriv.lean
f : ℕ → ℂ
n : ℕ
s : ℂ
hn : n ≠ 0
\u22a2 HasDerivAt (fun z => f n * ↑n ^ (-z)) (f n * (-log ↑n * ↑n ^ (-s))) s
-- Mathlib/MeasureTheory/Measure/Hausdorff.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m : ℝ≥0∞ → ℝ≥0∞
c : ℝ≥0
hc : c ≠ 0
\u22a2 mkMetric (c • m) = c • mkMetric m
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
f : LocallyConstant ↑C ℤ
hf : (LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f - (LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f = 0
\u22a2 ∃ y, (πs C o) y = f
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
f : LocallyConstant ↑C ℤ
hf : ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f) = ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f)
\u22a2 ∃ y, (πs C o) y = f
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
f : LocallyConstant ↑C ℤ
hf : ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f) = ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f)
C₀C : ↑(C0 C ho) → ↑C := fun x => ⟨↑x, ⋯⟩
\u22a2 ∃ y, (πs C o) y = f
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
f : LocallyConstant ↑C ℤ
hf : ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f) = ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f)
C₀C : ↑(C0 C ho) → ↑C := fun x => ⟨↑x, ⋯⟩
h₀ : Continuous C₀C
\u22a2 ∃ y, (πs C o) y = f
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
f : LocallyConstant ↑C ℤ
hf : ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f) = ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f)
C₀C : ↑(C0 C ho) → ↑C := fun x => ⟨↑x, ⋯⟩
h₀ : Continuous C₀C
C₁C : ↑(π (C1 C ho) fun x => ord I x < o) → ↑C := fun x => ⟨SwapTrue o ↑x, ⋯⟩
\u22a2 ∃ y, (πs C o) y = f
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
f : LocallyConstant ↑C ℤ
hf : ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f) = ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f)
C₀C : ↑(C0 C ho) → ↑C := fun x => ⟨↑x, ⋯⟩
h₀ : Continuous C₀C
C₁C : ↑(π (C1 C ho) fun x => ord I x < o) → ↑C := fun x => ⟨SwapTrue o ↑x, ⋯⟩
h₁ : Continuous C₁C
\u22a2 ∃ y, (πs C o) y = f
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
f : LocallyConstant ↑C ℤ
hf : ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f) = ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f)
C₀C : ↑(C0 C ho) → ↑C := fun x => ⟨↑x, ⋯⟩
h₀ : Continuous C₀C
C₁C : ↑(π (C1 C ho) fun x => ord I x < o) → ↑C := fun x => ⟨SwapTrue o ↑x, ⋯⟩
h₁ : Continuous C₁C
\u22a2 (π C fun x => ord I x < o) ⊆ C0 C ho ∪ π (C1 C ho) fun x => ord I x < o
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
f : LocallyConstant ↑C ℤ
hf : ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f) = ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f)
C₀C : ↑(C0 C ho) → ↑C := fun x => ⟨↑x, ⋯⟩
h₀ : Continuous C₀C
C₁C : ↑(π (C1 C ho) fun x => ord I x < o) → ↑C := fun x => ⟨SwapTrue o ↑x, ⋯⟩
h₁ : Continuous C₁C
\u22a2 ∀ (x : I → Bool) (hx : x ∈ C0 C ho ∩ π (C1 C ho) fun x => ord I x < o), (LocallyConstant.comap { toFun := C₀C, continuous_toFun := h₀ } f) ⟨x, ⋯⟩ = (LocallyConstant.comap { toFun := C₁C, continuous_toFun := h₁ } f) ⟨x, ⋯⟩
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
f : LocallyConstant ↑C ℤ
hf : ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f) = ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f)
C₀C : ↑(C0 C ho) → ↑C := fun x => ⟨↑x, ⋯⟩
h₀ : Continuous C₀C
C₁C : ↑(π (C1 C ho) fun x => ord I x < o) → ↑C := fun x => ⟨SwapTrue o ↑x, ⋯⟩
h₁ : Continuous C₁C
\u22a2 (πs C o) (LocallyConstant.piecewise' ?refine_1 ⋯ ⋯ (LocallyConstant.comap { toFun := C₀C, continuous_toFun := h₀ } f) (LocallyConstant.comap { toFun := C₁C, continuous_toFun := h₁ } f) ?refine_2) = f
-- Mathlib/Geometry/Manifold/ContMDiff/Basic.lean
𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
inst✝¹¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁷ : TopologicalSpace M'
inst✝⁶ : ChartedSpace H' M'
inst✝⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n : ℕ∞
c : M'
x : M
\u22a2 ContMDiffAt I I' n (fun x => c) x
-- Mathlib/Geometry/Manifold/ContMDiff/Basic.lean
𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
inst✝¹¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁷ : TopologicalSpace M'
inst✝⁶ : ChartedSpace H' M'
inst✝⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n : ℕ∞
c : M'
x : M
\u22a2 ContDiffWithinAtProp I I' n (↑(chartAt H' c) ∘ (fun x => c) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' univ) (↑(chartAt H x) x)
-- Mathlib/Geometry/Manifold/ContMDiff/Basic.lean
𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
inst✝¹¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁷ : TopologicalSpace M'
inst✝⁶ : ChartedSpace H' M'
inst✝⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n : ℕ∞
c : M'
x : M
\u22a2 ContDiffWithinAt 𝕜 n (fun x => ↑I' (↑(chartAt H' c) c)) (↑I.symm ⁻¹' (↑(chartAt H x).symm ⁻¹' univ) ∩ range ↑I) (↑I (↑(chartAt H x) x))