/-- The left inverse to a formal multilinear series is indeed a left inverse, provided its linear
term is invertible. -/
theorem leftInv_comp (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
    (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) : (leftInv p i).comp p = id 𝕜 E := by
  ext (n v)
  match n with
  | 0 =>
    simp only [leftInv_coeff_zero, ContinuousMultilinearMap.zero_apply, id_apply_ne_one, Ne.def,
      not_false_iff, zero_ne_one, comp_coeff_zero']
  | 1 =>
    simp only [leftInv_coeff_one, comp_coeff_one, h, id_apply_one, ContinuousLinearEquiv.coe_apply,
      ContinuousLinearEquiv.symm_apply_apply, continuousMultilinearCurryFin1_symm_apply]
  | n + 2 =>
    have A :
      (Finset.univ : Finset (Composition (n + 2))) =
        {c | Composition.length c < n + 2}.toFinset ∪ {Composition.ones (n + 2)} := by
      refine' Subset.antisymm (fun c _ => _) (subset_univ _)
      by_cases h : c.length < n + 2
      · simp [h, Set.mem_toFinset (s := {c | Composition.length c < n + 2})]
      · simp [Composition.eq_ones_iff_le_length.2 (not_lt.1 h)]
    have B :
      Disjoint ({c | Composition.length c < n + 2} : Set (Composition (n + 2))).toFinset
        {Composition.ones (n + 2)} := by
      simp [Set.mem_toFinset (s := {c | Composition.length c < n + 2})]
    have C :
      ((p.leftInv i (Composition.ones (n + 2)).length)
          fun j : Fin (Composition.ones n.succ.succ).length =>
          p 1 fun _ => v ((Fin.castLE (Composition.length_le _)) j)) =
        p.leftInv i (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j := by
      apply FormalMultilinearSeries.congr _ (Composition.ones_length _) fun j hj1 hj2 => ?_
      exact FormalMultilinearSeries.congr _ rfl fun k _ _ => by congr
    have D :
      (p.leftInv i (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j) =
        -∑ c : Composition (n + 2) in {c : Composition (n + 2) | c.length < n + 2}.toFinset,
            (p.leftInv i c.length) (p.applyComposition c v) := by
      simp only [leftInv, ContinuousMultilinearMap.neg_apply, neg_inj,
        ContinuousMultilinearMap.sum_apply]
      convert
        (sum_toFinset_eq_subtype
          (fun c : Composition (n + 2) => c.length < n + 2)
          (fun c : Composition (n + 2) =>
          (ContinuousMultilinearMap.compAlongComposition
            (p.compContinuousLinearMap (i.symm : F →L[𝕜] E)) c (p.leftInv i c.length))
            fun j : Fin (n + 2) => p 1 fun _ : Fin 1 => v j)).symm.trans
          _
      simp only [compContinuousLinearMap_applyComposition,
        ContinuousMultilinearMap.compAlongComposition_apply]
      congr
      ext c
      congr
      ext k
      simp [h, Function.comp]
    

Proof state:
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
h : p 1 = (LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 𝕜 E F)) ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
A : univ = Set.toFinset {c | Composition.length c < n + 2} ∪ {Composition.ones (n + 2)}
B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}
C :
  ((leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j =>
      (p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) ≤ n + 2) j)) =
    (leftInv p i (n + 2)) fun j => (p 1) fun x => v j
D :
  ((leftInv p i (n + 2)) fun j => (p 1) fun x => v j) =
    -∑ c in Set.toFinset {c | Composition.length c < n + 2},
        (leftInv p i (Composition.length c)) (applyComposition p c v)
⊢ (FormalMultilinearSeries.comp (leftInv p i) p (n + 2)) v = (id 𝕜 E (n + 2)) v

Next tactic:
simp [FormalMultilinearSeries.comp, show n + 2 ≠ 1 by norm_num, A, Finset.sum_union B,
      applyComposition_ones, C, D, -Set.toFinset_setOf]