ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay
theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay {f : ℝ → ℂ} (hc : Continuous f) {b : ℝ}
    (hb : 1 < b) (hf : f =O[cocompact ℝ] (|·| ^ (-b)))
    (hFf : (𝓕 f) =O[cocompact ℝ] (|·| ^ (-b))) (x : ℝ) :
    ∑' n : ℤ, f (x + n) = ∑' n : ℤ, 𝓕 f n * fourier n (x : UnitAddCircle) 
Arguments: {f : ℝ → ℂ} (hc : Continuous f) {b : ℝ} (hb : 1 < b) (hf : f =O[Filter.cocompact ℝ] fun x => |x| ^ (-b)) (hFf : 𝓕 f =O[Filter.cocompact ℝ] fun x => |x| ^ (-b)) (x : ℝ)
Goal: ∑' (n : ℤ), f (x + ↑n) = ∑' (n : ℤ), 𝓕 f ↑n * (fourier n) ↑x
-- SchwartzMap.tsum_eq_tsum_fourierIntegral
theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 ⇑f = ⇑g) (x : ℝ) :
    ∑' n : ℤ, f (x + n) = (∑' n : ℤ, g n * fourier n (x : UnitAddCircle)) 
Arguments: (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 ⇑f = ⇑g) (x : ℝ)
Goal: ∑' (n : ℤ), f (x + ↑n) = ∑' (n : ℤ), g ↑n * (fourier n) ↑x
-- Bornology.isVonNBounded_empty
@[simp]
theorem isVonNBounded_empty : IsVonNBounded 𝕜 (∅ : Set E) 
Arguments: (𝕜 : Type u_1) (E : Type u_3) [SeminormedRing 𝕜] [SMul 𝕜 E] [Zero E] [TopologicalSpace E]
Goal: Bornology.IsVonNBounded 𝕜 ∅
-- Bornology.isVonNBounded_iff
theorem isVonNBounded_iff (s : Set E) : IsVonNBounded 𝕜 s ↔ ∀ V ∈ 𝓝 (0 : E), Absorbs 𝕜 V s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] [Zero E] [TopologicalSpace E] (s : Set E)
Goal: Bornology.IsVonNBounded 𝕜 s ↔ ∀ V ∈ 𝓝 0, Absorbs 𝕜 V s
-- Filter.HasBasis.isVonNBounded_iff
theorem _root_.Filter.HasBasis.isVonNBounded_iff {q : ι → Prop} {s : ι → Set E} {A : Set E}
    (h : (𝓝 (0 : E)).HasBasis q s) : IsVonNBounded 𝕜 A ↔ ∀ i, q i → Absorbs 𝕜 (s i) A 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {ι : Type u_6} [SeminormedRing 𝕜] [SMul 𝕜 E] [Zero E] [TopologicalSpace E] {q : ι → Prop} {s : ι → Set E} {A : Set E} (h : (𝓝 0).HasBasis q s)
Goal: Bornology.IsVonNBounded 𝕜 A ↔ ∀ (i : ι), q i → Absorbs 𝕜 (s i) A
-- Bornology.IsVonNBounded.subset
theorem IsVonNBounded.subset {s₁ s₂ : Set E} (h : s₁ ⊆ s₂) (hs₂ : IsVonNBounded 𝕜 s₂) :
    IsVonNBounded 𝕜 s₁ 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] [Zero E] [TopologicalSpace E] {s₁ s₂ : Set E} (h : s₁ ⊆ s₂) (hs₂ : Bornology.IsVonNBounded 𝕜 s₂)
Goal: Bornology.IsVonNBounded 𝕜 s₁
-- Bornology.isVonNBounded_union
@[simp]
theorem isVonNBounded_union {s t : Set E} :
    IsVonNBounded 𝕜 (s ∪ t) ↔ IsVonNBounded 𝕜 s ∧ IsVonNBounded 𝕜 t 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] [Zero E] [TopologicalSpace E] {s t : Set E}
Goal: Bornology.IsVonNBounded 𝕜 (s ∪ t) ↔ Bornology.IsVonNBounded 𝕜 s ∧ Bornology.IsVonNBounded 𝕜 t
-- Bornology.IsVonNBounded.union
theorem IsVonNBounded.union {s₁ s₂ : Set E} (hs₁ : IsVonNBounded 𝕜 s₁) (hs₂ : IsVonNBounded 𝕜 s₂) :
    IsVonNBounded 𝕜 (s₁ ∪ s₂) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] [Zero E] [TopologicalSpace E] {s₁ s₂ : Set E} (hs₁ : Bornology.IsVonNBounded 𝕜 s₁) (hs₂ : Bornology.IsVonNBounded 𝕜 s₂)
Goal: Bornology.IsVonNBounded 𝕜 (s₁ ∪ s₂)
-- Bornology.IsVonNBounded.of_boundedSpace
theorem IsVonNBounded.of_boundedSpace [BoundedSpace 𝕜] {s : Set E} : IsVonNBounded 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] [Zero E] [TopologicalSpace E] [BoundedSpace 𝕜] {s : Set E}
Goal: Bornology.IsVonNBounded 𝕜 s
-- Bornology.isVonNBounded_iUnion
@[simp]
theorem isVonNBounded_iUnion {ι : Sort*} [Finite ι] {s : ι → Set E} :
    IsVonNBounded 𝕜 (⋃ i, s i) ↔ ∀ i, IsVonNBounded 𝕜 (s i) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] [Zero E] [TopologicalSpace E] {ι : Sort u_7} [Finite ι] {s : ι → Set E}
Goal: Bornology.IsVonNBounded 𝕜 (⋃ i, s i) ↔ ∀ (i : ι), Bornology.IsVonNBounded 𝕜 (s i)
-- Bornology.isVonNBounded_biUnion
theorem isVonNBounded_biUnion {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set E} :
    IsVonNBounded 𝕜 (⋃ i ∈ I, s i) ↔ ∀ i ∈ I, IsVonNBounded 𝕜 (s i) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] [Zero E] [TopologicalSpace E] {ι : Type u_7} {I : Set ι} (hI : I.Finite) {s : ι → Set E}
Goal: Bornology.IsVonNBounded 𝕜 (⋃ i ∈ I, s i) ↔ ∀ i ∈ I, Bornology.IsVonNBounded 𝕜 (s i)
-- Bornology.isVonNBounded_sUnion
theorem isVonNBounded_sUnion {S : Set (Set E)} (hS : S.Finite) :
    IsVonNBounded 𝕜 (⋃₀ S) ↔ ∀ s ∈ S, IsVonNBounded 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SMul 𝕜 E] [Zero E] [TopologicalSpace E] {S : Set (Set E)} (hS : S.Finite)
Goal: Bornology.IsVonNBounded 𝕜 (⋃₀ S) ↔ ∀ s ∈ S, Bornology.IsVonNBounded 𝕜 s
-- Bornology.IsVonNBounded.add
protected theorem IsVonNBounded.add (hs : IsVonNBounded 𝕜 s) (ht : IsVonNBounded 𝕜 t) :
    IsVonNBounded 𝕜 (s + t) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [AddZeroClass E] [TopologicalSpace E] [ContinuousAdd E] [DistribSMul 𝕜 E] {s t : Set E} (hs : Bornology.IsVonNBounded 𝕜 s) (ht : Bornology.IsVonNBounded 𝕜 t)
Goal: Bornology.IsVonNBounded 𝕜 (s + t)
-- Bornology.IsVonNBounded.neg
protected theorem IsVonNBounded.neg (hs : IsVonNBounded 𝕜 s) : IsVonNBounded 𝕜 (-s) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [AddGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [DistribMulAction 𝕜 E] {s : Set E} (hs : Bornology.IsVonNBounded 𝕜 s)
Goal: Bornology.IsVonNBounded 𝕜 (-s)
-- Bornology.isVonNBounded_neg
@[simp]
theorem isVonNBounded_neg : IsVonNBounded 𝕜 (-s) ↔ IsVonNBounded 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [AddGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [DistribMulAction 𝕜 E] {s : Set E}
Goal: Bornology.IsVonNBounded 𝕜 (-s) ↔ Bornology.IsVonNBounded 𝕜 s
-- Bornology.IsVonNBounded.sub
protected theorem IsVonNBounded.sub (hs : IsVonNBounded 𝕜 s) (ht : IsVonNBounded 𝕜 t) :
    IsVonNBounded 𝕜 (s - t) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [AddGroup E] [TopologicalSpace E] [TopologicalAddGroup E] [DistribMulAction 𝕜 E] {s t : Set E} (hs : Bornology.IsVonNBounded 𝕜 s) (ht : Bornology.IsVonNBounded 𝕜 t)
Goal: Bornology.IsVonNBounded 𝕜 (s - t)
-- Bornology.IsVonNBounded.of_topologicalSpace_le
theorem IsVonNBounded.of_topologicalSpace_le {t t' : TopologicalSpace E} (h : t ≤ t') {s : Set E}
    (hs : @IsVonNBounded 𝕜 E _ _ _ t s) : @IsVonNBounded 𝕜 E _ _ _ t' s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {t t' : TopologicalSpace E} (h : t ≤ t') {s : Set E} (hs : Bornology.IsVonNBounded 𝕜 s)
Goal: Bornology.IsVonNBounded 𝕜 s
-- Bornology.isVonNBounded_iff_tendsto_smallSets_nhds
lemma isVonNBounded_iff_tendsto_smallSets_nhds {𝕜 E : Type*} [NormedDivisionRing 𝕜]
    [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] {S : Set E} :
    IsVonNBounded 𝕜 S ↔ Tendsto (· • S : 𝕜 → Set E) (𝓝 0) (𝓝 0).smallSets 
Arguments: {𝕜 : Type u_7} {E : Type u_8} [NormedDivisionRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] {S : Set E}
Goal: Bornology.IsVonNBounded 𝕜 S ↔ Filter.Tendsto (fun x => x • S) (𝓝 0) (𝓝 0).smallSets
-- Bornology.isVonNBounded_pi_iff
lemma isVonNBounded_pi_iff {𝕜 ι : Type*} {E : ι → Type*} [NormedDivisionRing 𝕜]
    [∀ i, AddCommGroup (E i)] [∀ i, Module 𝕜 (E i)] [∀ i, TopologicalSpace (E i)]
    {S : Set (∀ i, E i)} : IsVonNBounded 𝕜 S ↔ ∀ i, IsVonNBounded 𝕜 (eval i '' S) 
Arguments: {𝕜 : Type u_7} {ι : Type u_8} {E : ι → Type u_9} [NormedDivisionRing 𝕜] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [(i : ι) → TopologicalSpace (E i)] {S : Set ((i : ι) → E i)}
Goal: Bornology.IsVonNBounded 𝕜 S ↔ ∀ (i : ι), Bornology.IsVonNBounded 𝕜 (Function.eval i '' S)
-- Bornology.IsVonNBounded.image
theorem IsVonNBounded.image {σ : 𝕜₁ →+* 𝕜₂} [RingHomSurjective σ] [RingHomIsometric σ] {s : Set E}
    (hs : IsVonNBounded 𝕜₁ s) (f : E →SL[σ] F) : IsVonNBounded 𝕜₂ (f '' s) 
Arguments: {E : Type u_3} {F : Type u_5} {𝕜₁ : Type u_7} {𝕜₂ : Type u_8} [NormedDivisionRing 𝕜₁] [NormedDivisionRing 𝕜₂] [AddCommGroup E] [Module 𝕜₁ E] [AddCommGroup F] [Module 𝕜₂ F] [TopologicalSpace E] [TopologicalSpace F] {σ : 𝕜₁ →+* 𝕜₂} [RingHomSurjective σ] [RingHomIsometric σ] {s : Set E} (hs : Bornology.IsVonNBounded 𝕜₁ s) (f : E →SL[σ] F)
Goal: Bornology.IsVonNBounded 𝕜₂ (⇑f '' s)
-- Bornology.IsVonNBounded.smul_tendsto_zero
theorem IsVonNBounded.smul_tendsto_zero [NormedField 𝕜]
    [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E]
    {S : Set E} {ε : ι → 𝕜} {x : ι → E} {l : Filter ι}
    (hS : IsVonNBounded 𝕜 S) (hxS : ∀ᶠ n in l, x n ∈ S) (hε : Tendsto ε l (𝓝 0)) :
    Tendsto (ε • x) l (𝓝 0) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {ι : Type u_6} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] {S : Set E} {ε : ι → 𝕜} {x : ι → E} {l : Filter ι} (hS : Bornology.IsVonNBounded 𝕜 S) (hxS : ∀ᶠ (n : ι) in l, x n ∈ S) (hε : Filter.Tendsto ε l (𝓝 0))
Goal: Filter.Tendsto (ε • x) l (𝓝 0)
-- Bornology.isVonNBounded_of_smul_tendsto_zero
theorem isVonNBounded_of_smul_tendsto_zero {ε : ι → 𝕜} {l : Filter ι} [l.NeBot]
    (hε : ∀ᶠ n in l, ε n ≠ 0) {S : Set E}
    (H : ∀ x : ι → E, (∀ n, x n ∈ S) → Tendsto (ε • x) l (𝓝 0)) : IsVonNBounded 𝕜 S 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {ι : Type u_6} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] {ε : ι → 𝕜} {l : Filter ι} [l.NeBot] (hε : ∀ᶠ (n : ι) in l, ε n ≠ 0) {S : Set E} (H : ∀ (x : ι → E), (∀ (n : ι), x n ∈ S) → Filter.Tendsto (ε • x) l (𝓝 0))
Goal: Bornology.IsVonNBounded 𝕜 S
-- Bornology.isVonNBounded_iff_smul_tendsto_zero
theorem isVonNBounded_iff_smul_tendsto_zero {ε : ι → 𝕜} {l : Filter ι} [l.NeBot]
    (hε : Tendsto ε l (𝓝[≠] 0)) {S : Set E} :
    IsVonNBounded 𝕜 S ↔ ∀ x : ι → E, (∀ n, x n ∈ S) → Tendsto (ε • x) l (𝓝 0) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} {ι : Type u_6} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] {ε : ι → 𝕜} {l : Filter ι} [l.NeBot] (hε : Filter.Tendsto ε l (𝓝[≠] 0)) {S : Set E}
Goal: Bornology.IsVonNBounded 𝕜 S ↔ ∀ (x : ι → E), (∀ (n : ι), x n ∈ S) → Filter.Tendsto (ε • x) l (𝓝 0)
-- Bornology.isVonNBounded_singleton
theorem isVonNBounded_singleton (x : E) : IsVonNBounded 𝕜 ({x} : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] (x : E)
Goal: Bornology.IsVonNBounded 𝕜 {x}
-- Bornology.isVonNBounded_insert
@[simp]
theorem isVonNBounded_insert (x : E) {s : Set E} :
    IsVonNBounded 𝕜 (insert x s) ↔ IsVonNBounded 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] (x : E) {s : Set E}
Goal: Bornology.IsVonNBounded 𝕜 (insert x s) ↔ Bornology.IsVonNBounded 𝕜 s
-- Bornology.IsVonNBounded.vadd
protected theorem IsVonNBounded.vadd (hs : IsVonNBounded 𝕜 s) (x : E) :
    IsVonNBounded 𝕜 (x +ᵥ s) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] [ContinuousAdd E] {s : Set E} (hs : Bornology.IsVonNBounded 𝕜 s) (x : E)
Goal: Bornology.IsVonNBounded 𝕜 (x +ᵥ s)
-- Bornology.isVonNBounded_vadd
@[simp]
theorem isVonNBounded_vadd (x : E) : IsVonNBounded 𝕜 (x +ᵥ s) ↔ IsVonNBounded 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] [ContinuousAdd E] {s : Set E} (x : E)
Goal: Bornology.IsVonNBounded 𝕜 (x +ᵥ s) ↔ Bornology.IsVonNBounded 𝕜 s
-- Bornology.IsVonNBounded.of_add_right
theorem IsVonNBounded.of_add_right (hst : IsVonNBounded 𝕜 (s + t)) (hs : s.Nonempty) :
    IsVonNBounded 𝕜 t 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] [ContinuousAdd E] {s t : Set E} (hst : Bornology.IsVonNBounded 𝕜 (s + t)) (hs : s.Nonempty)
Goal: Bornology.IsVonNBounded 𝕜 t
-- Bornology.IsVonNBounded.of_add_left
theorem IsVonNBounded.of_add_left (hst : IsVonNBounded 𝕜 (s + t)) (ht : t.Nonempty) :
    IsVonNBounded 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] [ContinuousAdd E] {s t : Set E} (hst : Bornology.IsVonNBounded 𝕜 (s + t)) (ht : t.Nonempty)
Goal: Bornology.IsVonNBounded 𝕜 s
-- Bornology.isVonNBounded_add_of_nonempty
theorem isVonNBounded_add_of_nonempty (hs : s.Nonempty) (ht : t.Nonempty) :
    IsVonNBounded 𝕜 (s + t) ↔ IsVonNBounded 𝕜 s ∧ IsVonNBounded 𝕜 t 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] [ContinuousAdd E] {s t : Set E} (hs : s.Nonempty) (ht : t.Nonempty)
Goal: Bornology.IsVonNBounded 𝕜 (s + t) ↔ Bornology.IsVonNBounded 𝕜 s ∧ Bornology.IsVonNBounded 𝕜 t
-- Bornology.isVonNBounded_add
theorem isVonNBounded_add :
    IsVonNBounded 𝕜 (s + t) ↔ s = ∅ ∨ t = ∅ ∨ IsVonNBounded 𝕜 s ∧ IsVonNBounded 𝕜 t 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] [ContinuousAdd E] {s t : Set E}
Goal: Bornology.IsVonNBounded 𝕜 (s + t) ↔ s = ∅ ∨ t = ∅ ∨ Bornology.IsVonNBounded 𝕜 s ∧ Bornology.IsVonNBounded 𝕜 t
-- Bornology.isVonNBounded_add_self
@[simp]
theorem isVonNBounded_add_self : IsVonNBounded 𝕜 (s + s) ↔ IsVonNBounded 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] [ContinuousAdd E] {s : Set E}
Goal: Bornology.IsVonNBounded 𝕜 (s + s) ↔ Bornology.IsVonNBounded 𝕜 s
-- Bornology.IsVonNBounded.of_sub_left
theorem IsVonNBounded.of_sub_left (hst : IsVonNBounded 𝕜 (s - t)) (ht : t.Nonempty) :
    IsVonNBounded 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] [ContinuousAdd E] {s t : Set E} (hst : Bornology.IsVonNBounded 𝕜 (s - t)) (ht : t.Nonempty)
Goal: Bornology.IsVonNBounded 𝕜 s
-- Bornology.IsVonNBounded.of_sub_right
theorem IsVonNBounded.of_sub_right (hst : IsVonNBounded 𝕜 (s - t)) (hs : s.Nonempty) :
    IsVonNBounded 𝕜 t 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] [TopologicalAddGroup E] {s t : Set E} (hst : Bornology.IsVonNBounded 𝕜 (s - t)) (hs : s.Nonempty)
Goal: Bornology.IsVonNBounded 𝕜 t
-- Bornology.isVonNBounded_sub_of_nonempty
theorem isVonNBounded_sub_of_nonempty (hs : s.Nonempty) (ht : t.Nonempty) :
    IsVonNBounded 𝕜 (s - t) ↔ IsVonNBounded 𝕜 s ∧ IsVonNBounded 𝕜 t 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] [TopologicalAddGroup E] {s t : Set E} (hs : s.Nonempty) (ht : t.Nonempty)
Goal: Bornology.IsVonNBounded 𝕜 (s - t) ↔ Bornology.IsVonNBounded 𝕜 s ∧ Bornology.IsVonNBounded 𝕜 t
-- Bornology.isVonNBounded_sub
theorem isVonNBounded_sub :
    IsVonNBounded 𝕜 (s - t) ↔ s = ∅ ∨ t = ∅ ∨ IsVonNBounded 𝕜 s ∧ IsVonNBounded 𝕜 t 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] [TopologicalAddGroup E] {s t : Set E}
Goal: Bornology.IsVonNBounded 𝕜 (s - t) ↔ s = ∅ ∨ t = ∅ ∨ Bornology.IsVonNBounded 𝕜 s ∧ Bornology.IsVonNBounded 𝕜 t
-- Bornology.isVonNBounded_covers
theorem isVonNBounded_covers : ⋃₀ setOf (IsVonNBounded 𝕜) = (Set.univ : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E]
Goal: ⋃₀ setOf (Bornology.IsVonNBounded 𝕜) = Set.univ
-- Bornology.isBounded_iff_isVonNBounded
@[simp]
theorem isBounded_iff_isVonNBounded {s : Set E} :
    @IsBounded _ (vonNBornology 𝕜 E) s ↔ IsVonNBounded 𝕜 s 
Arguments: (𝕜 : Type u_1) {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] {s : Set E}
Goal: Bornology.IsBounded s ↔ Bornology.IsVonNBounded 𝕜 s
-- TotallyBounded.isVonNBounded
theorem TotallyBounded.isVonNBounded {s : Set E} (hs : TotallyBounded s) :
    Bornology.IsVonNBounded 𝕜 s 
Arguments: (𝕜 : Type u_1) {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [UniformSpace E] [UniformAddGroup E] [ContinuousSMul 𝕜 E] {s : Set E} (hs : TotallyBounded s)
Goal: Bornology.IsVonNBounded 𝕜 s
-- Filter.Tendsto.isVonNBounded_range
theorem Filter.Tendsto.isVonNBounded_range [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
    [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E]
    {f : ℕ → E} {x : E} (hf : Tendsto f atTop (𝓝 x)) : Bornology.IsVonNBounded 𝕜 (range f) 
Arguments: (𝕜 : Type u_1) {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] {f : ℕ → E} {x : E} (hf : Filter.Tendsto f Filter.atTop (𝓝 x))
Goal: Bornology.IsVonNBounded 𝕜 (Set.range f)
-- NormedSpace.isVonNBounded_of_isBounded
theorem isVonNBounded_of_isBounded {s : Set E} (h : Bornology.IsBounded s) :
    Bornology.IsVonNBounded 𝕜 s 
Arguments: (𝕜 : Type u_1) {E : Type u_3} [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} (h : Bornology.IsBounded s)
Goal: Bornology.IsVonNBounded 𝕜 s
-- NormedSpace.isVonNBounded_ball
theorem isVonNBounded_ball (r : ℝ) : Bornology.IsVonNBounded 𝕜 (Metric.ball (0 : E) r) 
Arguments: (𝕜 : Type u_1) (E : Type u_3) [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (r : ℝ)
Goal: Bornology.IsVonNBounded 𝕜 (Metric.ball 0 r)
-- NormedSpace.isVonNBounded_closedBall
theorem isVonNBounded_closedBall (r : ℝ) :
    Bornology.IsVonNBounded 𝕜 (Metric.closedBall (0 : E) r) 
Arguments: (𝕜 : Type u_1) (E : Type u_3) [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (r : ℝ)
Goal: Bornology.IsVonNBounded 𝕜 (Metric.closedBall 0 r)
-- NormedSpace.isVonNBounded_iff
theorem isVonNBounded_iff {s : Set E} : Bornology.IsVonNBounded 𝕜 s ↔ Bornology.IsBounded s 
Arguments: (𝕜 : Type u_1) {E : Type u_3} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E}
Goal: Bornology.IsVonNBounded 𝕜 s ↔ Bornology.IsBounded s
-- NormedSpace.isVonNBounded_iff'
theorem isVonNBounded_iff' {s : Set E} :
    Bornology.IsVonNBounded 𝕜 s ↔ ∃ r : ℝ, ∀ x ∈ s, ‖x‖ ≤ r 
Arguments: (𝕜 : Type u_1) {E : Type u_3} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E}
Goal: Bornology.IsVonNBounded 𝕜 s ↔ ∃ r, ∀ x ∈ s, ‖x‖ ≤ r
-- NormedSpace.image_isVonNBounded_iff
theorem image_isVonNBounded_iff {α : Type*} {f : α → E} {s : Set α} :
    Bornology.IsVonNBounded 𝕜 (f '' s) ↔ ∃ r : ℝ, ∀ x ∈ s, ‖f x‖ ≤ r 
Arguments: (𝕜 : Type u_1) {E : Type u_3} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {α : Type u_7} {f : α → E} {s : Set α}
Goal: Bornology.IsVonNBounded 𝕜 (f '' s) ↔ ∃ r, ∀ x ∈ s, ‖f x‖ ≤ r
-- NormedSpace.vonNBornology_eq
theorem vonNBornology_eq : Bornology.vonNBornology 𝕜 E = PseudoMetricSpace.toBornology 
Arguments: (𝕜 : Type u_1) {E : Type u_3} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: Bornology.vonNBornology 𝕜 E = PseudoMetricSpace.toBornology
-- NormedSpace.isBounded_iff_subset_smul_ball
theorem isBounded_iff_subset_smul_ball {s : Set E} :
    Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.ball (0 : E) 1 
Arguments: (𝕜 : Type u_1) {E : Type u_3} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E}
Goal: Bornology.IsBounded s ↔ ∃ a, s ⊆ a • Metric.ball 0 1
-- NormedSpace.isBounded_iff_subset_smul_closedBall
theorem isBounded_iff_subset_smul_closedBall {s : Set E} :
    Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.closedBall (0 : E) 1 
Arguments: (𝕜 : Type u_1) {E : Type u_3} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E}
Goal: Bornology.IsBounded s ↔ ∃ a, s ⊆ a • Metric.closedBall 0 1
-- Asymptotics.IsBigO.antisymm
theorem IsBigO.antisymm (h₁ : f =O[l] g) (h₂ : g =O[l] f) : f =Θ[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} {l : Filter α} (h₁ : f =O[l] g) (h₂ : g =O[l] f)
Goal: f =Θ[l] g
-- Asymptotics.IsTheta.isBigO
lemma IsTheta.isBigO (h : f =Θ[l] g) : f =O[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} {l : Filter α} (h : f =Θ[l] g)
Goal: f =O[l] g
-- Asymptotics.IsTheta.isBigO_symm
lemma IsTheta.isBigO_symm (h : f =Θ[l] g) : g =O[l] f 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} {l : Filter α} (h : f =Θ[l] g)
Goal: g =O[l] f
-- Asymptotics.isTheta_refl
@[refl]
theorem isTheta_refl (f : α → E) (l : Filter α) : f =Θ[l] f 
Arguments: {α : Type u_1} {E : Type u_3} [Norm E] (f : α → E) (l : Filter α)
Goal: f =Θ[l] f
-- Asymptotics.isTheta_rfl
theorem isTheta_rfl : f =Θ[l] f 
Arguments: {α : Type u_1} {E : Type u_3} [Norm E] {f : α → E} {l : Filter α}
Goal: f =Θ[l] f
-- Asymptotics.IsTheta.symm
@[symm]
nonrec theorem IsTheta.symm (h : f =Θ[l] g) : g =Θ[l] f 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} {l : Filter α} (h : f =Θ[l] g)
Goal: g =Θ[l] f
-- Asymptotics.isTheta_comm
theorem isTheta_comm : f =Θ[l] g ↔ g =Θ[l] f 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {f : α → E} {g : α → F} {l : Filter α}
Goal: f =Θ[l] g ↔ g =Θ[l] f
-- Asymptotics.IsTheta.trans
@[trans]
theorem IsTheta.trans {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =Θ[l] g) (h₂ : g =Θ[l] k) :
    f =Θ[l] k 
Arguments: {α : Type u_1} {E : Type u_3} {G : Type u_5} {F' : Type u_7} [Norm E] [Norm G] [SeminormedAddCommGroup F'] {l : Filter α} {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =Θ[l] g) (h₂ : g =Θ[l] k)
Goal: f =Θ[l] k
-- Asymptotics.IsBigO.trans_isTheta
@[trans]
theorem IsBigO.trans_isTheta {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =O[l] g)
    (h₂ : g =Θ[l] k) : f =O[l] k 
Arguments: {α : Type u_1} {E : Type u_3} {G : Type u_5} {F' : Type u_7} [Norm E] [Norm G] [SeminormedAddCommGroup F'] {l : Filter α} {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =O[l] g) (h₂ : g =Θ[l] k)
Goal: f =O[l] k
-- Asymptotics.IsTheta.trans_isBigO
@[trans]
theorem IsTheta.trans_isBigO {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =Θ[l] g)
    (h₂ : g =O[l] k) : f =O[l] k 
Arguments: {α : Type u_1} {E : Type u_3} {G : Type u_5} {F' : Type u_7} [Norm E] [Norm G] [SeminormedAddCommGroup F'] {l : Filter α} {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =Θ[l] g) (h₂ : g =O[l] k)
Goal: f =O[l] k
-- Asymptotics.IsLittleO.trans_isTheta
@[trans]
theorem IsLittleO.trans_isTheta {f : α → E} {g : α → F} {k : α → G'} (h₁ : f =o[l] g)
    (h₂ : g =Θ[l] k) : f =o[l] k 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} {G' : Type u_8} [Norm E] [Norm F] [SeminormedAddCommGroup G'] {l : Filter α} {f : α → E} {g : α → F} {k : α → G'} (h₁ : f =o[l] g) (h₂ : g =Θ[l] k)
Goal: f =o[l] k
-- Asymptotics.IsTheta.trans_isLittleO
@[trans]
theorem IsTheta.trans_isLittleO {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =Θ[l] g)
    (h₂ : g =o[l] k) : f =o[l] k 
Arguments: {α : Type u_1} {E : Type u_3} {G : Type u_5} {F' : Type u_7} [Norm E] [Norm G] [SeminormedAddCommGroup F'] {l : Filter α} {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =Θ[l] g) (h₂ : g =o[l] k)
Goal: f =o[l] k
-- Asymptotics.IsTheta.trans_eventuallyEq
@[trans]
theorem IsTheta.trans_eventuallyEq {f : α → E} {g₁ g₂ : α → F} (h : f =Θ[l] g₁) (hg : g₁ =ᶠ[l] g₂) :
    f =Θ[l] g₂ 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {l : Filter α} {f : α → E} {g₁ g₂ : α → F} (h : f =Θ[l] g₁) (hg : g₁ =ᶠ[l] g₂)
Goal: f =Θ[l] g₂
-- Filter.EventuallyEq.trans_isTheta
@[trans]
theorem _root_.Filter.EventuallyEq.trans_isTheta {f₁ f₂ : α → E} {g : α → F} (hf : f₁ =ᶠ[l] f₂)
    (h : f₂ =Θ[l] g) : f₁ =Θ[l] g 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {l : Filter α} {f₁ f₂ : α → E} {g : α → F} (hf : f₁ =ᶠ[l] f₂) (h : f₂ =Θ[l] g)
Goal: f₁ =Θ[l] g
-- Filter.EventuallyEq.isTheta
lemma _root_.Filter.EventuallyEq.isTheta {f g : α → E} (h : f =ᶠ[l] g) : f =Θ[l] g 
Arguments: {α : Type u_1} {E : Type u_3} [Norm E] {l : Filter α} {f g : α → E} (h : f =ᶠ[l] g)
Goal: f =Θ[l] g
-- Asymptotics.isTheta_norm_left
@[simp]
theorem isTheta_norm_left : (fun x ↦ ‖f' x‖) =Θ[l] g ↔ f' =Θ[l] g 
Arguments: {α : Type u_1} {F : Type u_4} {E' : Type u_6} [Norm F] [SeminormedAddCommGroup E'] {g : α → F} {f' : α → E'} {l : Filter α}
Goal: (fun x => ‖f' x‖) =Θ[l] g ↔ f' =Θ[l] g