ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/ModelTheory/Definability.lean
M : Type w
A : Set M
L : Language
inst✝ : L.Structure M
α : Type u₁
β : Type u_1
B : Set M
s : Set (α → M)
φ : L.BoundedFormula (↑A ⊕ α) 0
x✝ : α → M
n✝ : ℕ
t✝ : L.Term ((↑A ⊕ α) ⊕ Fin n✝)
xs✝ : Fin n✝ → M
\u22a2 Term.realize (Sum.elim x✝ xs✝) (Term.constantsVarsEquivLeft.symm t✝) = Term.realize (Sum.elim (Sum.elim Subtype.val x✝) xs✝) t✝
-- Mathlib/ModelTheory/Definability.lean
M : Type w
A : Set M
L : Language
inst✝ : L.Structure M
α : Type u₁
β : Type u_1
B : Set M
s : Set (α → M)
φ : L.BoundedFormula (↑A ⊕ α) 0
x✝ : α → M
n✝ : ℕ
t✝ : L.Term ((↑A ⊕ α) ⊕ Fin n✝)
xs✝ : Fin n✝ → M
\u22a2 Term.realize (Sum.elim (fun a => ↑a) (Sum.elim x✝ xs✝) ∘ ⇑(Equiv.sumAssoc (↑A) α (Fin n✝))) t✝ = Term.realize (Sum.elim (Sum.elim Subtype.val x✝) xs✝) t✝
-- Mathlib/Analysis/Convex/Side.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
x : P
hx : x ∉ s
p : P
hp : p ∈ ↑s
this : Nonempty ↥s
\u22a2 IsConnected {y | s.SOppSide x y}
-- Mathlib/Analysis/Convex/Side.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
x : P
hx : x ∉ s
p : P
hp : p ∈ ↑s
this : Nonempty ↥s
\u22a2 IsConnected ((fun x_1 => x_1.1 • (x -ᵥ p) +ᵥ x_1.2) '' Set.Iio 0 ×ˢ ↑s)
-- Mathlib/CategoryTheory/GradedObject.lean
I : Type u_1
J : Type u_2
K : Type u_3
C : Type u_4
inst✝⁴ : Category.{u_5, u_4} C
X Y Z : GradedObject I C
φ : X ⟶ Y
e : X ≅ Y
ψ : Y ⟶ Z
p : I → J
j : J
inst✝³ : X.HasMap p
inst✝² : Y.HasMap p
inst✝¹ : Z.HasMap p
c : X.CofanMapObjFun p j
hc : IsColimit c
inst✝ : X.HasMap p
i : I
hi : p i = j
\u22a2 X.ιMapObj p i j hi ≫ (iso hc).inv = Cofan.inj c ⟨i, hi⟩
-- Mathlib/Computability/Partrec.lean
α : Type u_1
β : Type u_2
γ : Type u_3
σ : Type u_4
inst✝³ : Primcodable α
inst✝² : Primcodable β
inst✝¹ : Primcodable γ
inst✝ : Primcodable σ
x✝¹ : Fin 0 → α → σ
x✝ : ∀ (i : Fin 0), Computable (x✝¹ i)
\u22a2 Computable fun a => List.ofFn fun i => x✝¹ i a
-- Mathlib/Computability/Partrec.lean
α : Type u_1
β : Type u_2
γ : Type u_3
σ : Type u_4
inst✝³ : Primcodable α
inst✝² : Primcodable β
inst✝¹ : Primcodable γ
inst✝ : Primcodable σ
x✝¹ : Fin 0 → α → σ
x✝ : ∀ (i : Fin 0), Computable (x✝¹ i)
\u22a2 Computable fun a => []
-- Mathlib/Computability/Partrec.lean
α : Type u_1
β : Type u_2
γ : Type u_3
σ : Type u_4
inst✝³ : Primcodable α
inst✝² : Primcodable β
inst✝¹ : Primcodable γ
inst✝ : Primcodable σ
n : ℕ
f : Fin (n + 1) → α → σ
hf : ∀ (i : Fin (n + 1)), Computable (f i)
\u22a2 Computable fun a => List.ofFn fun i => f i a
-- Mathlib/Computability/Partrec.lean
α : Type u_1
β : Type u_2
γ : Type u_3
σ : Type u_4
inst✝³ : Primcodable α
inst✝² : Primcodable β
inst✝¹ : Primcodable γ
inst✝ : Primcodable σ
n : ℕ
f : Fin (n + 1) → α → σ
hf : ∀ (i : Fin (n + 1)), Computable (f i)
\u22a2 Computable fun a => f 0 a :: List.ofFn fun i => f i.succ a
-- Mathlib/Algebra/Polynomial/Degree/Definitions.lean
R : Type u
S : Type v
a b c d : R
n✝ m : ℕ
inst✝ : Semiring R
p q r : R[X]
n : ℕ
ha : a ≠ 0
\u22a2 ((monomial n) a).degree = ↑n
-- .lake/packages/batteries/Batteries/Data/List/Lemmas.lean
α : Type u_1
a : α
n m : Nat
l : List α
h : m < n
i : Nat
\u22a2 (take m (l.set n a))[i]? = (take m l)[i]?
-- .lake/packages/batteries/Batteries/Data/List/Lemmas.lean
α : Type u_1
a : α
n m : Nat
l : List α
h : m < n
i : Nat
\u22a2 (if i < m then (l.set n a)[i]? else none) = if i < m then l[i]? else none
-- Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean
C : Type u
inst✝¹ : Category.{v, u} C
R X Y Z : C
f : X ⟶ Y
a b : R ⟶ X
f₁ : X ⟶ Y
f₂ : Y ⟶ Z
inst✝ : Mono f₂
small_k : IsKernelPair f₁ a b
\u22a2 IsLimit (PullbackCone.mk a b ⋯)
-- Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean
C : Type u
inst✝¹ : Category.{v, u} C
R X Y Z : C
f : X ⟶ Y
a b : R ⟶ X
f₁ : X ⟶ Y
f₂ : Y ⟶ Z
inst✝ : Mono f₂
small_k : IsKernelPair f₁ a b
\u22a2 ∀ (s : PullbackCone (f₁ ≫ f₂) (f₁ ≫ f₂)) (m : s.pt ⟶ (PullbackCone.mk a b ⋯).pt), (∀ (j : WalkingCospan), m ≫ (PullbackCone.mk a b ⋯).π.app j = s.π.app j) → m = (fun s => small_k.lift s.fst s.snd ⋯) s
-- Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean
C : Type u
inst✝¹ : Category.{v, u} C
R X Y Z : C
f : X ⟶ Y
a b : R ⟶ X
f₁ : X ⟶ Y
f₂ : Y ⟶ Z
inst✝ : Mono f₂
small_k : IsKernelPair f₁ a b
s : PullbackCone (f₁ ≫ f₂) (f₁ ≫ f₂)
m : s.pt ⟶ (PullbackCone.mk a b ⋯).pt
hm : ∀ (j : WalkingCospan), m ≫ (PullbackCone.mk a b ⋯).π.app j = s.π.app j
\u22a2 m = (fun s => small_k.lift s.fst s.snd ⋯) s
-- Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean
C : Type u
inst✝¹ : Category.{v, u} C
R X Y Z : C
f : X ⟶ Y
a b : R ⟶ X
f₁ : X ⟶ Y
f₂ : Y ⟶ Z
inst✝ : Mono f₂
small_k : IsKernelPair f₁ a b
s : PullbackCone (f₁ ≫ f₂) (f₁ ≫ f₂)
m : s.pt ⟶ (PullbackCone.mk a b ⋯).pt
hm : ∀ (j : WalkingCospan), m ≫ (PullbackCone.mk a b ⋯).π.app j = s.π.app j
\u22a2 ∀ (j : WalkingCospan), m ≫ (IsPullback.cone small_k).π.app j = (fun s => small_k.lift s.fst s.snd ⋯) s ≫ (IsPullback.cone small_k).π.app j
-- Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean
C : Type u
inst✝¹ : Category.{v, u} C
R X Y Z : C
f : X ⟶ Y
a b : R ⟶ X
f₁ : X ⟶ Y
f₂ : Y ⟶ Z
inst✝ : Mono f₂
small_k : IsKernelPair f₁ a b
s : PullbackCone (f₁ ≫ f₂) (f₁ ≫ f₂)
m : s.pt ⟶ (PullbackCone.mk a b ⋯).pt
hm : ∀ (j : WalkingCospan), m ≫ (PullbackCone.mk a b ⋯).π.app j = s.π.app j
\u22a2 m ≫ (IsPullback.cone small_k).fst = (fun s => small_k.lift s.fst s.snd ⋯) s ≫ (IsPullback.cone small_k).fst
-- Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean
C : Type u
inst✝¹ : Category.{v, u} C
R X Y Z : C
f : X ⟶ Y
a b : R ⟶ X
f₁ : X ⟶ Y
f₂ : Y ⟶ Z
inst✝ : Mono f₂
small_k : IsKernelPair f₁ a b
s : PullbackCone (f₁ ≫ f₂) (f₁ ≫ f₂)
m : s.pt ⟶ (PullbackCone.mk a b ⋯).pt
hm : ∀ (j : WalkingCospan), m ≫ (PullbackCone.mk a b ⋯).π.app j = s.π.app j
\u22a2 m ≫ (IsPullback.cone small_k).snd = (fun s => small_k.lift s.fst s.snd ⋯) s ≫ (IsPullback.cone small_k).snd
-- Mathlib/SetTheory/Ordinal/Exponential.lean
a b c : Ordinal.{u_1}
a0 : a ≠ 0
h : b.IsLimit
\u22a2 a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c
-- Mathlib/Probability/Distributions/Gaussian.lean
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
\u22a2 (gaussianReal μ v) s = ENNReal.ofReal (∫ (x : ℝ) in s, gaussianPDFReal μ v x)
-- Mathlib/Probability/Distributions/Gaussian.lean
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
\u22a2 ∫⁻ (x : ℝ) in s, gaussianPDF μ v x = ∫⁻ (x : ℝ) in s, ENNReal.ofReal (gaussianPDFReal μ v x)
-- Mathlib/Probability/Distributions/Gaussian.lean
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
\u22a2 Integrable (gaussianPDFReal μ v) (ℙ.restrict s)
-- Mathlib/Probability/Distributions/Gaussian.lean
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
\u22a2 0 ≤ᶠ[ae (ℙ.restrict s)] gaussianPDFReal μ v
-- Mathlib/Analysis/Calculus/TangentCone.lean
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
ht : y ∈ closure t
v : E
c : ℕ → 𝕜
d : ℕ → E
hd : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 v)
\u22a2 (LinearMap.inl 𝕜 E F) v ∈ tangentConeAt 𝕜 (s ×ˢ t) (x, y)
-- Mathlib/Analysis/Calculus/TangentCone.lean
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y✝ : E
s t✝ : Set E
t : Set F
y : F
ht : y ∈ closure t
v : E
c : ℕ → 𝕜
d : ℕ → E
hd : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s
hc : Tendsto (fun n => ‖c n‖) atTop atTop
hy : Tendsto (fun n => c n • d n) atTop (𝓝 v)
this : ∀ (n : ℕ), ∃ d', y + d' ∈ t ∧ ‖c n • d'‖ < (1 / 2) ^ n
\u22a2 (LinearMap.inl 𝕜 E F) v ∈ tangentConeAt 𝕜 (s ×ˢ t) (x, y)
-- Mathlib/GroupTheory/Coset.lean
α : Type u_1
inst✝ : Group α
s : Subgroup α
x y : α
\u22a2 x • ↑s = y • ↑s ↔ x⁻¹ * y ∈ s
-- Mathlib/GroupTheory/Coset.lean
α : Type u_1
inst✝ : Group α
s : Subgroup α
x y : α
\u22a2 (∀ (x_1 : α), x_1 ∈ x • ↑s ↔ x_1 ∈ y • ↑s) ↔ x⁻¹ * y ∈ s
-- Mathlib/GroupTheory/Coset.lean
α : Type u_1
inst✝ : Group α
s : Subgroup α
x y : α
\u22a2 (∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s) ↔ x⁻¹ * y ∈ s
-- Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s✝ t s : Set E
x : E
hx : x ∈ balancedCoreAux 𝕜 s
\u22a2 x ∈ s
-- Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
a : UnitAddCircle
n : ℕ
\u22a2 sinZeta a (-2 * ↑n - 1) = 0
-- Mathlib/Data/List/ProdSigma.lean
α : Type u_1
β : Type u_2
head✝ : α
l : List α
\u22a2 (head✝ :: l) ×ˢ [] = []
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∃ᶠ (x : ℝ) in atTop, f x = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
\u22a2 ∀ᶠ (x : ℝ) in atTop, f x = 0
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
\u22a2 ∀ᶠ (x : ℝ) in atTop, f x = 0
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
\u22a2 f x = 0
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ max x 1
hx₀ : f x₀ = 0
\u22a2 f x = 0
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ max x 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
\u22a2 f x = 0
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ max x 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
\u22a2 f x = 0
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ max x 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
\u22a2 2 ^ (-↑⌊-logb 2 (x / x₀)⌋₊ - 1) * x₀ ≤ x
-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ max x 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
\u22a2 x ≤ 2 ^ (-↑⌊-logb 2 (x / x₀)⌋₊) * x₀
-- Mathlib/Analysis/Analytic/CPolynomial.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → F
p✝ pf pg : FormalMultilinearSeries 𝕜 E F
x : E
r r' : ℝ≥0∞
n✝ m : ℕ
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
\u22a2 Continuous p.sum
-- Mathlib/Analysis/Analytic/CPolynomial.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → F
p✝ pf pg : FormalMultilinearSeries 𝕜 E F
x : E
r r' : ℝ≥0∞
n✝ m : ℕ
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
\u22a2 ContinuousOn p.sum (EMetric.ball ?m.404882 ⊤)
-- Mathlib/Analysis/Analytic/CPolynomial.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → F
p✝ pf pg : FormalMultilinearSeries 𝕜 E F
x : E
r r' : ℝ≥0∞
n✝ m : ℕ
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
\u22a2 E
-- Mathlib/Order/Filter/AtTopBot.lean
ι : Type u_1
ι' : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
inst✝¹ : SemilatticeSup α
inst✝ : Nonempty α
p : α × α → Prop
\u22a2 (∀ᶠ (x : α × α) in atTop, p x) ↔ ∃ a, ∀ k ≥ a, ∀ l ≥ a, p (k, l)
-- Mathlib/SetTheory/Cardinal/Ordinal.lean
a b : Cardinal.{u_1}
h : b ≠ 0
\u22a2 a = 1 * a
-- Mathlib/Data/Seq/Computation.lean
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
\u22a2 ((pure a).thinkN n).Results a n
-- Mathlib/Data/Seq/Computation.lean
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
this : ((pure a).thinkN n).Results a (0 + n)
\u22a2 ((pure a).thinkN n).Results a n
-- Mathlib/Algebra/Group/Subgroup/Pointwise.lean
α : Type u_1
G : Type u_2
A✝ : Type u_3
S : Type u_4
inst✝¹ : Group G
inst✝ : AddGroup A✝
s : Set G
A B C : Subgroup G
h : A ≤ C
x✝ : G
\u22a2 x✝ ∈ ↑A * ↑(B ⊓ C) ↔ x✝ ∈ ↑A * ↑B ∩ ↑C
-- Mathlib/Algebra/Group/Subgroup/Pointwise.lean
α : Type u_1
G : Type u_2
A✝ : Type u_3
S : Type u_4
inst✝¹ : Group G
inst✝ : AddGroup A✝
s : Set G
A B C : Subgroup G
h : A ≤ C
x✝ : G
\u22a2 (∃ x ∈ ↑A, ∃ y, (y ∈ ↑B ∧ y ∈ ↑C) ∧ x * y = x✝) ↔ (∃ x ∈ ↑A, ∃ y ∈ ↑B, x * y = x✝) ∧ x✝ ∈ ↑C
-- Mathlib/Algebra/Group/Subgroup/Pointwise.lean
α : Type u_1
G : Type u_2
A✝ : Type u_3
S : Type u_4
inst✝¹ : Group G
inst✝ : AddGroup A✝
s : Set G
A B C : Subgroup G
h : A ≤ C
y : G
hy : y ∈ ↑A
z : G
hz : z ∈ ↑B
hyz : y * z ∈ ↑C
\u22a2 y⁻¹ * (y * z) ∈ C
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x y z : ℝ
hx : 0 ≤ x
hy : y ≠ 0
\u22a2 x ^ y = 0 ↔ x = 0
-- Mathlib/CategoryTheory/Limits/Lattice.lean
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
x y z : α
f : z ⟶ x
g : z ⟶ y
\u22a2 colimit (span f g) = Finset.univ.sup (span f g).obj
-- Mathlib/CategoryTheory/Limits/Lattice.lean
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
x y z : α
f : z ⟶ x
g : z ⟶ y
\u22a2 z ⊔ (x ⊔ (y ⊔ ⊥)) = z ⊔ (x ⊔ y)
-- Mathlib/Data/ENat/Basic.lean
m n : ℕ∞
x✝ : ℕ
\u22a2 ↑(↑x✝).toNat = ↑x✝ ↔ ↑x✝ ≠ ⊤
-- Mathlib/Algebra/MonoidAlgebra/Support.lean
k : Type u₁
G : Type u₂
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : DecidableEq G
a b : MonoidAlgebra k G
\u22a2 (a * b).support ⊆ a.support * b.support
-- Mathlib/Algebra/MonoidAlgebra/Support.lean
k : Type u₁
G : Type u₂
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : DecidableEq G
a b : MonoidAlgebra k G
\u22a2 (sum a fun a₁ b₁ => sum b fun a₂ b₂ => single (a₁ * a₂) (b₁ * b₂)).support ⊆ a.support * b.support
-- Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
a : ℝ
s : ℂ
hs : 1 < s.re
\u22a2 HasSum (fun n => ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2) (hurwitzZetaOdd (↑a) s)
-- Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
a : ℝ
s : ℂ
hs : 1 < s.re
n : ℤ
\u22a2 ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2 = (s + 1).Gammaℝ * ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2 / (s + 1).Gammaℝ
-- Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
a : ℝ
s : ℂ
hs : 1 < s.re
n : ℤ
this : 0 < (s + 1).re
\u22a2 ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2 = (s + 1).Gammaℝ * ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2 / (s + 1).Gammaℝ
-- Mathlib/Algebra/Field/Basic.lean
α : Type u_1
β : Type u_2
K : Type u_3
inst✝ : DivisionSemiring α
a b c d : α
hab : Commute a b
ha : a ≠ 0
hb : b ≠ 0
\u22a2 1 / a + 1 / b = (a + b) / (a * b)
-- Mathlib/Data/ENNReal/Real.lean
a b c d : ℝ≥0∞
r p✝ q✝ : ℝ≥0
p q : ℝ
hp : 0 ≤ p
\u22a2 ENNReal.ofReal (p * q) = ENNReal.ofReal p * ENNReal.ofReal q
-- Mathlib/SetTheory/Cardinal/ToNat.lean
α : Type u
c✝ d : Cardinal.{u}
c : Cardinal.{v}
\u22a2 toNat (lift.{u, v} c) = toNat c
-- Mathlib/Data/Matroid/Map.lean
α : Type u_1
β : Type u_2
f : α → β
E I s : Set α
M : Matroid α
N✝ N : Matroid β
\u22a2 ∀ I ⊆ (N.comap id).E, (N.comap id).Indep I ↔ N.Indep I
-- Mathlib/Order/Bounded.lean
α : Type u_1
r : α → α → Prop
s t : Set α
inst✝¹ : LinearOrder α
inst✝ : NoMaxOrder α
a : α
\u22a2 Unbounded (fun x x_1 => x < x_1) (s ∩ {b | a < b}) ↔ Unbounded (fun x x_1 => x < x_1) s
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g
\u22a2 P (pullback.fst f g)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g
\u22a2 ∀ (i : ((S.affineCover.pullbackCover f).pullbackCover (pullback.fst f g)).J), P ((S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g
i : ((S.affineCover.pullbackCover f).pullbackCover (pullback.fst f g)).J
\u22a2 P ((S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g
i : ((S.affineCover.pullbackCover f).pullbackCover (pullback.fst f g)).J
e : pullback (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≅ pullback (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) := pullbackSymmetry (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≪≫ pullbackRightPullbackFstIso f g ((S.affineCover.pullbackCover f).map i) ≪≫ asIso (pullback.map ((S.affineCover.pullbackCover f).map i ≫ f) g (pullback.snd f (S.affineCover.map i) ≫ S.affineCover.map i) g (𝟙 ((S.affineCover.pullbackCover f).obj i)) (𝟙 Y) (𝟙 S) ⋯ ⋯) ≪≫ (pullbackRightPullbackFstIso (S.affineCover.map i) g (pullback.snd f (S.affineCover.map i))).symm
\u22a2 P ((S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g
i : ((S.affineCover.pullbackCover f).pullbackCover (pullback.fst f g)).J
e : pullback (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≅ pullback (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) := pullbackSymmetry (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≪≫ pullbackRightPullbackFstIso f g ((S.affineCover.pullbackCover f).map i) ≪≫ asIso (pullback.map ((S.affineCover.pullbackCover f).map i ≫ f) g (pullback.snd f (S.affineCover.map i) ≫ S.affineCover.map i) g (𝟙 ((S.affineCover.pullbackCover f).obj i)) (𝟙 Y) (𝟙 S) ⋯ ⋯) ≪≫ (pullbackRightPullbackFstIso (S.affineCover.map i) g (pullback.snd f (S.affineCover.map i))).symm
this : e.hom ≫ pullback.fst (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) = (S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i
\u22a2 P ((S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g
i : ((S.affineCover.pullbackCover f).pullbackCover (pullback.fst f g)).J
e : pullback (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≅ pullback (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) := pullbackSymmetry (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≪≫ pullbackRightPullbackFstIso f g ((S.affineCover.pullbackCover f).map i) ≪≫ asIso (pullback.map ((S.affineCover.pullbackCover f).map i ≫ f) g (pullback.snd f (S.affineCover.map i) ≫ S.affineCover.map i) g (𝟙 ((S.affineCover.pullbackCover f).obj i)) (𝟙 Y) (𝟙 S) ⋯ ⋯) ≪≫ (pullbackRightPullbackFstIso (S.affineCover.map i) g (pullback.snd f (S.affineCover.map i))).symm
this : e.hom ≫ pullback.fst (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) = (S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i
\u22a2 P (pullback.fst (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g))
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g
i : ((S.affineCover.pullbackCover f).pullbackCover (pullback.fst f g)).J
e : pullback (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≅ pullback (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) := pullbackSymmetry (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≪≫ pullbackRightPullbackFstIso f g ((S.affineCover.pullbackCover f).map i) ≪≫ asIso (pullback.map ((S.affineCover.pullbackCover f).map i ≫ f) g (pullback.snd f (S.affineCover.map i) ≫ S.affineCover.map i) g (𝟙 ((S.affineCover.pullbackCover f).obj i)) (𝟙 Y) (𝟙 S) ⋯ ⋯) ≪≫ (pullbackRightPullbackFstIso (S.affineCover.map i) g (pullback.snd f (S.affineCover.map i))).symm
this : e.hom ≫ pullback.fst (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) = (S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i
\u22a2 Q (pullback.fst (S.affineCover.map i) g)