-- Module: Mathlib.LinearAlgebra.Multilinear.Basic
Goal:
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst : Semiring R
inst_1 : (i : ι) → AddCommMonoid (M₁ i)
inst_2 : AddCommMonoid M₂
inst_3 : (i : ι) → Module.{uR, v₁} R (M₁ i)
inst_4 : Module R M₂
f : MultilinearMap R M₁ M₂
α : ι → Type u_1
g : (i : ι) → α i → M₁ i
A : (i : ι) → Finset (α i)
inst_5 : DecidableEq ι
inst_6 : Fintype ι
n : ℕ
h : ∑ i : ι, #(A i) = n
this : (i : ι) → DecidableEq (α i) := fun (i : ι) ↦ Classical.decEq (α i)
h₁ :
  ∀ (n : ℕ)
    (IH :
      ∀ m < n,
        ∀ (A : (i : ι) → Finset (α i)) (h : ∑ i : ι, #(A i) = m),
          Eq.{v₂ + 1} (α := M₂) ((f : (a : (i : ι) → M₁ i) → M₂) fun (i : ι) ↦ ∑ j ∈ A i, g i j : M₂)
            (∑ r ∈ piFinset (δ := α) A, (f : (a : (i : ι) → M₁ i) → M₂) fun (i : ι) ↦ g i (r i) : M₂))
    (A : (i : ι) → Finset (α i)) (h : ∑ i : ι, #(A i) = n),
    Eq.{v₂ + 1} (α := M₂) ((f : (a : (i : ι) → M₁ i) → M₂) fun (i : ι) ↦ ∑ j ∈ A i, g i j : M₂)
      (∑ r ∈ piFinset (δ := α) A, (f : (a : (i : ι) → M₁ i) → M₂) fun (i : ι) ↦ g i (r i) : M₂)
⊢ Eq.{v₂ + 1} (α := M₂) ((f : (a : (i : ι) → M₁ i) → M₂) fun (i : ι) ↦ ∑ j ∈ A i, g i j : M₂)
    (∑ r ∈ piFinset (δ := α) A, (f : (a : (i : ι) → M₁ i) → M₂) fun (i : ι) ↦ g i (r i) : M₂)

Tactic:
induction n using Nat.strong_induction_on generalizing A with | h n IH =>
  -- If one of the sets is empty, then all the sums are zero
  by_cases! Ai_empty : ∃ i, A i = ∅
  · obtain ⟨i, hi⟩ : ∃ i, ∑ j ∈ A i, g i j = 0 := Ai_empty.imp fun i hi ↦ by simp [hi]
    have hpi : piFinset A = ∅ := by simpa
    rw [f.map_coord_zero i hi, hpi, Finset.sum_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) ≤ 1
  · have Ai_card : ∀ i, #(A i) = 1 := by
      intro i
      have pos : #(A i) ≠ 0 := by rw [Finset.card_ne_zero]; exact Ai_empty i
      have : #(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 ∈ 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.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`.
  obtain ⟨i₀, hi₀⟩ : ∃ i, 1 < #(A i) := 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 [B, sdiff_subset, update_self]
    · simp only [B, hi, update_of_ne, Ne, 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 [C, hj₂, Finset.singleton_subset_iff, update_self]
    · simp only [C, hi, update_of_ne, Ne, 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 ∈ A i, g i j) =
      Function.update (fun i => ∑ j ∈ A i, g i j) i₀
        ((∑ j ∈ B i₀, g i₀ j) + ∑ j ∈ C i₀, g i₀ j) := by
    ext i
    by_cases hi : i = i₀
    · rw [hi, update_self]
      have : A i₀ = B i₀ ∪ C i₀ := by
        simp only [B, C, Function.update_self, 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 [C] using hj
      rw [this]
      simp only [B, mem_sdiff, not_true, not_false_iff, Finset.mem_singleton,
        update_self, and_false]
    · simp [hi]
  have Beq :
    Function.update (fun i => ∑ j ∈ A i, g i j) i₀ (∑ j ∈ B i₀, g i₀ j) = fun i =>
      ∑ j ∈ B i, g i j := by
    ext i
    by_cases hi : i = i₀
    · rw [hi]
      simp only [update_self]
    · simp only [B, hi, update_of_ne, Ne, not_false_iff]
  have Ceq :
    Function.update (fun i => ∑ j ∈ A i, g i j) i₀ (∑ j ∈ C i₀, g i₀ j) = fun i =>
      ∑ j ∈ C i, g i j := by
    ext i
    by_cases hi : i = i₀
    · rw [hi]
      simp only [update_self]
    · simp only [C, hi, update_of_ne, Ne, not_false_iff]
  -- Express the inductive assumption for `B`
  have Brec : (f fun i => ∑ j ∈ B i, g i j) = ∑ r ∈ piFinset B, f fun i => g i (r i) := by
    have : ∑ i, #(B i) < ∑ i, #(A i) := by
      refine sum_lt_sum (fun i _ => card_le_card (B_subset_A i)) ⟨i₀, mem_univ _, ?_⟩
      have : {j₂} ⊆ A i₀ := by simp [hj₂]
      simp only [B, Finset.card_sdiff_of_subset this, Function.update_self, 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 ∈ C i, g i j) = ∑ r ∈ piFinset C, f fun i => g i (r i) := by
    have : (∑ i, #(C i)) < ∑ i, #(A i) :=
      Finset.sum_lt_sum (fun i _ => Finset.card_le_card (C_subset_A i))
        ⟨i₀, Finset.mem_univ _, by simp [C, hi₀]⟩
    rw [h] at this
    exact IH _ this C rfl
  have D : Disjoint (piFinset B) (piFinset C) :=
    haveI : Disjoint (B i₀) (C i₀) := by simp [B, C]
    piFinset_disjoint_of_disjoint B C this
  have pi_BC : piFinset A = piFinset B ∪ piFinset C := by
    apply Finset.Subset.antisymm
    · intro r hr
      by_cases hri₀ : r i₀ = j₂
      · apply Finset.mem_union_right
        refine mem_piFinset.2 fun i => ?_
        by_cases hi : i = i₀
        · have : r i₀ ∈ C i₀ := by simp [C, hri₀]
          rwa [hi]
        · simp [C, hi, mem_piFinset.1 hr i]
      · apply Finset.mem_union_left
        refine mem_piFinset.2 fun i => ?_
        by_cases hi : i = i₀
        · have : r i₀ ∈ B i₀ := by simp [B, hri₀, mem_piFinset.1 hr i₀]
          rwa [hi]
        · simp [B, hi, mem_piFinset.1 hr i]
    · exact
        Finset.union_subset (piFinset_subset _ _ fun i => B_subset_A i)
          (piFinset_subset _ _ fun i => C_subset_A i)
  rw [A_eq_BC]
  simp only [MultilinearMap.map_update_add, Beq, Ceq, Brec, Crec, pi_BC]
  rw [← Finset.sum_union D]

Premises used: MultilinearMap.map_update_add, Fintype.card_piFinset, Finset.prod_const_one, Finset.sum_empty, one_nsmul, Finset.sum_congr, rfl, Finset.sum_const