set_option maxHeartbeats 300000 in
/-- If two formal multilinear series have positive radius of convergence, then the terms appearing
in the definition of their composition are also summable (when multiplied by a suitable positive
geometric term). -/
theorem comp_summable_nnreal (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
    (hq : 0 < q.radius) (hp : 0 < p.radius) :
    ∃ r > (0 : ℝ≥0),
      Summable fun i : Σ n, Composition n => ‖q.compAlongComposition p i.2‖₊ * r ^ i.1 := by
  /- This follows from the fact that the growth rate of `‖qₙ‖` and `‖pₙ‖` is at most geometric,
    giving a geometric bound on each `‖q.comp_along_composition p op‖`, together with the
    fact that there are `2^(n-1)` compositions of `n`, giving at most a geometric loss. -/
  rcases ENNReal.lt_iff_exists_nnreal_btwn.1 (lt_min zero_lt_one hq) with ⟨rq, rq_pos, hrq⟩
  rcases ENNReal.lt_iff_exists_nnreal_btwn.1 (lt_min zero_lt_one hp) with ⟨rp, rp_pos, hrp⟩
  simp only [lt_min_iff, ENNReal.coe_lt_one_iff, ENNReal.coe_pos] at hrp hrq rp_pos rq_pos
  obtain ⟨Cq, _hCq0, hCq⟩ : ∃ Cq > 0, ∀ n, ‖q n‖₊ * rq ^ n ≤ Cq :=
    q.nnnorm_mul_pow_le_of_lt_radius hrq.2
  obtain ⟨Cp, hCp1, hCp⟩ : ∃ Cp ≥ 1, ∀ n, ‖p n‖₊ * rp ^ n ≤ Cp := by
    rcases p.nnnorm_mul_pow_le_of_lt_radius hrp.2 with ⟨Cp, -, hCp⟩
    exact ⟨max Cp 1, le_max_right _ _, fun n => (hCp n).trans (le_max_left _ _)⟩
  let r0 : ℝ≥0 := (4 * Cp)⁻¹
  have r0_pos : 0 < r0 := inv_pos.2 (mul_pos zero_lt_four (zero_lt_one.trans_le hCp1))
  set r : ℝ≥0 := rp * rq * r0
  have r_pos : 0 < r := mul_pos (mul_pos rp_pos rq_pos) r0_pos
  have I :
    ∀ i : Σ n : ℕ, Composition n, ‖q.compAlongComposition p i.2‖₊ * r ^ i.1 ≤ Cq / 4 ^ i.1 := by
    rintro ⟨n, c⟩
    have A
    calc
      ‖q c.length‖₊ * rq ^ n ≤ ‖q c.length‖₊ * rq ^ c.length :=
        mul_le_mul' le_rfl (pow_le_pow_of_le_one rq.2 hrq.1.le c.length_le)
      _ ≤ Cq := hCq _
    have B
    calc
      (∏ i, ‖p (c.blocksFun i)‖₊) * rp ^ n = ∏ i, ‖p (c.blocksFun i)‖₊ * rp ^ c.blocksFun i := by
        simp only [Finset.prod_mul_distrib, Finset.prod_pow_eq_pow_sum, c.sum_blocksFun]
      _ ≤ ∏ _i : Fin c.length, Cp := (Finset.prod_le_prod' fun i _ => hCp _)
      _ = Cp ^ c.length := by simp
      _ ≤ Cp ^ n := pow_le_pow_right hCp1 c.length_le
    calc
      ‖q.compAlongComposition p c‖₊ * r ^ n ≤
          (‖q c.length‖₊ * ∏ i, ‖p (c.blocksFun i)‖₊) * r ^ n :=
        mul_le_mul' (q.compAlongComposition_nnnorm p c) le_rfl
      _ = ‖q c.length‖₊ * rq ^ n * ((∏ i, ‖p (c.blocksFun i)‖₊) * rp ^ n) * r0 ^ n := by
        simp only [mul_pow]; ring
      _ ≤ Cq * Cp ^ n * r0 ^ n := (mul_le_mul' (mul_le_mul' A B) le_rfl)
      _ = Cq / 4 ^ n := by
        simp only
        field_simp [mul_pow, (zero_lt_one.trans_le hCp1).ne']
        ring
  refine' ⟨r, r_pos, NNReal.summable_of_le I _⟩
  simp_rw [div_eq_mul_inv]
  refine' Summable.mul_left _ _
  have : ∀ n : ℕ, HasSum (fun c : Composition n => (4 ^ n : ℝ≥0)⁻¹) (2 ^ (n - 1) / 4 ^ n) := by
    intro n
    convert hasSum_fintype fun c : Composition n => (4 ^ n : ℝ≥0)⁻¹
    simp [Finset.card_univ, composition_card, div_eq_mul_inv]
  refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, (NNReal.summable_nat_add_iff 1).1 _⟩
  convert (NNReal.summable_geometric (NNReal.div_lt_one_of_lt one_lt_two)).mul_left (1 / 4) using 1
  ext1 n
  rw [(this _).tsum_eq, add_tsub_cancel_right]
  

Proof state:
case h.e'_5.h
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
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
hq : 0 < radius q
hp : 0 < radius p
rq rp : ℝ≥0
hrp : rp < 1 ∧ ↑rp < radius p
hrq : rq < 1 ∧ ↑rq < radius q
rp_pos : 0 < rp
rq_pos : 0 < rq
Cq : ℝ≥0
_hCq0 : Cq > 0
hCq : ∀ (n : ℕ), ‖q n‖₊ * rq ^ n ≤ Cq
Cp : ℝ≥0
hCp1 : Cp ≥ 1
hCp : ∀ (n : ℕ), ‖p n‖₊ * rp ^ n ≤ Cp
r0 : ℝ≥0 := (4 * Cp)⁻¹
r0_pos : 0 < r0
r : ℝ≥0 := rp * rq * r0
r_pos : 0 < r
I : ∀ (i : (n : ℕ) × Composition n), ‖compAlongComposition q p i.snd‖₊ * r ^ i.fst ≤ Cq / 4 ^ i.fst
this : ∀ (n : ℕ), HasSum (fun c => (4 ^ n)⁻¹) (2 ^ (n - 1) / 4 ^ n)
n : ℕ
⊢ 2 ^ n / 4 ^ (n + 1) = 1 / 4 * (1 / 2) ^ n

Next tactic:
field_simp [← mul_assoc, pow_succ', mul_pow, show (4 : ℝ≥0) = 2 * 2 by norm_num,
    mul_right_comm]