Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
hl : l.bRiemann = false
B : ι →ᵇᵃ[↑I] ℝ
hB0 : ∀ (J : Box ι), 0 ≤ B J
g : ι →ᵇᵃ[↑I] F
s : Set (ι → ℝ)
hs : s.Countable
hlH : s.Nonempty → l.bHenstock = true
ε : ℝ
ε0 : 0 < ε
δ₁ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₁ :
  ∀ (c : ℝ≥0),
    ∀ x ∈ Box.Icc I ∩ s,
      ∀ (ε : ℝ),
        0 < ε →
          ∀ J ≤ I,
            Box.Icc J ⊆ Metric.closedBall x ↑(δ₁ c x ε) →
              x ∈ Box.Icc J → (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε
δ₂ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₂ :
  ∀ (c : ℝ≥0),
    ∀ x ∈ Box.Icc I \ s,
      ∀ (ε : ℝ),
        0 < ε →
          ∀ J ≤ I,
            Box.Icc J ⊆ Metric.closedBall x ↑(δ₂ c x ε) →
              (l.bHenstock = true → x ∈ Box.Icc J) →
                (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε * B J
ε0' : 0 < ε / 2
H0 : 0 < 2 ^ Fintype.card ι
εs : (ι → ℝ) → ℝ
hεs0 : ∀ (i : ι → ℝ), 0 < εs i
hεs : ∀ (t : Finset (ι → ℝ)), ↑t ⊆ s → ∑ i ∈ t, 2 ^ Fintype.card ι * εs i ≤ ε / 2
ε' : ℝ
ε'0 : 0 < ε'
hεI : B I * ε' < ε / 2
⊢ ∃ ia,
    (∀ (c : ℝ≥0), l.RCond (ia c)) ∧
      ∀ x ∈ {π | ∃ c, l.MemBaseSet I c (ia c) π ∧ π.IsPartition}, integralSum f vol x ∈ Metric.closedBall (g I) ε

Tactic:
classical
  set δ : ℝ≥0 → ℝⁿ → Ioi (0 : ℝ) := fun c x => if x ∈ s then δ₁ c x (εs x) else (δ₂ c) x ε'
  refine ⟨δ, fun c => l.rCond_of_bRiemann_eq_false hl, ?_⟩
  simp only [mem_ofPred_eq]
  rintro π ⟨c, hπδ, hπp⟩
  -- Now we split the sum into two parts based on whether `π.tag J` belongs to `s` or not.
  rw [← g.sum_partition_boxes le_rfl hπp, Metric.mem_closedBall, integralSum,
    ← sum_filter_add_sum_filter_not π.boxes fun J => π.tag J ∈ s,
    ← sum_filter_add_sum_filter_not π.boxes fun J => π.tag J ∈ s, ← add_halves ε]
  refine dist_add_add_le_of_le ?_ ?_
  · rcases s.eq_empty_or_nonempty with (rfl | hsne); · simp [ε0'.le]
    /- For the boxes such that `π.tag J ∈ s`, we use the fact that at most `2 ^ #ι` boxes have the
        same tag. -/
    specialize hlH hsne
    have : ∀ J ∈ {J ∈ π.boxes | π.tag J ∈ s},
        dist (vol J (f <| π.tag J)) (g J) ≤ εs (π.tag J) := fun J hJ ↦ by
      rw [Finset.mem_filter] at hJ; obtain ⟨hJ, hJs⟩ := hJ
      refine Hδ₁ c _ ⟨π.tag_mem_Icc _, hJs⟩ _ (hεs0 _) _ (π.le_of_mem' _ hJ) ?_
        (hπδ.2 hlH J hJ) fun hD => (Finset.le_sup hJ).trans (hπδ.3 hD)
      convert! hπδ.1 J hJ using 3; exact (if_pos hJs).symm
    refine (dist_sum_sum_le_of_le _ this).trans ?_
    rw [sum_comp]
    refine (sum_le_sum ?_).trans (hεs _ ?_)
    · rintro b -
      rw [← Nat.cast_two, ← Nat.cast_pow, ← nsmul_eq_mul]
      refine nsmul_le_nsmul_left (hεs0 _).le ?_
      refine (Finset.card_le_card ?_).trans ((hπδ.isHenstock hlH).card_filter_tag_eq_le b)
      exact filter_subset_filter _ (filter_subset _ _)
    · rw [Finset.coe_image, Set.image_subset_iff]
      exact fun J hJ => (Finset.mem_filter.1 hJ).2
  /- Now we deal with boxes such that `π.tag J ∉ s`.
    In this case the estimate is straightforward. -/
  calc
    dist (∑ J ∈ π.boxes with tag π J ∉ s, vol J (f (tag π J)))
      (∑ J ∈ π.boxes with tag π J ∉ s, g J)
      ≤ ∑ J ∈ π.boxes with tag π J ∉ s, ε' * B J := dist_sum_sum_le_of_le _ fun J hJ ↦ by
      rw [Finset.mem_filter] at hJ; obtain ⟨hJ, hJs⟩ := hJ
      refine Hδ₂ c _ ⟨π.tag_mem_Icc _, hJs⟩ _ ε'0 _ (π.le_of_mem' _ hJ) ?_ (fun hH => hπδ.2 hH J hJ)
        fun hD => (Finset.le_sup hJ).trans (hπδ.3 hD)
      convert! hπδ.1 J hJ using 3; exact (if_neg hJs).symm
    _ ≤ ∑ J ∈ π.boxes, ε' * B J := by
      gcongr
      · exact fun _ _ _ ↦ mul_nonneg ε'0.le (hB0 _)
      · apply filter_subset
    _ = B I * ε' := by rw [← mul_sum, B.sum_partition_boxes le_rfl hπp, mul_comm]
    _ ≤ ε / 2 := hεI.le

Goal state after: