ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.SimpleFunc.measure_support_lt_top_of_memℒp
theorem measure_support_lt_top_of_memℒp (f : α →ₛ E) (hf : Memℒp f p μ) (hp_ne_zero : p ≠ 0)
    (hp_ne_top : p ≠ ∞) : μ (support f) < ∞ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Memℒp (↑f) p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤)
Goal: μ (Function.support ↑f) < ⊤
-- MeasureTheory.SimpleFunc.measure_support_lt_top_of_integrable
theorem measure_support_lt_top_of_integrable (f : α →ₛ E) (hf : Integrable f μ) :
    μ (support f) < ∞ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Integrable (↑f) μ)
Goal: μ (Function.support ↑f) < ⊤
-- MeasureTheory.SimpleFunc.measure_lt_top_of_memℒp_indicator
theorem measure_lt_top_of_memℒp_indicator (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ∞) {c : E} (hc : c ≠ 0)
    {s : Set α} (hs : MeasurableSet s) (hcs : Memℒp ((const α c).piecewise s hs (const α 0)) p μ) :
    μ s < ⊤ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ⊤) {c : E} (hc : c ≠ 0) {s : Set α} (hs : MeasurableSet s) (hcs :
  MeasureTheory.Memℒp
    (↑(MeasureTheory.SimpleFunc.piecewise s hs (MeasureTheory.SimpleFunc.const α c)
        (MeasureTheory.SimpleFunc.const α 0)))
    p μ)
Goal: μ s < ⊤
-- MeasureTheory.Lp.simpleFunc.eq'
protected theorem eq' {f g : Lp.simpleFunc E p μ} : (f : α →ₘ[μ] E) = (g : α →ₘ[μ] E) → f = g 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {f g : ↥(MeasureTheory.Lp.simpleFunc E p μ)}
Goal: ↑↑f = ↑↑g → f = g
-- MeasureTheory.Lp.simpleFunc.coe_smul
@[simp, norm_cast]
theorem coe_smul (c : 𝕜) (f : Lp.simpleFunc E p μ) :
    ((c • f : Lp.simpleFunc E p μ) : Lp E p μ) = c • (f : Lp E p μ) 
Arguments: {α : Type u_1} {E : Type u_4} {𝕜 : Type u_6} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (c : 𝕜) (f : ↥(MeasureTheory.Lp.simpleFunc E p μ))
Goal: ↑(c • f) = c • ↑f
-- MeasureTheory.Lp.simpleFunc.boundedSMul
protected theorem boundedSMul [Fact (1 ≤ p)] : BoundedSMul 𝕜 (Lp.simpleFunc E p μ) 
Arguments: {α : Type u_1} {E : Type u_4} {𝕜 : Type u_6} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] [Fact (1 ≤ p)]
Goal: BoundedSMul 𝕜 ↥(MeasureTheory.Lp.simpleFunc E p μ)
-- MeasureTheory.Lp.simpleFunc.toLp_eq_toLp
theorem toLp_eq_toLp (f : α →ₛ E) (hf : Memℒp f p μ) : (toLp f hf : Lp E p μ) = hf.toLp f 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Memℒp (↑f) p μ)
Goal: ↑(MeasureTheory.Lp.simpleFunc.toLp f hf) = MeasureTheory.Memℒp.toLp (↑f) hf
-- MeasureTheory.Lp.simpleFunc.toLp_eq_mk
theorem toLp_eq_mk (f : α →ₛ E) (hf : Memℒp f p μ) :
    (toLp f hf : α →ₘ[μ] E) = AEEqFun.mk f f.aestronglyMeasurable 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Memℒp (↑f) p μ)
Goal: ↑↑(MeasureTheory.Lp.simpleFunc.toLp f hf) = MeasureTheory.AEEqFun.mk ↑f ⋯
-- MeasureTheory.Lp.simpleFunc.toLp_zero
theorem toLp_zero : toLp (0 : α →ₛ E) zero_memℒp = (0 : Lp.simpleFunc E p μ) 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.Lp.simpleFunc.toLp 0 ⋯ = 0
-- MeasureTheory.Lp.simpleFunc.toLp_add
theorem toLp_add (f g : α →ₛ E) (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
    toLp (f + g) (hf.add hg) = toLp f hf + toLp g hg 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f g : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Memℒp (↑f) p μ) (hg : MeasureTheory.Memℒp (↑g) p μ)
Goal: MeasureTheory.Lp.simpleFunc.toLp (f + g) ⋯ =
  MeasureTheory.Lp.simpleFunc.toLp f hf + MeasureTheory.Lp.simpleFunc.toLp g hg
-- MeasureTheory.Lp.simpleFunc.toLp_neg
theorem toLp_neg (f : α →ₛ E) (hf : Memℒp f p μ) : toLp (-f) hf.neg = -toLp f hf 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Memℒp (↑f) p μ)
Goal: MeasureTheory.Lp.simpleFunc.toLp (-f) ⋯ = -MeasureTheory.Lp.simpleFunc.toLp f hf
-- MeasureTheory.Lp.simpleFunc.toLp_sub
theorem toLp_sub (f g : α →ₛ E) (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
    toLp (f - g) (hf.sub hg) = toLp f hf - toLp g hg 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f g : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Memℒp (↑f) p μ) (hg : MeasureTheory.Memℒp (↑g) p μ)
Goal: MeasureTheory.Lp.simpleFunc.toLp (f - g) ⋯ =
  MeasureTheory.Lp.simpleFunc.toLp f hf - MeasureTheory.Lp.simpleFunc.toLp g hg
-- MeasureTheory.Lp.simpleFunc.toLp_smul
theorem toLp_smul (f : α →ₛ E) (hf : Memℒp f p μ) (c : 𝕜) :
    toLp (c • f) (hf.const_smul c) = c • toLp f hf 
Arguments: {α : Type u_1} {E : Type u_4} {𝕜 : Type u_6} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Memℒp (↑f) p μ) (c : 𝕜)
Goal: MeasureTheory.Lp.simpleFunc.toLp (c • f) ⋯ = c • MeasureTheory.Lp.simpleFunc.toLp f hf
-- MeasureTheory.Lp.simpleFunc.norm_toLp
nonrec theorem norm_toLp [Fact (1 ≤ p)] (f : α →ₛ E) (hf : Memℒp f p μ) :
    ‖toLp f hf‖ = ENNReal.toReal (eLpNorm f p μ) 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)] (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Memℒp (↑f) p μ)
Goal: ‖MeasureTheory.Lp.simpleFunc.toLp f hf‖ = (MeasureTheory.eLpNorm (↑f) p μ).toReal
-- MeasureTheory.Lp.simpleFunc.measurable
@[measurability]
protected theorem measurable [MeasurableSpace E] (f : Lp.simpleFunc E p μ) :
    Measurable (toSimpleFunc f) 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [MeasurableSpace E] (f : ↥(MeasureTheory.Lp.simpleFunc E p μ))
Goal: Measurable ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f)
-- MeasureTheory.Lp.simpleFunc.stronglyMeasurable
protected theorem stronglyMeasurable (f : Lp.simpleFunc E p μ) :
    StronglyMeasurable (toSimpleFunc f) 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp.simpleFunc E p μ))
Goal: MeasureTheory.StronglyMeasurable ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f)
-- MeasureTheory.Lp.simpleFunc.aemeasurable
@[measurability]
protected theorem aemeasurable [MeasurableSpace E] (f : Lp.simpleFunc E p μ) :
    AEMeasurable (toSimpleFunc f) μ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [MeasurableSpace E] (f : ↥(MeasureTheory.Lp.simpleFunc E p μ))
Goal: AEMeasurable (↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f)) μ
-- MeasureTheory.Lp.simpleFunc.aestronglyMeasurable
protected theorem aestronglyMeasurable (f : Lp.simpleFunc E p μ) :
    AEStronglyMeasurable (toSimpleFunc f) μ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp.simpleFunc E p μ))
Goal: MeasureTheory.AEStronglyMeasurable (↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f)) μ
-- MeasureTheory.Lp.simpleFunc.toSimpleFunc_eq_toFun
theorem toSimpleFunc_eq_toFun (f : Lp.simpleFunc E p μ) : toSimpleFunc f =ᵐ[μ] f 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp.simpleFunc E p μ))
Goal: ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f) =ᵐ[μ] ↑↑↑f
-- MeasureTheory.Lp.simpleFunc.memℒp
protected theorem memℒp (f : Lp.simpleFunc E p μ) : Memℒp (toSimpleFunc f) p μ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp.simpleFunc E p μ))
Goal: MeasureTheory.Memℒp (↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f)) p μ
-- MeasureTheory.Lp.simpleFunc.toLp_toSimpleFunc
theorem toLp_toSimpleFunc (f : Lp.simpleFunc E p μ) :
    toLp (toSimpleFunc f) (simpleFunc.memℒp f) = f 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp.simpleFunc E p μ))
Goal: MeasureTheory.Lp.simpleFunc.toLp (MeasureTheory.Lp.simpleFunc.toSimpleFunc f) ⋯ = f
-- MeasureTheory.Lp.simpleFunc.toSimpleFunc_toLp
theorem toSimpleFunc_toLp (f : α →ₛ E) (hfi : Memℒp f p μ) : toSimpleFunc (toLp f hfi) =ᵐ[μ] f 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α E) (hfi : MeasureTheory.Memℒp (↑f) p μ)
Goal: ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc (MeasureTheory.Lp.simpleFunc.toLp f hfi)) =ᵐ[μ] ↑f
-- MeasureTheory.Lp.simpleFunc.zero_toSimpleFunc
theorem zero_toSimpleFunc : toSimpleFunc (0 : Lp.simpleFunc E p μ) =ᵐ[μ] 0 
Arguments: {α : Type u_1} (E : Type u_4) [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} (μ : MeasureTheory.Measure α)
Goal: ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc 0) =ᵐ[μ] 0
-- MeasureTheory.Lp.simpleFunc.add_toSimpleFunc
theorem add_toSimpleFunc (f g : Lp.simpleFunc E p μ) :
    toSimpleFunc (f + g) =ᵐ[μ] toSimpleFunc f + toSimpleFunc g 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f g : ↥(MeasureTheory.Lp.simpleFunc E p μ))
Goal: ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc (f + g)) =ᵐ[μ]
  ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f) + ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc g)
-- MeasureTheory.Lp.simpleFunc.neg_toSimpleFunc
theorem neg_toSimpleFunc (f : Lp.simpleFunc E p μ) : toSimpleFunc (-f) =ᵐ[μ] -toSimpleFunc f 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp.simpleFunc E p μ))
Goal: ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc (-f)) =ᵐ[μ] -↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f)
-- MeasureTheory.Lp.simpleFunc.sub_toSimpleFunc
theorem sub_toSimpleFunc (f g : Lp.simpleFunc E p μ) :
    toSimpleFunc (f - g) =ᵐ[μ] toSimpleFunc f - toSimpleFunc g 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (f g : ↥(MeasureTheory.Lp.simpleFunc E p μ))
Goal: ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc (f - g)) =ᵐ[μ]
  ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f) - ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc g)
-- MeasureTheory.Lp.simpleFunc.smul_toSimpleFunc
theorem smul_toSimpleFunc (k : 𝕜) (f : Lp.simpleFunc E p μ) :
    toSimpleFunc (k • f) =ᵐ[μ] k • ⇑(toSimpleFunc f) 
Arguments: {α : Type u_1} {E : Type u_4} {𝕜 : Type u_6} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (k : 𝕜) (f : ↥(MeasureTheory.Lp.simpleFunc E p μ))
Goal: ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc (k • f)) =ᵐ[μ] k • ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f)
-- MeasureTheory.Lp.simpleFunc.norm_toSimpleFunc
theorem norm_toSimpleFunc [Fact (1 ≤ p)] (f : Lp.simpleFunc E p μ) :
    ‖f‖ = ENNReal.toReal (eLpNorm (toSimpleFunc f) p μ) 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)] (f : ↥(MeasureTheory.Lp.simpleFunc E p μ))
Goal: ‖f‖ = (MeasureTheory.eLpNorm (↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f)) p μ).toReal
-- MeasureTheory.Lp.simpleFunc.coe_indicatorConst
@[simp]
theorem coe_indicatorConst {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (c : E) :
    (↑(indicatorConst p hs hμs c) : Lp E p μ) = indicatorConstLp p hs hμs c 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (c : E)
Goal: ↑(MeasureTheory.Lp.simpleFunc.indicatorConst p hs hμs c) = MeasureTheory.indicatorConstLp p hs hμs c
-- MeasureTheory.Lp.simpleFunc.toSimpleFunc_indicatorConst
theorem toSimpleFunc_indicatorConst {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (c : E) :
    toSimpleFunc (indicatorConst p hs hμs c) =ᵐ[μ]
      (SimpleFunc.const _ c).piecewise s hs (SimpleFunc.const _ 0) 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (c : E)
Goal: ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc (MeasureTheory.Lp.simpleFunc.indicatorConst p hs hμs c)) =ᵐ[μ]
  ↑(MeasureTheory.SimpleFunc.piecewise s hs (MeasureTheory.SimpleFunc.const α c) (MeasureTheory.SimpleFunc.const α 0))
-- MeasureTheory.Lp.simpleFunc.induction
@[elab_as_elim]
protected theorem induction (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ∞) {P : Lp.simpleFunc E p μ → Prop}
    (h_ind :
      ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞),
        P (Lp.simpleFunc.indicatorConst p hs hμs.ne c))
    (h_add :
      ∀ ⦃f g : α →ₛ E⦄,
        ∀ hf : Memℒp f p μ,
          ∀ hg : Memℒp g p μ,
            Disjoint (support f) (support g) →
              P (Lp.simpleFunc.toLp f hf) →
                P (Lp.simpleFunc.toLp g hg) → P (Lp.simpleFunc.toLp f hf + Lp.simpleFunc.toLp g hg))
    (f : Lp.simpleFunc E p μ) : P f 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ⊤) {P : ↥(MeasureTheory.Lp.simpleFunc E p μ) → Prop} (h_ind :
  ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P (MeasureTheory.Lp.simpleFunc.indicatorConst p hs ⋯ c)) (h_add :
  ∀ ⦃f g : MeasureTheory.SimpleFunc α E⦄ (hf : MeasureTheory.Memℒp (↑f) p μ) (hg : MeasureTheory.Memℒp (↑g) p μ),
    Disjoint (Function.support ↑f) (Function.support ↑g) →
      P (MeasureTheory.Lp.simpleFunc.toLp f hf) →
        P (MeasureTheory.Lp.simpleFunc.toLp g hg) →
          P (MeasureTheory.Lp.simpleFunc.toLp f hf + MeasureTheory.Lp.simpleFunc.toLp g hg)) (f : ↥(MeasureTheory.Lp.simpleFunc E p μ))
Goal: P f
-- MeasureTheory.Lp.simpleFunc.uniformContinuous
protected theorem uniformContinuous : UniformContinuous ((↑) : Lp.simpleFunc E p μ → Lp E p μ) 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)]
Goal: UniformContinuous Subtype.val
-- MeasureTheory.Lp.simpleFunc.uniformEmbedding
protected theorem uniformEmbedding : UniformEmbedding ((↑) : Lp.simpleFunc E p μ → Lp E p μ) 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)]
Goal: UniformEmbedding Subtype.val
-- MeasureTheory.Lp.simpleFunc.uniformInducing
protected theorem uniformInducing : UniformInducing ((↑) : Lp.simpleFunc E p μ → Lp E p μ) 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)]
Goal: UniformInducing Subtype.val
-- MeasureTheory.Lp.simpleFunc.denseEmbedding
protected theorem denseEmbedding (hp_ne_top : p ≠ ∞) :
    DenseEmbedding ((↑) : Lp.simpleFunc E p μ → Lp E p μ) 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)] (hp_ne_top : p ≠ ⊤)
Goal: DenseEmbedding Subtype.val
-- MeasureTheory.Lp.simpleFunc.denseInducing
protected theorem denseInducing (hp_ne_top : p ≠ ∞) :
    DenseInducing ((↑) : Lp.simpleFunc E p μ → Lp E p μ) 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)] (hp_ne_top : p ≠ ⊤)
Goal: DenseInducing Subtype.val
-- MeasureTheory.Lp.simpleFunc.denseRange
protected theorem denseRange (hp_ne_top : p ≠ ∞) :
    DenseRange ((↑) : Lp.simpleFunc E p μ → Lp E p μ) 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)] (hp_ne_top : p ≠ ⊤)
Goal: DenseRange Subtype.val
-- MeasureTheory.Lp.simpleFunc.dense
protected theorem dense (hp_ne_top : p ≠ ∞) : Dense (Lp.simpleFunc E p μ : Set (Lp E p μ)) 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [Fact (1 ≤ p)] (hp_ne_top : p ≠ ⊤)
Goal: Dense ↑(MeasureTheory.Lp.simpleFunc E p μ)
-- MeasureTheory.Lp.simpleFunc.coeFn_le
theorem coeFn_le (f g : Lp.simpleFunc G p μ) : (f : α → G) ≤ᵐ[μ] g ↔ f ≤ g 
Arguments: {α : Type u_1} [MeasurableSpace α] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {G : Type u_7} [NormedLatticeAddCommGroup G] (f g : ↥(MeasureTheory.Lp.simpleFunc G p μ))
Goal: ↑↑↑f ≤ᵐ[μ] ↑↑↑g ↔ f ≤ g
-- MeasureTheory.Lp.simpleFunc.coeFn_zero
theorem coeFn_zero : (0 : Lp.simpleFunc G p μ) =ᵐ[μ] (0 : α → G) 
Arguments: {α : Type u_1} [MeasurableSpace α] (p : ℝ≥0∞) (μ : MeasureTheory.Measure α) (G : Type u_7) [NormedLatticeAddCommGroup G]
Goal: ↑↑↑0 =ᵐ[μ] 0
-- MeasureTheory.Lp.simpleFunc.coeFn_nonneg
theorem coeFn_nonneg (f : Lp.simpleFunc G p μ) : (0 : α → G) ≤ᵐ[μ] f ↔ 0 ≤ f 
Arguments: {α : Type u_1} [MeasurableSpace α] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {G : Type u_7} [NormedLatticeAddCommGroup G] (f : ↥(MeasureTheory.Lp.simpleFunc G p μ))
Goal: 0 ≤ᵐ[μ] ↑↑↑f ↔ 0 ≤ f
-- MeasureTheory.Lp.simpleFunc.exists_simpleFunc_nonneg_ae_eq
theorem exists_simpleFunc_nonneg_ae_eq {f : Lp.simpleFunc G p μ} (hf : 0 ≤ f) :
    ∃ f' : α →ₛ G, 0 ≤ f' ∧ f =ᵐ[μ] f' 
Arguments: {α : Type u_1} [MeasurableSpace α] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {G : Type u_7} [NormedLatticeAddCommGroup G] {f : ↥(MeasureTheory.Lp.simpleFunc G p μ)} (hf : 0 ≤ f)
Goal: ∃ f', 0 ≤ f' ∧ ↑↑↑f =ᵐ[μ] ↑f'
-- MeasureTheory.Lp.simpleFunc.denseRange_coeSimpleFuncNonnegToLpNonneg
theorem denseRange_coeSimpleFuncNonnegToLpNonneg [hp : Fact (1 ≤ p)] (hp_ne_top : p ≠ ∞) :
    DenseRange (coeSimpleFuncNonnegToLpNonneg p μ G) 
Arguments: {α : Type u_1} [MeasurableSpace α] (p : ℝ≥0∞) (μ : MeasureTheory.Measure α) (G : Type u_7) [NormedLatticeAddCommGroup G] [hp : Fact (1 ≤ p)] (hp_ne_top : p ≠ ⊤)
Goal: DenseRange (MeasureTheory.Lp.simpleFunc.coeSimpleFuncNonnegToLpNonneg p μ G)
-- MeasureTheory.Lp.induction
@[elab_as_elim]
theorem Lp.induction [_i : Fact (1 ≤ p)] (hp_ne_top : p ≠ ∞) (P : Lp E p μ → Prop)
    (h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞),
      P (Lp.simpleFunc.indicatorConst p hs hμs.ne c))
    (h_add : ∀ ⦃f g⦄, ∀ hf : Memℒp f p μ, ∀ hg : Memℒp g p μ, Disjoint (support f) (support g) →
      P (hf.toLp f) → P (hg.toLp g) → P (hf.toLp f + hg.toLp g))
    (h_closed : IsClosed { f : Lp E p μ | P f }) : ∀ f : Lp E p μ, P f 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [_i : Fact (1 ≤ p)] (hp_ne_top : p ≠ ⊤) (P : ↥(MeasureTheory.Lp E p μ) → Prop) (h_ind :
  ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤),
    P ↑(MeasureTheory.Lp.simpleFunc.indicatorConst p hs ⋯ c)) (h_add :
  ∀ ⦃f g : α → E⦄ (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ),
    Disjoint (Function.support f) (Function.support g) →
      P (MeasureTheory.Memℒp.toLp f hf) →
        P (MeasureTheory.Memℒp.toLp g hg) → P (MeasureTheory.Memℒp.toLp f hf + MeasureTheory.Memℒp.toLp g hg)) (h_closed : IsClosed {f | P f}) (f : ↥(MeasureTheory.Lp E p μ))
Goal: P f
-- MeasureTheory.Memℒp.induction
@[elab_as_elim]
theorem Memℒp.induction [_i : Fact (1 ≤ p)] (hp_ne_top : p ≠ ∞) (P : (α → E) → Prop)
    (h_ind : ∀ (c : E) ⦃s⦄, MeasurableSet s → μ s < ∞ → P (s.indicator fun _ => c))
    (h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p μ → Memℒp g p μ →
      P f → P g → P (f + g))
    (h_closed : IsClosed { f : Lp E p μ | P f })
    (h_ae : ∀ ⦃f g⦄, f =ᵐ[μ] g → Memℒp f p μ → P f → P g) :
    ∀ ⦃f : α → E⦄, Memℒp f p μ → P f 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [_i : Fact (1 ≤ p)] (hp_ne_top : p ≠ ⊤) (P : (α → E) → Prop) (h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → P (s.indicator fun x => c)) (h_add :
  ∀ ⦃f g : α → E⦄,
    Disjoint (Function.support f) (Function.support g) →
      MeasureTheory.Memℒp f p μ → MeasureTheory.Memℒp g p μ → P f → P g → P (f + g)) (h_closed : IsClosed {f | P ↑↑f}) (h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MeasureTheory.Memℒp f p μ → P f → P g) ⦃f : α → E⦄
Goal: MeasureTheory.Memℒp f p μ → P f
-- MeasureTheory.Memℒp.induction_dense
theorem Memℒp.induction_dense (hp_ne_top : p ≠ ∞) (P : (α → E) → Prop)
    (h0P :
      ∀ (c : E) ⦃s : Set α⦄,
        MeasurableSet s →
          μ s < ∞ →
            ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g : α → E, eLpNorm (g - s.indicator fun _ => c) p μ ≤ ε ∧ P g)
    (h1P : ∀ f g, P f → P g → P (f + g)) (h2P : ∀ f, P f → AEStronglyMeasurable f μ) {f : α → E}
    (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ g : α → E, eLpNorm (f - g) p μ ≤ ε ∧ P g 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} (hp_ne_top : p ≠ ⊤) (P : (α → E) → Prop) (h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s →
      μ s < ⊤ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, MeasureTheory.eLpNorm (g - s.indicator fun x => c) p μ ≤ ε ∧ P g) (h1P : ∀ (f g : α → E), P f → P g → P (f + g)) (h2P : ∀ (f : α → E), P f → MeasureTheory.AEStronglyMeasurable f μ) {f : α → E} (hf : MeasureTheory.Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ g, MeasureTheory.eLpNorm (f - g) p μ ≤ ε ∧ P g
-- MeasureTheory.L1.SimpleFunc.toLp_one_eq_toL1
theorem L1.SimpleFunc.toLp_one_eq_toL1 (f : α →ₛ E) (hf : Integrable f μ) :
    (Lp.simpleFunc.toLp f (memℒp_one_iff_integrable.2 hf) : α →₁[μ] E) = hf.toL1 f 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Integrable (↑f) μ)
Goal: ↑(MeasureTheory.Lp.simpleFunc.toLp f ⋯) = MeasureTheory.Integrable.toL1 (↑f) hf
-- MeasureTheory.L1.SimpleFunc.integrable
protected theorem L1.SimpleFunc.integrable (f : α →₁ₛ[μ] E) :
    Integrable (Lp.simpleFunc.toSimpleFunc f) μ 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.Integrable (↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f)) μ
-- MeasureTheory.Integrable.induction
@[elab_as_elim]
theorem Integrable.induction (P : (α → E) → Prop)
    (h_ind : ∀ (c : E) ⦃s⦄, MeasurableSet s → μ s < ∞ → P (s.indicator fun _ => c))
    (h_add :
      ∀ ⦃f g : α → E⦄,
        Disjoint (support f) (support g) → Integrable f μ → Integrable g μ → P f → P g → P (f + g))
    (h_closed : IsClosed { f : α →₁[μ] E | P f })
    (h_ae : ∀ ⦃f g⦄, f =ᵐ[μ] g → Integrable f μ → P f → P g) :
    ∀ ⦃f : α → E⦄, Integrable f μ → P f 
Arguments: {α : Type u_1} {E : Type u_4} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} (P : (α → E) → Prop) (h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → P (s.indicator fun x => c)) (h_add :
  ∀ ⦃f g : α → E⦄,
    Disjoint (Function.support f) (Function.support g) →
      MeasureTheory.Integrable f μ → MeasureTheory.Integrable g μ → P f → P g → P (f + g)) (h_closed : IsClosed {f | P ↑↑f}) (h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MeasureTheory.Integrable f μ → P f → P g) ⦃f : α → E⦄
Goal: MeasureTheory.Integrable f μ → P f