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
    

Proof state:
case a
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
this : y ∈ EMetric.ball 0 δ
⊢ x + y ∈ EMetric.ball x δ

Next tactic:
simpa [edist_eq_coe_nnnorm_sub, edist_eq_coe_nnnorm]