/-- If a a formal multilinear series has a positive radius of convergence, then its right inverse
also has a positive radius of convergence. -/
theorem radius_rightInv_pos_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
    (hp : 0 < p.radius) : 0 < (p.rightInv i).radius := by
  obtain ⟨C, r, Cpos, rpos, ple⟩ :
    ∃ (C r : _) (_ : 0 < C) (_ : 0 < r), ∀ n : ℕ, ‖p n‖ ≤ C * r ^ n :=
    le_mul_pow_of_radius_pos p hp
  let I := ‖(i.symm : F →L[𝕜] E)‖
  -- choose `a` small enough to make sure that `∑_{k ≤ n} aᵏ Qₖ` will be controllable by
  -- induction
  obtain ⟨a, apos, ha1, ha2⟩ :
    ∃ (a : _) (apos : 0 < a),
      2 * I * C * r ^ 2 * (I + 1) ^ 2 * a ≤ 1 ∧ r * (I + 1) * a ≤ 1 / 2 := by
    have :
      Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (𝓝 0)
        (𝓝 (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0)) :=
      tendsto_const_nhds.mul tendsto_id
    have A : ∀ᶠ a in 𝓝 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1 := by
      apply (tendsto_order.1 this).2; simp [zero_lt_one]
    have : Tendsto (fun a => r * (I + 1) * a) (𝓝 0) (𝓝 (r * (I + 1) * 0)) :=
      tendsto_const_nhds.mul tendsto_id
    have B : ∀ᶠ a in 𝓝 0, r * (I + 1) * a < 1 / 2 := by
      apply (tendsto_order.1 this).2; simp [zero_lt_one]
    have C : ∀ᶠ a in 𝓝[>] (0 : ℝ), (0 : ℝ) < a := by
      filter_upwards [self_mem_nhdsWithin] with _ ha using ha
    rcases (C.and ((A.and B).filter_mono inf_le_left)).exists with ⟨a, ha⟩
    exact ⟨a, ha.1, ha.2.1.le, ha.2.2.le⟩
  -- check by induction that the partial sums are suitably bounded, using the choice of `a` and the
  -- inductive control from Lemma `radius_rightInv_pos_of_radius_pos_aux2`.
  let S n := ∑ k in Ico 1 n, a ^ k * ‖p.rightInv i k‖
  have IRec : ∀ n, 1 ≤ n → S n ≤ (I + 1) * a := by
    apply Nat.le_induction
    · simp only
      rw [Ico_eq_empty_of_le (le_refl 1), sum_empty]
      exact mul_nonneg (add_nonneg (norm_nonneg _) zero_le_one) apos.le
    · intro n one_le_n hn
      have In : 2 ≤ n + 1 := by linarith only [one_le_n]
      have Snonneg : 0 ≤ S n :=
        sum_nonneg fun x _ => mul_nonneg (pow_nonneg apos.le _) (norm_nonneg _)
      have rSn : r * S n ≤ 1 / 2 :=
        calc
          r * S n ≤ r * ((I + 1) * a) := by gcongr
          _ ≤ 1 / 2 := by rwa [← mul_assoc]
      calc
        S (n + 1) ≤ I * a + I * C * ∑ k in Ico 2 (n + 1), (r * S n) ^ k :=
          radius_rightInv_pos_of_radius_pos_aux2 In p i rpos.le apos.le Cpos.le ple
        _ = I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) := by
          rw [geom_sum_Ico' _ In]; exact ne_of_lt (rSn.trans_lt (by norm_num))
        _ ≤ I * a + I * C * ((r * S n) ^ 2 / (1 / 2)) := by
          gcongr
          · simp only [sub_le_self_iff]
            positivity
          · linarith only [rSn]
        _ = I * a + 2 * I * C * (r * S n) ^ 2 := by ring
        _ ≤ I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2 := by gcongr
        _ = (I + 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) * a := by ring
        _ ≤ (I + 1) * a := by gcongr
  -- conclude that all coefficients satisfy `aⁿ Qₙ ≤ (I + 1) a`.
  let a' : NNReal := ⟨a, apos.le⟩
  suffices H : (a' : ENNReal) ≤ (p.rightInv i).radius
  · apply lt_of_lt_of_le _ H
    -- Prior to leanprover/lean4#2734, this was `exact_mod_cast apos`.
    simpa only [ENNReal.coe_pos]
  apply le_radius_of_bound _ ((I + 1) * a) fun n => ?_
  by_cases hn : n = 0
  · have : ‖p.rightInv i n‖ = ‖p.rightInv i 0‖ := by congr <;> try rw [hn]
    

Proof state:
case pos
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
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
hp : 0 < radius p
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖↑(ContinuousLinearEquiv.symm i)‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : ℕ → ℝ := fun n => ∑ k in Ico 1 n, a ^ k * ‖rightInv p i k‖
IRec : ∀ (n : ℕ), 1 ≤ n → S n ≤ (I + 1) * a
a' : NNReal := { val := a, property := (_ : 0 ≤ a) }
n : ℕ
hn : n = 0
this : ‖rightInv p i n‖ = ‖rightInv p i 0‖
⊢ ‖rightInv p i n‖ * ↑a' ^ n ≤ (I + 1) * a

Next tactic:
simp only [this, norm_zero, zero_mul, rightInv_coeff_zero]