ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Topology/AlexandrovDiscrete.lean
ι : Sort u_1
κ : ι → Sort u_2
α : Type u_3
β : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
s t : Set α
a x y : α
\u22a2 s ⊆ exterior t ↔ ∀ (U : Set α), IsOpen U → t ⊆ U → s ⊆ U
-- Mathlib/CategoryTheory/Preadditive/FunctorCategory.lean
C : Type u_1
D : Type u_2
inst✝² : Category.{u_5, u_1} C
inst✝¹ : Category.{u_4, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
ι : Type u_3
s : Finset ι
X : C
α : ι → (F ⟶ G)
\u22a2 (∑ i ∈ s, α i).app X = ∑ i ∈ s, (α i).app X
-- Mathlib/Topology/Order/DenselyOrdered.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b
\u22a2 closure (Ioo a b) = Icc a b
-- Mathlib/Topology/Order/DenselyOrdered.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b
\u22a2 closure (Ioo a b) ⊆ Icc a b
-- Mathlib/Topology/Order/DenselyOrdered.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b
\u22a2 Icc a b ⊆ closure (Ioo a b)
-- Mathlib/Data/Nat/ModEq.lean
m n a b c d : ℕ
\u22a2 a ≡ b [MOD n] ↔ ↑n ∣ ↑b - ↑a
-- Mathlib/Data/List/Nodup.lean
α : Type u
β : Type v
l l₁ l₂✝ : List α
r : α → α → Prop
a b : α
σ : α → Type u_1
l₂ : (a : α) → List (σ a)
d₁ : l₁.Nodup
d₂ : ∀ (a : α), (l₂ a).Nodup
a₁ a₂ : α
n : a₁ ≠ a₂
x : (a : α) × σ a
h₁ : x ∈ map (Sigma.mk a₁) (l₂ a₁)
h₂ : x ∈ map (Sigma.mk a₂) (l₂ a₂)
\u22a2 False
-- Mathlib/Data/List/Nodup.lean
α : Type u
β : Type v
l l₁ l₂✝ : List α
r : α → α → Prop
a b : α
σ : α → Type u_1
l₂ : (a : α) → List (σ a)
d₁ : l₁.Nodup
d₂ : ∀ (a : α), (l₂ a).Nodup
a₁ a₂ : α
n : a₁ ≠ a₂
b₁ : σ a₁
left✝ : b₁ ∈ l₂ a₁
h₁ : ⟨a₁, b₁⟩ ∈ map (Sigma.mk a₁) (l₂ a₁)
h₂ : ⟨a₁, b₁⟩ ∈ map (Sigma.mk a₂) (l₂ a₂)
\u22a2 False
-- Mathlib/Data/List/Nodup.lean
α : Type u
β : Type v
l l₁ l₂✝ : List α
r : α → α → Prop
a b : α
σ : α → Type u_1
l₂ : (a : α) → List (σ a)
d₁ : l₁.Nodup
d₂ : ∀ (a : α), (l₂ a).Nodup
a₁ : α
b₁ : σ a₁
left✝ : b₁ ∈ l₂ a₁
h₁ : ⟨a₁, b₁⟩ ∈ map (Sigma.mk a₁) (l₂ a₁)
n : a₁ ≠ a₁
h₂ : ⟨a₁, b₁⟩ ∈ map (Sigma.mk a₁) (l₂ a₁)
mb₂ : b₁ ∈ l₂ a₁
\u22a2 False
-- Mathlib/Analysis/Complex/Basic.lean
x : ℝ
\u22a2 ↑x ∈ slitPlane ↔ 0 < x
-- Mathlib/Algebra/Order/ToIntervalMod.lean
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a✝ b✝ c : α
n : ℤ
a b : α
\u22a2 toIcoMod hp a b = toIcoMod hp 0 (b - a) + a
-- Mathlib/LinearAlgebra/AffineSpace/Combination.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
S : AffineSpace V P
ι : Type u_4
s : Finset ι
ι₂ : Type u_5
s₂ : Finset ι₂
inst✝ : DecidableEq ι
p : ι → P
i : ι
hi : i ∈ s
\u22a2 (affineCombination k s p) (affineCombinationSingleWeights k i) = p i
-- Mathlib/LinearAlgebra/AffineSpace/Combination.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
S : AffineSpace V P
ι : Type u_4
s : Finset ι
ι₂ : Type u_5
s₂ : Finset ι₂
inst✝ : DecidableEq ι
p : ι → P
i : ι
hi : i ∈ s
\u22a2 ∀ i2 ∈ s, i2 ≠ i → affineCombinationSingleWeights k i i2 = 0
-- Mathlib/Data/Real/Pi/Bounds.lean
\u22a2 π < 3.15
-- Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasBinaryBiproducts C
W X Y Z : C
f : W ⟶ Y
g : X ⟶ Z
inst✝ : IsIso (biprod.map f g)
\u22a2 f ≫ biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst = 𝟙 W
-- Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasBinaryBiproducts C
W X Y Z : C
f : W ⟶ Y
g : X ⟶ Z
inst✝ : IsIso (biprod.map f g)
t : (fun p => biprod.inl ≫ p ≫ biprod.fst) (biprod.map f g ≫ inv (biprod.map f g)) = (fun p => biprod.inl ≫ p ≫ biprod.fst) (𝟙 (W ⊞ X))
\u22a2 f ≫ biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst = 𝟙 W
-- Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasBinaryBiproducts C
W X Y Z : C
f : W ⟶ Y
g : X ⟶ Z
inst✝ : IsIso (biprod.map f g)
t : f ≫ biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst = biprod.inl ≫ biprod.fst
\u22a2 f ≫ biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst = 𝟙 W
-- Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasBinaryBiproducts C
W X Y Z : C
f : W ⟶ Y
g : X ⟶ Z
inst✝ : IsIso (biprod.map f g)
\u22a2 (biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst) ≫ f = 𝟙 Y
-- Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasBinaryBiproducts C
W X Y Z : C
f : W ⟶ Y
g : X ⟶ Z
inst✝ : IsIso (biprod.map f g)
t : (fun p => biprod.inl ≫ p ≫ biprod.fst) (inv (biprod.map f g) ≫ biprod.map f g) = (fun p => biprod.inl ≫ p ≫ biprod.fst) (𝟙 (Y ⊞ Z))
\u22a2 (biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst) ≫ f = 𝟙 Y
-- Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasBinaryBiproducts C
W X Y Z : C
f : W ⟶ Y
g : X ⟶ Z
inst✝ : IsIso (biprod.map f g)
t : biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst ≫ f = biprod.inl ≫ biprod.fst
\u22a2 (biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst) ≫ f = 𝟙 Y
-- Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasBinaryBiproducts C
W X Y Z : C
f : W ⟶ Y
g : X ⟶ Z
inst✝ : IsIso (biprod.map f g)
t : biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst ≫ f = biprod.inl ≫ biprod.fst
\u22a2 biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst ≫ f = 𝟙 Y
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
x : ℝ
\u22a2 cos (arctan x) ^ 2 = 1 / (1 + x ^ 2)
-- Mathlib/Order/Filter/Lift.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
f✝ f₁ f₂ : Filter α
h h₁ h₂ : Set α → Set β
inst✝ : Nonempty ι
f : ι → Filter α
g : Set α → Set β
hg : ∀ (s t : Set α), g (s ∩ t) = g s ∩ g t
s t : Set α
\u22a2 (𝓟 ∘ g) (s ∩ t) = (𝓟 ∘ g) s ⊓ (𝓟 ∘ g) t
-- Mathlib/Probability/Kernel/MeasureCompProd.lean
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ η : Kernel α β
inst✝² : SFinite μ
inst✝¹ : IsSFiniteKernel κ
E : Type u_3
inst✝ : NormedAddCommGroup E
f : α × β → E
hf : AEStronglyMeasurable f (μ ⊗ₘ κ)
\u22a2 Integrable f (μ ⊗ₘ κ) ↔ (∀ᵐ (x : α) ∂μ, Integrable (fun y => f (x, y)) (κ x)) ∧ Integrable (fun x => ∫ (y : β), ‖f (x, y)‖ ∂κ x) μ
-- Mathlib/Probability/Kernel/MeasureCompProd.lean
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ η : Kernel α β
inst✝² : SFinite μ
inst✝¹ : IsSFiniteKernel κ
E : Type u_3
inst✝ : NormedAddCommGroup E
f : α × β → E
hf : AEStronglyMeasurable f (μ ⊗ₘ κ)
\u22a2 (∀ᵐ (x : α) ∂(Kernel.const Unit μ) (), Integrable (fun y => f (x, y)) ((Kernel.prodMkLeft Unit κ) ((), x))) ∧ Integrable (fun x => ∫ (y : β), ‖f (x, y)‖ ∂(Kernel.prodMkLeft Unit κ) ((), x)) ((Kernel.const Unit μ) ()) ↔ (∀ᵐ (x : α) ∂μ, Integrable (fun y => f (x, y)) (κ x)) ∧ Integrable (fun x => ∫ (y : β), ‖f (x, y)‖ ∂κ x) μ
-- Mathlib/Data/Real/EReal.lean
x : ℝ
\u22a2 Real.toEReal ⁻¹' Ico ↑x ⊤ = Ici x
-- Mathlib/Data/Real/EReal.lean
x : ℝ
\u22a2 Real.toEReal ⁻¹' (Ici ↑x ∩ Iio ⊤) = Ici x
-- Mathlib/Data/Complex/Abs.lean
n : ℤ
\u22a2 abs ↑n = |↑n|
-- Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean
s : ℝ
hs : s ≠ 0
\u22a2 (s + 1).doublingGamma = s * s.doublingGamma
-- Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean
s : ℝ
hs : s ≠ 0
\u22a2 Gamma (s / 2 + 1 / 2) * (s / 2 * Gamma (s / 2)) * (2 ^ (s - 1) * 2) / √π = s * (Gamma (s / 2) * Gamma (s / 2 + 1 / 2) * 2 ^ (s - 1) / √π)
-- Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
p : ι → ℝ
F : ℝ → ℂ
s : ℂ
hp : ∀ (i : ι), 0 ≤ p i
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => if p i = 0 then 0 else a i * ↑(rexp (-π * p i * t))) (F t)
h_sum : Summable fun i => ‖a i‖ / p i ^ s.re
\u22a2 HasSum (fun i => ↑π ^ (-s) * Complex.Gamma s * a i / ↑(p i) ^ s) (mellin F s)
-- Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
p : ι → ℝ
F : ℝ → ℂ
s : ℂ
hp : ∀ (i : ι), 0 ≤ p i
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => if p i = 0 then 0 else a i * ↑(rexp (-π * p i * t))) (F t)
h_sum : Summable fun i => ‖a i‖ / p i ^ s.re
hs' : s ≠ 0
\u22a2 HasSum (fun i => ↑π ^ (-s) * Complex.Gamma s * a i / ↑(p i) ^ s) (mellin F s)
-- Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
p : ι → ℝ
F : ℝ → ℂ
s : ℂ
hp : ∀ (i : ι), 0 ≤ p i
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => if p i = 0 then 0 else a i * ↑(rexp (-π * p i * t))) (F t)
h_sum : Summable fun i => ‖a i‖ / p i ^ s.re
hs' : s ≠ 0
a' : ι → ℂ := fun i => if p i = 0 then 0 else a i
\u22a2 HasSum (fun i => ↑π ^ (-s) * Complex.Gamma s * a i / ↑(p i) ^ s) (mellin F s)
-- Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
p : ι → ℝ
F : ℝ → ℂ
s : ℂ
hp : ∀ (i : ι), 0 ≤ p i
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => if p i = 0 then 0 else a i * ↑(rexp (-π * p i * t))) (F t)
h_sum : Summable fun i => ‖a i‖ / p i ^ s.re
hs' : s ≠ 0
a' : ι → ℂ := fun i => if p i = 0 then 0 else a i
hp' : ∀ (i : ι), a' i = 0 ∨ 0 < p i
\u22a2 HasSum (fun i => ↑π ^ (-s) * Complex.Gamma s * a i / ↑(p i) ^ s) (mellin F s)
-- Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
p : ι → ℝ
F : ℝ → ℂ
s : ℂ
hp : ∀ (i : ι), 0 ≤ p i
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => if p i = 0 then 0 else a i * ↑(rexp (-π * p i * t))) (F t)
h_sum : Summable fun i => ‖a i‖ / p i ^ s.re
hs' : s ≠ 0
a' : ι → ℂ := fun i => if p i = 0 then 0 else a i
hp' : ∀ (i : ι), a' i = 0 ∨ 0 < p i
this : ∀ (i : ι) (t : ℝ), (if p i = 0 then 0 else a i * ↑(rexp (-π * p i * t))) = a' i * ↑(rexp (-π * p i * t))
\u22a2 HasSum (fun i => ↑π ^ (-s) * Complex.Gamma s * a i / ↑(p i) ^ s) (mellin F s)
-- Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
p : ι → ℝ
F : ℝ → ℂ
s : ℂ
hp : ∀ (i : ι), 0 ≤ p i
hs : 0 < s.re
h_sum : Summable fun i => ‖a i‖ / p i ^ s.re
hs' : s ≠ 0
a' : ι → ℂ := fun i => if p i = 0 then 0 else a i
hp' : ∀ (i : ι), a' i = 0 ∨ 0 < p i
this : ∀ (i : ι) (t : ℝ), (if p i = 0 then 0 else a i * ↑(rexp (-π * p i * t))) = a' i * ↑(rexp (-π * p i * t))
hF : ∀ t ∈ Ioi 0, HasSum (fun i => a' i * ↑(rexp (-π * p i * t))) (F t)
\u22a2 HasSum (fun i => ↑π ^ (-s) * Complex.Gamma s * a i / ↑(p i) ^ s) (mellin F s)
-- Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
p : ι → ℝ
F : ℝ → ℂ
s : ℂ
hp : ∀ (i : ι), 0 ≤ p i
hs : 0 < s.re
h_sum : Summable fun i => ‖a i‖ / p i ^ s.re
hs' : s ≠ 0
a' : ι → ℂ := fun i => if p i = 0 then 0 else a i
hp' : ∀ (i : ι), a' i = 0 ∨ 0 < p i
this : ∀ (i : ι) (t : ℝ), (if p i = 0 then 0 else a i * ↑(rexp (-π * p i * t))) = a' i * ↑(rexp (-π * p i * t))
hF : ∀ t ∈ Ioi 0, HasSum (fun i => a' i * ↑(rexp (-π * p i * t))) (F t)
i : ι
\u22a2 ↑π ^ (-s) * Complex.Gamma s * a i / ↑(p i) ^ s = ↑π ^ (-s) * Complex.Gamma s * a' i / ↑(p i) ^ s
-- Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
p : ι → ℝ
F : ℝ → ℂ
s : ℂ
hp : ∀ (i : ι), 0 ≤ p i
hs : 0 < s.re
h_sum : Summable fun i => ‖a i‖ / p i ^ s.re
hs' : s ≠ 0
a' : ι → ℂ := fun i => if p i = 0 then 0 else a i
hp' : ∀ (i : ι), a' i = 0 ∨ 0 < p i
this : ∀ (i : ι) (t : ℝ), (if p i = 0 then 0 else a i * ↑(rexp (-π * p i * t))) = a' i * ↑(rexp (-π * p i * t))
hF : ∀ t ∈ Ioi 0, HasSum (fun i => a' i * ↑(rexp (-π * p i * t))) (F t)
\u22a2 Summable fun i => ‖a' i‖ / p i ^ s.re
-- Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms C
X : C
hX : IsInitial X
\u22a2 IsZero X
-- Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms C
X : C
hX : IsInitial X
\u22a2 𝟙 X = 0
-- Mathlib/CategoryTheory/GradedObject/Unitor.lean
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst✝⁷ : Category.{u_6, u_1} C
inst✝⁶ : Category.{u_5, u_2} D
inst✝⁵ : Zero I
inst✝⁴ : DecidableEq I
inst✝³ : HasInitial C
F : D ⥤ C ⥤ D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst✝² : (X : D) → PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
inst✝¹ : (((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y)).HasMap p
inst✝ : (((mapBifunctor F J I).obj X').obj ((single₀ I).obj Y)).HasMap p
Y' : ?m.145927
\u22a2 mapBifunctorMapMap F p φ (𝟙 ((single₀ I).obj Y)) ≫ (mapBifunctorRightUnitor F Y e p hp X').hom = (mapBifunctorRightUnitor F Y e p hp X).hom ≫ φ
-- Mathlib/Data/Complex/Module.lean
A : Type u_1
inst✝³ : AddCommGroup A
inst✝² : Module ℂ A
inst✝¹ : StarAddMonoid A
inst✝ : StarModule ℂ A
z : ℂ
\u22a2 ↑(ℜ z) = ?m.419985
-- Mathlib/Data/Complex/Module.lean
A : Type u_1
inst✝³ : AddCommGroup A
inst✝² : Module ℂ A
inst✝¹ : StarAddMonoid A
inst✝ : StarModule ℂ A
z : ℂ
\u22a2 ↑(ℜ (↑z.re + ↑z.im * I)) = ↑z.re
-- Mathlib/Data/Complex/Module.lean
A : Type u_1
inst✝³ : AddCommGroup A
inst✝² : Module ℂ A
inst✝¹ : StarAddMonoid A
inst✝ : StarModule ℂ A
z : ℂ
\u22a2 ↑(ℜ ↑z.re) + ↑(-ℑ ↑z.im) = ↑z.re
-- Mathlib/Data/Finset/Pairwise.lean
α : Type u_1
ι : Type u_2
ι' : Type u_3
inst✝² : SemilatticeInf α
inst✝¹ : OrderBot α
s✝ : Finset ι
f✝ : ι → α
inst✝ : DecidableEq ι
s : Finset ι
f : ι → α
hs : (↑s).PairwiseDisjoint f
g : ι → ι
hf : ∀ (a : ι), f (g a) ≤ f a
\u22a2 (↑(Finset.image g s)).PairwiseDisjoint f
-- Mathlib/Data/Finset/Pairwise.lean
α : Type u_1
ι : Type u_2
ι' : Type u_3
inst✝² : SemilatticeInf α
inst✝¹ : OrderBot α
s✝ : Finset ι
f✝ : ι → α
inst✝ : DecidableEq ι
s : Finset ι
f : ι → α
hs : (↑s).PairwiseDisjoint f
g : ι → ι
hf : ∀ (a : ι), f (g a) ≤ f a
\u22a2 (g '' ↑s).PairwiseDisjoint f
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a x : α
hx : x ∈ factors a
\u22a2 Prime x
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a x : α
hx : x ∈ factors a
ane0 : a ≠ 0
\u22a2 Prime x
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a x : α
ane0 : a ≠ 0
hx : x ∈ Classical.choose ⋯
\u22a2 Prime x
-- Mathlib/Algebra/CharP/Basic.lean
R : Type u_1
inst✝² : CommSemiring R
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP R p
n : ℕ
x y : R
\u22a2 (x + y) ^ n = (x + y) ^ (n % p) * (x ^ p + y ^ p) ^ (n / p)
-- Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
ι : Type u_1
α : Type u_2
inst✝ : PseudoMetricSpace α
x : α
u : Set α
hu : u ∈ 𝓝 x
\u22a2 ∀ᶠ (r : ℝ) in 𝓝 0, closedBall x r ⊆ u
-- Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
ι : Type u_1
α : Type u_2
inst✝ : PseudoMetricSpace α
x : α
u : Set α
hu : u ∈ 𝓝 x
ε : ℝ
εpos : 0 < ε
hε : closedBall x ε ⊆ u
\u22a2 ∀ᶠ (r : ℝ) in 𝓝 0, closedBall x r ⊆ u
-- Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
ι : Type u_1
α : Type u_2
inst✝ : PseudoMetricSpace α
x : α
u : Set α
hu : u ∈ 𝓝 x
ε : ℝ
εpos : 0 < ε
hε : closedBall x ε ⊆ u
this : Iic ε ∈ 𝓝 0
\u22a2 ∀ᶠ (r : ℝ) in 𝓝 0, closedBall x r ⊆ u
-- Mathlib/GroupTheory/CoprodI.lean
ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
N : Type u_3
inst✝¹ : Monoid N
inst✝ : (i : ι) → DecidableEq (M i)
i : ι
m : M i
w : Word M
h1 : m ≠ 1
h2 : w.fstIdx ≠ some i
\u22a2 (cons m w h2 h1).prod = of m * w.prod
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
x : ℝ≥0∞
z : ℝ
\u22a2 x.toNNReal ^ z = (x ^ z).toNNReal
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
x : ℝ≥0∞
z : ℝ
H : z < 0
\u22a2 x.toNNReal ^ z = (x ^ z).toNNReal
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
x : ℝ≥0∞
z : ℝ
H : z = 0
\u22a2 x.toNNReal ^ z = (x ^ z).toNNReal
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
x : ℝ≥0∞
z : ℝ
H : 0 < z
\u22a2 x.toNNReal ^ z = (x ^ z).toNNReal
-- Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
l : Type u_1
m : Type u
n : Type u'
α : Type v
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A B : Matrix n n α
\u22a2 A⁻¹ᵀ = Aᵀ⁻¹
-- Mathlib/RingTheory/PiTensorProduct.lean
ι : Type u_1
R' : Type u_2
R : Type u_3
A : ι → Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : (i : ι) → NonAssocSemiring (A i)
inst✝² : (i : ι) → Module R (A i)
inst✝¹ : ∀ (i : ι), SMulCommClass R (A i) (A i)
inst✝ : ∀ (i : ι), IsScalarTower R (A i) (A i)
x : ⨂[R] (i : ι), A i
\u22a2 (mul ((tprod R) 1)) x = x
-- Mathlib/Order/Interval/Set/Basic.lean
α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : OrderTop α
a : α
\u22a2 Icc a ⊤ = Ici a
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
inst✝ : CompleteLattice α
f : Filter β
\u22a2 limsup (fun x => ⊥) f = ⊥
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
inst✝ : CompleteLattice α
f : Filter β
\u22a2 sInf {a | ∀ᶠ (n : β) in f, ⊥ ≤ a} ≤ ⊥
-- Mathlib/Geometry/Euclidean/MongePoint.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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
\u22a2 ⟪s.mongePoint -ᵥ centroid ℝ {i₁, i₂}ᶜ s.points, s.points i₁ -ᵥ s.points i₂⟫_ℝ = 0
-- Mathlib/Geometry/Euclidean/MongePoint.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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
\u22a2 ⟪(univ.weightedVSub s.pointsWithCircumcenter) (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂), (univ.weightedVSub s.pointsWithCircumcenter) (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂)⟫_ℝ = 0
-- Mathlib/Geometry/Euclidean/MongePoint.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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs : ∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
\u22a2 ⟪(univ.weightedVSub s.pointsWithCircumcenter) (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂), (univ.weightedVSub s.pointsWithCircumcenter) (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂)⟫_ℝ = 0
-- Mathlib/Geometry/Euclidean/MongePoint.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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs : ∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
\u22a2 -(∑ i : Fin (n + 2 + 1), ∑ i₂_1 : PointsWithCircumcenterIndex (n + 2), mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ (pointIndex i) * (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i₂_1 * (dist (s.pointsWithCircumcenter (pointIndex i)) (s.pointsWithCircumcenter i₂_1) * dist (s.pointsWithCircumcenter (pointIndex i)) (s.pointsWithCircumcenter i₂_1)) + ∑ i₂_1 : PointsWithCircumcenterIndex (n + 2), mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ circumcenterIndex * (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i₂_1 * (dist s.circumcenter (s.pointsWithCircumcenter i₂_1) * dist s.circumcenter (s.pointsWithCircumcenter i₂_1))) / 2 = 0
-- Mathlib/Geometry/Euclidean/MongePoint.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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs : ∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
\u22a2 -(∑ x : Fin (n + 2 + 1), ∑ x_1 : PointsWithCircumcenterIndex (n + 2), (if x = i₁ ∨ x = i₂ then (↑(n + 1))⁻¹ else 0) * (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 * (dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)) + ∑ x : PointsWithCircumcenterIndex (n + 2), -2 / ↑(n + 1) * (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x * (dist s.circumcenter (s.pointsWithCircumcenter x) * dist s.circumcenter (s.pointsWithCircumcenter x))) / 2 = 0
-- Mathlib/Geometry/Euclidean/MongePoint.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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs : ∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
\u22a2 -(∑ x : Fin (n + 2 + 1), ∑ x_1 : PointsWithCircumcenterIndex (n + 2), (if x = i₁ ∨ x = i₂ then (↑(n + 1))⁻¹ else 0) * (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 * (dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)) + ∑ x : PointsWithCircumcenterIndex (n + 2), -2 / ↑(n + 1) * (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x * (dist s.circumcenter (s.pointsWithCircumcenter x) * dist s.circumcenter (s.pointsWithCircumcenter x))) / 2 = 0
-- Mathlib/Geometry/Euclidean/MongePoint.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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs : ∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
\u22a2 -(∑ x ∈ fs, ∑ x_1 : PointsWithCircumcenterIndex (n + 2), (if x = i₁ ∨ x = i₂ then (↑(n + 1))⁻¹ else 0) * (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 * (dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)) + ∑ x : PointsWithCircumcenterIndex (n + 2), -2 / ↑(n + 1) * (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x * (dist s.circumcenter (s.pointsWithCircumcenter x) * dist s.circumcenter (s.pointsWithCircumcenter x))) / 2 = 0
-- Mathlib/Geometry/Euclidean/MongePoint.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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs : ∑ i : PointsWithCircumcenterIndex (n + 2), (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
\u22a2 ∀ x ∈ univ, x ∉ fs → ∑ x_1 : PointsWithCircumcenterIndex (n + 2), (if x = i₁ ∨ x = i₂ then (↑(n + 1))⁻¹ else 0) * (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) x_1 * (dist (s.points x) (s.pointsWithCircumcenter x_1) * dist (s.points x) (s.pointsWithCircumcenter x_1)) = 0
-- 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✝ : Finite n
c : R
\u22a2 updateRow 1 i (1 i + c • 1 j) = transvection i j c
-- 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✝ : Finite n
c : R
val✝ : Fintype n
\u22a2 updateRow 1 i (1 i + c • 1 j) = transvection i j c
-- Mathlib/Analysis/Convex/SpecificFunctions/Pow.lean
\u22a2 StrictConcaveOn ℝ (Ici 0) fun x => √x
-- Mathlib/Analysis/Convex/SpecificFunctions/Pow.lean
\u22a2 StrictConcaveOn ℝ (Ici 0) fun x => (fun x => x ^ (1 / 2)) x