/-- If `f` is multilinear, then `f (Σ_{j₁ ∈ A₁} g₁ j₁, ..., Σ_{jₙ ∈ Aₙ} gₙ jₙ)` is the sum of
`f (g₁ (r 1), ..., gₙ (r n))` where `r` ranges over all functions with `r 1 ∈ A₁`, ...,
`r n ∈ Aₙ`. This follows from multilinearity by expanding successively with respect to each
coordinate. Here, we give an auxiliary statement tailored for an inductive proof. Use instead
`map_sum_finset`. -/
theorem map_sum_finset_aux [DecidableEq ι] [Fintype ι] {n : ℕ} (h : (∑ i, (A i).card) = n) :
    (f fun i => ∑ j in A i, g i j) = ∑ r in piFinset A, f fun i => g i (r i) := by
  letI := fun i => Classical.decEq (α i)
  induction' n using Nat.strong_induction_on with n IH generalizing A
  -- If one of the sets is empty, then all the sums are zero
  by_cases Ai_empty : ∃ i, A i = ∅
  · rcases Ai_empty with ⟨i, hi⟩
    have : ∑ j in A i, g i j = 0 := by rw [hi, Finset.sum_empty]
    rw [f.map_coord_zero i this]
    have : piFinset A = ∅ := by
      refine Finset.eq_empty_of_forall_not_mem fun r hr => ?_
      have : r i ∈ A i := mem_piFinset.mp hr i
      simp [hi] at this
    rw [this, Finset.sum_empty]
  push_neg at Ai_empty
  -- Otherwise, if all sets are at most singletons, then they are exactly singletons and the result
  -- is again straightforward
  by_cases Ai_singleton : ∀ i, (A i).card ≤ 1
  · have Ai_card : ∀ i, (A i).card = 1 := by
      intro i
      have pos : Finset.card (A i) ≠ 0 := by simp [Finset.card_eq_zero, Ai_empty i]
      have : Finset.card (A i) ≤ 1 := Ai_singleton i
      exact le_antisymm this (Nat.succ_le_of_lt (_root_.pos_iff_ne_zero.mpr pos))
    have :
      ∀ r : ∀ i, α i, r ∈ piFinset A → (f fun i => g i (r i)) = f fun i => ∑ j in A i, g i j := by
      intro r hr
      congr with i
      have : ∀ j ∈ A i, g i j = g i (r i) := by
        intro j hj
        congr
        apply Finset.card_le_one_iff.1 (Ai_singleton i) hj
        exact mem_piFinset.mp hr i
      simp only [Finset.sum_congr rfl this, Finset.mem_univ, Finset.sum_const, Ai_card i, one_nsmul]
    simp only [Finset.sum_congr rfl this, Ai_card, card_piFinset, prod_const_one, one_nsmul,
      Finset.sum_const]
  -- Remains the interesting case where one of the `A i`, say `A i₀`, has cardinality at least 2.
  -- We will split into two parts `B i₀` and `C i₀` of smaller cardinality, let `B i = C i = A i`
  -- for `i ≠ i₀`, apply the inductive assumption to `B` and `C`, and add up the corresponding
  -- parts to get the sum for `A`.
  push_neg at Ai_singleton
  obtain ⟨i₀, hi₀⟩ : ∃ i, 1 < (A i).card := Ai_singleton
  obtain ⟨j₁, j₂, _, hj₂, _⟩ : ∃ j₁ j₂, j₁ ∈ A i₀ ∧ j₂ ∈ A i₀ ∧ j₁ ≠ j₂ :=
    Finset.one_lt_card_iff.1 hi₀
  let B := Function.update A i₀ (A i₀ \ {j₂})
  let C := Function.update A i₀ {j₂}
  have B_subset_A : ∀ i, B i ⊆ A i := by
    intro i
    by_cases hi : i = i₀
    · rw [hi]
      simp only [sdiff_subset, update_same]
    · simp only [hi, update_noteq, Ne.def, not_false_iff, Finset.Subset.refl]
  have C_subset_A : ∀ i, C i ⊆ A i := by
    intro i
    by_cases hi : i = i₀
    · rw [hi]
      simp only [hj₂, Finset.singleton_subset_iff, update_same]
    · simp only [hi, update_noteq, Ne.def, not_false_iff, Finset.Subset.refl]
  -- split the sum at `i₀` as the sum over `B i₀` plus the sum over `C i₀`, to use additivity.
  have A_eq_BC :
    (fun i => ∑ j in A i, g i j) =
      Function.update (fun i => ∑ j in A i, g i j) i₀
        ((∑ j in B i₀, g i₀ j) + ∑ j in C i₀, g i₀ j) := by
    ext i
    by_cases hi : i = i₀
    · rw [hi, update_same]
      have : A i₀ = B i₀ ∪ C i₀ := by
        simp only [Function.update_same, Finset.sdiff_union_self_eq_union]
        symm
        simp only [hj₂, Finset.singleton_subset_iff, Finset.union_eq_left]
      rw [this]
      refine Finset.sum_union <| Finset.disjoint_right.2 fun j hj => ?_
      have : j = j₂ := by
        simpa using hj
      rw [this]
      simp only [mem_sdiff, eq_self_iff_true, not_true, not_false_iff, Finset.mem_singleton,
        update_same, and_false_iff]
    · simp [hi]
  have Beq :
    Function.update (fun i => ∑ j in A i, g i j) i₀ (∑ j in B i₀, g i₀ j) = fun i =>
      ∑ j in B i, g i j := by
    ext i
    by_cases hi : i = i₀
    · rw [hi]
      simp only [update_same]
    · simp only [hi, update_noteq, Ne.def, not_false_iff]
  have Ceq :
    Function.update (fun i => ∑ j in A i, g i j) i₀ (∑ j in C i₀, g i₀ j) = fun i =>
      ∑ j in C i, g i j := by
    ext i
    by_cases hi : i = i₀
    · rw [hi]
      simp only [update_same]
    · simp only [hi, update_noteq, Ne.def, not_false_iff]
  -- Express the inductive assumption for `B`
  have Brec : (f fun i => ∑ j in B i, g i j) = ∑ r in piFinset B, f fun i => g i (r i) := by
    have : (∑ i, Finset.card (B i)) < ∑ i, Finset.card (A i) := by
      refine'
        Finset.sum_lt_sum (fun i _ => Finset.card_le_of_subset (B_subset_A i))
          ⟨i₀, Finset.mem_univ _, _⟩
      have : {j₂} ⊆ A i₀ := by simp [hj₂]
      simp only [Finset.card_sdiff this, Function.update_same, Finset.card_singleton]
      exact Nat.pred_lt (ne_of_gt (lt_trans Nat.zero_lt_one hi₀))
    rw [h] at this
    exact IH _ this B rfl
  -- Express the inductive assumption for `C`
  have Crec : (f fun i => ∑ j in C i, g i j) = ∑ r in piFinset C, f fun i => g i (r i) := by
    have : (∑ i, Finset.card (C i)) < ∑ i, Finset.card (A i) :=
      Finset.sum_lt_sum (fun i _ => Finset.card_le_of_subset (C_subset_A i))
        ⟨i₀, Finset.mem_univ _, by simp [hi₀]⟩
    rw [h] at this
    

Proof state:
R : Type uR
S : Type uS
ι : Type uι
n✝ : ℕ
M : Fin (Nat.succ n✝) → Type v
M₁ : ι → Type v₁
M₂ : Type v₂
M₃ : Type v₃
M' : Type v'
inst✝¹² : Semiring R
inst✝¹¹ : (i : Fin (Nat.succ n✝)) → AddCommMonoid (M i)
inst✝¹⁰ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁹ : AddCommMonoid M₂
inst✝⁸ : AddCommMonoid M₃
inst✝⁷ : AddCommMonoid M'
inst✝⁶ : (i : Fin (Nat.succ n✝)) → Module R (M i)
inst✝⁵ : (i : ι) → Module R (M₁ i)
inst✝⁴ : Module R M₂
inst✝³ : Module R M₃
inst✝² : Module R M'
f f' : MultilinearMap R M₁ M₂
α : ι → Type u_1
g : (i : ι) → α i → M₁ i
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
this✝ : (i : ι) → DecidableEq (α i) := fun i => Classical.decEq (α i)
n : ℕ
IH :
  ∀ m < n,
    ∀ (A : (i : ι) → Finset (α i)),
      ∑ i : ι, Finset.card (A i) = m → (f fun i => ∑ j in A i, g i j) = ∑ r in piFinset A, f fun i => g i (r i)
A : (i : ι) → Finset (α i)
h : ∑ i : ι, Finset.card (A i) = n
Ai_empty : ∀ (i : ι), A i ≠ ∅
i₀ : ι
hi₀ : 1 < Finset.card (A i₀)
j₁ j₂ : α i₀
left✝ : j₁ ∈ A i₀
hj₂ : j₂ ∈ A i₀
right✝ : j₁ ≠ j₂
B : (a : ι) → Finset (α a) := update A i₀ (A i₀ \ {j₂})
C : (a : ι) → Finset (α a) := update A i₀ {j₂}
B_subset_A : ∀ (i : ι), B i ⊆ A i
C_subset_A : ∀ (i : ι), C i ⊆ A i
A_eq_BC :
  (fun i => ∑ j in A i, g i j) = update (fun i => ∑ j in A i, g i j) i₀ (∑ j in B i₀, g i₀ j + ∑ j in C i₀, g i₀ j)
Beq : update (fun i => ∑ j in A i, g i j) i₀ (∑ j in B i₀, g i₀ j) = fun i => ∑ j in B i, g i j
Ceq : update (fun i => ∑ j in A i, g i j) i₀ (∑ j in C i₀, g i₀ j) = fun i => ∑ j in C i, g i j
Brec : (f fun i => ∑ j in B i, g i j) = ∑ r in piFinset B, f fun i => g i (r i)
this : ∑ i : ι, Finset.card (C i) < n
⊢ (f fun i => ∑ j in C i, g i j) = ∑ r in piFinset C, f fun i => g i (r i)

Next tactic:
exact IH _ this C rfl