-- Lean 4 proof step from Mathlib.RingTheory.KrullDimension.Regular (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsNoetherianRing R
M : Type u_2
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : Module.Finite R M
x : R
h : Membership.mem (γ := Ideal R) (Ideal.jacobson (annihilator R M)) x
h_1 : Nontrivial M
p : LTSeries (↑(support R M) : Type u_1)
hxp :
  x ∈
    (↑(RelSeries.last (r :=
            {x : (↑(support R M) : Type u_1) × (↑(support R M) : Type u_1) |
              match (motive := (↑(support R M) : Type u_1) × (↑(support R M) : Type u_1) → Prop) x with
              | (a, b) => a < b})
            p) :
        PrimeSpectrum R).asIdeal
q : LTSeries (PrimeSpectrum R)
hxq :
  x ∈
    (q.toFun (r :=
        {x : PrimeSpectrum R × PrimeSpectrum R |
          match (motive := PrimeSpectrum R × PrimeSpectrum R → Prop) x with
          | (a, b) => a < b})
        1).asIdeal
hq :
  RelSeries.length (r :=
      {x : PrimeSpectrum R × PrimeSpectrum R |
        match (motive := PrimeSpectrum R × PrimeSpectrum R → Prop) x with
        | (a, b) => a < b})
      (p.map
        (Subtype.val (p := fun (x : PrimeSpectrum R) ↦ Membership.mem (γ := Set (PrimeSpectrum R)) (support R M) x))
        ⋯) =
    q.length (r :=
      {x : PrimeSpectrum R × PrimeSpectrum R |
        match (motive := PrimeSpectrum R × PrimeSpectrum R → Prop) x with
        | (a, b) => a < b})
h0 :
  (↑(RelSeries.head (r :=
          {x : (↑(support R M) : Type u_1) × (↑(support R M) : Type u_1) |
            match (motive := (↑(support R M) : Type u_1) × (↑(support R M) : Type u_1) → Prop) x with
            | (a, b) => a < b})
          p) :
      PrimeSpectrum R) =
    RelSeries.head (r :=
      {x : PrimeSpectrum R × PrimeSpectrum R |
        match (motive := PrimeSpectrum R × PrimeSpectrum R → Prop) x with
        | (a, b) => a < b})
      q
right :
  RelSeries.last (r :=
      {x : PrimeSpectrum R × PrimeSpectrum R |
        match (motive := PrimeSpectrum R × PrimeSpectrum R → Prop) x with
        | (a, b) => a < b})
      (p.map
        (Subtype.val (p := fun (x : PrimeSpectrum R) ↦ Membership.mem (γ := Set (PrimeSpectrum R)) (support R M) x))
        ⋯) =
    RelSeries.last (r :=
      {x : PrimeSpectrum R × PrimeSpectrum R |
        match (motive := PrimeSpectrum R × PrimeSpectrum R → Prop) x with
        | (a, b) => a < b})
      q
pos :
  ∀
    (hp0 :
      p.length (r :=
          {x : (↑(support R M) : Type u_1) × (↑(support R M) : Type u_1) |
            match (motive := (↑(support R M) : Type u_1) × (↑(support R M) : Type u_1) → Prop) x with
            | (a, b) => a < b}) =
        0),
    (↑(p.length (r :=
            {x : (↑(support R M) : Type u_1) × (↑(support R M) : Type u_1) |
              match (motive := (↑(support R M) : Type u_1) × (↑(support R M) : Type u_1) → Prop) x with
              | (a, b) => a < b})) :
        WithBot ℕ∞) ≤
      supportDim.{u_1, u_2} R (QuotSMulTop x M) + 1
⊢ (↑(p.length (r :=
          {x : (↑(support R M) : Type u_1) × (↑(support R M) : Type u_1) |
            match (motive := (↑(support R M) : Type u_1) × (↑(support R M) : Type u_1) → Prop) x with
            | (a, b) => a < b})) :
      WithBot ℕ∞) ≤
    supportDim.{u_1, u_2} R (QuotSMulTop x M) + 1
have neg : ∀
  (hp0 :
    ¬p.length (r :=
          {x : (↑(support R M) : Type u_1) × (↑(support R M) : Type u_1) |
            match (motive := (↑(support R M) : Type u_1) × (↑(support R M) : Type u_1) → Prop) x with
            | (a, b) => a < b}) =
        0),
  (↑(p.length (r :=
          {x : (↑(support R M) : Type u_1) × (↑(support R M) : Type u_1) |
            match (motive := (↑(support R M) : Type u_1) × (↑(support R M) : Type u_1) → Prop) x with
            | (a, b) => a < b})) :
      WithBot ℕ∞) ≤
    supportDim.{u_1, u_2} R (QuotSMulTop x M) + 1
Tactic: by_cases hp0 : p.length = 0