theorem of.zero_exact_aux [Nonempty ι] [IsDirected ι (· ≤ ·)] {x : FreeCommRing (Σi, G i)}
    (H : (Ideal.Quotient.mk _ x : DirectLimit G fun i j h => f' i j h)
        = (0 : DirectLimit G fun i j h => f' i j h)) :
    ∃ j s,
      ∃ H : ∀ k : Σi, G i, k ∈ s → k.1 ≤ j,
        IsSupported x s ∧
          lift (fun ix : s => f' ix.1.1 j (H ix ix.2) ix.1.2) (restriction s x) = (0 : G j) := by
  refine' span_induction (Ideal.Quotient.eq_zero_iff_mem.1 H) _ _ _ _
  · rintro x (⟨i, j, hij, x, rfl⟩ | ⟨i, rfl⟩ | ⟨i, x, y, rfl⟩ | ⟨i, x, y, rfl⟩)
    · refine'
        ⟨j, {⟨i, x⟩, ⟨j, f' i j hij x⟩}, _,
          isSupported_sub (isSupported_of.2 <| Or.inr rfl) (isSupported_of.2 <| Or.inl rfl), _⟩
      · rintro k (rfl | ⟨rfl | _⟩)
        exact hij
        rfl
      · rw [(restriction _).map_sub, RingHom.map_sub, restriction_of, dif_pos,
          restriction_of, dif_pos, lift_of, lift_of]
        dsimp only
        have := DirectedSystem.map_map (fun i j h => f' i j h) hij (le_refl j : j ≤ j)
        rw [this]
        exact sub_self _
        exacts [Or.inr rfl, Or.inl rfl]
    · refine' ⟨i, {⟨i, 1⟩}, _, isSupported_sub (isSupported_of.2 rfl) isSupported_one, _⟩
      · rintro k (rfl | h)
        rfl
        -- porting note: the Lean3 proof contained `rw [restriction_of]`, but this
        -- lemma does not seem to work here
      · rw [RingHom.map_sub, RingHom.map_sub]
        erw [lift_of, dif_pos rfl, RingHom.map_one, lift_of, RingHom.map_one, sub_self]
    · refine'
        ⟨i, {⟨i, x + y⟩, ⟨i, x⟩, ⟨i, y⟩}, _,
          isSupported_sub (isSupported_of.2 <| Or.inl rfl)
            (isSupported_add (isSupported_of.2 <| Or.inr <| Or.inl rfl)
              (isSupported_of.2 <| Or.inr <| Or.inr rfl)),
          _⟩
      · rintro k (rfl | ⟨rfl | ⟨rfl | hk⟩⟩) <;> rfl
      · rw [(restriction _).map_sub, (restriction _).map_add, restriction_of, restriction_of,
          restriction_of, dif_pos, dif_pos, dif_pos, RingHom.map_sub,
          (FreeCommRing.lift _).map_add, lift_of, lift_of, lift_of]
        dsimp only
        rw [(f' i i _).map_add]
        exact sub_self _
        all_goals tauto
    · refine'
        ⟨i, {⟨i, x * y⟩, ⟨i, x⟩, ⟨i, y⟩}, _,
          isSupported_sub (isSupported_of.2 <| Or.inl rfl)
            (isSupported_mul (isSupported_of.2 <| Or.inr <| Or.inl rfl)
              (isSupported_of.2 <| Or.inr <| Or.inr rfl)), _⟩
      · rintro k (rfl | ⟨rfl | ⟨rfl | hk⟩⟩) <;> rfl
      · rw [(restriction _).map_sub, (restriction _).map_mul, restriction_of, restriction_of,
          restriction_of, dif_pos, dif_pos, dif_pos, RingHom.map_sub,
          (FreeCommRing.lift _).map_mul, lift_of, lift_of, lift_of]
        dsimp only
        rw [(f' i i _).map_mul]
        exact sub_self _
        all_goals tauto
        -- porting note: was
        --exacts [sub_self _, Or.inl rfl, Or.inr (Or.inr rfl), Or.inr (Or.inl rfl)]
  · refine' Nonempty.elim (by infer_instance) fun ind : ι => _
    refine' ⟨ind, ∅, fun _ => False.elim, isSupported_zero, _⟩
    -- porting note: `RingHom.map_zero` was `(restriction _).map_zero`
    rw [RingHom.map_zero, (FreeCommRing.lift _).map_zero]
  · rintro x y ⟨i, s, hi, hxs, ihs⟩ ⟨j, t, hj, hyt, iht⟩
    obtain ⟨k, hik, hjk⟩ := exists_ge_ge i j
    have : ∀ z : Σi, G i, z ∈ s ∪ t → z.1 ≤ k := by
      rintro z (hz | hz)
      exact le_trans (hi z hz) hik
      exact le_trans (hj z hz) hjk
    refine'
      ⟨k, s ∪ t, this,
        isSupported_add (isSupported_upwards hxs <| Set.subset_union_left s t)
          (isSupported_upwards hyt <| Set.subset_union_right s t), _⟩
    · -- porting note: was `(restriction _).map_add`
      rw [RingHom.map_add, (FreeCommRing.lift _).map_add, ←
        of.zero_exact_aux2 G f' hxs hi this hik (Set.subset_union_left s t), ←
        of.zero_exact_aux2 G f' hyt hj this hjk (Set.subset_union_right s t), ihs,
        (f' i k hik).map_zero, iht, (f' j k hjk).map_zero, zero_add]
  · rintro x y ⟨j, t, hj, hyt, iht⟩
    rw [smul_eq_mul]
    rcases exists_finset_support x with ⟨s, hxs⟩
    rcases (s.image Sigma.fst).exists_le with ⟨i, hi⟩
    obtain ⟨k, hik, hjk⟩ := exists_ge_ge i j
    have : ∀ z : Σi, G i, z ∈ ↑s ∪ t → z.1 ≤ k := by
      rintro z (hz | hz)
      exacts [(hi z.1 <| Finset.mem_image.2 ⟨z, hz, rfl⟩).trans hik, (hj z hz).trans hjk]
    

Proof state:
case refine'_4.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u
inst✝⁵ : Ring R
ι : Type v
dec_ι : DecidableEq ι
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → CommRing (G i)
f : (i j : ι) → i ≤ j → G i → G j
f' : (i j : ι) → i ≤ j → G i →+* G j
inst✝² : DirectedSystem G fun i j h => ⇑(f' i j h)
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
x✝ : FreeCommRing ((i : ι) × G i)
H :
  (Ideal.Quotient.mk
        (Ideal.span
          {a |
            (∃ i j,
                ∃ (H : i ≤ j),
                  ∃ x,
                    FreeCommRing.of { fst := j, snd := (fun i j h => ⇑(f' i j h)) i j H x } -
                        FreeCommRing.of { fst := i, snd := x } =
                      a) ∨
              (∃ i, FreeCommRing.of { fst := i, snd := 1 } - 1 = a) ∨
                (∃ i x y,
                    FreeCommRing.of { fst := i, snd := x + y } -
                        (FreeCommRing.of { fst := i, snd := x } + FreeCommRing.of { fst := i, snd := y }) =
                      a) ∨
                  ∃ i x y,
                    FreeCommRing.of { fst := i, snd := x * y } -
                        FreeCommRing.of { fst := i, snd := x } * FreeCommRing.of { fst := i, snd := y } =
                      a}))
      x✝ =
    0
x y : FreeCommRing ((i : ι) × G i)
j : ι
t : Set ((i : ι) × G i)
hj : ∀ k ∈ t, k.fst ≤ j
hyt : IsSupported y t
iht : (lift fun ix => (f' (↑ix).fst j (_ : (↑ix).fst ≤ j)) (↑ix).snd) ((restriction t) y) = 0
s : Finset ((i : ι) × G i)
hxs : IsSupported x ↑s
i : ι
hi : ∀ i_1 ∈ Finset.image Sigma.fst s, i_1 ≤ i
k : ι
hik : i ≤ k
hjk : j ≤ k
this : ∀ z ∈ ↑s ∪ t, z.fst ≤ k
⊢ ∃ j s,
    ∃ (H : ∀ k ∈ s, k.fst ≤ j),
      IsSupported (x * y) s ∧
        (lift fun ix => (f' (↑ix).fst j (_ : (↑ix).fst ≤ j)) (↑ix).snd) ((restriction s) (x * y)) = 0

Next tactic:
refine'
      ⟨k, ↑s ∪ t, this,
        isSupported_mul (isSupported_upwards hxs <| Set.subset_union_left (↑s) t)
          (isSupported_upwards hyt <| Set.subset_union_right (↑s) t), _⟩