/-- The induction hypothesis of `Submodule.basisOfPid` and `Submodule.smithNormalForm`.

Basically, it says: let `N ≤ M` be a pair of submodules, then we can find a pair of
submodules `N' ≤ M'` of strictly smaller rank, whose basis we can extend to get a basis
of `N` and `M`. Moreover, if the basis for `M'` is up to scalars a basis for `N'`,
then the basis we find for `M` is up to scalars a basis for `N`.

For `basis_of_pid` we only need the first half and can fix `M = ⊤`,
for `smith_normal_form` we need the full statement,
but must also feed in a basis for `M` using `basis_of_pid` to keep the induction going.
-/
theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Module R O]
    (M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
    ∃ y ∈ M,
      ∃ (a : R) (_ : a • y ∈ N),
        ∃ M' ≤ M,
          ∃ N' ≤ N,
            ∃ (_N'_le_M' : N' ≤ M') (_y_ortho_M' :
              ∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0) (_ay_ortho_N' :
              ∀ (c : R) (z : O), z ∈ N' → c • a • y + z = 0 → c = 0),
              ∀ (n') (bN' : Basis (Fin n') R N'),
                ∃ bN : Basis (Fin (n' + 1)) R N,
                  ∀ (m') (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R M'),
                    ∃ (hnm : n' + 1 ≤ m' + 1) (bM : Basis (Fin (m' + 1)) R M),
                      ∀ (as : Fin n' → R)
                        (_h : ∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLE hn'm' i) : O)),
                        ∃ as' : Fin (n' + 1) → R,
                          ∀ i : Fin (n' + 1), (bN i : O) = as' i • (bM (Fin.castLE hnm i) : O) := by
  -- Let `ϕ` be a maximal projection of `M` onto `R`, in the sense that there is
  -- no `ψ` whose image of `N` is larger than `ϕ`'s image of `N`.
  have : ∃ ϕ : M →ₗ[R] R, ∀ ψ : M →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N := by
    obtain ⟨P, P_eq, P_max⟩ :=
      set_has_maximal_iff_noetherian.mpr (inferInstance : IsNoetherian R R) _
        (show (Set.range fun ψ : M →ₗ[R] R ↦ ψ.submoduleImage N).Nonempty from
          ⟨_, Set.mem_range.mpr ⟨0, rfl⟩⟩)
    obtain ⟨ϕ, rfl⟩ := Set.mem_range.mp P_eq
    exact ⟨ϕ, fun ψ hψ ↦ P_max _ ⟨_, rfl⟩ hψ⟩
  let ϕ := this.choose
  have ϕ_max := this.choose_spec
  -- Since `ϕ(N)` is an `R`-submodule of the PID `R`,
  -- it is principal and generated by some `a`.
  let a := generator (ϕ.submoduleImage N)
  have a_mem : a ∈ ϕ.submoduleImage N := generator_mem _
  -- If `a` is zero, then the submodule is trivial. So let's assume `a ≠ 0`, `N ≠ ⊥`.
  by_cases a_zero : a = 0
  · have := eq_bot_of_generator_maximal_submoduleImage_eq_zero b'M N_le_M ϕ_max a_zero
    contradiction
  -- We claim that `ϕ⁻¹ a = y` can be taken as basis element of `N`.
  obtain ⟨y, yN, ϕy_eq⟩ := (LinearMap.mem_submoduleImage_of_le N_le_M).mp a_mem
  have _ϕy_ne_zero : ϕ ⟨y, N_le_M yN⟩ ≠ 0 := fun h ↦ a_zero (ϕy_eq.symm.trans h)
  -- Write `y` as `a • y'` for some `y'`.
  have hdvd : ∀ i, a ∣ b'M.coord i ⟨y, N_le_M yN⟩ := fun i ↦
    generator_maximal_submoduleImage_dvd N_le_M ϕ_max y yN ϕy_eq (b'M.coord i)
  choose c hc using hdvd
  cases nonempty_fintype ι
  let y' : O := ∑ i, c i • b'M i
  have y'M : y' ∈ M := M.sum_mem fun i _ ↦ M.smul_mem (c i) (b'M i).2
  have mk_y' : (⟨y', y'M⟩ : M) = ∑ i, c i • b'M i :=
    Subtype.ext
      (show y' = M.subtype _ by
        simp only [map_sum, map_smul]
        rfl)
  have a_smul_y' : a • y' = y := by
    refine Subtype.mk_eq_mk.mp (show (a • ⟨y', y'M⟩ : M) = ⟨y, N_le_M yN⟩ from ?_)
    rw [← b'M.sum_repr ⟨y, N_le_M yN⟩, mk_y', Finset.smul_sum]
    refine' Finset.sum_congr rfl fun i _ ↦ _
    rw [← mul_smul, ← hc]
    rfl
  -- We found a `y` and an `a`!
  

Proof state:
case neg.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
M✝ : Type u_3
inst✝⁴ : AddCommGroup M✝
inst✝³ : Module R M✝
b : ι → M✝
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬LinearMap.submoduleImage ϕ N < LinearMap.submoduleImage ψ N
ϕ : ↥M →ₗ[R] R := Exists.choose this
ϕ_max : ∀ (ψ : ↥M →ₗ[R] R), ¬LinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage ψ N
a : R := generator (LinearMap.submoduleImage ϕ N)
a_mem : a ∈ LinearMap.submoduleImage ϕ N
a_zero : ¬a = 0
y : O
yN : y ∈ N
ϕy_eq : ϕ { val := y, property := (_ : y ∈ M) } = a
_ϕy_ne_zero : ϕ { val := y, property := (_ : y ∈ M) } ≠ 0
c : ι → R
hc : ∀ (i : ι), (Basis.coord b'M i) { val := y, property := (_ : y ∈ M) } = a * c i
val✝ : Fintype ι
y' : O := ∑ i : ι, c i • ↑(b'M i)
y'M : y' ∈ M
mk_y' : { val := y', property := y'M } = ∑ i : ι, c i • b'M i
a_smul_y' : a • y' = y
⊢ ∃ y ∈ M,
    ∃ a,
      ∃ (_ : a • y ∈ N),
        ∃ M' ≤ M,
          ∃ N' ≤ N,
            ∃ (_ : N' ≤ M') (_ : ∀ (c : R), ∀ z ∈ M', c • y + z = 0 → c = 0) (_ :
              ∀ (c : R), ∀ z ∈ N', c • a • y + z = 0 → c = 0),
              ∀ (n' : ℕ) (bN' : Basis (Fin n') R ↥N'),
                ∃ bN,
                  ∀ (m' : ℕ) (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R ↥M'),
                    ∃ (hnm : n' + 1 ≤ m' + 1),
                      ∃ bM,
                        ∀ (as : Fin n' → R),
                          (∀ (i : Fin n'), ↑(bN' i) = as i • ↑(bM' (Fin.castLE hn'm' i))) →
                            ∃ as', ∀ (i : Fin (n' + 1)), ↑(bN i) = as' i • ↑(bM (Fin.castLE hnm i))

Next tactic:
refine' ⟨y', y'M, a, a_smul_y'.symm ▸ yN, _⟩