ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- DomMulAct.smul_Lp_ae_eq
@[to_additive]
theorem smul_Lp_ae_eq (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • f =ᵐ[μ] (f <| mk.symm c • ·) 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [SMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [MeasurableSMul M α] (c : Mᵈᵐᵃ) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ↑↑(c • f) =ᵐ[μ] fun x => ↑↑f (DomMulAct.mk.symm c • x)
-- DomAddAct.vadd_Lp_ae_eq
@[to_additive]
theorem smul_Lp_ae_eq (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • f =ᵐ[μ] (f <| mk.symm c • ·) 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [VAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] [MeasurableVAdd M α] (c : Mᵈᵃᵃ) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ↑↑(c +ᵥ f) =ᵐ[μ] fun x => ↑↑f (DomAddAct.mk.symm c +ᵥ x)
-- DomAddAct.mk_vadd_toLp
@[to_additive]
theorem mk_smul_toLp (c : M) {f : α → E} (hf : Memℒp f p μ) :
    mk c • hf.toLp f =
      (hf.comp_measurePreserving <| measurePreserving_smul c μ).toLp (f <| c • ·) 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [VAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] [MeasurableVAdd M α] (c : M) {f : α → E} (hf : MeasureTheory.Memℒp f p μ)
Goal: DomAddAct.mk c +ᵥ MeasureTheory.Memℒp.toLp f hf = MeasureTheory.Memℒp.toLp (fun x => f (c +ᵥ x)) ⋯
-- DomMulAct.mk_smul_toLp
@[to_additive]
theorem mk_smul_toLp (c : M) {f : α → E} (hf : Memℒp f p μ) :
    mk c • hf.toLp f =
      (hf.comp_measurePreserving <| measurePreserving_smul c μ).toLp (f <| c • ·) 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [SMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [MeasurableSMul M α] (c : M) {f : α → E} (hf : MeasureTheory.Memℒp f p μ)
Goal: DomMulAct.mk c • MeasureTheory.Memℒp.toLp f hf = MeasureTheory.Memℒp.toLp (fun x => f (c • x)) ⋯
-- DomMulAct.smul_Lp_const
@[to_additive (attr := simp)]
theorem smul_Lp_const [IsFiniteMeasure μ] (c : Mᵈᵐᵃ) (a : E) :
    c • Lp.const p μ a = Lp.const p μ a 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [SMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [MeasurableSMul M α] [MeasureTheory.IsFiniteMeasure μ] (c : Mᵈᵐᵃ) (a : E)
Goal: c • (MeasureTheory.Lp.const p μ) a = (MeasureTheory.Lp.const p μ) a
-- DomAddAct.vadd_Lp_const
@[to_additive (attr := simp)]
theorem smul_Lp_const [IsFiniteMeasure μ] (c : Mᵈᵐᵃ) (a : E) :
    c • Lp.const p μ a = Lp.const p μ a 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [VAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] [MeasurableVAdd M α] [MeasureTheory.IsFiniteMeasure μ] (c : Mᵈᵃᵃ) (a : E)
Goal: c +ᵥ (MeasureTheory.Lp.const p μ) a = (MeasureTheory.Lp.const p μ) a
-- DomMulAct.mk_smul_indicatorConstLp
@[to_additive]
theorem mk_smul_indicatorConstLp (c : M)
    {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (b : E) :
    mk c • indicatorConstLp p hs hμs b =
      indicatorConstLp p (hs.preimage <| measurable_const_smul c)
        (by rwa [SMulInvariantMeasure.measure_preimage_smul c hs]) b 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [SMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [MeasurableSMul M α] (c : M) {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (b : E)
Goal: DomMulAct.mk c • MeasureTheory.indicatorConstLp p hs hμs b = MeasureTheory.indicatorConstLp p ⋯ ⋯ b
-- DomAddAct.mk_vadd_indicatorConstLp
@[to_additive]
theorem mk_smul_indicatorConstLp (c : M)
    {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (b : E) :
    mk c • indicatorConstLp p hs hμs b =
      indicatorConstLp p (hs.preimage <| measurable_const_smul c)
        (by rwa [SMulInvariantMeasure.measure_preimage_smul c hs]) b 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [VAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] [MeasurableVAdd M α] (c : M) {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (b : E)
Goal: DomAddAct.mk c +ᵥ MeasureTheory.indicatorConstLp p hs hμs b = MeasureTheory.indicatorConstLp p ⋯ ⋯ b
-- DomAddAct.vadd_Lp_add
@[to_additive]
theorem smul_Lp_add (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f + g) = c • f + c • g 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [VAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] [MeasurableVAdd M α] (c : Mᵈᵃᵃ) (f g : ↥(MeasureTheory.Lp E p μ))
Goal: c +ᵥ (f + g) = c +ᵥ f + (c +ᵥ g)
-- DomMulAct.smul_Lp_add
@[to_additive]
theorem smul_Lp_add (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f + g) = c • f + c • g 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [SMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [MeasurableSMul M α] (c : Mᵈᵐᵃ) (f g : ↥(MeasureTheory.Lp E p μ))
Goal: c • (f + g) = c • f + c • g
-- DomMulAct.smul_Lp_zero
@[to_additive (attr := simp 1001)]
theorem smul_Lp_zero (c : Mᵈᵐᵃ) : c • (0 : Lp E p μ) = 0 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [SMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [MeasurableSMul M α] (c : Mᵈᵐᵃ)
Goal: c • 0 = 0
-- DomAddAct.vadd_Lp_zero
@[to_additive (attr := simp 1001)]
theorem smul_Lp_zero (c : Mᵈᵐᵃ) : c • (0 : Lp E p μ) = 0 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [VAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] [MeasurableVAdd M α] (c : Mᵈᵃᵃ)
Goal: c +ᵥ 0 = 0
-- DomAddAct.vadd_Lp_neg
@[to_additive]
theorem smul_Lp_neg (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • (-f) = -(c • f) 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [VAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] [MeasurableVAdd M α] (c : Mᵈᵃᵃ) (f : ↥(MeasureTheory.Lp E p μ))
Goal: c +ᵥ -f = -(c +ᵥ f)
-- DomMulAct.smul_Lp_neg
@[to_additive]
theorem smul_Lp_neg (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • (-f) = -(c • f) 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [SMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [MeasurableSMul M α] (c : Mᵈᵐᵃ) (f : ↥(MeasureTheory.Lp E p μ))
Goal: c • -f = -(c • f)
-- DomAddAct.vadd_Lp_sub
@[to_additive]
theorem smul_Lp_sub (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f - g) = c • f - c • g 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [VAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] [MeasurableVAdd M α] (c : Mᵈᵃᵃ) (f g : ↥(MeasureTheory.Lp E p μ))
Goal: c +ᵥ (f - g) = c +ᵥ f - (c +ᵥ g)
-- DomMulAct.smul_Lp_sub
@[to_additive]
theorem smul_Lp_sub (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f - g) = c • f - c • g 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [SMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [MeasurableSMul M α] (c : Mᵈᵐᵃ) (f g : ↥(MeasureTheory.Lp E p μ))
Goal: c • (f - g) = c • f - c • g
-- DomAddAct.norm_vadd_Lp
@[to_additive (attr := simp)]
theorem norm_smul_Lp (c : Mᵈᵐᵃ) (f : Lp E p μ) : ‖c • f‖ = ‖f‖ 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [VAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] [MeasurableVAdd M α] (c : Mᵈᵃᵃ) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ‖c +ᵥ f‖ = ‖f‖
-- DomMulAct.norm_smul_Lp
@[to_additive (attr := simp)]
theorem norm_smul_Lp (c : Mᵈᵐᵃ) (f : Lp E p μ) : ‖c • f‖ = ‖f‖ 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [SMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [MeasurableSMul M α] (c : Mᵈᵐᵃ) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ‖c • f‖ = ‖f‖
-- DomMulAct.nnnorm_smul_Lp
@[to_additive (attr := simp)]
theorem nnnorm_smul_Lp (c : Mᵈᵐᵃ) (f : Lp E p μ) : ‖c • f‖₊ = ‖f‖₊ 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [SMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [MeasurableSMul M α] (c : Mᵈᵐᵃ) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ‖c • f‖₊ = ‖f‖₊
-- DomAddAct.nnnorm_vadd_Lp
@[to_additive (attr := simp)]
theorem nnnorm_smul_Lp (c : Mᵈᵐᵃ) (f : Lp E p μ) : ‖c • f‖₊ = ‖f‖₊ 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [VAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] [MeasurableVAdd M α] (c : Mᵈᵃᵃ) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ‖c +ᵥ f‖₊ = ‖f‖₊
-- DomMulAct.dist_smul_Lp
@[to_additive (attr := simp)]
theorem dist_smul_Lp (c : Mᵈᵐᵃ) (f g : Lp E p μ) : dist (c • f) (c • g) = dist f g 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [SMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [MeasurableSMul M α] (c : Mᵈᵐᵃ) (f g : ↥(MeasureTheory.Lp E p μ))
Goal: dist (c • f) (c • g) = dist f g
-- DomAddAct.dist_vadd_Lp
@[to_additive (attr := simp)]
theorem dist_smul_Lp (c : Mᵈᵐᵃ) (f g : Lp E p μ) : dist (c • f) (c • g) = dist f g 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [VAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] [MeasurableVAdd M α] (c : Mᵈᵃᵃ) (f g : ↥(MeasureTheory.Lp E p μ))
Goal: dist (c +ᵥ f) (c +ᵥ g) = dist f g
-- DomAddAct.edist_vadd_Lp
@[to_additive (attr := simp)]
theorem edist_smul_Lp (c : Mᵈᵐᵃ) (f g : Lp E p μ) : edist (c • f) (c • g) = edist f g 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [VAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] [MeasurableVAdd M α] (c : Mᵈᵃᵃ) (f g : ↥(MeasureTheory.Lp E p μ))
Goal: edist (c +ᵥ f) (c +ᵥ g) = edist f g
-- DomMulAct.edist_smul_Lp
@[to_additive (attr := simp)]
theorem edist_smul_Lp (c : Mᵈᵐᵃ) (f g : Lp E p μ) : edist (c • f) (c • g) = edist f g 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [SMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [MeasurableSMul M α] (c : Mᵈᵐᵃ) (f g : ↥(MeasureTheory.Lp E p μ))
Goal: edist (c • f) (c • g) = edist f g
-- MeasureTheory.SimpleFunc.nearestPtInd_zero
@[simp]
theorem nearestPtInd_zero (e : ℕ → α) : nearestPtInd e 0 = const α 0 
Arguments: {α : Type u_1} [MeasurableSpace α] [PseudoEMetricSpace α] [OpensMeasurableSpace α] (e : ℕ → α)
Goal: MeasureTheory.SimpleFunc.nearestPtInd e 0 = MeasureTheory.SimpleFunc.const α 0
-- MeasureTheory.SimpleFunc.nearestPt_zero
@[simp]
theorem nearestPt_zero (e : ℕ → α) : nearestPt e 0 = const α (e 0) 
Arguments: {α : Type u_1} [MeasurableSpace α] [PseudoEMetricSpace α] [OpensMeasurableSpace α] (e : ℕ → α)
Goal: MeasureTheory.SimpleFunc.nearestPt e 0 = MeasureTheory.SimpleFunc.const α (e 0)
-- MeasureTheory.SimpleFunc.nearestPtInd_succ
theorem nearestPtInd_succ (e : ℕ → α) (N : ℕ) (x : α) :
    nearestPtInd e (N + 1) x =
      if ∀ k ≤ N, edist (e (N + 1)) x < edist (e k) x then N + 1 else nearestPtInd e N x 
Arguments: {α : Type u_1} [MeasurableSpace α] [PseudoEMetricSpace α] [OpensMeasurableSpace α] (e : ℕ → α) (N : ℕ) (x : α)
Goal: ↑(MeasureTheory.SimpleFunc.nearestPtInd e (N + 1)) x =
  if ∀ k ≤ N, edist (e (N + 1)) x < edist (e k) x then N + 1 else ↑(MeasureTheory.SimpleFunc.nearestPtInd e N) x
-- MeasureTheory.SimpleFunc.nearestPtInd_le
theorem nearestPtInd_le (e : ℕ → α) (N : ℕ) (x : α) : nearestPtInd e N x ≤ N 
Arguments: {α : Type u_1} [MeasurableSpace α] [PseudoEMetricSpace α] [OpensMeasurableSpace α] (e : ℕ → α) (N : ℕ) (x : α)
Goal: ↑(MeasureTheory.SimpleFunc.nearestPtInd e N) x ≤ N
-- MeasureTheory.SimpleFunc.edist_nearestPt_le
theorem edist_nearestPt_le (e : ℕ → α) (x : α) {k N : ℕ} (hk : k ≤ N) :
    edist (nearestPt e N x) x ≤ edist (e k) x 
Arguments: {α : Type u_1} [MeasurableSpace α] [PseudoEMetricSpace α] [OpensMeasurableSpace α] (e : ℕ → α) (x : α) {k N : ℕ} (hk : k ≤ N)
Goal: edist (↑(MeasureTheory.SimpleFunc.nearestPt e N) x) x ≤ edist (e k) x
-- MeasureTheory.SimpleFunc.tendsto_nearestPt
theorem tendsto_nearestPt {e : ℕ → α} {x : α} (hx : x ∈ closure (range e)) :
    Tendsto (fun N => nearestPt e N x) atTop (𝓝 x) 
Arguments: {α : Type u_1} [MeasurableSpace α] [PseudoEMetricSpace α] [OpensMeasurableSpace α] {e : ℕ → α} {x : α} (hx : x ∈ closure (Set.range e))
Goal: Filter.Tendsto (fun N => ↑(MeasureTheory.SimpleFunc.nearestPt e N) x) Filter.atTop (𝓝 x)
-- MeasureTheory.SimpleFunc.approxOn_zero
@[simp]
theorem approxOn_zero {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s)
    [SeparableSpace s] (x : β) : approxOn f hf s y₀ h₀ 0 x = y₀ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [PseudoEMetricSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [TopologicalSpace.SeparableSpace ↑s] (x : β)
Goal: ↑(MeasureTheory.SimpleFunc.approxOn f hf s y₀ h₀ 0) x = y₀
-- MeasureTheory.SimpleFunc.approxOn_mem
theorem approxOn_mem {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s)
    [SeparableSpace s] (n : ℕ) (x : β) : approxOn f hf s y₀ h₀ n x ∈ s 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [PseudoEMetricSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [TopologicalSpace.SeparableSpace ↑s] (n : ℕ) (x : β)
Goal: ↑(MeasureTheory.SimpleFunc.approxOn f hf s y₀ h₀ n) x ∈ s
-- MeasureTheory.SimpleFunc.approxOn_comp
@[simp, nolint simpNF] -- Porting note: LHS doesn't simplify.
theorem approxOn_comp {γ : Type*} [MeasurableSpace γ] {f : β → α} (hf : Measurable f) {g : γ → β}
    (hg : Measurable g) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [SeparableSpace s] (n : ℕ) :
    approxOn (f ∘ g) (hf.comp hg) s y₀ h₀ n = (approxOn f hf s y₀ h₀ n).comp g hg 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [PseudoEMetricSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] {γ : Type u_7} [MeasurableSpace γ] {f : β → α} (hf : Measurable f) {g : γ → β} (hg : Measurable g) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [TopologicalSpace.SeparableSpace ↑s] (n : ℕ)
Goal: MeasureTheory.SimpleFunc.approxOn (f ∘ g) ⋯ s y₀ h₀ n = (MeasureTheory.SimpleFunc.approxOn f hf s y₀ h₀ n).comp g hg
-- MeasureTheory.SimpleFunc.tendsto_approxOn
theorem tendsto_approxOn {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s)
    [SeparableSpace s] {x : β} (hx : f x ∈ closure s) :
    Tendsto (fun n => approxOn f hf s y₀ h₀ n x) atTop (𝓝 <| f x) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [PseudoEMetricSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [TopologicalSpace.SeparableSpace ↑s] {x : β} (hx : f x ∈ closure s)
Goal: Filter.Tendsto (fun n => ↑(MeasureTheory.SimpleFunc.approxOn f hf s y₀ h₀ n) x) Filter.atTop (𝓝 (f x))
-- MeasureTheory.SimpleFunc.edist_approxOn_mono
theorem edist_approxOn_mono {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s)
    [SeparableSpace s] (x : β) {m n : ℕ} (h : m ≤ n) :
    edist (approxOn f hf s y₀ h₀ n x) (f x) ≤ edist (approxOn f hf s y₀ h₀ m x) (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [PseudoEMetricSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [TopologicalSpace.SeparableSpace ↑s] (x : β) {m n : ℕ} (h : m ≤ n)
Goal: edist (↑(MeasureTheory.SimpleFunc.approxOn f hf s y₀ h₀ n) x) (f x) ≤
  edist (↑(MeasureTheory.SimpleFunc.approxOn f hf s y₀ h₀ m) x) (f x)
-- MeasureTheory.SimpleFunc.edist_approxOn_le
theorem edist_approxOn_le {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s)
    [SeparableSpace s] (x : β) (n : ℕ) : edist (approxOn f hf s y₀ h₀ n x) (f x) ≤ edist y₀ (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [PseudoEMetricSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [TopologicalSpace.SeparableSpace ↑s] (x : β) (n : ℕ)
Goal: edist (↑(MeasureTheory.SimpleFunc.approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist y₀ (f x)
-- MeasureTheory.SimpleFunc.edist_approxOn_y0_le
theorem edist_approxOn_y0_le {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s)
    [SeparableSpace s] (x : β) (n : ℕ) :
    edist y₀ (approxOn f hf s y₀ h₀ n x) ≤ edist y₀ (f x) + edist y₀ (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [PseudoEMetricSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s) [TopologicalSpace.SeparableSpace ↑s] (x : β) (n : ℕ)
Goal: edist y₀ (↑(MeasureTheory.SimpleFunc.approxOn f hf s y₀ h₀ n) x) ≤ edist y₀ (f x) + edist y₀ (f x)
-- HasCompactSupport.exists_simpleFunc_approx_of_prod
lemma HasCompactSupport.exists_simpleFunc_approx_of_prod [PseudoMetricSpace α]
    {f : X × Y → α} (hf : Continuous f) (h'f : HasCompactSupport f)
    {ε : ℝ} (hε : 0 < ε) :
    ∃ (g : SimpleFunc (X × Y) α), ∀ x, dist (f x) (g x) < ε 
Arguments: {X : Type u_7} {Y : Type u_8} {α : Type u_9} [Zero α] [TopologicalSpace X] [TopologicalSpace Y] [MeasurableSpace X] [MeasurableSpace Y] [OpensMeasurableSpace X] [OpensMeasurableSpace Y] [PseudoMetricSpace α] {f : X × Y → α} (hf : Continuous f) (h'f : HasCompactSupport f) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ g, ∀ (x : X × Y), dist (f x) (↑g x) < ε
-- HasCompactSupport.measurable_of_prod
lemma HasCompactSupport.measurable_of_prod
    [TopologicalSpace α] [PseudoMetrizableSpace α] [MeasurableSpace α] [BorelSpace α]
    {f : X × Y → α} (hf : Continuous f) (h'f : HasCompactSupport f) :
    Measurable f 
Arguments: {X : Type u_7} {Y : Type u_8} {α : Type u_9} [Zero α] [TopologicalSpace X] [TopologicalSpace Y] [MeasurableSpace X] [MeasurableSpace Y] [OpensMeasurableSpace X] [OpensMeasurableSpace Y] [TopologicalSpace α] [TopologicalSpace.PseudoMetrizableSpace α] [MeasurableSpace α] [BorelSpace α] {f : X × Y → α} (hf : Continuous f) (h'f : HasCompactSupport f)
Goal: Measurable f
-- MeasureTheory.Memℒp.finStronglyMeasurable_of_stronglyMeasurable
theorem Memℒp.finStronglyMeasurable_of_stronglyMeasurable (hf : Memℒp f p μ)
    (hf_meas : StronglyMeasurable f) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
    FinStronglyMeasurable f μ 
Arguments: {α : Type u_1} {G : Type u_2} {p : ℝ≥0∞} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup G] {f : α → G} (hf : MeasureTheory.Memℒp f p μ) (hf_meas : MeasureTheory.StronglyMeasurable f) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤)
Goal: MeasureTheory.FinStronglyMeasurable f μ
-- MeasureTheory.Memℒp.aefinStronglyMeasurable
theorem Memℒp.aefinStronglyMeasurable (hf : Memℒp f p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
    AEFinStronglyMeasurable f μ 
Arguments: {α : Type u_1} {G : Type u_2} {p : ℝ≥0∞} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup G] {f : α → G} (hf : MeasureTheory.Memℒp f p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤)
Goal: MeasureTheory.AEFinStronglyMeasurable f μ
-- MeasureTheory.Integrable.aefinStronglyMeasurable
theorem Integrable.aefinStronglyMeasurable (hf : Integrable f μ) : AEFinStronglyMeasurable f μ 
Arguments: {α : Type u_1} {G : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup G] {f : α → G} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.AEFinStronglyMeasurable f μ
-- MeasureTheory.Lp.finStronglyMeasurable
theorem Lp.finStronglyMeasurable (f : Lp G p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
    FinStronglyMeasurable f μ 
Arguments: {α : Type u_1} {G : Type u_2} {p : ℝ≥0∞} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup G] (f : ↥(MeasureTheory.Lp G p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤)
Goal: MeasureTheory.FinStronglyMeasurable (↑↑f) μ
-- intervalIntegral.FTCFilter.meas_gen
class FTCFilter (a : outParam ℝ) (outer : Filter ℝ) (inner : outParam <| Filter ℝ) extends
    TendstoIxxClass Ioc outer inner : Prop where
  pure_le : pure a ≤ outer
  le_nhds : inner ≤ 𝓝 a
  [meas_gen : IsMeasurablyGenerated inner]
Arguments: {a : outParam ℝ} (outer : Filter ℝ) {inner : outParam (Filter ℝ)} [self : intervalIntegral.FTCFilter a outer inner]
Goal: Filter.IsMeasurablyGenerated inner
-- intervalIntegral.FTCFilter.le_nhds
class FTCFilter (a : outParam ℝ) (outer : Filter ℝ) (inner : outParam <| Filter ℝ) extends
    TendstoIxxClass Ioc outer inner : Prop where
  pure_le : pure a ≤ outer
  le_nhds : inner ≤ 𝓝 a
  [meas_gen : IsMeasurablyGenerated inner]
Arguments: {a : outParam ℝ} (outer : Filter ℝ) {inner : outParam (Filter ℝ)} [self : intervalIntegral.FTCFilter a outer inner]
Goal: inner ≤ 𝓝 a
-- intervalIntegral.FTCFilter.pure_le
class FTCFilter (a : outParam ℝ) (outer : Filter ℝ) (inner : outParam <| Filter ℝ) extends
    TendstoIxxClass Ioc outer inner : Prop where
  pure_le : pure a ≤ outer
  le_nhds : inner ≤ 𝓝 a
  [meas_gen : IsMeasurablyGenerated inner]
Arguments: {a : outParam ℝ} {outer : Filter ℝ} {inner : outParam (Filter ℝ)} [self : intervalIntegral.FTCFilter a outer inner]
Goal: pure a ≤ outer
-- intervalIntegral.FTCFilter.finiteAt_inner
theorem finiteAt_inner {a : ℝ} (l : Filter ℝ) {l'} [h : FTCFilter a l l'] {μ : Measure ℝ}
    [IsLocallyFiniteMeasure μ] : μ.FiniteAtFilter l' 
Arguments: {a : ℝ} (l : Filter ℝ) {l' : Filter ℝ} [h : intervalIntegral.FTCFilter a l l'] {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: μ.FiniteAtFilter l'
-- intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae'
theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae' [IsMeasurablyGenerated l']
    [TendstoIxxClass Ioc l l'] (hfm : StronglyMeasurableAtFilter f l' μ)
    (hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)) (hl : μ.FiniteAtFilter l') (hu : Tendsto u lt l)
    (hv : Tendsto v lt l) :
    (fun t => (∫ x in u t..v t, f x ∂μ) - ∫ _ in u t..v t, c ∂μ) =o[lt] fun t =>
      ∫ _ in u t..v t, (1 : ℝ) ∂μ 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {c : E} {l l' : Filter ℝ} {lt : Filter ι} {μ : MeasureTheory.Measure ℝ} {u v : ι → ℝ} [l'.IsMeasurablyGenerated] [Filter.TendstoIxxClass Set.Ioc l l'] (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Filter.Tendsto f (l' ⊓ MeasureTheory.ae μ) (𝓝 c)) (hl : μ.FiniteAtFilter l') (hu : Filter.Tendsto u lt l) (hv : Filter.Tendsto v lt l)
Goal: (fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ - ∫ (x : ℝ) in u t..v t, c ∂μ) =o[lt] fun t => ∫ (x : ℝ) in u t..v t, 1 ∂μ