ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- variationOnFromTo.sub_self_monotoneOn
protected theorem sub_self_monotoneOn {f : α → ℝ} {s : Set α} (hf : LocallyBoundedVariationOn f s)
    {a : α} (as : a ∈ s) : MonotoneOn (variationOnFromTo f s a - f) s 
Arguments: {α : Type u_1} [LinearOrder α] {f : α → ℝ} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a : α} (as : a ∈ s)
Goal: MonotoneOn (variationOnFromTo f s a - f) s
-- variationOnFromTo.comp_eq_of_monotoneOn
protected theorem comp_eq_of_monotoneOn {β : Type*} [LinearOrder β] (f : α → E) {t : Set β}
    (φ : β → α) (hφ : MonotoneOn φ t) {x y : β} (hx : x ∈ t) (hy : y ∈ t) :
    variationOnFromTo (f ∘ φ) t x y = variationOnFromTo f (φ '' t) (φ x) (φ y) 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {β : Type u_3} [LinearOrder β] (f : α → E) {t : Set β} (φ : β → α) (hφ : MonotoneOn φ t) {x y : β} (hx : x ∈ t) (hy : y ∈ t)
Goal: variationOnFromTo (f ∘ φ) t x y = variationOnFromTo f (φ '' t) (φ x) (φ y)
-- LocallyBoundedVariationOn.exists_monotoneOn_sub_monotoneOn
theorem LocallyBoundedVariationOn.exists_monotoneOn_sub_monotoneOn {f : α → ℝ} {s : Set α}
    (h : LocallyBoundedVariationOn f s) :
    ∃ p q : α → ℝ, MonotoneOn p s ∧ MonotoneOn q s ∧ f = p - q 
Arguments: {α : Type u_1} [LinearOrder α] {f : α → ℝ} {s : Set α} (h : LocallyBoundedVariationOn f s)
Goal: ∃ p q, MonotoneOn p s ∧ MonotoneOn q s ∧ f = p - q
-- LipschitzOnWith.comp_eVariationOn_le
theorem LipschitzOnWith.comp_eVariationOn_le {f : E → F} {C : ℝ≥0} {t : Set E}
    (h : LipschitzOnWith C f t) {g : α → E} {s : Set α} (hg : MapsTo g s t) :
    eVariationOn (f ∘ g) s ≤ C * eVariationOn g s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {F : Type u_3} [PseudoEMetricSpace F] {f : E → F} {C : ℝ≥0} {t : Set E} (h : LipschitzOnWith C f t) {g : α → E} {s : Set α} (hg : Set.MapsTo g s t)
Goal: eVariationOn (f ∘ g) s ≤ ↑C * eVariationOn g s
-- LipschitzOnWith.comp_boundedVariationOn
theorem LipschitzOnWith.comp_boundedVariationOn {f : E → F} {C : ℝ≥0} {t : Set E}
    (hf : LipschitzOnWith C f t) {g : α → E} {s : Set α} (hg : MapsTo g s t)
    (h : BoundedVariationOn g s) : BoundedVariationOn (f ∘ g) s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {F : Type u_3} [PseudoEMetricSpace F] {f : E → F} {C : ℝ≥0} {t : Set E} (hf : LipschitzOnWith C f t) {g : α → E} {s : Set α} (hg : Set.MapsTo g s t) (h : BoundedVariationOn g s)
Goal: BoundedVariationOn (f ∘ g) s
-- LipschitzOnWith.comp_locallyBoundedVariationOn
theorem LipschitzOnWith.comp_locallyBoundedVariationOn {f : E → F} {C : ℝ≥0} {t : Set E}
    (hf : LipschitzOnWith C f t) {g : α → E} {s : Set α} (hg : MapsTo g s t)
    (h : LocallyBoundedVariationOn g s) : LocallyBoundedVariationOn (f ∘ g) s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {F : Type u_3} [PseudoEMetricSpace F] {f : E → F} {C : ℝ≥0} {t : Set E} (hf : LipschitzOnWith C f t) {g : α → E} {s : Set α} (hg : Set.MapsTo g s t) (h : LocallyBoundedVariationOn g s)
Goal: LocallyBoundedVariationOn (f ∘ g) s
-- LipschitzWith.comp_boundedVariationOn
theorem LipschitzWith.comp_boundedVariationOn {f : E → F} {C : ℝ≥0} (hf : LipschitzWith C f)
    {g : α → E} {s : Set α} (h : BoundedVariationOn g s) : BoundedVariationOn (f ∘ g) s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {F : Type u_3} [PseudoEMetricSpace F] {f : E → F} {C : ℝ≥0} (hf : LipschitzWith C f) {g : α → E} {s : Set α} (h : BoundedVariationOn g s)
Goal: BoundedVariationOn (f ∘ g) s
-- LipschitzWith.comp_locallyBoundedVariationOn
theorem LipschitzWith.comp_locallyBoundedVariationOn {f : E → F} {C : ℝ≥0}
    (hf : LipschitzWith C f) {g : α → E} {s : Set α} (h : LocallyBoundedVariationOn g s) :
    LocallyBoundedVariationOn (f ∘ g) s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {F : Type u_3} [PseudoEMetricSpace F] {f : E → F} {C : ℝ≥0} (hf : LipschitzWith C f) {g : α → E} {s : Set α} (h : LocallyBoundedVariationOn g s)
Goal: LocallyBoundedVariationOn (f ∘ g) s
-- LipschitzOnWith.locallyBoundedVariationOn
theorem LipschitzOnWith.locallyBoundedVariationOn {f : ℝ → E} {C : ℝ≥0} {s : Set ℝ}
    (hf : LipschitzOnWith C f s) : LocallyBoundedVariationOn f s 
Arguments: {E : Type u_2} [PseudoEMetricSpace E] {f : ℝ → E} {C : ℝ≥0} {s : Set ℝ} (hf : LipschitzOnWith C f s)
Goal: LocallyBoundedVariationOn f s
-- LipschitzWith.locallyBoundedVariationOn
theorem LipschitzWith.locallyBoundedVariationOn {f : ℝ → E} {C : ℝ≥0} (hf : LipschitzWith C f)
    (s : Set ℝ) : LocallyBoundedVariationOn f s 
Arguments: {E : Type u_2} [PseudoEMetricSpace E] {f : ℝ → E} {C : ℝ≥0} (hf : LipschitzWith C f) (s : Set ℝ)
Goal: LocallyBoundedVariationOn f s
-- LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem_real
theorem ae_differentiableWithinAt_of_mem_real {f : ℝ → ℝ} {s : Set ℝ}
    (h : LocallyBoundedVariationOn f s) : ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x 
Arguments: {f : ℝ → ℝ} {s : Set ℝ} (h : LocallyBoundedVariationOn f s)
Goal: ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x
-- LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem_pi
theorem ae_differentiableWithinAt_of_mem_pi {ι : Type*} [Fintype ι] {f : ℝ → ι → ℝ} {s : Set ℝ}
    (h : LocallyBoundedVariationOn f s) : ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x 
Arguments: {ι : Type u_4} [Fintype ι] {f : ℝ → ι → ℝ} {s : Set ℝ} (h : LocallyBoundedVariationOn f s)
Goal: ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x
-- LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem
theorem ae_differentiableWithinAt_of_mem {f : ℝ → V} {s : Set ℝ}
    (h : LocallyBoundedVariationOn f s) : ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x 
Arguments: {V : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] {f : ℝ → V} {s : Set ℝ} (h : LocallyBoundedVariationOn f s)
Goal: ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x
-- LocallyBoundedVariationOn.ae_differentiableWithinAt
theorem ae_differentiableWithinAt {f : ℝ → V} {s : Set ℝ} (h : LocallyBoundedVariationOn f s)
    (hs : MeasurableSet s) : ∀ᵐ x ∂volume.restrict s, DifferentiableWithinAt ℝ f s x 
Arguments: {V : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] {f : ℝ → V} {s : Set ℝ} (h : LocallyBoundedVariationOn f s) (hs : MeasurableSet s)
Goal: ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict s, DifferentiableWithinAt ℝ f s x
-- LocallyBoundedVariationOn.ae_differentiableAt
theorem ae_differentiableAt {f : ℝ → V} (h : LocallyBoundedVariationOn f univ) :
    ∀ᵐ x, DifferentiableAt ℝ f x 
Arguments: {V : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] {f : ℝ → V} (h : LocallyBoundedVariationOn f Set.univ)
Goal: ∀ᵐ (x : ℝ), DifferentiableAt ℝ f x
-- LipschitzOnWith.ae_differentiableWithinAt_of_mem_real
theorem LipschitzOnWith.ae_differentiableWithinAt_of_mem_real {C : ℝ≥0} {f : ℝ → V} {s : Set ℝ}
    (h : LipschitzOnWith C f s) : ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x 
Arguments: {V : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] {C : ℝ≥0} {f : ℝ → V} {s : Set ℝ} (h : LipschitzOnWith C f s)
Goal: ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x
-- LipschitzOnWith.ae_differentiableWithinAt_real
theorem LipschitzOnWith.ae_differentiableWithinAt_real {C : ℝ≥0} {f : ℝ → V} {s : Set ℝ}
    (h : LipschitzOnWith C f s) (hs : MeasurableSet s) :
    ∀ᵐ x ∂volume.restrict s, DifferentiableWithinAt ℝ f s x 
Arguments: {V : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] {C : ℝ≥0} {f : ℝ → V} {s : Set ℝ} (h : LipschitzOnWith C f s) (hs : MeasurableSet s)
Goal: ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict s, DifferentiableWithinAt ℝ f s x
-- LipschitzWith.ae_differentiableAt_real
theorem LipschitzWith.ae_differentiableAt_real {C : ℝ≥0} {f : ℝ → V} (h : LipschitzWith C f) :
    ∀ᵐ x, DifferentiableAt ℝ f x 
Arguments: {V : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] {C : ℝ≥0} {f : ℝ → V} (h : LipschitzWith C f)
Goal: ∀ᵐ (x : ℝ), DifferentiableAt ℝ f x
-- SemiNormedGrp.completion_obj
@[simps]
def completion : SemiNormedGrp.{u} ⥤ SemiNormedGrp.{u} 
Arguments: (V : SemiNormedGrp)
Goal: SemiNormedGrp.completion.obj V = SemiNormedGrp.of (UniformSpace.Completion ↑V)
-- SemiNormedGrp.completion_map
@[simps]
def completion : SemiNormedGrp.{u} ⥤ SemiNormedGrp.{u} 
Goal: ∀ {X Y : SemiNormedGrp} (f : X ⟶ Y), SemiNormedGrp.completion.map f = NormedAddGroupHom.completion f
-- SemiNormedGrp.completion.incl_apply
@[simps]
def completion.incl {V : SemiNormedGrp} : V ⟶ completion.obj V 
Arguments: {V : SemiNormedGrp} (v : ↑V)
Goal: SemiNormedGrp.completion.incl v = ↑↑V v
-- SemiNormedGrp.completion.norm_incl_eq
theorem completion.norm_incl_eq {V : SemiNormedGrp} {v : V} : ‖completion.incl v‖ = ‖v‖ 
Arguments: {V : SemiNormedGrp} {v : ↑V}
Goal: ‖SemiNormedGrp.completion.incl v‖ = ‖v‖
-- SemiNormedGrp.completion.map_normNoninc
theorem completion.map_normNoninc {V W : SemiNormedGrp} {f : V ⟶ W} (hf : f.NormNoninc) :
    (completion.map f).NormNoninc 
Arguments: {V W : SemiNormedGrp} {f : V ⟶ W} (hf : NormedAddGroupHom.NormNoninc f)
Goal: NormedAddGroupHom.NormNoninc (SemiNormedGrp.completion.map f)
-- SemiNormedGrp.completion.map_zero
theorem completion.map_zero (V W : SemiNormedGrp) : completion.map (0 : V ⟶ W) = 0 
Arguments: (V W : SemiNormedGrp)
Goal: SemiNormedGrp.completion.map 0 = 0
-- SemiNormedGrp.completion.lift_comp_incl
theorem completion.lift_comp_incl {V W : SemiNormedGrp} [CompleteSpace W] [T0Space W]
    (f : V ⟶ W) : completion.incl ≫ completion.lift f = f 
Arguments: {V W : SemiNormedGrp} [CompleteSpace ↑W] [T0Space ↑W] (f : V ⟶ W)
Goal: SemiNormedGrp.completion.incl ≫ SemiNormedGrp.completion.lift f = f
-- SemiNormedGrp.completion.lift_unique
theorem completion.lift_unique {V W : SemiNormedGrp} [CompleteSpace W] [T0Space W]
    (f : V ⟶ W) (g : completion.obj V ⟶ W) : completion.incl ≫ g = f → g = completion.lift f 
Arguments: {V W : SemiNormedGrp} [CompleteSpace ↑W] [T0Space ↑W] (f : V ⟶ W) (g : SemiNormedGrp.completion.obj V ⟶ W)
Goal: SemiNormedGrp.completion.incl ≫ g = f → g = SemiNormedGrp.completion.lift f
-- Submodule.coe_norm
@[simp]
theorem coe_norm [Ring 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] {s : Submodule 𝕜 E}
    (x : s) : ‖x‖ = ‖(x : E)‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [Ring 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] {s : Submodule 𝕜 E} (x : ↥s)
Goal: ‖x‖ = ‖↑x‖
-- Submodule.norm_coe
@[norm_cast]
theorem norm_coe [Ring 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] {s : Submodule 𝕜 E}
    (x : s) : ‖(x : E)‖ = ‖x‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [Ring 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] {s : Submodule 𝕜 E} (x : ↥s)
Goal: ‖↑x‖ = ‖x‖
-- cauchySeq_finset_iff_vanishing_norm
theorem cauchySeq_finset_iff_vanishing_norm {f : ι → E} :
    (CauchySeq fun s : Finset ι => ∑ i ∈ s, f i) ↔
      ∀ ε > (0 : ℝ), ∃ s : Finset ι, ∀ t, Disjoint t s → ‖∑ i ∈ t, f i‖ < ε 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] {f : ι → E}
Goal: (CauchySeq fun s => ∑ i ∈ s, f i) ↔ ∀ ε > 0, ∃ s, ∀ (t : Finset ι), Disjoint t s → ‖∑ i ∈ t, f i‖ < ε
-- summable_iff_vanishing_norm
theorem summable_iff_vanishing_norm [CompleteSpace E] {f : ι → E} :
    Summable f ↔ ∀ ε > (0 : ℝ), ∃ s : Finset ι, ∀ t, Disjoint t s → ‖∑ i ∈ t, f i‖ < ε 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] [CompleteSpace E] {f : ι → E}
Goal: Summable f ↔ ∀ ε > 0, ∃ s, ∀ (t : Finset ι), Disjoint t s → ‖∑ i ∈ t, f i‖ < ε
-- cauchySeq_finset_of_norm_bounded_eventually
theorem cauchySeq_finset_of_norm_bounded_eventually {f : ι → E} {g : ι → ℝ} (hg : Summable g)
    (h : ∀ᶠ i in cofinite, ‖f i‖ ≤ g i) : CauchySeq fun s => ∑ i ∈ s, f i 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] {f : ι → E} {g : ι → ℝ} (hg : Summable g) (h : ∀ᶠ (i : ι) in Filter.cofinite, ‖f i‖ ≤ g i)
Goal: CauchySeq fun s => ∑ i ∈ s, f i
-- cauchySeq_finset_of_norm_bounded
theorem cauchySeq_finset_of_norm_bounded {f : ι → E} (g : ι → ℝ) (hg : Summable g)
    (h : ∀ i, ‖f i‖ ≤ g i) : CauchySeq fun s : Finset ι => ∑ i ∈ s, f i 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] {f : ι → E} (g : ι → ℝ) (hg : Summable g) (h : ∀ (i : ι), ‖f i‖ ≤ g i)
Goal: CauchySeq fun s => ∑ i ∈ s, f i
-- cauchySeq_range_of_norm_bounded
theorem cauchySeq_range_of_norm_bounded {f : ℕ → E} (g : ℕ → ℝ)
    (hg : CauchySeq fun n => ∑ i ∈ range n, g i) (hf : ∀ i, ‖f i‖ ≤ g i) :
    CauchySeq fun n => ∑ i ∈ range n, f i 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] {f : ℕ → E} (g : ℕ → ℝ) (hg : CauchySeq fun n => ∑ i ∈ Finset.range n, g i) (hf : ∀ (i : ℕ), ‖f i‖ ≤ g i)
Goal: CauchySeq fun n => ∑ i ∈ Finset.range n, f i
-- cauchySeq_finset_of_summable_norm
theorem cauchySeq_finset_of_summable_norm {f : ι → E} (hf : Summable fun a => ‖f a‖) :
    CauchySeq fun s : Finset ι => ∑ a ∈ s, f a 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] {f : ι → E} (hf : Summable fun a => ‖f a‖)
Goal: CauchySeq fun s => ∑ a ∈ s, f a
-- hasSum_of_subseq_of_summable
theorem hasSum_of_subseq_of_summable {f : ι → E} (hf : Summable fun a => ‖f a‖) {s : α → Finset ι}
    {p : Filter α} [NeBot p] (hs : Tendsto s p atTop) {a : E}
    (ha : Tendsto (fun b => ∑ i ∈ s b, f i) p (𝓝 a)) : HasSum f a 
Arguments: {ι : Type u_1} {α : Type u_2} {E : Type u_3} [SeminormedAddCommGroup E] {f : ι → E} (hf : Summable fun a => ‖f a‖) {s : α → Finset ι} {p : Filter α} [p.NeBot] (hs : Filter.Tendsto s p Filter.atTop) {a : E} (ha : Filter.Tendsto (fun b => ∑ i ∈ s b, f i) p (𝓝 a))
Goal: HasSum f a
-- hasSum_iff_tendsto_nat_of_summable_norm
theorem hasSum_iff_tendsto_nat_of_summable_norm {f : ℕ → E} {a : E} (hf : Summable fun i => ‖f i‖) :
    HasSum f a ↔ Tendsto (fun n : ℕ => ∑ i ∈ range n, f i) atTop (𝓝 a) 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] {f : ℕ → E} {a : E} (hf : Summable fun i => ‖f i‖)
Goal: HasSum f a ↔ Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop (𝓝 a)
-- Summable.of_norm_bounded
theorem Summable.of_norm_bounded [CompleteSpace E] {f : ι → E} (g : ι → ℝ) (hg : Summable g)
    (h : ∀ i, ‖f i‖ ≤ g i) : Summable f 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] [CompleteSpace E] {f : ι → E} (g : ι → ℝ) (hg : Summable g) (h : ∀ (i : ι), ‖f i‖ ≤ g i)
Goal: Summable f
-- HasSum.norm_le_of_bounded
theorem HasSum.norm_le_of_bounded {f : ι → E} {g : ι → ℝ} {a : E} {b : ℝ} (hf : HasSum f a)
    (hg : HasSum g b) (h : ∀ i, ‖f i‖ ≤ g i) : ‖a‖ ≤ b 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] {f : ι → E} {g : ι → ℝ} {a : E} {b : ℝ} (hf : HasSum f a) (hg : HasSum g b) (h : ∀ (i : ι), ‖f i‖ ≤ g i)
Goal: ‖a‖ ≤ b
-- tsum_of_norm_bounded
theorem tsum_of_norm_bounded {f : ι → E} {g : ι → ℝ} {a : ℝ} (hg : HasSum g a)
    (h : ∀ i, ‖f i‖ ≤ g i) : ‖∑' i : ι, f i‖ ≤ a 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] {f : ι → E} {g : ι → ℝ} {a : ℝ} (hg : HasSum g a) (h : ∀ (i : ι), ‖f i‖ ≤ g i)
Goal: ‖∑' (i : ι), f i‖ ≤ a
-- norm_tsum_le_tsum_norm
theorem norm_tsum_le_tsum_norm {f : ι → E} (hf : Summable fun i => ‖f i‖) :
    ‖∑' i, f i‖ ≤ ∑' i, ‖f i‖ 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] {f : ι → E} (hf : Summable fun i => ‖f i‖)
Goal: ‖∑' (i : ι), f i‖ ≤ ∑' (i : ι), ‖f i‖
-- tsum_of_nnnorm_bounded
theorem tsum_of_nnnorm_bounded {f : ι → E} {g : ι → ℝ≥0} {a : ℝ≥0} (hg : HasSum g a)
    (h : ∀ i, ‖f i‖₊ ≤ g i) : ‖∑' i : ι, f i‖₊ ≤ a 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] {f : ι → E} {g : ι → ℝ≥0} {a : ℝ≥0} (hg : HasSum g a) (h : ∀ (i : ι), ‖f i‖₊ ≤ g i)
Goal: ‖∑' (i : ι), f i‖₊ ≤ a
-- nnnorm_tsum_le
theorem nnnorm_tsum_le {f : ι → E} (hf : Summable fun i => ‖f i‖₊) : ‖∑' i, f i‖₊ ≤ ∑' i, ‖f i‖₊ 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] {f : ι → E} (hf : Summable fun i => ‖f i‖₊)
Goal: ‖∑' (i : ι), f i‖₊ ≤ ∑' (i : ι), ‖f i‖₊
-- Summable.of_norm_bounded_eventually
theorem Summable.of_norm_bounded_eventually {f : ι → E} (g : ι → ℝ) (hg : Summable g)
    (h : ∀ᶠ i in cofinite, ‖f i‖ ≤ g i) : Summable f 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] [CompleteSpace E] {f : ι → E} (g : ι → ℝ) (hg : Summable g) (h : ∀ᶠ (i : ι) in Filter.cofinite, ‖f i‖ ≤ g i)
Goal: Summable f
-- Summable.of_norm_bounded_eventually_nat
theorem Summable.of_norm_bounded_eventually_nat {f : ℕ → E} (g : ℕ → ℝ) (hg : Summable g)
    (h : ∀ᶠ i in atTop, ‖f i‖ ≤ g i) : Summable f 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] [CompleteSpace E] {f : ℕ → E} (g : ℕ → ℝ) (hg : Summable g) (h : ∀ᶠ (i : ℕ) in Filter.atTop, ‖f i‖ ≤ g i)
Goal: Summable f
-- Summable.of_nnnorm_bounded
theorem Summable.of_nnnorm_bounded {f : ι → E} (g : ι → ℝ≥0) (hg : Summable g)
    (h : ∀ i, ‖f i‖₊ ≤ g i) : Summable f 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] [CompleteSpace E] {f : ι → E} (g : ι → ℝ≥0) (hg : Summable g) (h : ∀ (i : ι), ‖f i‖₊ ≤ g i)
Goal: Summable f
-- Summable.of_norm
theorem Summable.of_norm {f : ι → E} (hf : Summable fun a => ‖f a‖) : Summable f 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] [CompleteSpace E] {f : ι → E} (hf : Summable fun a => ‖f a‖)
Goal: Summable f
-- Summable.of_nnnorm
theorem Summable.of_nnnorm {f : ι → E} (hf : Summable fun a => ‖f a‖₊) : Summable f 
Arguments: {ι : Type u_1} {E : Type u_3} [SeminormedAddCommGroup E] [CompleteSpace E] {f : ι → E} (hf : Summable fun a => ‖f a‖₊)
Goal: Summable f
-- AddSubgroup.quotient_norm_eq
theorem AddSubgroup.quotient_norm_eq {S : AddSubgroup M} (x : M ⧸ S) :
    ‖x‖ = sInf (norm '' { m : M | (m : M ⧸ S) = x }) 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {S : AddSubgroup M} (x : M ⧸ S)
Goal: ‖x‖ = sInf (norm '' {m | ↑m = x})
-- QuotientAddGroup.norm_eq_infDist
theorem QuotientAddGroup.norm_eq_infDist {S : AddSubgroup M} (x : M ⧸ S) :
    ‖x‖ = infDist 0 { m : M | (m : M ⧸ S) = x } 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {S : AddSubgroup M} (x : M ⧸ S)
Goal: ‖x‖ = Metric.infDist 0 {m | ↑m = x}
-- QuotientAddGroup.norm_mk
theorem QuotientAddGroup.norm_mk {S : AddSubgroup M} (x : M) :
    ‖(x : M ⧸ S)‖ = infDist x S 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {S : AddSubgroup M} (x : M)
Goal: ‖↑x‖ = Metric.infDist x ↑S
-- image_norm_nonempty
theorem image_norm_nonempty {S : AddSubgroup M} (x : M ⧸ S) :
    (norm '' { m | mk' S m = x }).Nonempty 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {S : AddSubgroup M} (x : M ⧸ S)
Goal: (norm '' {m | (QuotientAddGroup.mk' S) m = x}).Nonempty
-- bddBelow_image_norm
theorem bddBelow_image_norm (s : Set M) : BddBelow (norm '' s) 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (s : Set M)
Goal: BddBelow (norm '' s)
-- isGLB_quotient_norm
theorem isGLB_quotient_norm {S : AddSubgroup M} (x : M ⧸ S) :
    IsGLB (norm '' { m | mk' S m = x }) (‖x‖) 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {S : AddSubgroup M} (x : M ⧸ S)
Goal: IsGLB (norm '' {m | (QuotientAddGroup.mk' S) m = x}) ‖x‖
-- quotient_norm_neg
theorem quotient_norm_neg {S : AddSubgroup M} (x : M ⧸ S) : ‖-x‖ = ‖x‖ 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {S : AddSubgroup M} (x : M ⧸ S)
Goal: ‖-x‖ = ‖x‖
-- quotient_norm_sub_rev
theorem quotient_norm_sub_rev {S : AddSubgroup M} (x y : M ⧸ S) : ‖x - y‖ = ‖y - x‖ 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {S : AddSubgroup M} (x y : M ⧸ S)
Goal: ‖x - y‖ = ‖y - x‖
-- quotient_norm_mk_le
theorem quotient_norm_mk_le (S : AddSubgroup M) (m : M) : ‖mk' S m‖ ≤ ‖m‖ 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M) (m : M)
Goal: ‖(QuotientAddGroup.mk' S) m‖ ≤ ‖m‖
-- quotient_norm_mk_le'
theorem quotient_norm_mk_le' (S : AddSubgroup M) (m : M) : ‖(m : M ⧸ S)‖ ≤ ‖m‖ 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M) (m : M)
Goal: ‖↑m‖ ≤ ‖m‖
-- quotient_norm_mk_eq
theorem quotient_norm_mk_eq (S : AddSubgroup M) (m : M) :
    ‖mk' S m‖ = sInf ((‖m + ·‖) '' S) 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M) (m : M)
Goal: ‖(QuotientAddGroup.mk' S) m‖ = sInf ((fun x => ‖m + x‖) '' ↑S)
-- quotient_norm_nonneg
theorem quotient_norm_nonneg (S : AddSubgroup M) (x : M ⧸ S) : 0 ≤ ‖x‖ 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M) (x : M ⧸ S)
Goal: 0 ≤ ‖x‖
-- norm_mk_nonneg
theorem norm_mk_nonneg (S : AddSubgroup M) (m : M) : 0 ≤ ‖mk' S m‖ 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M) (m : M)
Goal: 0 ≤ ‖(QuotientAddGroup.mk' S) m‖
-- quotient_norm_eq_zero_iff
theorem quotient_norm_eq_zero_iff (S : AddSubgroup M) (m : M) :
    ‖mk' S m‖ = 0 ↔ m ∈ closure (S : Set M) 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M) (m : M)
Goal: ‖(QuotientAddGroup.mk' S) m‖ = 0 ↔ m ∈ closure ↑S
-- QuotientAddGroup.norm_lt_iff
theorem QuotientAddGroup.norm_lt_iff {S : AddSubgroup M} {x : M ⧸ S} {r : ℝ} :
    ‖x‖ < r ↔ ∃ m : M, ↑m = x ∧ ‖m‖ < r 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {S : AddSubgroup M} {x : M ⧸ S} {r : ℝ}
Goal: ‖x‖ < r ↔ ∃ m, ↑m = x ∧ ‖m‖ < r
-- norm_mk_lt
theorem norm_mk_lt {S : AddSubgroup M} (x : M ⧸ S) {ε : ℝ} (hε : 0 < ε) :
    ∃ m : M, mk' S m = x ∧ ‖m‖ < ‖x‖ + ε 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] {S : AddSubgroup M} (x : M ⧸ S) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ m, (QuotientAddGroup.mk' S) m = x ∧ ‖m‖ < ‖x‖ + ε
-- norm_mk_lt'
theorem norm_mk_lt' (S : AddSubgroup M) (m : M) {ε : ℝ} (hε : 0 < ε) :
    ∃ s ∈ S, ‖m + s‖ < ‖mk' S m‖ + ε 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M) (m : M) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ s ∈ S, ‖m + s‖ < ‖(QuotientAddGroup.mk' S) m‖ + ε
-- quotient_norm_add_le
theorem quotient_norm_add_le (S : AddSubgroup M) (x y : M ⧸ S) : ‖x + y‖ ≤ ‖x‖ + ‖y‖ 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M) (x y : M ⧸ S)
Goal: ‖x + y‖ ≤ ‖x‖ + ‖y‖
-- norm_mk_zero
theorem norm_mk_zero (S : AddSubgroup M) : ‖(0 : M ⧸ S)‖ = 0 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M)
Goal: ‖0‖ = 0
-- norm_mk_eq_zero
theorem norm_mk_eq_zero (S : AddSubgroup M) (hS : IsClosed (S : Set M)) (m : M)
    (h : ‖mk' S m‖ = 0) : m ∈ S 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M) (hS : IsClosed ↑S) (m : M) (h : ‖(QuotientAddGroup.mk' S) m‖ = 0)
Goal: m ∈ S
-- quotient_nhd_basis
theorem quotient_nhd_basis (S : AddSubgroup M) :
    (𝓝 (0 : M ⧸ S)).HasBasis (fun ε ↦ 0 < ε) fun ε ↦ { x | ‖x‖ < ε } 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M)
Goal: (𝓝 0).HasBasis (fun ε => 0 < ε) fun ε => {x | ‖x‖ < ε}
-- AddSubgroup.normedMk.apply
@[simp]
theorem normedMk.apply (S : AddSubgroup M) (m : M) : normedMk S m = QuotientAddGroup.mk' S m 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M) (m : M)
Goal: S.normedMk m = (QuotientAddGroup.mk' S) m
-- AddSubgroup.surjective_normedMk
theorem surjective_normedMk (S : AddSubgroup M) : Function.Surjective (normedMk S) 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M)
Goal: Function.Surjective ⇑S.normedMk
-- AddSubgroup.ker_normedMk
theorem ker_normedMk (S : AddSubgroup M) : S.normedMk.ker = S 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M)
Goal: S.normedMk.ker = S
-- AddSubgroup.norm_normedMk_le
theorem norm_normedMk_le (S : AddSubgroup M) : ‖S.normedMk‖ ≤ 1 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M)
Goal: ‖S.normedMk‖ ≤ 1
-- QuotientAddGroup.norm_lift_apply_le
theorem _root_.QuotientAddGroup.norm_lift_apply_le {S : AddSubgroup M} (f : NormedAddGroupHom M N)
    (hf : ∀ x ∈ S, f x = 0) (x : M ⧸ S) : ‖lift S f.toAddMonoidHom hf x‖ ≤ ‖f‖ * ‖x‖ 
Arguments: {M : Type u_1} {N : Type u_2} [SeminormedAddCommGroup M] [SeminormedAddCommGroup N] {S : AddSubgroup M} (f : NormedAddGroupHom M N) (hf : ∀ x ∈ S, f x = 0) (x : M ⧸ S)
Goal: ‖(QuotientAddGroup.lift S f.toAddMonoidHom hf) x‖ ≤ ‖f‖ * ‖x‖
-- AddSubgroup.norm_normedMk
theorem norm_normedMk (S : AddSubgroup M) (h : (S.topologicalClosure : Set M) ≠ univ) :
    ‖S.normedMk‖ = 1 
Arguments: {M : Type u_1} [SeminormedAddCommGroup M] (S : AddSubgroup M) (h : ↑S.topologicalClosure ≠ Set.univ)
Goal: ‖S.normedMk‖ = 1