simp only [Multiset.not_mem_zero, Set.mem_setOf_eq, IsEmpty.forall_iff, forall_const,
  Multiset.prod_zero, forall_true_left, forall_eq']
case intro.intro.refine'_1
R : Type u_1
inst✝ : CommMonoidWithZero R
l : Multiset R
⊢ (fun l =>
      ∀ (a : R),
        (∀ y ∈ l, y ∈ {r | IsUnit r ∨ Prime r}) →
          Multiset.prod l = a → ∃ f, (∀ b ∈ f, Prime b) ∧ Associated (Multiset.prod f) a)
    0
case intro.intro.refine'_1
R : Type u_1
inst✝ : CommMonoidWithZero R
l : Multiset R
⊢ ∃ f, (∀ b ∈ f, Prime b) ∧ Associated (Multiset.prod f) 1