ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Egorov.iUnionNotConvergentSeq_subset
theorem iUnionNotConvergentSeq_subset (hε : 0 < ε) (hf : ∀ n, StronglyMeasurable (f n))
    (hg : StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ∞)
    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
    iUnionNotConvergentSeq hε hf hg hsm hs hfg ⊆ s 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} [MetricSpace β] {μ : MeasureTheory.Measure α} {s : Set α} {ε : ℝ} {f : ι → α → β} {g : α → β} [SemilatticeSup ι] [Nonempty ι] [Countable ι] (hε : 0 < ε) (hf : ∀ (n : ι), MeasureTheory.StronglyMeasurable (f n)) (hg : MeasureTheory.StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ⊤) (hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x)))
Goal: MeasureTheory.Egorov.iUnionNotConvergentSeq hε hf hg hsm hs hfg ⊆ s
-- MeasureTheory.Egorov.tendstoUniformlyOn_diff_iUnionNotConvergentSeq
theorem tendstoUniformlyOn_diff_iUnionNotConvergentSeq (hε : 0 < ε)
    (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g) (hsm : MeasurableSet s)
    (hs : μ s ≠ ∞) (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
    TendstoUniformlyOn f g atTop (s \ Egorov.iUnionNotConvergentSeq hε hf hg hsm hs hfg) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} [MetricSpace β] {μ : MeasureTheory.Measure α} {s : Set α} {ε : ℝ} {f : ι → α → β} {g : α → β} [SemilatticeSup ι] [Nonempty ι] [Countable ι] (hε : 0 < ε) (hf : ∀ (n : ι), MeasureTheory.StronglyMeasurable (f n)) (hg : MeasureTheory.StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ⊤) (hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x)))
Goal: TendstoUniformlyOn f g Filter.atTop (s \ MeasureTheory.Egorov.iUnionNotConvergentSeq hε hf hg hsm hs hfg)
-- MeasureTheory.tendstoUniformlyOn_of_ae_tendsto
theorem tendstoUniformlyOn_of_ae_tendsto (hf : ∀ n, StronglyMeasurable (f n))
    (hg : StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ∞)
    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) {ε : ℝ} (hε : 0 < ε) :
    ∃ t ⊆ s, MeasurableSet t ∧ μ t ≤ ENNReal.ofReal ε ∧ TendstoUniformlyOn f g atTop (s \ t) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} [MetricSpace β] {μ : MeasureTheory.Measure α} [SemilatticeSup ι] [Nonempty ι] [Countable ι] {f : ι → α → β} {g : α → β} {s : Set α} (hf : ∀ (n : ι), MeasureTheory.StronglyMeasurable (f n)) (hg : MeasureTheory.StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ⊤) (hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x))) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ t ⊆ s, MeasurableSet t ∧ μ t ≤ ENNReal.ofReal ε ∧ TendstoUniformlyOn f g Filter.atTop (s \ t)
-- MeasureTheory.tendstoUniformlyOn_of_ae_tendsto'
theorem tendstoUniformlyOn_of_ae_tendsto' [IsFiniteMeasure μ] (hf : ∀ n, StronglyMeasurable (f n))
    (hg : StronglyMeasurable g) (hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) {ε : ℝ}
    (hε : 0 < ε) :
    ∃ t, MeasurableSet t ∧ μ t ≤ ENNReal.ofReal ε ∧ TendstoUniformlyOn f g atTop tᶜ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} [MetricSpace β] {μ : MeasureTheory.Measure α} [SemilatticeSup ι] [Nonempty ι] [Countable ι] {f : ι → α → β} {g : α → β} [MeasureTheory.IsFiniteMeasure μ] (hf : ∀ (n : ι), MeasureTheory.StronglyMeasurable (f n)) (hg : MeasureTheory.StronglyMeasurable g) (hfg : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x))) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ t, MeasurableSet t ∧ μ t ≤ ENNReal.ofReal ε ∧ TendstoUniformlyOn f g Filter.atTop tᶜ
-- MeasureTheory.tendstoInMeasure_iff_norm
theorem tendstoInMeasure_iff_norm [SeminormedAddCommGroup E] {l : Filter ι} {f : ι → α → E}
    {g : α → E} :
    TendstoInMeasure μ f l g ↔
      ∀ ε, 0 < ε → Tendsto (fun i => μ { x | ε ≤ ‖f i x - g x‖ }) l (𝓝 0) 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [SeminormedAddCommGroup E] {l : Filter ι} {f : ι → α → E} {g : α → E}
Goal: MeasureTheory.TendstoInMeasure μ f l g ↔ ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun i => μ {x | ε ≤ ‖f i x - g x‖}) l (𝓝 0)
-- MeasureTheory.TendstoInMeasure.congr'
protected theorem congr' (h_left : ∀ᶠ i in l, f i =ᵐ[μ] f' i) (h_right : g =ᵐ[μ] g')
    (h_tendsto : TendstoInMeasure μ f l g) : TendstoInMeasure μ f' l g' 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Dist E] {l : Filter ι} {f f' : ι → α → E} {g g' : α → E} (h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i) (h_right : g =ᵐ[μ] g') (h_tendsto : MeasureTheory.TendstoInMeasure μ f l g)
Goal: MeasureTheory.TendstoInMeasure μ f' l g'
-- MeasureTheory.TendstoInMeasure.congr
protected theorem congr (h_left : ∀ i, f i =ᵐ[μ] f' i) (h_right : g =ᵐ[μ] g')
    (h_tendsto : TendstoInMeasure μ f l g) : TendstoInMeasure μ f' l g' 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Dist E] {l : Filter ι} {f f' : ι → α → E} {g g' : α → E} (h_left : ∀ (i : ι), f i =ᵐ[μ] f' i) (h_right : g =ᵐ[μ] g') (h_tendsto : MeasureTheory.TendstoInMeasure μ f l g)
Goal: MeasureTheory.TendstoInMeasure μ f' l g'
-- MeasureTheory.TendstoInMeasure.congr_left
theorem congr_left (h : ∀ i, f i =ᵐ[μ] f' i) (h_tendsto : TendstoInMeasure μ f l g) :
    TendstoInMeasure μ f' l g 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Dist E] {l : Filter ι} {f f' : ι → α → E} {g : α → E} (h : ∀ (i : ι), f i =ᵐ[μ] f' i) (h_tendsto : MeasureTheory.TendstoInMeasure μ f l g)
Goal: MeasureTheory.TendstoInMeasure μ f' l g
-- MeasureTheory.TendstoInMeasure.congr_right
theorem congr_right (h : g =ᵐ[μ] g') (h_tendsto : TendstoInMeasure μ f l g) :
    TendstoInMeasure μ f l g' 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Dist E] {l : Filter ι} {f : ι → α → E} {g g' : α → E} (h : g =ᵐ[μ] g') (h_tendsto : MeasureTheory.TendstoInMeasure μ f l g)
Goal: MeasureTheory.TendstoInMeasure μ f l g'
-- MeasureTheory.tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable
theorem tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable [IsFiniteMeasure μ]
    (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
    (hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) : TendstoInMeasure μ f atTop g 
Arguments: {α : Type u_1} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MetricSpace E] {f : ℕ → α → E} {g : α → E} [MeasureTheory.IsFiniteMeasure μ] (hf : ∀ (n : ℕ), MeasureTheory.StronglyMeasurable (f n)) (hg : MeasureTheory.StronglyMeasurable g) (hfg : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x)))
Goal: MeasureTheory.TendstoInMeasure μ f Filter.atTop g
-- MeasureTheory.tendstoInMeasure_of_tendsto_ae
theorem tendstoInMeasure_of_tendsto_ae [IsFiniteMeasure μ] (hf : ∀ n, AEStronglyMeasurable (f n) μ)
    (hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) : TendstoInMeasure μ f atTop g 
Arguments: {α : Type u_1} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MetricSpace E] {f : ℕ → α → E} {g : α → E} [MeasureTheory.IsFiniteMeasure μ] (hf : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (f n) μ) (hfg : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x)))
Goal: MeasureTheory.TendstoInMeasure μ f Filter.atTop g
-- MeasureTheory.ExistsSeqTendstoAe.exists_nat_measure_lt_two_inv
theorem exists_nat_measure_lt_two_inv (hfg : TendstoInMeasure μ f atTop g) (n : ℕ) :
    ∃ N, ∀ m ≥ N, μ { x | (2 : ℝ)⁻¹ ^ n ≤ dist (f m x) (g x) } ≤ (2⁻¹ : ℝ≥0∞) ^ n 
Arguments: {α : Type u_1} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MetricSpace E] {f : ℕ → α → E} {g : α → E} (hfg : MeasureTheory.TendstoInMeasure μ f Filter.atTop g) (n : ℕ)
Goal: ∃ N, ∀ m_1 ≥ N, μ {x | 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n
-- MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_succ
theorem seqTendstoAeSeq_succ (hfg : TendstoInMeasure μ f atTop g) {n : ℕ} :
    seqTendstoAeSeq hfg (n + 1) =
      max (seqTendstoAeSeqAux hfg (n + 1)) (seqTendstoAeSeq hfg n + 1) 
Arguments: {α : Type u_1} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MetricSpace E] {f : ℕ → α → E} {g : α → E} (hfg : MeasureTheory.TendstoInMeasure μ f Filter.atTop g) {n : ℕ}
Goal: MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq hfg (n + 1) =
  max (MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeqAux hfg (n + 1))
    (MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq hfg n + 1)
-- MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_spec
theorem seqTendstoAeSeq_spec (hfg : TendstoInMeasure μ f atTop g) (n k : ℕ)
    (hn : seqTendstoAeSeq hfg n ≤ k) :
    μ { x | (2 : ℝ)⁻¹ ^ n ≤ dist (f k x) (g x) } ≤ (2 : ℝ≥0∞)⁻¹ ^ n 
Arguments: {α : Type u_1} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MetricSpace E] {f : ℕ → α → E} {g : α → E} (hfg : MeasureTheory.TendstoInMeasure μ f Filter.atTop g) (n k : ℕ) (hn : MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq hfg n ≤ k)
Goal: μ {x | 2⁻¹ ^ n ≤ dist (f k x) (g x)} ≤ 2⁻¹ ^ n
-- MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono
theorem seqTendstoAeSeq_strictMono (hfg : TendstoInMeasure μ f atTop g) :
    StrictMono (seqTendstoAeSeq hfg) 
Arguments: {α : Type u_1} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MetricSpace E] {f : ℕ → α → E} {g : α → E} (hfg : MeasureTheory.TendstoInMeasure μ f Filter.atTop g)
Goal: StrictMono (MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq hfg)
-- MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae
theorem TendstoInMeasure.exists_seq_tendsto_ae (hfg : TendstoInMeasure μ f atTop g) :
    ∃ ns : ℕ → ℕ, StrictMono ns ∧ ∀ᵐ x ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) 
Arguments: {α : Type u_1} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MetricSpace E] {f : ℕ → α → E} {g : α → E} (hfg : MeasureTheory.TendstoInMeasure μ f Filter.atTop g)
Goal: ∃ ns, StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun i => f (ns i) x) Filter.atTop (𝓝 (g x))
-- MeasureTheory.TendstoInMeasure.exists_seq_tendstoInMeasure_atTop
theorem TendstoInMeasure.exists_seq_tendstoInMeasure_atTop {u : Filter ι} [NeBot u]
    [IsCountablyGenerated u] {f : ι → α → E} {g : α → E} (hfg : TendstoInMeasure μ f u g) :
    ∃ ns : ℕ → ι, TendstoInMeasure μ (fun n => f (ns n)) atTop g 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MetricSpace E] {u : Filter ι} [u.NeBot] [u.IsCountablyGenerated] {f : ι → α → E} {g : α → E} (hfg : MeasureTheory.TendstoInMeasure μ f u g)
Goal: ∃ ns, MeasureTheory.TendstoInMeasure μ (fun n => f (ns n)) Filter.atTop g
-- MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae'
theorem TendstoInMeasure.exists_seq_tendsto_ae' {u : Filter ι} [NeBot u] [IsCountablyGenerated u]
    {f : ι → α → E} {g : α → E} (hfg : TendstoInMeasure μ f u g) :
    ∃ ns : ℕ → ι, ∀ᵐ x ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MetricSpace E] {u : Filter ι} [u.NeBot] [u.IsCountablyGenerated] {f : ι → α → E} {g : α → E} (hfg : MeasureTheory.TendstoInMeasure μ f u g)
Goal: ∃ ns, ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun i => f (ns i) x) Filter.atTop (𝓝 (g x))
-- MeasureTheory.TendstoInMeasure.aemeasurable
theorem TendstoInMeasure.aemeasurable {u : Filter ι} [NeBot u] [IsCountablyGenerated u]
    {f : ι → α → E} {g : α → E} (hf : ∀ n, AEMeasurable (f n) μ)
    (h_tendsto : TendstoInMeasure μ f u g) : AEMeasurable g μ 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace E] [NormedAddCommGroup E] [BorelSpace E] {u : Filter ι} [u.NeBot] [u.IsCountablyGenerated] {f : ι → α → E} {g : α → E} (hf : ∀ (n : ι), AEMeasurable (f n) μ) (h_tendsto : MeasureTheory.TendstoInMeasure μ f u g)
Goal: AEMeasurable g μ
-- MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_of_stronglyMeasurable
theorem tendstoInMeasure_of_tendsto_eLpNorm_of_stronglyMeasurable (hp_ne_zero : p ≠ 0)
    (hp_ne_top : p ≠ ∞) (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
    {l : Filter ι} (hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) l (𝓝 0)) :
    TendstoInMeasure μ f l g 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {p : ℝ≥0∞} {f : ι → α → E} {g : α → E} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (hf : ∀ (n : ι), MeasureTheory.StronglyMeasurable (f n)) (hg : MeasureTheory.StronglyMeasurable g) {l : Filter ι} (hfg : Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) l (𝓝 0))
Goal: MeasureTheory.TendstoInMeasure μ f l g
-- MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_of_ne_top
theorem tendstoInMeasure_of_tendsto_eLpNorm_of_ne_top (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : AEStronglyMeasurable g μ) {l : Filter ι}
    (hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) l (𝓝 0)) : TendstoInMeasure μ f l g 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {p : ℝ≥0∞} {f : ι → α → E} {g : α → E} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (hf : ∀ (n : ι), MeasureTheory.AEStronglyMeasurable (f n) μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) {l : Filter ι} (hfg : Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) l (𝓝 0))
Goal: MeasureTheory.TendstoInMeasure μ f l g
-- MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_top
theorem tendstoInMeasure_of_tendsto_eLpNorm_top {E} [NormedAddCommGroup E] {f : ι → α → E}
    {g : α → E} {l : Filter ι} (hfg : Tendsto (fun n => eLpNorm (f n - g) ∞ μ) l (𝓝 0)) :
    TendstoInMeasure μ f l g 
Arguments: {α : Type u_1} {ι : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_4} [NormedAddCommGroup E] {f : ι → α → E} {g : α → E} {l : Filter ι} (hfg : Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) ⊤ μ) l (𝓝 0))
Goal: MeasureTheory.TendstoInMeasure μ f l g
-- MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm
theorem tendstoInMeasure_of_tendsto_eLpNorm {l : Filter ι} (hp_ne_zero : p ≠ 0)
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : AEStronglyMeasurable g μ)
    (hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) l (𝓝 0)) : TendstoInMeasure μ f l g 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {p : ℝ≥0∞} {f : ι → α → E} {g : α → E} {l : Filter ι} (hp_ne_zero : p ≠ 0) (hf : ∀ (n : ι), MeasureTheory.AEStronglyMeasurable (f n) μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) (hfg : Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) l (𝓝 0))
Goal: MeasureTheory.TendstoInMeasure μ f l g
-- MeasureTheory.tendstoInMeasure_of_tendsto_Lp
theorem tendstoInMeasure_of_tendsto_Lp [hp : Fact (1 ≤ p)] {f : ι → Lp E p μ} {g : Lp E p μ}
    {l : Filter ι} (hfg : Tendsto f l (𝓝 g)) : TendstoInMeasure μ (fun n => f n) l g 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {p : ℝ≥0∞} [hp : Fact (1 ≤ p)] {f : ι → ↥(MeasureTheory.Lp E p μ)} {g : ↥(MeasureTheory.Lp E p μ)} {l : Filter ι} (hfg : Filter.Tendsto f l (𝓝 g))
Goal: MeasureTheory.TendstoInMeasure μ (fun n => ↑↑(f n)) l ↑↑g
-- MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁
theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₁ (I : Box (Fin (n + 1)))
    (f : ℝⁿ⁺¹ → Eⁿ⁺¹)
    (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹) (s : Set ℝⁿ⁺¹)
    (hs : s.Countable) (Hc : ContinuousOn f (Box.Icc I))
    (Hd : ∀ x ∈ (Box.Icc I) \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x)
    (Hi : IntegrableOn (fun x => ∑ i, f' x (e i) i) (Box.Icc I)) :
    (∫ x in Box.Icc I, ∑ i, f' x (e i) i) =
      ∑ i : Fin (n + 1),
        ((∫ x in Box.Icc (I.face i), f (i.insertNth (I.upper i) x) i) -
          ∫ x in Box.Icc (I.face i), f (i.insertNth (I.lower i) x) i) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {n : ℕ} (I : BoxIntegral.Box (Fin (n + 1))) (f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E) (f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E) (s : Set (Fin (n + 1) → ℝ)) (hs : s.Countable) (Hc : ContinuousOn f (BoxIntegral.Box.Icc I)) (Hd : ∀ x ∈ BoxIntegral.Box.Icc I \ s, HasFDerivWithinAt f (f' x) (BoxIntegral.Box.Icc I) x) (Hi :
  MeasureTheory.IntegrableOn (fun x => ∑ i : Fin (n + 1), (f' x) (Pi.single i 1) i) (BoxIntegral.Box.Icc I)
    MeasureTheory.volume)
Goal: ∫ (x : Fin (n + 1) → ℝ) in BoxIntegral.Box.Icc I, ∑ i : Fin (n + 1), (f' x) (Pi.single i 1) i =
  ∑ i : Fin (n + 1),
    ((∫ (x : Fin n → ℝ) in BoxIntegral.Box.Icc (I.face i), f (i.insertNth (I.upper i) x) i) -
      ∫ (x : Fin n → ℝ) in BoxIntegral.Box.Icc (I.face i), f (i.insertNth (I.lower i) x) i)
-- MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂
theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (Fin (n + 1)))
    (f : ℝⁿ⁺¹ → Eⁿ⁺¹)
    (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹)
    (s : Set ℝⁿ⁺¹) (hs : s.Countable) (Hc : ContinuousOn f (Box.Icc I))
    (Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x)
    (Hi : IntegrableOn (∑ i, f' · (e i) i) (Box.Icc I)) :
    (∫ x in Box.Icc I, ∑ i, f' x (e i) i) =
      ∑ i : Fin (n + 1),
        ((∫ x in Box.Icc (I.face i), f (i.insertNth (I.upper i) x) i) -
          ∫ x in Box.Icc (I.face i), f (i.insertNth (I.lower i) x) i) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {n : ℕ} (I : BoxIntegral.Box (Fin (n + 1))) (f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E) (f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E) (s : Set (Fin (n + 1) → ℝ)) (hs : s.Countable) (Hc : ContinuousOn f (BoxIntegral.Box.Icc I)) (Hd : ∀ x ∈ BoxIntegral.Box.Ioo I \ s, HasFDerivAt f (f' x) x) (Hi :
  MeasureTheory.IntegrableOn (fun x => ∑ i : Fin (n + 1), (f' x) (Pi.single i 1) i) (BoxIntegral.Box.Icc I)
    MeasureTheory.volume)
Goal: ∫ (x : Fin (n + 1) → ℝ) in BoxIntegral.Box.Icc I, ∑ i : Fin (n + 1), (f' x) (Pi.single i 1) i =
  ∑ i : Fin (n + 1),
    ((∫ (x : Fin n → ℝ) in BoxIntegral.Box.Icc (I.face i), f (i.insertNth (I.upper i) x) i) -
      ∫ (x : Fin n → ℝ) in BoxIntegral.Box.Icc (I.face i), f (i.insertNth (I.lower i) x) i)
-- MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable
theorem integral_divergence_of_hasFDerivWithinAt_off_countable (hle : a ≤ b)
    (f : ℝⁿ⁺¹ → Eⁿ⁺¹)
    (f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹)
    (s : Set ℝⁿ⁺¹) (hs : s.Countable) (Hc : ContinuousOn f (Icc a b))
    (Hd : ∀ x ∈ (Set.pi univ fun i => Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x)
    (Hi : IntegrableOn (fun x => ∑ i, f' x (e i) i) (Icc a b)) :
    (∫ x in Icc a b, ∑ i, f' x (e i) i) =
      ∑ i : Fin (n + 1),
        ((∫ x in face i, f (frontFace i x) i) - ∫ x in face i, f (backFace i x) i) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {n : ℕ} (a b : Fin (n + 1) → ℝ) (hle : a ≤ b) (f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E) (f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E) (s : Set (Fin (n + 1) → ℝ)) (hs : s.Countable) (Hc : ContinuousOn f (Set.Icc a b)) (Hd : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x) (Hi :
  MeasureTheory.IntegrableOn (fun x => ∑ i : Fin (n + 1), (f' x) (Pi.single i 1) i) (Set.Icc a b) MeasureTheory.volume)
Goal: ∫ (x : Fin (n + 1) → ℝ) in Set.Icc a b, ∑ i : Fin (n + 1), (f' x) (Pi.single i 1) i =
  ∑ i : Fin (n + 1),
    ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f (i.insertNth (b i) x) i) -
      ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f (i.insertNth (a i) x) i)
-- MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable'
theorem integral_divergence_of_hasFDerivWithinAt_off_countable' (hle : a ≤ b)
    (f : Fin (n + 1) → ℝⁿ⁺¹ → E)
    (f' : Fin (n + 1) → ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] E) (s : Set ℝⁿ⁺¹)
    (hs : s.Countable) (Hc : ∀ i, ContinuousOn (f i) (Icc a b))
    (Hd : ∀ x ∈ (pi Set.univ fun i => Ioo (a i) (b i)) \ s, ∀ (i), HasFDerivAt (f i) (f' i x) x)
    (Hi : IntegrableOn (fun x => ∑ i, f' i x (e i)) (Icc a b)) :
    (∫ x in Icc a b, ∑ i, f' i x (e i)) =
      ∑ i : Fin (n + 1), ((∫ x in face i, f i (frontFace i x)) -
        ∫ x in face i, f i (backFace i x)) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {n : ℕ} (a b : Fin (n + 1) → ℝ) (hle : a ≤ b) (f : Fin (n + 1) → (Fin (n + 1) → ℝ) → E) (f' : Fin (n + 1) → (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E) (s : Set (Fin (n + 1) → ℝ)) (hs : s.Countable) (Hc : ∀ (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)) (Hd : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ s, ∀ (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x) (Hi :
  MeasureTheory.IntegrableOn (fun x => ∑ i : Fin (n + 1), (f' i x) (Pi.single i 1)) (Set.Icc a b) MeasureTheory.volume)
Goal: ∫ (x : Fin (n + 1) → ℝ) in Set.Icc a b, ∑ i : Fin (n + 1), (f' i x) (Pi.single i 1) =
  ∑ i : Fin (n + 1),
    ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f i (i.insertNth (b i) x)) -
      ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove), f i (i.insertNth (a i) x))
-- MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv
theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Type*}
    [NormedAddCommGroup F] [NormedSpace ℝ F] [PartialOrder F] [MeasureSpace F] [BorelSpace F]
    (eL : F ≃L[ℝ] ℝⁿ⁺¹) (he_ord : ∀ x y, eL x ≤ eL y ↔ x ≤ y)
    (he_vol : MeasurePreserving eL volume volume) (f : Fin (n + 1) → F → E)
    (f' : Fin (n + 1) → F → F →L[ℝ] E) (s : Set F) (hs : s.Countable) (a b : F) (hle : a ≤ b)
    (Hc : ∀ i, ContinuousOn (f i) (Icc a b))
    (Hd : ∀ x ∈ interior (Icc a b) \ s, ∀ (i), HasFDerivAt (f i) (f' i x) x) (DF : F → E)
    (hDF : ∀ x, DF x = ∑ i, f' i x (eL.symm <| e i)) (Hi : IntegrableOn DF (Icc a b)) :
    ∫ x in Icc a b, DF x =
      ∑ i : Fin (n + 1),
        ((∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
            f i (eL.symm <| i.insertNth (eL b i) x)) -
          ∫ x in Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove),
            f i (eL.symm <| i.insertNth (eL a i) x)) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {n : ℕ} {F : Type u_1} [NormedAddCommGroup F] [NormedSpace ℝ F] [PartialOrder F] [MeasureTheory.MeasureSpace F] [BorelSpace F] (eL : F ≃L[ℝ] Fin (n + 1) → ℝ) (he_ord : ∀ (x y : F), eL x ≤ eL y ↔ x ≤ y) (he_vol : MeasureTheory.MeasurePreserving (⇑eL) MeasureTheory.volume MeasureTheory.volume) (f : Fin (n + 1) → F → E) (f' : Fin (n + 1) → F → F →L[ℝ] E) (s : Set F) (hs : s.Countable) (a b : F) (hle : a ≤ b) (Hc : ∀ (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)) (Hd : ∀ x ∈ interior (Set.Icc a b) \ s, ∀ (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x) (DF : F → E) (hDF : ∀ (x : F), DF x = ∑ i : Fin (n + 1), (f' i x) (eL.symm (Pi.single i 1))) (Hi : MeasureTheory.IntegrableOn DF (Set.Icc a b) MeasureTheory.volume)
Goal: ∫ (x : F) in Set.Icc a b, DF x =
  ∑ i : Fin (n + 1),
    ((∫ (x : Fin n → ℝ) in Set.Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove), f i (eL.symm (i.insertNth (eL b i) x))) -
      ∫ (x : Fin n → ℝ) in Set.Icc (eL a ∘ i.succAbove) (eL b ∘ i.succAbove), f i (eL.symm (i.insertNth (eL a i) x)))
-- MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable_of_le
theorem integral_eq_of_hasDerivWithinAt_off_countable_of_le (f f' : ℝ → E) {a b : ℝ}
    (hle : a ≤ b) {s : Set ℝ} (hs : s.Countable) (Hc : ContinuousOn f (Icc a b))
    (Hd : ∀ x ∈ Ioo a b \ s, HasDerivAt f (f' x) x) (Hi : IntervalIntegrable f' volume a b) :
    ∫ x in a..b, f' x = f b - f a 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (f f' : ℝ → E) {a b : ℝ} (hle : a ≤ b) {s : Set ℝ} (hs : s.Countable) (Hc : ContinuousOn f (Set.Icc a b)) (Hd : ∀ x ∈ Set.Ioo a b \ s, HasDerivAt f (f' x) x) (Hi : IntervalIntegrable f' MeasureTheory.volume a b)
Goal: ∫ (x : ℝ) in a..b, f' x = f b - f a