ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearMap.comp_memℒp'
theorem comp_memℒp' (L : E →L[𝕜] F) {f : α → E} (hf : Memℒp f p μ) : Memℒp (L ∘ f) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_5} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (L : E →L[𝕜] F) {f : α → E} (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp (⇑L ∘ f) p μ
-- MeasureTheory.Memℒp.ofReal
theorem _root_.MeasureTheory.Memℒp.ofReal {f : α → ℝ} (hf : Memℒp f p μ) :
    Memℒp (fun x => (f x : K)) p μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {K : Type u_6} [RCLike K] {f : α → ℝ} (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp (fun x => ↑(f x)) p μ
-- MeasureTheory.memℒp_re_im_iff
theorem _root_.MeasureTheory.memℒp_re_im_iff {f : α → K} :
    Memℒp (fun x ↦ RCLike.re (f x)) p μ ∧ Memℒp (fun x ↦ RCLike.im (f x)) p μ ↔
      Memℒp f p μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {K : Type u_6} [RCLike K] {f : α → K}
Goal: MeasureTheory.Memℒp (fun x => RCLike.re (f x)) p μ ∧ MeasureTheory.Memℒp (fun x => RCLike.im (f x)) p μ ↔
  MeasureTheory.Memℒp f p μ
-- ContinuousLinearMap.add_compLp
theorem add_compLp (L L' : E →L[𝕜] F) (f : Lp E p μ) :
    (L + L').compLp f = L.compLp f + L'.compLp f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_5} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (L L' : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ))
Goal: (L + L').compLp f = L.compLp f + L'.compLp f
-- ContinuousLinearMap.smul_compLp
theorem smul_compLp {𝕜'} [NormedRing 𝕜'] [Module 𝕜' F] [BoundedSMul 𝕜' F] [SMulCommClass 𝕜 𝕜' F]
    (c : 𝕜') (L : E →L[𝕜] F) (f : Lp E p μ) : (c • L).compLp f = c • L.compLp f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_5} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {𝕜' : Type u_6} [NormedRing 𝕜'] [Module 𝕜' F] [BoundedSMul 𝕜' F] [SMulCommClass 𝕜 𝕜' F] (c : 𝕜') (L : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ))
Goal: (c • L).compLp f = c • L.compLp f
-- ContinuousLinearMap.norm_compLp_le
theorem norm_compLp_le (L : E →L[𝕜] F) (f : Lp E p μ) : ‖L.compLp f‖ ≤ ‖L‖ * ‖f‖ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_5} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (L : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ‖L.compLp f‖ ≤ ‖L‖ * ‖f‖
-- ContinuousLinearMap.coeFn_compLpL
theorem coeFn_compLpL [Fact (1 ≤ p)] (L : E →L[𝕜] F) (f : Lp E p μ) :
    L.compLpL p μ f =ᵐ[μ] fun a => L (f a) 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_5} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [Fact (1 ≤ p)] (L : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ↑↑((ContinuousLinearMap.compLpL p μ L) f) =ᵐ[μ] fun a => L (↑↑f a)
-- ContinuousLinearMap.add_compLpL
theorem add_compLpL [Fact (1 ≤ p)] (L L' : E →L[𝕜] F) :
    (L + L').compLpL p μ = L.compLpL p μ + L'.compLpL p μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_5} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [Fact (1 ≤ p)] (L L' : E →L[𝕜] F)
Goal: ContinuousLinearMap.compLpL p μ (L + L') = ContinuousLinearMap.compLpL p μ L + ContinuousLinearMap.compLpL p μ L'
-- ContinuousLinearMap.smul_compLpL
theorem smul_compLpL [Fact (1 ≤ p)] {𝕜'} [NormedRing 𝕜'] [Module 𝕜' F] [BoundedSMul 𝕜' F]
    [SMulCommClass 𝕜 𝕜' F] (c : 𝕜') (L : E →L[𝕜] F) : (c • L).compLpL p μ = c • L.compLpL p μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_5} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [Fact (1 ≤ p)] {𝕜' : Type u_6} [NormedRing 𝕜'] [Module 𝕜' F] [BoundedSMul 𝕜' F] [SMulCommClass 𝕜 𝕜' F] (c : 𝕜') (L : E →L[𝕜] F)
Goal: ContinuousLinearMap.compLpL p μ (c • L) = c • ContinuousLinearMap.compLpL p μ L
-- ContinuousLinearMap.norm_compLpL_le
theorem norm_compLpL_le [Fact (1 ≤ p)] (L : E →L[𝕜] F) : ‖L.compLpL p μ‖ ≤ ‖L‖ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_5} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [Fact (1 ≤ p)] (L : E →L[𝕜] F)
Goal: ‖ContinuousLinearMap.compLpL p μ L‖ ≤ ‖L‖
-- MeasureTheory.indicatorConstLp_eq_toSpanSingleton_compLp
theorem indicatorConstLp_eq_toSpanSingleton_compLp {s : Set α} [NormedSpace ℝ F]
    (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : F) :
    indicatorConstLp 2 hs hμs x =
      (ContinuousLinearMap.toSpanSingleton ℝ x).compLp (indicatorConstLp 2 hs hμs (1 : ℝ)) 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {s : Set α} [NormedSpace ℝ F] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : F)
Goal: MeasureTheory.indicatorConstLp 2 hs hμs x =
  (ContinuousLinearMap.toSpanSingleton ℝ x).compLp (MeasureTheory.indicatorConstLp 2 hs hμs 1)
-- MeasureTheory.Lp.lipschitzWith_pos_part
theorem lipschitzWith_pos_part : LipschitzWith 1 fun x : ℝ => max x 0 
Goal: LipschitzWith 1 fun x => max x 0
-- MeasureTheory.Memℒp.pos_part
theorem _root_.MeasureTheory.Memℒp.pos_part {f : α → ℝ} (hf : Memℒp f p μ) :
    Memℒp (fun x => max (f x) 0) p μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp (fun x => max (f x) 0) p μ
-- MeasureTheory.Memℒp.neg_part
theorem _root_.MeasureTheory.Memℒp.neg_part {f : α → ℝ} (hf : Memℒp f p μ) :
    Memℒp (fun x => max (-f x) 0) p μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp (fun x => max (-f x) 0) p μ
-- MeasureTheory.Lp.coe_posPart
@[norm_cast]
theorem coe_posPart (f : Lp ℝ p μ) : (posPart f : α →ₘ[μ] ℝ) = (f : α →ₘ[μ] ℝ).posPart 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp ℝ p μ))
Goal: ↑(MeasureTheory.Lp.posPart f) = (↑f).posPart
-- MeasureTheory.Lp.coeFn_posPart
theorem coeFn_posPart (f : Lp ℝ p μ) : ⇑(posPart f) =ᵐ[μ] fun a => max (f a) 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp ℝ p μ))
Goal: ↑↑(MeasureTheory.Lp.posPart f) =ᵐ[μ] fun a => max (↑↑f a) 0
-- MeasureTheory.Lp.coeFn_negPart_eq_max
theorem coeFn_negPart_eq_max (f : Lp ℝ p μ) : ∀ᵐ a ∂μ, negPart f a = max (-f a) 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp ℝ p μ))
Goal: ∀ᵐ (a : α) ∂μ, ↑↑(MeasureTheory.Lp.negPart f) a = max (-↑↑f a) 0
-- MeasureTheory.Lp.coeFn_negPart
theorem coeFn_negPart (f : Lp ℝ p μ) : ∀ᵐ a ∂μ, negPart f a = -min (f a) 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp ℝ p μ))
Goal: ∀ᵐ (a : α) ∂μ, ↑↑(MeasureTheory.Lp.negPart f) a = -min (↑↑f a) 0
-- MeasureTheory.Lp.continuous_posPart
theorem continuous_posPart [Fact (1 ≤ p)] : Continuous fun f : Lp ℝ p μ => posPart f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)]
Goal: Continuous fun f => MeasureTheory.Lp.posPart f
-- MeasureTheory.Lp.continuous_negPart
theorem continuous_negPart [Fact (1 ≤ p)] : Continuous fun f : Lp ℝ p μ => negPart f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)]
Goal: Continuous fun f => MeasureTheory.Lp.negPart f
-- MeasureTheory.Lp.eLpNorm'_lim_eq_lintegral_liminf
theorem eLpNorm'_lim_eq_lintegral_liminf {ι} [Nonempty ι] [LinearOrder ι] {f : ι → α → G} {p : ℝ}
    (hp_nonneg : 0 ≤ p) {f_lim : α → G}
    (h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
    eLpNorm' f_lim p μ = (∫⁻ a, atTop.liminf fun m => (‖f m a‖₊ : ℝ≥0∞) ^ p ∂μ) ^ (1 / p) 
Arguments: {α : Type u_1} {G : Type u_4} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup G] {ι : Type u_5} [Nonempty ι] [LinearOrder ι] {f : ι → α → G} {p : ℝ} (hp_nonneg : 0 ≤ p) {f_lim : α → G} (h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (f_lim x)))
Goal: MeasureTheory.eLpNorm' f_lim p μ = (∫⁻ (a : α), Filter.liminf (fun m => ↑‖f m a‖₊ ^ p) Filter.atTop ∂μ) ^ (1 / p)
-- MeasureTheory.Lp.eLpNorm'_lim_le_liminf_eLpNorm'
theorem eLpNorm'_lim_le_liminf_eLpNorm' {E} [NormedAddCommGroup E] {f : ℕ → α → E} {p : ℝ}
    (hp_pos : 0 < p) (hf : ∀ n, AEStronglyMeasurable (f n) μ) {f_lim : α → E}
    (h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
    eLpNorm' f_lim p μ ≤ atTop.liminf fun n => eLpNorm' (f n) p μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedAddCommGroup E] {f : ℕ → α → E} {p : ℝ} (hp_pos : 0 < p) (hf : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (f n) μ) {f_lim : α → E} (h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (f_lim x)))
Goal: MeasureTheory.eLpNorm' f_lim p μ ≤ Filter.liminf (fun n => MeasureTheory.eLpNorm' (f n) p μ) Filter.atTop
-- MeasureTheory.Lp.eLpNorm_exponent_top_lim_eq_essSup_liminf
theorem eLpNorm_exponent_top_lim_eq_essSup_liminf {ι} [Nonempty ι] [LinearOrder ι] {f : ι → α → G}
    {f_lim : α → G} (h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
    eLpNorm f_lim ∞ μ = essSup (fun x => atTop.liminf fun m => (‖f m x‖₊ : ℝ≥0∞)) μ 
Arguments: {α : Type u_1} {G : Type u_4} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup G] {ι : Type u_5} [Nonempty ι] [LinearOrder ι] {f : ι → α → G} {f_lim : α → G} (h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (f_lim x)))
Goal: MeasureTheory.eLpNorm f_lim ⊤ μ = essSup (fun x => Filter.liminf (fun m => ↑‖f m x‖₊) Filter.atTop) μ
-- MeasureTheory.Lp.eLpNorm_exponent_top_lim_le_liminf_eLpNorm_exponent_top
theorem eLpNorm_exponent_top_lim_le_liminf_eLpNorm_exponent_top {ι} [Nonempty ι] [Countable ι]
    [LinearOrder ι] {f : ι → α → F} {f_lim : α → F}
    (h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
    eLpNorm f_lim ∞ μ ≤ atTop.liminf fun n => eLpNorm (f n) ∞ μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {ι : Type u_5} [Nonempty ι] [Countable ι] [LinearOrder ι] {f : ι → α → F} {f_lim : α → F} (h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (f_lim x)))
Goal: MeasureTheory.eLpNorm f_lim ⊤ μ ≤ Filter.liminf (fun n => MeasureTheory.eLpNorm (f n) ⊤ μ) Filter.atTop
-- MeasureTheory.Lp.eLpNorm_lim_le_liminf_eLpNorm
theorem eLpNorm_lim_le_liminf_eLpNorm {E} [NormedAddCommGroup E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (f_lim : α → E)
    (h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
    eLpNorm f_lim p μ ≤ atTop.liminf fun n => eLpNorm (f n) p μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedAddCommGroup E] {f : ℕ → α → E} (hf : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (f n) μ) (f_lim : α → E) (h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (f_lim x)))
Goal: MeasureTheory.eLpNorm f_lim p μ ≤ Filter.liminf (fun n => MeasureTheory.eLpNorm (f n) p μ) Filter.atTop
-- MeasureTheory.Lp.tendsto_Lp_iff_tendsto_ℒp'
theorem tendsto_Lp_iff_tendsto_ℒp' {ι} {fi : Filter ι} [Fact (1 ≤ p)] (f : ι → Lp E p μ)
    (f_lim : Lp E p μ) :
    fi.Tendsto f (𝓝 f_lim) ↔ fi.Tendsto (fun n => eLpNorm (⇑(f n) - ⇑f_lim) p μ) (𝓝 0) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {ι : Type u_5} {fi : Filter ι} [Fact (1 ≤ p)] (f : ι → ↥(MeasureTheory.Lp E p μ)) (f_lim : ↥(MeasureTheory.Lp E p μ))
Goal: Filter.Tendsto f fi (𝓝 f_lim) ↔ Filter.Tendsto (fun n => MeasureTheory.eLpNorm (↑↑(f n) - ↑↑f_lim) p μ) fi (𝓝 0)
-- MeasureTheory.Lp.tendsto_Lp_iff_tendsto_ℒp
theorem tendsto_Lp_iff_tendsto_ℒp {ι} {fi : Filter ι} [Fact (1 ≤ p)] (f : ι → Lp E p μ)
    (f_lim : α → E) (f_lim_ℒp : Memℒp f_lim p μ) :
    fi.Tendsto f (𝓝 (f_lim_ℒp.toLp f_lim)) ↔
      fi.Tendsto (fun n => eLpNorm (⇑(f n) - f_lim) p μ) (𝓝 0) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {ι : Type u_5} {fi : Filter ι} [Fact (1 ≤ p)] (f : ι → ↥(MeasureTheory.Lp E p μ)) (f_lim : α → E) (f_lim_ℒp : MeasureTheory.Memℒp f_lim p μ)
Goal: Filter.Tendsto f fi (𝓝 (MeasureTheory.Memℒp.toLp f_lim f_lim_ℒp)) ↔
  Filter.Tendsto (fun n => MeasureTheory.eLpNorm (↑↑(f n) - f_lim) p μ) fi (𝓝 0)
-- MeasureTheory.Lp.tendsto_Lp_iff_tendsto_ℒp''
theorem tendsto_Lp_iff_tendsto_ℒp'' {ι} {fi : Filter ι} [Fact (1 ≤ p)] (f : ι → α → E)
    (f_ℒp : ∀ n, Memℒp (f n) p μ) (f_lim : α → E) (f_lim_ℒp : Memℒp f_lim p μ) :
    fi.Tendsto (fun n => (f_ℒp n).toLp (f n)) (𝓝 (f_lim_ℒp.toLp f_lim)) ↔
      fi.Tendsto (fun n => eLpNorm (f n - f_lim) p μ) (𝓝 0) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {ι : Type u_5} {fi : Filter ι} [Fact (1 ≤ p)] (f : ι → α → E) (f_ℒp : ∀ (n : ι), MeasureTheory.Memℒp (f n) p μ) (f_lim : α → E) (f_lim_ℒp : MeasureTheory.Memℒp f_lim p μ)
Goal: Filter.Tendsto (fun n => MeasureTheory.Memℒp.toLp (f n) ⋯) fi (𝓝 (MeasureTheory.Memℒp.toLp f_lim f_lim_ℒp)) ↔
  Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - f_lim) p μ) fi (𝓝 0)
-- MeasureTheory.Lp.tendsto_Lp_of_tendsto_ℒp
theorem tendsto_Lp_of_tendsto_ℒp {ι} {fi : Filter ι} [Fact (1 ≤ p)] {f : ι → Lp E p μ}
    (f_lim : α → E) (f_lim_ℒp : Memℒp f_lim p μ)
    (h_tendsto : fi.Tendsto (fun n => eLpNorm (⇑(f n) - f_lim) p μ) (𝓝 0)) :
    fi.Tendsto f (𝓝 (f_lim_ℒp.toLp f_lim)) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {ι : Type u_5} {fi : Filter ι} [Fact (1 ≤ p)] {f : ι → ↥(MeasureTheory.Lp E p μ)} (f_lim : α → E) (f_lim_ℒp : MeasureTheory.Memℒp f_lim p μ) (h_tendsto : Filter.Tendsto (fun n => MeasureTheory.eLpNorm (↑↑(f n) - f_lim) p μ) fi (𝓝 0))
Goal: Filter.Tendsto f fi (𝓝 (MeasureTheory.Memℒp.toLp f_lim f_lim_ℒp))
-- MeasureTheory.Lp.cauchySeq_Lp_iff_cauchySeq_ℒp
theorem cauchySeq_Lp_iff_cauchySeq_ℒp {ι} [Nonempty ι] [SemilatticeSup ι] [hp : Fact (1 ≤ p)]
    (f : ι → Lp E p μ) :
    CauchySeq f ↔ Tendsto (fun n : ι × ι => eLpNorm (⇑(f n.fst) - ⇑(f n.snd)) p μ) atTop (𝓝 0) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {ι : Type u_5} [Nonempty ι] [SemilatticeSup ι] [hp : Fact (1 ≤ p)] (f : ι → ↥(MeasureTheory.Lp E p μ))
Goal: CauchySeq f ↔ Filter.Tendsto (fun n => MeasureTheory.eLpNorm (↑↑(f n.1) - ↑↑(f n.2)) p μ) Filter.atTop (𝓝 0)
-- MeasureTheory.Lp.completeSpace_lp_of_cauchy_complete_ℒp
theorem completeSpace_lp_of_cauchy_complete_ℒp [hp : Fact (1 ≤ p)]
    (H :
      ∀ (f : ℕ → α → E) (hf : ∀ n, Memℒp (f n) p μ) (B : ℕ → ℝ≥0∞) (hB : ∑' i, B i < ∞)
        (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N),
        ∃ (f_lim : α → E), Memℒp f_lim p μ ∧
          atTop.Tendsto (fun n => eLpNorm (f n - f_lim) p μ) (𝓝 0)) :
    CompleteSpace (Lp E p μ) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [hp : Fact (1 ≤ p)] (H :
  ∀ (f : ℕ → α → E),
    (∀ (n : ℕ), MeasureTheory.Memℒp (f n) p μ) →
      ∀ (B : ℕ → ℝ≥0∞),
        ∑' (i : ℕ), B i < ⊤ →
          (∀ (N n m : ℕ), N ≤ n → N ≤ m → MeasureTheory.eLpNorm (f n - f m) p μ < B N) →
            ∃ f_lim,
              MeasureTheory.Memℒp f_lim p μ ∧
                Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - f_lim) p μ) Filter.atTop (𝓝 0))
Goal: CompleteSpace ↥(MeasureTheory.Lp E p μ)
-- _private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.eLpNorm'_sum_norm_sub_le_tsum_of_cauchy_eLpNorm'
private theorem eLpNorm'_sum_norm_sub_le_tsum_of_cauchy_eLpNorm' {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞}
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → eLpNorm' (f n - f m) p μ < B N) (n : ℕ) :
    eLpNorm' (fun x => ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' i, B i 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : ℕ → α → E} (hf : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (f n) μ) {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → MeasureTheory.eLpNorm' (f n - f m) p μ < B N) (n : ℕ)
Goal: MeasureTheory.eLpNorm' (fun x => ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i
-- _private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum
private theorem lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum
    {f : ℕ → α → E} {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (n : ℕ)
    (hn : eLpNorm' (fun x => ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' i, B i) :
    (∫⁻ a, (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖₊ : ℝ≥0∞) ^ p ∂μ) ≤
      (∑' i, B i) ^ p 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : ℕ → α → E} {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (n : ℕ) (hn : MeasureTheory.eLpNorm' (fun x => ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i)
Goal: ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
-- _private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum
private theorem lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞}
    (h :
      ∀ n,
        (∫⁻ a, (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖₊ : ℝ≥0∞) ^ p ∂μ) ≤
          (∑' i, B i) ^ p) :
    (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖₊ : ℝ≥0∞) ^ p ∂μ) ^ (1 / p) ≤ ∑' i, B i 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : ℕ → α → E} (hf : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (f n) μ) {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (h : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p)
Goal: (∫⁻ (a : α), (∑' (i : ℕ), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
-- _private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.tsum_nnnorm_sub_ae_lt_top
private theorem tsum_nnnorm_sub_ae_lt_top {f : ℕ → α → E} (hf : ∀ n, AEStronglyMeasurable (f n) μ)
    {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h : (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖₊ : ℝ≥0∞) ^ p ∂μ) ^ (1 / p) ≤ ∑' i, B i) :
    ∀ᵐ x ∂μ, (∑' i, ‖f (i + 1) x - f i x‖₊ : ℝ≥0∞) < ∞ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : ℕ → α → E} (hf : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (f n) μ) {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' (i : ℕ), B i ≠ ⊤) (h : (∫⁻ (a : α), (∑' (i : ℕ), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i)
Goal: ∀ᵐ (x : α) ∂μ, ∑' (i : ℕ), ↑‖f (i + 1) x - f i x‖₊ < ⊤
-- MeasureTheory.Lp.ae_tendsto_of_cauchy_eLpNorm'
theorem ae_tendsto_of_cauchy_eLpNorm' [CompleteSpace E] {f : ℕ → α → E} {p : ℝ}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → eLpNorm' (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [CompleteSpace E] {f : ℕ → α → E} {p : ℝ} (hf : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (f n) μ) (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' (i : ℕ), B i ≠ ⊤) (h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → MeasureTheory.eLpNorm' (f n - f m) p μ < B N)
Goal: ∀ᵐ (x : α) ∂μ, ∃ l, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 l)
-- MeasureTheory.Lp.ae_tendsto_of_cauchy_eLpNorm
theorem ae_tendsto_of_cauchy_eLpNorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [CompleteSpace E] {f : ℕ → α → E} (hf : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' (i : ℕ), B i ≠ ⊤) (h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → MeasureTheory.eLpNorm (f n - f m) p μ < B N)
Goal: ∀ᵐ (x : α) ∂μ, ∃ l, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 l)
-- MeasureTheory.Lp.cauchy_tendsto_of_tendsto
theorem cauchy_tendsto_of_tendsto {f : ℕ → α → E} (hf : ∀ n, AEStronglyMeasurable (f n) μ)
    (f_lim : α → E) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N)
    (h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
    atTop.Tendsto (fun n => eLpNorm (f n - f_lim) p μ) (𝓝 0) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : ℕ → α → E} (hf : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (f n) μ) (f_lim : α → E) {B : ℕ → ℝ≥0∞} (hB : ∑' (i : ℕ), B i ≠ ⊤) (h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → MeasureTheory.eLpNorm (f n - f m) p μ < B N) (h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (f_lim x)))
Goal: Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - f_lim) p μ) Filter.atTop (𝓝 0)
-- MeasureTheory.Lp.memℒp_of_cauchy_tendsto
theorem memℒp_of_cauchy_tendsto (hp : 1 ≤ p) {f : ℕ → α → E} (hf : ∀ n, Memℒp (f n) p μ)
    (f_lim : α → E) (h_lim_meas : AEStronglyMeasurable f_lim μ)
    (h_tendsto : atTop.Tendsto (fun n => eLpNorm (f n - f_lim) p μ) (𝓝 0)) : Memℒp f_lim p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (hp : 1 ≤ p) {f : ℕ → α → E} (hf : ∀ (n : ℕ), MeasureTheory.Memℒp (f n) p μ) (f_lim : α → E) (h_lim_meas : MeasureTheory.AEStronglyMeasurable f_lim μ) (h_tendsto : Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - f_lim) p μ) Filter.atTop (𝓝 0))
Goal: MeasureTheory.Memℒp f_lim p μ
-- MeasureTheory.Lp.cauchy_complete_ℒp
theorem cauchy_complete_ℒp [CompleteSpace E] (hp : 1 ≤ p) {f : ℕ → α → E}
    (hf : ∀ n, Memℒp (f n) p μ) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) :
    ∃ (f_lim : α → E), Memℒp f_lim p μ ∧
      atTop.Tendsto (fun n => eLpNorm (f n - f_lim) p μ) (𝓝 0) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [CompleteSpace E] (hp : 1 ≤ p) {f : ℕ → α → E} (hf : ∀ (n : ℕ), MeasureTheory.Memℒp (f n) p μ) {B : ℕ → ℝ≥0∞} (hB : ∑' (i : ℕ), B i ≠ ⊤) (h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → MeasureTheory.eLpNorm (f n - f m) p μ < B N)
Goal: ∃ f_lim,
  MeasureTheory.Memℒp f_lim p μ ∧ Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - f_lim) p μ) Filter.atTop (𝓝 0)