/-- Let `p` be a predicate on `Box ι`, let `I` be a box. Suppose that the following two properties
hold true.

* `H_ind` : Consider a smaller box `J ≤ I`. The hyperplanes passing through the center of `J` split
  it into `2 ^ n` boxes. If `p` holds true on each of these boxes, then it true on `J`.

* `H_nhds` : For each `z` in the closed box `I.Icc` there exists a neighborhood `U` of `z` within
  `I.Icc` such that for every box `J ≤ I` such that `z ∈ J.Icc ⊆ U`, if `J` is homothetic to `I`
  with a coefficient of the form `1 / 2 ^ m`, then `p` is true on `J`.

Then `p I` is true. See also `BoxIntegral.Box.subbox_induction_on` for a version using
`BoxIntegral.Prepartition.splitCenter` instead of `BoxIntegral.Box.splitCenterBox`.

The proof still works if we assume `H_ind` only for subboxes `J ≤ I` that are homothetic to `I` with
a coefficient of the form `2⁻ᵐ` but we do not need this generalization yet. -/
@[elab_as_elim]
theorem subbox_induction_on' {p : Box ι → Prop} (I : Box ι)
    (H_ind : ∀ J ≤ I, (∀ s, p (splitCenterBox J s)) → p J)
    (H_nhds : ∀ z ∈ Box.Icc I, ∃ U ∈ 𝓝[Box.Icc I] z, ∀ J ≤ I, ∀ (m : ℕ), z ∈ Box.Icc J →
      Box.Icc J ⊆ U → (∀ i, J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J) :
    p I := by
  by_contra hpI
  -- First we use `H_ind` to construct a decreasing sequence of boxes such that `∀ m, ¬p (J m)`.
  replace H_ind := fun J hJ ↦ not_imp_not.2 (H_ind J hJ)
  simp only [exists_imp, not_forall] at H_ind
  choose! s hs using H_ind
  set J : ℕ → Box ι := fun m ↦ (fun J ↦ splitCenterBox J (s J))^[m] I
  have J_succ : ∀ m, J (m + 1) = splitCenterBox (J m) (s <| J m) :=
    fun m ↦ iterate_succ_apply' _ _ _
  -- Now we prove some properties of `J`
  have hJmono : Antitone J :=
    antitone_nat_of_succ_le fun n ↦ by simpa [J_succ] using splitCenterBox_le _ _
  have hJle : ∀ m, J m ≤ I := fun m ↦ hJmono (zero_le m)
  have hJp : ∀ m, ¬p (J m) :=
    fun m ↦ Nat.recOn m hpI fun m ↦ by simpa only [J_succ] using hs (J m) (hJle m)
  have hJsub : ∀ m i, (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m := by
    intro m i
    induction' m with m ihm
    · simp [Nat.zero_eq]
    simp only [pow_succ', J_succ, upper_sub_lower_splitCenterBox, ihm, div_div]
  have h0 : J 0 = I := rfl
  clear_value J
  clear hpI hs J_succ s
  -- Let `z` be the unique common point of all `(J m).Icc`. Then `H_nhds` proves `p (J m)` for
  -- sufficiently large `m`. This contradicts `hJp`.
  set z : ι → ℝ := ⨆ m, (J m).lower
  have hzJ : ∀ m, z ∈ Box.Icc (J m) :=
    mem_iInter.1 (ciSup_mem_iInter_Icc_of_antitone_Icc
      ((@Box.Icc ι).monotone.comp_antitone hJmono) fun m ↦ (J m).lower_le_upper)
  have hJl_mem : ∀ m, (J m).lower ∈ Box.Icc I := fun m ↦ le_iff_Icc.1 (hJle m) (J m).lower_mem_Icc
  have hJu_mem : ∀ m, (J m).upper ∈ Box.Icc I := fun m ↦ le_iff_Icc.1 (hJle m) (J m).upper_mem_Icc
  have hJlz : Tendsto (fun m ↦ (J m).lower) atTop (𝓝 z) :=
    tendsto_atTop_ciSup (antitone_lower.comp hJmono) ⟨I.upper, fun x ⟨m, hm⟩ ↦ hm ▸ (hJl_mem m).2⟩
  have hJuz : Tendsto (fun m ↦ (J m).upper) atTop (𝓝 z) := by
    suffices Tendsto (fun m ↦ (J m).upper - (J m).lower) atTop (𝓝 0) by simpa using hJlz.add this
    refine' tendsto_pi_nhds.2 fun i ↦ _
    simpa [hJsub] using
      tendsto_const_nhds.div_atTop (tendsto_pow_atTop_atTop_of_one_lt _root_.one_lt_two)
  replace hJlz : Tendsto (fun m ↦ (J m).lower) atTop (𝓝[Icc I.lower I.upper] z)
  · exact
      tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ hJlz (eventually_of_forall hJl_mem)
  replace hJuz : Tendsto (fun m ↦ (J m).upper) atTop (𝓝[Icc I.lower I.upper] z)
  · exact
      tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ hJuz (eventually_of_forall hJu_mem)
  rcases H_nhds z (h0 ▸ hzJ 0) with ⟨U, hUz, hU⟩
  

Proof state:
case intro.intro
ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), upper J i - lower J i = (upper I i - lower I i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), upper (J m) i - lower (J m) i = (upper I i - lower I i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
hJlz : Tendsto (fun m => (J m).lower) atTop (𝓝[Set.Icc I.lower I.upper] z)
hJuz : Tendsto (fun m => (J m).upper) atTop (𝓝[Set.Icc I.lower I.upper] z)
U : Set (ι → ℝ)
hUz : U ∈ 𝓝[Box.Icc I] z
hU :
  ∀ J ≤ I,
    ∀ (m : ℕ),
      z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), upper J i - lower J i = (upper I i - lower I i) / 2 ^ m) → p J
⊢ False

Next tactic:
rcases (tendsto_lift'.1 (hJlz.Icc hJuz) U hUz).exists with ⟨m, hUm⟩