-- Module: Mathlib.RingTheory.Finiteness.Nakayama
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
I : Ideal R
N : Submodule R M
s : Set M
hfs : s.Finite
H :
  ∃ (r : R),
    r - 1 ∈ I ∧
      N ≤
          comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
            (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
        s ⊆ (↑N : Set M)
empty :
  ∀
    (H :
      ∃ (r : R),
        r - 1 ∈ I ∧
          N ≤
              comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                (HSMul.hSMul (β := Submodule R M) I (span R ∅)) ∧
            ∅ ⊆ (↑N : Set M)),
    ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
insert :
  ∀ {i : M} {s : Set M},
    i ∉ s →
      s.Finite →
        ∀
          (ih :
            ∀
              (H :
                ∃ (r : R),
                  r - 1 ∈ I ∧
                    N ≤
                        comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                          (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
                      s ⊆ (↑N : Set M)),
              ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0)
          (H :
            ∃ (r : R),
              r - 1 ∈ I ∧
                N ≤
                    comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                      (HSMul.hSMul (β := Submodule R M) I (span R (Insert.insert i s))) ∧
                  Insert.insert i s ⊆ (↑N : Set M)),
          ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
⊢ ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0

Tactic:
induction s, hfs using Set.Finite.induction_on with
  | empty =>
    rcases H with ⟨r, hr1, hrn, _⟩
    refine ⟨r, hr1, fun n hn => ?_⟩
    specialize hrn hn
    rwa [mem_comap, span_empty, smul_bot, mem_bot] at hrn
  | @insert i s _ _ ih =>
  apply ih
  rcases H with ⟨r, hr1, hrn, hs⟩
  rw [← Set.singleton_union, span_union, smul_sup] at hrn
  rw [Set.insert_subset_iff] at hs
  have : ∃ c : R, c - 1 ∈ I ∧ c • i ∈ I • span R s := by
    specialize hrn hs.1
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨c, hci, rfl⟩
    use r - c
    constructor
    · rw [sub_right_comm]
      exact I.sub_mem hr1 hci
    · rw [sub_smul, ← hyz, add_sub_cancel_left]
      exact hz
  rcases this with ⟨c, hc1, hci⟩
  refine ⟨c * r, ?_, ?_, hs.2⟩
  · simpa only [mul_sub, mul_one, sub_add_sub_cancel] using I.add_mem (I.mul_mem_left c hr1) hc1
  · intro n hn
    specialize hrn hn
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨d, _, rfl⟩
    simp only [mem_comap, LinearMap.lsmul_apply]
    rw [mul_smul, ← hyz, smul_add, smul_smul, mul_comm, mul_smul]
    exact add_mem (smul_mem _ _ hci) (smul_mem _ _ hz)

Premises used: Submodule.smul_bot, sub_add_sub_cancel, Submodule.mem_smul_span_singleton, mul_sub, Submodule.mem_bot, Submodule.mem_comap, Set.singleton_union, smul_smul, Insert.insert, LinearMap.lsmul_apply, Set.insert_subset_iff, Submodule.smul_sup, Submodule.span_union, mul_comm, Submodule.span_empty, Submodule.mem_sup, mul_one, SemigroupAction.mul_smul, smul_add, rfl