/-- First 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 a general abstract setup. -/
theorem radius_right_inv_pos_of_radius_pos_aux1 (n : ℕ) (p : ℕ → ℝ) (hp : ∀ k, 0 ≤ p k) {r a : ℝ}
    (hr : 0 ≤ r) (ha : 0 ≤ a) :
    ∑ k in Ico 2 (n + 1),
        a ^ k *
          ∑ c in ({c | 1 < Composition.length c}.toFinset : Finset (Composition k)),
            r ^ c.length * ∏ j, p (c.blocksFun j) ≤
      ∑ j in Ico 2 (n + 1), r ^ j * (∑ k in Ico 1 n, a ^ k * p k) ^ j :=
  calc
    ∑ k in Ico 2 (n + 1),
          a ^ k *
            ∑ c in ({c | 1 < Composition.length c}.toFinset : Finset (Composition k)),
              r ^ c.length * ∏ j, p (c.blocksFun j) =
        ∑ k in Ico 2 (n + 1),
          ∑ c in ({c | 1 < Composition.length c}.toFinset : Finset (Composition k)),
            ∏ j, r * (a ^ c.blocksFun j * p (c.blocksFun j)) := by
      simp_rw [mul_sum]
      congr! with k _ c
      rw [prod_mul_distrib, prod_mul_distrib, prod_pow_eq_pow_sum, Composition.sum_blocksFun,
        prod_const, card_fin]
      ring
    _ ≤
        ∑ d in compPartialSumTarget 2 (n + 1) n,
          ∏ j : Fin d.2.length, r * (a ^ d.2.blocksFun j * p (d.2.blocksFun j)) := by
      rw [sum_sigma']
      refine'
        sum_le_sum_of_subset_of_nonneg _ fun x _ _ =>
          prod_nonneg fun j _ => mul_nonneg hr (mul_nonneg (pow_nonneg ha _) (hp _))
      rintro ⟨k, c⟩ hd
      simp only [Set.mem_toFinset (s := {c | 1 < Composition.length c}), mem_Ico, mem_sigma,
        Set.mem_setOf_eq] at hd
      simp only [mem_compPartialSumTarget_iff]
      refine' ⟨hd.2, c.length_le.trans_lt hd.1.2, fun j => _⟩
      have : c ≠ Composition.single k (zero_lt_two.trans_le hd.1.1) := by
        simp [Composition.eq_single_iff_length, ne_of_gt hd.2]
      rw [Composition.ne_single_iff] at this
      exact (this j).trans_le (Nat.lt_succ_iff.mp hd.1.2)
    _ = ∑ e in compPartialSumSource 2 (n + 1) n, ∏ j : Fin e.1, r * (a ^ e.2 j * p (e.2 j)) := by
      symm
      apply compChangeOfVariables_sum
      rintro ⟨k, blocks_fun⟩ H
      have K : (compChangeOfVariables 2 (n + 1) n ⟨k, blocks_fun⟩ H).snd.length = k := by simp
      congr 2 <;> try rw [K]
      rw [Fin.heq_fun_iff K.symm]
      intro j
      rw [compChangeOfVariables_blocksFun]
    _ = ∑ j in Ico 2 (n + 1), r ^ j * (∑ k in Ico 1 n, a ^ k * p k) ^ j := by
      rw [compPartialSumSource,
        ← sum_sigma' (Ico 2 (n + 1))
          (fun k : ℕ => (Fintype.piFinset fun _ : Fin k => Ico 1 n : Finset (Fin k → ℕ)))
          (fun n e => ∏ j : Fin n, r * (a ^ e j * p (e j)))]
      congr! with j
      simp only [← @MultilinearMap.mkPiAlgebra_apply ℝ (Fin j) _ ℝ]
      simp only [←
        MultilinearMap.map_sum_finset (MultilinearMap.mkPiAlgebra ℝ (Fin j) ℝ) fun _ (m : ℕ) =>
          r * (a ^ m * p m)]
      

Proof state:
case a
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
p : ℕ → ℝ
hp : ∀ (k : ℕ), 0 ≤ p k
r a : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
j : ℕ
a✝ : j ∈ Ico 2 (n + 1)
⊢ ((MultilinearMap.mkPiAlgebra ℝ (Fin j) ℝ) fun i => ∑ j in Ico 1 n, r * (a ^ j * p j)) =
    r ^ j * (∑ k in Ico 1 n, a ^ k * p k) ^ j

Next tactic:
simp only [MultilinearMap.mkPiAlgebra_apply]