-- Module: Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness
Goal:
A : Type u
inst : CommRing A
inst_1 : IsNoetherianRing A
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module A M
h_1 : Module.Finite A M
motive : (N : Type v) → [inst_4 : AddCommGroup N] → [inst_5 : Module A N] → [Module.Finite A N] → Prop
subsingleton :
  ∀ (N : Type v) [inst_4 : AddCommGroup N] [inst_5 : Module A N] [inst : Module.Finite A N] [Subsingleton N], motive N
quotient :
  ∀ (N : Type v) [inst_4 : AddCommGroup N] [inst_5 : Module A N] [inst_6 : Module.Finite A N] (p : PrimeSpectrum A)
    (a : N ≃ₗ[A] A ⧸ p.asIdeal), motive N
exact :
  ∀ (N₁ : Type v) [inst_4 : AddCommGroup N₁] [inst_5 : Module A N₁] [inst_6 : Module.Finite A N₁] (N₂ : Type v)
    [inst_7 : AddCommGroup N₂] [inst_8 : Module A N₂] [inst_9 : Module.Finite A N₂] (N₃ : Type v)
    [inst_10 : AddCommGroup N₃] [inst_11 : Module A N₃] [inst_12 : Module.Finite A N₃] (f : N₁ →ₗ[A] N₂)
    (g : N₂ →ₗ[A] N₃),
    Function.Injective (α := N₁) (β := N₂) (⇑f : (a : N₁) → N₂) →
      Function.Surjective (α := N₂) (β := N₃) (⇑g : (a : N₂) → N₃) →
        Function.Exact (M := N₁) (N := N₂) (P := N₃) (⇑f : (a : N₁) → N₂) (⇑g : (a : N₂) → N₃) →
          motive N₁ → motive N₃ → motive N₂
equiv :
  ∀ (N₁ : Type v) [inst_4 : AddCommGroup N₁] [inst_5 : Module A N₁] [inst_6 : Module.Finite A N₁] (N₂ : Type v)
    [inst_7 : AddCommGroup N₂] [inst_8 : Module A N₂] [inst_9 : Module.Finite A N₂] (f : N₁ ≃ₗ[A] N₂) (h : motive N₁),
    motive N₂
s :
  RelSeries (α := Submodule A M)
    {x : Submodule A M × Submodule A M |
      match (motive := Submodule A M × Submodule A M → Prop) x with
      | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂}
hs1 :
  s.head (r :=
      {x : Submodule A M × Submodule A M |
        match (motive := Submodule A M × Submodule A M → Prop) x with
        | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂}) =
    ⊥
hs2 :
  s.last (r :=
      {x : Submodule A M × Submodule A M |
        match (motive := Submodule A M × Submodule A M → Prop) x with
        | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂}) =
    ⊤
n : ℕ
h :
  n <
    s.length (r :=
        {x : Submodule A M × Submodule A M |
          match (motive := Submodule A M × Submodule A M → Prop) x with
          | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂}) +
      1
zero :
  ∀
    (h :
      0 <
        s.length (r :=
            {x : Submodule A M × Submodule A M |
              match (motive := Submodule A M × Submodule A M → Prop) x with
              | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂}) +
          1),
    motive
      ↥(s.toFun (r :=
          {x : Submodule A M × Submodule A M |
            match (motive := Submodule A M × Submodule A M → Prop) x with
            | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂})
          (⟨0, h⟩ :
            Fin
              (s.length (r :=
                  {x : Submodule A M × Submodule A M |
                    match (motive := Submodule A M × Submodule A M → Prop) x with
                    | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂}) +
                1)))
succ :
  ∀ (n : ℕ)
    (ih :
      ∀
        (h :
          n <
            s.length (r :=
                {x : Submodule A M × Submodule A M |
                  match (motive := Submodule A M × Submodule A M → Prop) x with
                  | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂}) +
              1),
        motive
          ↥(s.toFun (r :=
              {x : Submodule A M × Submodule A M |
                match (motive := Submodule A M × Submodule A M → Prop) x with
                | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂})
              (⟨n, h⟩ :
                Fin
                  (s.length (r :=
                      {x : Submodule A M × Submodule A M |
                        match (motive := Submodule A M × Submodule A M → Prop) x with
                        | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂}) +
                    1))))
    (h :
      n + 1 <
        s.length (r :=
            {x : Submodule A M × Submodule A M |
              match (motive := Submodule A M × Submodule A M → Prop) x with
              | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂}) +
          1),
    motive
      ↥(s.toFun (r :=
          {x : Submodule A M × Submodule A M |
            match (motive := Submodule A M × Submodule A M → Prop) x with
            | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂})
          (⟨n + 1, h⟩ :
            Fin
              (s.length (r :=
                  {x : Submodule A M × Submodule A M |
                    match (motive := Submodule A M × Submodule A M → Prop) x with
                    | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂}) +
                1)))
⊢ motive
    ↥(s.toFun (r :=
        {x : Submodule A M × Submodule A M |
          match (motive := Submodule A M × Submodule A M → Prop) x with
          | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂})
        (⟨n, h⟩ :
          Fin
            (s.length (r :=
                {x : Submodule A M × Submodule A M |
                  match (motive := Submodule A M × Submodule A M → Prop) x with
                  | (N₁, N₂) => N₁.IsQuotientEquivQuotientPrime N₂}) +
              1)))

Tactic:
induction n with
  | zero =>
    change motive s.head
    rw [hs1]
    exact subsingleton _
  | succ n ih =>
    specialize ih (n.lt_add_one.trans h)
    obtain ⟨hle, p, ⟨f⟩⟩ := s.step ⟨n, (add_lt_add_iff_right _).1 h⟩
    replace ih := equiv _ _ (Submodule.submoduleOfEquivOfLe hle).symm ih
    exact exact _ _ _ _ _ (Submodule.injective_subtype _) (Submodule.mkQ_surjective _)
      (LinearMap.exact_subtype_mkQ _) ih (quotient _ p f)

Premises used: Membership.mem, Fin.mk, Subtype, Iff.mp, RelSeries.head, RelSeries.step, add_lt_add_iff_right