set_option maxHeartbeats 300000 in
/-- If two functions `g` and `f` have power series `q` and `p` respectively at `f x` and `x`, then
`g ∘ f` admits the power series `q.comp p` at `x`. -/
theorem HasFPowerSeriesAt.comp {g : F → G} {f : E → F} {q : FormalMultilinearSeries 𝕜 F G}
    {p : FormalMultilinearSeries 𝕜 E F} {x : E} (hg : HasFPowerSeriesAt g q (f x))
    (hf : HasFPowerSeriesAt f p x) : HasFPowerSeriesAt (g ∘ f) (q.comp p) x := by
  /- Consider `rf` and `rg` such that `f` and `g` have power series expansion on the disks
    of radius `rf` and `rg`. -/
  rcases hg with ⟨rg, Hg⟩
  rcases hf with ⟨rf, Hf⟩
  -- The terms defining `q.comp p` are geometrically summable in a disk of some radius `r`.
  rcases q.comp_summable_nnreal p Hg.radius_pos Hf.radius_pos with ⟨r, r_pos : 0 < r, hr⟩
  /- We will consider `y` which is smaller than `r` and `rf`, and also small enough that
    `f (x + y)` is close enough to `f x` to be in the disk where `g` is well behaved. Let
    `min (r, rf, δ)` be this new radius.-/
  obtain ⟨δ, δpos, hδ⟩ :
    ∃ (δ : ℝ≥0∞) (_H : 0 < δ), ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg := by
    have : EMetric.ball (f x) rg ∈ 𝓝 (f x) := EMetric.ball_mem_nhds _ Hg.r_pos
    rcases EMetric.mem_nhds_iff.1 (Hf.analyticAt.continuousAt this) with ⟨δ, δpos, Hδ⟩
    exact ⟨δ, δpos, fun hz => Hδ hz⟩
  let rf' := min rf δ
  have min_pos : 0 < min rf' r := by
    simp only [r_pos, Hf.r_pos, δpos, lt_min_iff, ENNReal.coe_pos, and_self_iff]
  /- We will show that `g ∘ f` admits the power series `q.comp p` in the disk of
    radius `min (r, rf', δ)`. -/
  refine' ⟨min rf' r, _⟩
  refine'
    ⟨le_trans (min_le_right rf' r) (FormalMultilinearSeries.le_comp_radius_of_summable q p r hr),
      min_pos, @fun y hy => _⟩
  /- Let `y` satisfy `‖y‖ < min (r, rf', δ)`. We want to show that `g (f (x + y))` is the sum of
    `q.comp p` applied to `y`. -/
  -- First, check that `y` is small enough so that estimates for `f` and `g` apply.
  have y_mem : y ∈ EMetric.ball (0 : E) rf :=
    (EMetric.ball_subset_ball (le_trans (min_le_left _ _) (min_le_left _ _))) hy
  have fy_mem : f (x + y) ∈ EMetric.ball (f x) rg := by
    apply hδ
    have : y ∈ EMetric.ball (0 : E) δ :=
      (EMetric.ball_subset_ball (le_trans (min_le_left _ _) (min_le_right _ _))) hy
    simpa [edist_eq_coe_nnnorm_sub, edist_eq_coe_nnnorm]
  /- Now the proof starts. To show that the sum of `q.comp p` at `y` is `g (f (x + y))`,
    we will write `q.comp p` applied to `y` as a big sum over all compositions.
    Since the sum is summable, to get its convergence it suffices to get
    the convergence along some increasing sequence of sets.
    We will use the sequence of sets `comp_partial_sum_target 0 n n`,
    along which the sum is exactly the composition of the partial sums of `q` and `p`, by design.
    To show that it converges to `g (f (x + y))`, pointwise convergence would not be enough,
    but we have uniform convergence to save the day. -/
  -- First step: the partial sum of `p` converges to `f (x + y)`.
  have A : Tendsto (fun n => ∑ a in Finset.Ico 1 n, p a fun _b => y)
      atTop (𝓝 (f (x + y) - f x)) := by
    have L :
      ∀ᶠ n in atTop, (∑ a in Finset.range n, p a fun _b => y) - f x
        = ∑ a in Finset.Ico 1 n, p a fun _b => y := by
      rw [eventually_atTop]
      refine' ⟨1, fun n hn => _⟩
      symm
      

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
rg : ℝ≥0∞
Hg : HasFPowerSeriesOnBall g q (f x) rg
rf : ℝ≥0∞
Hf : HasFPowerSeriesOnBall f p x rf
r : ℝ≥0
r_pos : 0 < r
hr : Summable fun i => ‖compAlongComposition q p i.snd‖₊ * r ^ i.fst
δ : ℝ≥0∞
δpos : 0 < δ
hδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg
rf' : ℝ≥0∞ := min rf δ
min_pos : 0 < min rf' ↑r
y : E
hy : y ∈ EMetric.ball 0 (min rf' ↑r)
y_mem : y ∈ EMetric.ball 0 rf
fy_mem : f (x + y) ∈ EMetric.ball (f x) rg
n : ℕ
hn : n ≥ 1
⊢ (∑ a in Finset.Ico 1 n, (p a) fun _b => y) = (∑ a in Finset.range n, (p a) fun _b => y) - f x

Next tactic:
rw [eq_sub_iff_add_eq', Finset.range_eq_Ico, ← Hf.coeff_zero fun _i => y,
        Finset.sum_eq_sum_Ico_succ_bot hn]