ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsCompactOperator.codRestrict
theorem IsCompactOperator.codRestrict {f : M₁ → M₂} (hf : IsCompactOperator f) {V : Submodule R₂ M₂}
    (hV : ∀ x, f x ∈ V) (h_closed : IsClosed (V : Set M₂)) :
    IsCompactOperator (Set.codRestrict f V hV) 
Arguments: {R₂ : Type u_2} [Semiring R₂] {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [TopologicalSpace M₂] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R₂ M₂] {f : M₁ → M₂} (hf : IsCompactOperator f) {V : Submodule R₂ M₂} (hV : ∀ (x : M₁), f x ∈ V) (h_closed : IsClosed ↑V)
Goal: IsCompactOperator (Set.codRestrict f (↑V) hV)
-- IsCompactOperator.restrict
theorem IsCompactOperator.restrict {f : M₁ →ₗ[R₁] M₁} (hf : IsCompactOperator f)
    {V : Submodule R₁ M₁} (hV : ∀ v ∈ V, f v ∈ V) (h_closed : IsClosed (V : Set M₁)) :
    IsCompactOperator (f.restrict hV) 
Arguments: {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] {f : M₁ →ₗ[R₁] M₁} (hf : IsCompactOperator ⇑f) {V : Submodule R₁ M₁} (hV : ∀ v ∈ V, f v ∈ V) (h_closed : IsClosed ↑V)
Goal: IsCompactOperator ⇑(f.restrict hV)
-- IsCompactOperator.restrict'
theorem IsCompactOperator.restrict' [T0Space M₂] {f : M₂ →ₗ[R₂] M₂}
    (hf : IsCompactOperator f) {V : Submodule R₂ M₂} (hV : ∀ v ∈ V, f v ∈ V)
    [hcomplete : CompleteSpace V] : IsCompactOperator (f.restrict hV) 
Arguments: {R₂ : Type u_2} [Semiring R₂] {M₂ : Type u_5} [UniformSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] [T0Space M₂] {f : M₂ →ₗ[R₂] M₂} (hf : IsCompactOperator ⇑f) {V : Submodule R₂ M₂} (hV : ∀ v ∈ V, f v ∈ V) [hcomplete : CompleteSpace ↥V]
Goal: IsCompactOperator ⇑(f.restrict hV)
-- IsCompactOperator.continuous
@[continuity]
theorem IsCompactOperator.continuous {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator f) :
    Continuous f 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} [RingHomIsometric σ₁₂] {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommGroup M₁] [TopologicalSpace M₂] [AddCommGroup M₂] [Module 𝕜₁ M₁] [Module 𝕜₂ M₂] [TopologicalAddGroup M₁] [ContinuousConstSMul 𝕜₁ M₁] [TopologicalAddGroup M₂] [ContinuousSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f)
Goal: Continuous ⇑f
-- ContinuousLinearMap.mkOfIsCompactOperator_to_linearMap
@[simp]
theorem ContinuousLinearMap.mkOfIsCompactOperator_to_linearMap {f : M₁ →ₛₗ[σ₁₂] M₂}
    (hf : IsCompactOperator f) :
    (ContinuousLinearMap.mkOfIsCompactOperator hf : M₁ →ₛₗ[σ₁₂] M₂) = f 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} [RingHomIsometric σ₁₂] {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommGroup M₁] [TopologicalSpace M₂] [AddCommGroup M₂] [Module 𝕜₁ M₁] [Module 𝕜₂ M₂] [TopologicalAddGroup M₁] [ContinuousConstSMul 𝕜₁ M₁] [TopologicalAddGroup M₂] [ContinuousSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f)
Goal: ↑(ContinuousLinearMap.mkOfIsCompactOperator hf) = f
-- ContinuousLinearMap.coe_mkOfIsCompactOperator
@[simp]
theorem ContinuousLinearMap.coe_mkOfIsCompactOperator {f : M₁ →ₛₗ[σ₁₂] M₂}
    (hf : IsCompactOperator f) : (ContinuousLinearMap.mkOfIsCompactOperator hf : M₁ → M₂) = f 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} [RingHomIsometric σ₁₂] {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommGroup M₁] [TopologicalSpace M₂] [AddCommGroup M₂] [Module 𝕜₁ M₁] [Module 𝕜₂ M₂] [TopologicalAddGroup M₁] [ContinuousConstSMul 𝕜₁ M₁] [TopologicalAddGroup M₂] [ContinuousSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f)
Goal: ⇑(ContinuousLinearMap.mkOfIsCompactOperator hf) = ⇑f
-- ContinuousLinearMap.mkOfIsCompactOperator_mem_compactOperator
theorem ContinuousLinearMap.mkOfIsCompactOperator_mem_compactOperator {f : M₁ →ₛₗ[σ₁₂] M₂}
    (hf : IsCompactOperator f) :
    ContinuousLinearMap.mkOfIsCompactOperator hf ∈ compactOperator σ₁₂ M₁ M₂ 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} [RingHomIsometric σ₁₂] {M₁ : Type u_3} {M₂ : Type u_4} [TopologicalSpace M₁] [AddCommGroup M₁] [TopologicalSpace M₂] [AddCommGroup M₂] [Module 𝕜₁ M₁] [Module 𝕜₂ M₂] [TopologicalAddGroup M₁] [ContinuousConstSMul 𝕜₁ M₁] [TopologicalAddGroup M₂] [ContinuousSMul 𝕜₂ M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator ⇑f)
Goal: ContinuousLinearMap.mkOfIsCompactOperator hf ∈ compactOperator σ₁₂ M₁ M₂
-- isClosed_setOf_isCompactOperator
theorem isClosed_setOf_isCompactOperator {𝕜₁ 𝕜₂ : Type*} [NontriviallyNormedField 𝕜₁]
    [NormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ M₂ : Type*} [SeminormedAddCommGroup M₁]
    [AddCommGroup M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [UniformSpace M₂] [UniformAddGroup M₂]
    [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] [CompleteSpace M₂] :
    IsClosed { f : M₁ →SL[σ₁₂] M₂ | IsCompactOperator f } 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [NormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [AddCommGroup M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [UniformSpace M₂] [UniformAddGroup M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] [CompleteSpace M₂]
Goal: IsClosed {f | IsCompactOperator ⇑f}
-- compactOperator_topologicalClosure
theorem compactOperator_topologicalClosure {𝕜₁ 𝕜₂ : Type*} [NontriviallyNormedField 𝕜₁]
    [NormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ M₂ : Type*} [SeminormedAddCommGroup M₁]
    [AddCommGroup M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [UniformSpace M₂] [UniformAddGroup M₂]
    [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] [CompleteSpace M₂] :
    (compactOperator σ₁₂ M₁ M₂).topologicalClosure = compactOperator σ₁₂ M₁ M₂ 
Arguments: {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₁] [NormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_3} {M₂ : Type u_4} [SeminormedAddCommGroup M₁] [AddCommGroup M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [UniformSpace M₂] [UniformAddGroup M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] [CompleteSpace M₂]
Goal: (compactOperator σ₁₂ M₁ M₂).topologicalClosure = compactOperator σ₁₂ M₁ M₂
-- isCompactOperator_of_tendsto
theorem isCompactOperator_of_tendsto {ι 𝕜₁ 𝕜₂ : Type*} [NontriviallyNormedField 𝕜₁]
    [NormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ M₂ : Type*} [SeminormedAddCommGroup M₁]
    [AddCommGroup M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [UniformSpace M₂] [UniformAddGroup M₂]
    [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] [CompleteSpace M₂] {l : Filter ι} [l.NeBot]
    {F : ι → M₁ →SL[σ₁₂] M₂} {f : M₁ →SL[σ₁₂] M₂} (hf : Tendsto F l (𝓝 f))
    (hF : ∀ᶠ i in l, IsCompactOperator (F i)) : IsCompactOperator f 
Arguments: {ι : Type u_1} {𝕜₁ : Type u_2} {𝕜₂ : Type u_3} [NontriviallyNormedField 𝕜₁] [NormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ : Type u_4} {M₂ : Type u_5} [SeminormedAddCommGroup M₁] [AddCommGroup M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [UniformSpace M₂] [UniformAddGroup M₂] [ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] [CompleteSpace M₂] {l : Filter ι} [l.NeBot] {F : ι → M₁ →SL[σ₁₂] M₂} {f : M₁ →SL[σ₁₂] M₂} (hf : Filter.Tendsto F l (𝓝 f)) (hF : ∀ᶠ (i : ι) in l, IsCompactOperator ⇑(F i))
Goal: IsCompactOperator ⇑f
-- Rat.norm_cast_real
@[norm_cast, simp 1001]
-- Porting note: increase priority to prevent the left-hand side from simplifying
theorem norm_cast_real (r : ℚ) : ‖(r : ℝ)‖ = ‖r‖ 
Arguments: (r : ℚ)
Goal: ‖↑r‖ = ‖r‖
-- Int.norm_cast_rat
@[norm_cast, simp]
theorem _root_.Int.norm_cast_rat (m : ℤ) : ‖(m : ℚ)‖ = ‖m‖ 
Arguments: (m : ℤ)
Goal: ‖↑m‖ = ‖m‖
-- fourierIntegral_half_period_translate
theorem fourierIntegral_half_period_translate {w : V} (hw : w ≠ 0) :
    (∫ v : V, 𝐞 (-⟪v, w⟫) • f (v + i w)) = -∫ v : V, 𝐞 (-⟪v, w⟫) • f v 
Arguments: {E : Type u_1} {V : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → E} [NormedAddCommGroup V] [MeasurableSpace V] [BorelSpace V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] {w : V} (hw : w ≠ 0)
Goal: ∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f (v + (fun w => (1 / (2 * ‖w‖ ^ 2)) • w) w) = -∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v
-- fourierIntegral_eq_half_sub_half_period_translate
theorem fourierIntegral_eq_half_sub_half_period_translate {w : V} (hw : w ≠ 0)
    (hf : Integrable f) :
    ∫ v : V, 𝐞 (-⟪v, w⟫) • f v = (1 / (2 : ℂ)) • ∫ v : V, 𝐞 (-⟪v, w⟫) • (f v - f (v + i w)) 
Arguments: {E : Type u_1} {V : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → E} [NormedAddCommGroup V] [MeasurableSpace V] [BorelSpace V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] {w : V} (hw : w ≠ 0) (hf : MeasureTheory.Integrable f MeasureTheory.volume)
Goal: ∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v =
  (1 / 2) • ∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • (f v - f (v + (fun w => (1 / (2 * ‖w‖ ^ 2)) • w) w))
-- tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support
theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
    (hf2 : HasCompactSupport f) :
    Tendsto (fun w : V => ∫ v : V, 𝐞 (-⟪v, w⟫) • f v) (cocompact V) (𝓝 0) 
Arguments: {E : Type u_1} {V : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → E} [NormedAddCommGroup V] [MeasurableSpace V] [BorelSpace V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] (hf1 : Continuous f) (hf2 : HasCompactSupport f)
Goal: Filter.Tendsto (fun w => ∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v) (Filter.cocompact V) (𝓝 0)
-- tendsto_integral_exp_inner_smul_cocompact
theorem tendsto_integral_exp_inner_smul_cocompact :
    Tendsto (fun w : V => ∫ v, 𝐞 (-⟪v, w⟫) • f v) (cocompact V) (𝓝 0) 
Arguments: {E : Type u_1} {V : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : V → E) [NormedAddCommGroup V] [MeasurableSpace V] [BorelSpace V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V]
Goal: Filter.Tendsto (fun w => ∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v) (Filter.cocompact V) (𝓝 0)
-- Real.tendsto_integral_exp_smul_cocompact
theorem Real.tendsto_integral_exp_smul_cocompact (f : ℝ → E) :
    Tendsto (fun w : ℝ => ∫ v : ℝ, 𝐞 (-(v * w)) • f v) (cocompact ℝ) (𝓝 0) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℝ → E)
Goal: Filter.Tendsto (fun w => ∫ (v : ℝ), 𝐞 (-(v * w)) • f v) (Filter.cocompact ℝ) (𝓝 0)
-- Real.zero_at_infty_fourierIntegral
theorem Real.zero_at_infty_fourierIntegral (f : ℝ → E) : Tendsto (𝓕 f) (cocompact ℝ) (𝓝 0) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℝ → E)
Goal: Filter.Tendsto (𝓕 f) (Filter.cocompact ℝ) (𝓝 0)
-- tendsto_integral_exp_smul_cocompact_of_inner_product
theorem tendsto_integral_exp_smul_cocompact_of_inner_product (μ : Measure V) [μ.IsAddHaarMeasure] :
    Tendsto (fun w : V →L[ℝ] ℝ => ∫ v, 𝐞 (-w v) • f v ∂μ) (cocompact (V →L[ℝ] ℝ)) (𝓝 0) 
Arguments: {E : Type u_1} {V : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : V → E) [NormedAddCommGroup V] [MeasurableSpace V] [BorelSpace V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] (μ : MeasureTheory.Measure V) [μ.IsAddHaarMeasure]
Goal: Filter.Tendsto (fun w => ∫ (v : V), 𝐞 (-w v) • f v ∂μ) (Filter.cocompact (V →L[ℝ] ℝ)) (𝓝 0)
-- tendsto_integral_exp_smul_cocompact
theorem tendsto_integral_exp_smul_cocompact (μ : Measure V) [μ.IsAddHaarMeasure] :
    Tendsto (fun w : V →L[ℝ] ℝ => ∫ v, 𝐞 (-w v) • f v ∂μ) (cocompact (V →L[ℝ] ℝ)) (𝓝 0) 
Arguments: {E : Type u_1} {V : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : V → E) [AddCommGroup V] [TopologicalSpace V] [TopologicalAddGroup V] [T2Space V] [MeasurableSpace V] [BorelSpace V] [Module ℝ V] [ContinuousSMul ℝ V] [FiniteDimensional ℝ V] (μ : MeasureTheory.Measure V) [μ.IsAddHaarMeasure]
Goal: Filter.Tendsto (fun w => ∫ (v : V), 𝐞 (-w v) • f v ∂μ) (Filter.cocompact (V →L[ℝ] ℝ)) (𝓝 0)
-- Real.zero_at_infty_vector_fourierIntegral
theorem Real.zero_at_infty_vector_fourierIntegral (μ : Measure V) [μ.IsAddHaarMeasure] :
    Tendsto (VectorFourier.fourierIntegral 𝐞 μ (topDualPairing ℝ V).flip f) (cocompact (V →L[ℝ] ℝ))
      (𝓝 0) 
Arguments: {E : Type u_1} {V : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : V → E) [AddCommGroup V] [TopologicalSpace V] [TopologicalAddGroup V] [T2Space V] [MeasurableSpace V] [BorelSpace V] [Module ℝ V] [ContinuousSMul ℝ V] [FiniteDimensional ℝ V] (μ : MeasureTheory.Measure V) [μ.IsAddHaarMeasure]
Goal: Filter.Tendsto (VectorFourier.fourierIntegral 𝐞 μ (topDualPairing ℝ V).flip f) (Filter.cocompact (V →L[ℝ] ℝ)) (𝓝 0)
-- tendsto_tsum_of_dominated_convergence
lemma tendsto_tsum_of_dominated_convergence {α β G : Type*} {𝓕 : Filter α}
    [NormedAddCommGroup G] [CompleteSpace G]
    {f : α → β → G} {g : β → G} {bound : β → ℝ} (h_sum : Summable bound)
    (hab : ∀ k : β, Tendsto (f · k) 𝓕 (𝓝 (g k)))
    (h_bound : ∀ᶠ n in 𝓕, ∀ k, ‖f n k‖ ≤ bound k) :
    Tendsto (∑' k, f · k) 𝓕 (𝓝 (∑' k, g k)) 
Arguments: {α : Type u_1} {β : Type u_2} {G : Type u_3} {𝓕 : Filter α} [NormedAddCommGroup G] [CompleteSpace G] {f : α → β → G} {g : β → G} {bound : β → ℝ} (h_sum : Summable bound) (hab : ∀ (k : β), Filter.Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))) (h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k)
Goal: Filter.Tendsto (fun x => ∑' (k : β), f x k) 𝓕 (𝓝 (∑' (k : β), g k))
-- CocompactMapClass.norm_le
theorem CocompactMapClass.norm_le [FunLike 𝓕 E F] [CocompactMapClass 𝓕 E F] (ε : ℝ) :
    ∃ r : ℝ, ∀ x : E, r < ‖x‖ → ε < ‖f x‖ 
Arguments: {E : Type u_2} {F : Type u_3} {𝓕 : Type u_4} [NormedAddCommGroup E] [NormedAddCommGroup F] [ProperSpace F] {f : 𝓕} [FunLike 𝓕 E F] [CocompactMapClass 𝓕 E F] (ε : ℝ)
Goal: ∃ r, ∀ (x : E), r < ‖x‖ → ε < ‖f x‖
-- Filter.tendsto_cocompact_cocompact_of_norm
theorem Filter.tendsto_cocompact_cocompact_of_norm {f : E → F}
    (h : ∀ ε : ℝ, ∃ r : ℝ, ∀ x : E, r < ‖x‖ → ε < ‖f x‖) :
    Tendsto f (cocompact E) (cocompact F) 
Arguments: {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedAddCommGroup F] [ProperSpace E] {f : E → F} (h : ∀ (ε : ℝ), ∃ r, ∀ (x : E), r < ‖x‖ → ε < ‖f x‖)
Goal: Filter.Tendsto f (Filter.cocompact E) (Filter.cocompact F)
-- ContinuousMapClass.toCocompactMapClass_of_norm
theorem ContinuousMapClass.toCocompactMapClass_of_norm [FunLike 𝓕 E F] [ContinuousMapClass 𝓕 E F]
    (h : ∀ (f : 𝓕) (ε : ℝ), ∃ r : ℝ, ∀ x : E, r < ‖x‖ → ε < ‖f x‖) :
    CocompactMapClass 𝓕 E F 
Arguments: {E : Type u_2} {F : Type u_3} {𝓕 : Type u_4} [NormedAddCommGroup E] [NormedAddCommGroup F] [ProperSpace E] [FunLike 𝓕 E F] [ContinuousMapClass 𝓕 E F] (h : ∀ (f : 𝓕) (ε : ℝ), ∃ r, ∀ (x : E), r < ‖x‖ → ε < ‖f x‖)
Goal: CocompactMapClass 𝓕 E F
-- NormedAddGroupHom.completion_def
theorem NormedAddGroupHom.completion_def (f : NormedAddGroupHom G H) (x : Completion G) :
    f.completion x = Completion.map f x 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H) (x : UniformSpace.Completion G)
Goal: f.completion x = UniformSpace.Completion.map (⇑f) x
-- NormedAddGroupHom.completion_coe_to_fun
@[simp]
theorem NormedAddGroupHom.completion_coe_to_fun (f : NormedAddGroupHom G H) :
    (f.completion : Completion G → Completion H) = Completion.map f 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H)
Goal: ⇑f.completion = UniformSpace.Completion.map ⇑f
-- NormedAddGroupHom.completion_coe
theorem NormedAddGroupHom.completion_coe (f : NormedAddGroupHom G H) (g : G) :
    f.completion g = f g 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H) (g : G)
Goal: f.completion (↑G g) = ↑H (f g)
-- NormedAddGroupHom.completion_coe'
@[simp]
theorem NormedAddGroupHom.completion_coe' (f : NormedAddGroupHom G H) (g : G) :
    Completion.map f g = f g 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H) (g : G)
Goal: UniformSpace.Completion.map (⇑f) (↑G g) = ↑H (f g)
-- normedAddGroupHomCompletionHom_apply
@[simps]
def normedAddGroupHomCompletionHom :
    NormedAddGroupHom G H →+ NormedAddGroupHom (Completion G) (Completion H) 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H)
Goal: normedAddGroupHomCompletionHom f = f.completion
-- NormedAddGroupHom.completion_id
@[simp]
theorem NormedAddGroupHom.completion_id :
    (NormedAddGroupHom.id G).completion = NormedAddGroupHom.id (Completion G) 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G]
Goal: (NormedAddGroupHom.id G).completion = NormedAddGroupHom.id (UniformSpace.Completion G)
-- NormedAddGroupHom.completion_comp
theorem NormedAddGroupHom.completion_comp (f : NormedAddGroupHom G H) (g : NormedAddGroupHom H K) :
    g.completion.comp f.completion = (g.comp f).completion 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] {K : Type u_3} [SeminormedAddCommGroup K] (f : NormedAddGroupHom G H) (g : NormedAddGroupHom H K)
Goal: g.completion.comp f.completion = (g.comp f).completion
-- NormedAddGroupHom.completion_neg
theorem NormedAddGroupHom.completion_neg (f : NormedAddGroupHom G H) :
    (-f).completion = -f.completion 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H)
Goal: (-f).completion = -f.completion
-- NormedAddGroupHom.completion_add
theorem NormedAddGroupHom.completion_add (f g : NormedAddGroupHom G H) :
    (f + g).completion = f.completion + g.completion 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f g : NormedAddGroupHom G H)
Goal: (f + g).completion = f.completion + g.completion
-- NormedAddGroupHom.completion_sub
theorem NormedAddGroupHom.completion_sub (f g : NormedAddGroupHom G H) :
    (f - g).completion = f.completion - g.completion 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f g : NormedAddGroupHom G H)
Goal: (f - g).completion = f.completion - g.completion
-- NormedAddGroupHom.zero_completion
@[simp]
theorem NormedAddGroupHom.zero_completion : (0 : NormedAddGroupHom G H).completion = 0 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H]
Goal: NormedAddGroupHom.completion 0 = 0
-- NormedAddCommGroup.toCompl_apply
@[simps] -- Porting note: added `@[simps]`
def NormedAddCommGroup.toCompl : NormedAddGroupHom G (Completion G) 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G]
Goal: ∀ (a : G), NormedAddCommGroup.toCompl a = ↑G a
-- NormedAddCommGroup.norm_toCompl
theorem NormedAddCommGroup.norm_toCompl (x : G) : ‖toCompl x‖ = ‖x‖ 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] (x : G)
Goal: ‖NormedAddCommGroup.toCompl x‖ = ‖x‖
-- NormedAddCommGroup.denseRange_toCompl
theorem NormedAddCommGroup.denseRange_toCompl : DenseRange (toCompl : G → Completion G) 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G]
Goal: DenseRange ⇑NormedAddCommGroup.toCompl
-- NormedAddGroupHom.completion_toCompl
@[simp]
theorem NormedAddGroupHom.completion_toCompl (f : NormedAddGroupHom G H) :
    f.completion.comp toCompl = toCompl.comp f 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H)
Goal: f.completion.comp NormedAddCommGroup.toCompl = NormedAddCommGroup.toCompl.comp f
-- NormedAddGroupHom.norm_completion
@[simp]
theorem NormedAddGroupHom.norm_completion (f : NormedAddGroupHom G H) : ‖f.completion‖ = ‖f‖ 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H)
Goal: ‖f.completion‖ = ‖f‖
-- NormedAddGroupHom.ker_le_ker_completion
theorem NormedAddGroupHom.ker_le_ker_completion (f : NormedAddGroupHom G H) :
    (toCompl.comp <| incl f.ker).range ≤ f.completion.ker 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H)
Goal: (NormedAddCommGroup.toCompl.comp (NormedAddGroupHom.incl f.ker)).range ≤ f.completion.ker
-- NormedAddGroupHom.ker_completion
theorem NormedAddGroupHom.ker_completion {f : NormedAddGroupHom G H} {C : ℝ}
    (h : f.SurjectiveOnWith f.range C) :
    (f.completion.ker : Set <| Completion G) = closure (toCompl.comp <| incl f.ker).range 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] {f : NormedAddGroupHom G H} {C : ℝ} (h : f.SurjectiveOnWith f.range C)
Goal: ↑f.completion.ker = closure ↑(NormedAddCommGroup.toCompl.comp (NormedAddGroupHom.incl f.ker)).range
-- NormedAddGroupHom.extension_def
theorem NormedAddGroupHom.extension_def (f : NormedAddGroupHom G H) (v : G) :
    f.extension v = Completion.extension f v 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] [T0Space H] [CompleteSpace H] (f : NormedAddGroupHom G H) (v : G)
Goal: f.extension (↑G v) = UniformSpace.Completion.extension (⇑f) (↑G v)
-- NormedAddGroupHom.extension_coe
@[simp]
theorem NormedAddGroupHom.extension_coe (f : NormedAddGroupHom G H) (v : G) : f.extension v = f v 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] [T0Space H] [CompleteSpace H] (f : NormedAddGroupHom G H) (v : G)
Goal: f.extension (↑G v) = f v
-- NormedAddGroupHom.extension_coe_to_fun
theorem NormedAddGroupHom.extension_coe_to_fun (f : NormedAddGroupHom G H) :
    (f.extension : Completion G → H) = Completion.extension f 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] [T0Space H] [CompleteSpace H] (f : NormedAddGroupHom G H)
Goal: ⇑f.extension = UniformSpace.Completion.extension ⇑f
-- NormedAddGroupHom.extension_unique
theorem NormedAddGroupHom.extension_unique (f : NormedAddGroupHom G H)
    {g : NormedAddGroupHom (Completion G) H} (hg : ∀ v, f v = g v) : f.extension = g 
Arguments: {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] [T0Space H] [CompleteSpace H] (f : NormedAddGroupHom G H) {g : NormedAddGroupHom (UniformSpace.Completion G) H} (hg : ∀ (v : G), f v = g (↑G v))
Goal: f.extension = g
-- Metric.exists_subseq_summable_dist_of_cauchySeq
lemma Metric.exists_subseq_summable_dist_of_cauchySeq (u : ℕ → α) (hu : CauchySeq u) :
    ∃ f : ℕ → ℕ, StrictMono f ∧ Summable fun i => dist (u (f (i+1))) (u (f i)) 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (u : ℕ → α) (hu : CauchySeq u)
Goal: ∃ f, StrictMono f ∧ Summable fun i => dist (u (f (i + 1))) (u (f i))
-- NormedAddCommGroup.completeSpace_of_summable_imp_tendsto
lemma NormedAddCommGroup.completeSpace_of_summable_imp_tendsto
    (h : ∀ u : ℕ → E,
      Summable (‖u ·‖) → ∃ a, Tendsto (fun n => ∑ i ∈ range n, u i) atTop (𝓝 a)) :
    CompleteSpace E 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (h :
  ∀ (u : ℕ → E),
    (Summable fun x => ‖u x‖) → ∃ a, Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, u i) Filter.atTop (𝓝 a))
Goal: CompleteSpace E
-- NormedAddCommGroup.summable_imp_tendsto_of_complete
lemma NormedAddCommGroup.summable_imp_tendsto_of_complete [CompleteSpace E] (u : ℕ → E)
    (hu : Summable (‖u ·‖)) : ∃ a, Tendsto (fun n => ∑ i ∈ range n, u i) atTop (𝓝 a) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [CompleteSpace E] (u : ℕ → E) (hu : Summable fun x => ‖u x‖)
Goal: ∃ a, Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, u i) Filter.atTop (𝓝 a)
-- NormedAddCommGroup.summable_imp_tendsto_iff_completeSpace
lemma NormedAddCommGroup.summable_imp_tendsto_iff_completeSpace :
    (∀ u : ℕ → E, Summable (‖u ·‖) → ∃ a, Tendsto (fun n => ∑ i ∈ range n, u i) atTop (𝓝 a))
     ↔ CompleteSpace E 
Arguments: {E : Type u_1} [NormedAddCommGroup E]
Goal: (∀ (u : ℕ → E),
    (Summable fun x => ‖u x‖) → ∃ a, Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, u i) Filter.atTop (𝓝 a)) ↔
  CompleteSpace E
-- Isometry.norm_map_of_map_zero
@[to_additive]
theorem Isometry.norm_map_of_map_one {f : E → F} (hi : Isometry f) (h₁ : f 1 = 1) (x : E) :
    ‖f x‖ = ‖x‖ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] {f : E → F} (hi : Isometry f) (h₁ : f 0 = 0) (x : E)
Goal: ‖f x‖ = ‖x‖
-- Isometry.norm_map_of_map_one
@[to_additive]
theorem Isometry.norm_map_of_map_one {f : E → F} (hi : Isometry f) (h₁ : f 1 = 1) (x : E) :
    ‖f x‖ = ‖x‖ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] {f : E → F} (hi : Isometry f) (h₁ : f 1 = 1) (x : E)
Goal: ‖f x‖ = ‖x‖