Module: Mathlib.LinearAlgebra.Multilinear.Basic

Goal state before:
pp: case neg
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁶ : Semiring R
inst✝⁵ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁴ : AddCommMonoid M₂
inst✝³ : (i : ι) → Module R (M₁ i)
inst✝² : Module R M₂
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, #(A i) = m → (f fun i ↦ ∑ j ∈ A i, g i j) = ∑ r ∈ piFinset A, f fun i ↦ g i (r i)
A : (i : ι) → Finset (α i)
h : ∑ i, #(A i) = n
Ai_empty : ∀ (i : ι), (A i).Nonempty
i₀ : ι
hi₀ : 1 < #(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 ∈ A i, g i j) = update (fun i ↦ ∑ j ∈ A i, g i j) i₀ (∑ j ∈ B i₀, g i₀ j + ∑ j ∈ C i₀, g i₀ j)
Beq : update (fun i ↦ ∑ j ∈ A i, g i j) i₀ (∑ j ∈ B i₀, g i₀ j) = fun i ↦ ∑ j ∈ B i, g i j
Ceq : update (fun i ↦ ∑ j ∈ A i, g i j) i₀ (∑ j ∈ C i₀, g i₀ j) = fun i ↦ ∑ j ∈ C i, g i j
Brec : (f fun i ↦ ∑ j ∈ B i, g i j) = ∑ r ∈ piFinset B, f fun i ↦ g i (r i)
Crec : (f fun i ↦ ∑ j ∈ C i, g i j) = ∑ r ∈ piFinset C, f fun i ↦ g i (r i)
D : Disjoint (piFinset B) (piFinset C)
⊢ (f fun i ↦ ∑ j ∈ A i, g i j) = ∑ r ∈ piFinset A, f fun i ↦ g i (r i)

Tactic:
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)

Goal state after:
case neg
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁶ : Semiring R
inst✝⁵ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁴ : AddCommMonoid M₂
inst✝³ : (i : ι) → Module R (M₁ i)
inst✝² : Module R M₂
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, #(A i) = m → (f fun i ↦ ∑ j ∈ A i, g i j) = ∑ r ∈ piFinset A, f fun i ↦ g i (r i)
A : (i : ι) → Finset (α i)
h : ∑ i, #(A i) = n
Ai_empty : ∀ (i : ι), (A i).Nonempty
i₀ : ι
hi₀ : 1 < #(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 ∈ A i, g i j) = update (fun i ↦ ∑ j ∈ A i, g i j) i₀ (∑ j ∈ B i₀, g i₀ j + ∑ j ∈ C i₀, g i₀ j)
Beq : update (fun i ↦ ∑ j ∈ A i, g i j) i₀ (∑ j ∈ B i₀, g i₀ j) = fun i ↦ ∑ j ∈ B i, g i j
Ceq : update (fun i ↦ ∑ j ∈ A i, g i j) i₀ (∑ j ∈ C i₀, g i₀ j) = fun i ↦ ∑ j ∈ C i, g i j
Brec : (f fun i ↦ ∑ j ∈ B i, g i j) = ∑ r ∈ piFinset B, f fun i ↦ g i (r i)
Crec : (f fun i ↦ ∑ j ∈ C i, g i j) = ∑ r ∈ piFinset C, f fun i ↦ g i (r i)
D : Disjoint (piFinset B) (piFinset C)
pi_BC : piFinset A = piFinset B ∪ piFinset C
⊢ (f fun i ↦ ∑ j ∈ A i, g i j) = ∑ r ∈ piFinset A, f fun i ↦ g i (r i)