-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
m : Multiset M₀
hm : ∀ y ∈ m, Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} y
hprod : m.prod = x * y
empty :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm :
      ∀ (y : M₀), Membership.mem (γ := Multiset M₀) 0 y → Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} y)
    (hprod : Multiset.prod 0 = x * y), Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
cons :
  ∀ (c : M₀) (s : Multiset M₀)
    (hind :
      ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
        (hm : ∀ y ∈ s, Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} y) (hprod : s.prod = x * y),
        Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x)
    (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ c ::ₘ s, Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} y) (hprod : (c ::ₘ s).prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x

Tactic:
induction m using Multiset.induction generalizing x y with
  | empty =>
    apply subset_closure
    push _ ∈ _
    simp only [Multiset.prod_zero] at hprod
    left; exact .of_mul_eq_one _ hprod.symm
  | cons c s hind =>
    simp only [Multiset.mem_cons, forall_eq_or_imp, Set.mem_setOf] at hm
    simp only [Multiset.prod_cons] at hprod
    simp only [Set.mem_setOf_eq] at hind
    obtain ⟨ha₁ | ha₂, hs⟩ := hm
    · rcases ha₁.exists_right_inv with ⟨k, hk⟩
      refine hind x (y * k) ?_ hs ?_
      · simp only [← mul_assoc, ← hprod, ← Multiset.prod_cons, mul_comm]
        refine multiset_prod_mem _ _ (Multiset.forall_mem_cons.2 ⟨subset_closure ?_,
          Multiset.forall_mem_cons.2 ⟨subset_closure ?_, fun t ht => subset_closure (hs t ht)⟩⟩)
        · left; exact .of_mul_eq_one_right _ hk
        · left; exact ha₁
      · rw [← mul_one s.prod, ← hk, ← mul_assoc, ← mul_assoc, mul_eq_mul_right_iff, mul_comm]
        left; exact hprod
    · rcases ha₂.dvd_mul.1 (Dvd.intro _ hprod) with ⟨c, hc⟩ | ⟨c, hc⟩
      · rw [hc]; rw [hc, mul_assoc] at hprod
        refine Submonoid.mul_mem _ (subset_closure ?_)
          (hind _ _ ?_ hs (mul_left_cancel₀ ha₂.ne_zero hprod))
        · right; exact ha₂
        rw [← mul_left_cancel₀ ha₂.ne_zero hprod]
        exact multiset_prod_mem _ _ (fun t ht => subset_closure (hs t ht))
      rw [hc, mul_comm x _, mul_assoc, mul_comm c _] at hprod
      refine hind x c ?_ hs (mul_left_cancel₀ ha₂.ne_zero hprod)
      rw [← mul_left_cancel₀ ha₂.ne_zero hprod]
      exact multiset_prod_mem _ _ (fun t ht => subset_closure (hs t ht))

Premises used: Multiset.prod_zero, Multiset.prod_cons, mul_assoc, mul_eq_mul_right_iff, Set.mem_setOf_eq, forall_eq_or_imp, Set.mem_setOf, Multiset.mem_cons, mul_comm, Submonoid.subset_closure