refine' l.induction (p := fun l => ∀ a, (∀ y ∈ l, y ∈ {r | IsUnit r ∨ Prime r}) →
  l.prod = a → ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod a) _ _
case intro.intro
R : Type u_1
inst✝ : CommMonoidWithZero R
l : Multiset R
⊢ ∀ (a : R),
    (∀ y ∈ l, y ∈ {r | IsUnit r ∨ Prime r}) →
      Multiset.prod l = a → ∃ f, (∀ b ∈ f, Prime b) ∧ Associated (Multiset.prod f) a
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'_2
R : Type u_1
inst✝ : CommMonoidWithZero R
l : Multiset R
⊢ ∀ ⦃a : R⦄ {s : 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)
        s →
      (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)
        (a ::ₘ s)