/-- Second technical lemma to control the growth of coefficients of the inverse. Bound the explicit
expression for `∑_{k<n+1} aᵏ Qₖ` in terms of a sum of powers of the same sum one step before,
in the specific setup we are interesting in, by reducing to the general bound in
`radius_rightInv_pos_of_radius_pos_aux1`. -/
theorem radius_rightInv_pos_of_radius_pos_aux2 {n : ℕ} (hn : 2 ≤ n + 1)
    (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) {r a C : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a)
    (hC : 0 ≤ C) (hp : ∀ n, ‖p n‖ ≤ C * r ^ n) :
    ∑ k in Ico 1 (n + 1), a ^ k * ‖p.rightInv i k‖ ≤
      ‖(i.symm : F →L[𝕜] E)‖ * a +
        ‖(i.symm : F →L[𝕜] E)‖ * C *
          ∑ k in Ico 2 (n + 1), (r * ∑ j in Ico 1 n, a ^ j * ‖p.rightInv i j‖) ^ k :=
  let I := ‖(i.symm : F →L[𝕜] E)‖
  calc
    ∑ k in Ico 1 (n + 1), a ^ k * ‖p.rightInv i k‖ =
        a * I + ∑ k in Ico 2 (n + 1), a ^ k * ‖p.rightInv i k‖ := by
      simp only [LinearIsometryEquiv.norm_map, pow_one, rightInv_coeff_one,
        show Ico (1 : ℕ) 2 = {1} from Nat.Ico_succ_singleton 1,
        sum_singleton, ← sum_Ico_consecutive _ one_le_two hn]
    _ =
        a * I +
          ∑ k in Ico 2 (n + 1),
            a ^ k *
              ‖(i.symm : F →L[𝕜] E).compContinuousMultilinearMap
                  (∑ c in ({c | 1 < Composition.length c}.toFinset : Finset (Composition k)),
                    p.compAlongComposition (p.rightInv i) c)‖ := by
      congr! 2 with j hj
      rw [rightInv_coeff _ _ _ (mem_Ico.1 hj).1, norm_neg]
    _ ≤
        a * ‖(i.symm : F →L[𝕜] E)‖ +
          ∑ k in Ico 2 (n + 1),
            a ^ k *
              (I *
                ∑ c in ({c | 1 < Composition.length c}.toFinset : Finset (Composition k)),
                  C * r ^ c.length * ∏ j, ‖p.rightInv i (c.blocksFun j)‖) := by
      gcongr with j
      apply (ContinuousLinearMap.norm_compContinuousMultilinearMap_le _ _).trans
      gcongr
      apply (norm_sum_le _ _).trans
      gcongr
      apply (compAlongComposition_norm _ _ _).trans
      gcongr
      · exact prod_nonneg fun j _ => norm_nonneg _
      · apply hp
    _ =
        I * a +
          I * C *
            ∑ k in Ico 2 (n + 1),
              a ^ k *
                ∑ c in ({c | 1 < Composition.length c}.toFinset : Finset (Composition k)),
                  r ^ c.length * ∏ j, ‖p.rightInv i (c.blocksFun j)‖ := by
      simp_rw [mul_assoc C, ← mul_sum, ← mul_assoc, mul_comm _ ‖(i.symm : F →L[𝕜] E)‖, mul_assoc,
        ← mul_sum, ← mul_assoc, mul_comm _ C, mul_assoc, ← mul_sum]
      ring
    _ ≤ I * a + I * C *
        ∑ k in Ico 2 (n + 1), (r * ∑ j in Ico 1 n, a ^ j * ‖p.rightInv i j‖) ^ k := by
      gcongr _ + _ * _ * ?_
      simp_rw [mul_pow]
      

Proof state:
case bc.h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
hn : 2 ≤ n + 1
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
r a C : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
hC : 0 ≤ C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖↑(ContinuousLinearEquiv.symm i)‖
⊢ ∑ x in Ico 2 (n + 1),
      a ^ x *
        ∑ x_1 in Set.toFinset {c | 1 < Composition.length c},
          r ^ Composition.length x_1 *
            ∏ x_2 : Fin (Composition.length x_1), ‖rightInv p i (Composition.blocksFun x_1 x_2)‖ ≤
    ∑ x in Ico 2 (n + 1), r ^ x * (∑ x in Ico 1 n, a ^ x * ‖rightInv p i x‖) ^ x

Next tactic:
apply
        radius_right_inv_pos_of_radius_pos_aux1 n (fun k => ‖p.rightInv i k‖)
          (fun k => norm_nonneg _) hr ha