/--
Removing the leading `o` from a term of `MaxProducts C` yields a list which `isGood` with respect to
`C'`.
-/
theorem maxTail_isGood (l : MaxProducts C ho)
    (h₁: ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (ord I · < o))))) :
    l.val.Tail.isGood (C' C ho) := by
  -- Write `l.Tail` as a linear combination of smaller products:
  intro h
  rw [Finsupp.mem_span_image_iff_total, ← max_eq_eval C hsC ho] at h
  obtain ⟨m, ⟨hmmem, hmsum⟩⟩ := h
  rw [Finsupp.total_apply] at hmsum

  -- Write the image of `l` under `Linear_CC'` as `Linear_CC'` applied to the linear combination
  -- above, with leading `term I ho`'s added to each term:
  have : (Linear_CC' C hsC ho) (l.val.eval C) = (Linear_CC' C hsC ho)
    (Finsupp.sum m fun i a ↦ a • ((term I ho :: i.1).map (e C)).prod)
  · rw [← hmsum]
    simp only [map_finsupp_sum]
    apply Finsupp.sum_congr
    intro q hq
    rw [LinearMap.map_smul]
    rw [Finsupp.mem_supported] at hmmem
    have hx'' : q < l.val.Tail := hmmem hq
    have : ∃ (p : Products I), p.val ≠ [] ∧ p.val.head! = term I ho ∧ q = p.Tail :=
      ⟨⟨term I ho :: q.val, chain'_cons_of_lt C hsC ho l q hx''⟩,
        ⟨List.cons_ne_nil _ _, by simp only [List.head!_cons],
        by simp only [Products.Tail, List.tail_cons, Subtype.coe_eta]⟩⟩
    obtain ⟨p, hp⟩ := this
    rw [hp.2.2, ← Products.max_eq_eval C hsC ho p hp.1 hp.2.1]
    dsimp [Products.eval]
    rw [Products.max_eq_o_cons_tail ho p hp.1 hp.2.1]
    rfl
  have hse := succ_exact C hC hsC ho
  rw [ShortComplex.moduleCat_exact_iff_range_eq_ker] at hse
  dsimp [ModuleCat.ofHom] at hse

  -- Rewrite `this` using exact sequence manipulations to conclude that a term is in the range of
  -- the linear map `πs`:
  rw [← LinearMap.sub_mem_ker_iff, ← hse] at this
  obtain ⟨(n : LocallyConstant (π C (ord I · < o)) ℤ), hn⟩ := this
  rw [eq_sub_iff_add_eq] at hn
  have hn' := h₁ (Submodule.mem_top : n ∈ ⊤)
  rw [Finsupp.mem_span_range_iff_exists_finsupp] at hn'
  obtain ⟨w,hc⟩ := hn'
  rw [← hc, map_finsupp_sum] at hn
  apply l.prop.1
  rw [← hn]

  -- Now we just need to prove that a sum of two terms belongs to a span:
  apply Submodule.add_mem
  · apply Submodule.finsupp_sum_mem
    intro q _
    erw [LinearMap.map_smul (fₗ := πs C o) (c := w q) (x := eval (π C (ord I · < o)) q)]
    apply Submodule.smul_mem
    

Proof state:
case intro.intro.intro.intro.h₁.h.h
I : Type u
inst✝² : Inhabited I
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
m : Products I →₀ ℤ
hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < Products.Tail ↑l}
hmsum : (Finsupp.sum m fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
hse : LinearMap.range (πs C o) = LinearMap.ker (Linear_CC' C hsC ho)
n : LocallyConstant ↑(π C fun x => ord I x < o) ℤ
w : { l // Products.isGood (π C fun x => ord I x < o) l } →₀ ℤ
hn :
  ((Finsupp.sum w fun a b => (πs C o) (b • eval (π C fun x => ord I x < o) a)) +
      Finsupp.sum m fun i a => a • List.prod (List.map (e C) (term I ho :: ↑i))) =
    Products.eval C ↑l
hc : (Finsupp.sum w fun i a => a • eval (π C fun x => ord I x < o) i) = n
q : { l // Products.isGood (π C fun x => ord I x < o) l }
a✝ : w q ≠ 0
⊢ (πs C o) (eval (π C fun x => ord I x < o) q) ∈ Submodule.span ℤ (Products.eval C '' {m | m < ↑l})

Next tactic:
apply Submodule.subset_span