-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
n : ℕ
P : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop
hP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }
hP₂ :
  ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),
    (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e
hP₃ :
  ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i j : Fin n),
    (e.val i).Monic →
      (e.val i).degree ≤ (e.val j).degree →
        i ≠ j →
          P R { val := Function.update (β := fun (a : Fin n) ↦ R[X]) e.val j (e.val j %ₘ e.val i) : InductionObj R n } →
            P R e
hP₄ :
  ∀ (R : Type u) [inst : CommRing R] (c : R) (i : Fin n) (e : InductionObj R n),
    c = (e.val i).leadingCoeff →
      c ≠ 0 →
        P (Away c)
            {
              val :=
                HSMul.hSMul (α := (Away c)[X]) (β := Fin n → (Away c)[X])
                  ((Polynomial.C : (a : Away c) → (Away c)[X]) (IsLocalization.Away.invSelf c))
                  ((⇑(mapRingHom (algebraMap R (Away c))) : (a : R[X]) → (Away c)[X]) ∘ e.val) :
              InductionObj (Away c) n } →
          P (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))
              {
                val :=
                  (⇑(mapRingHom (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c})))) :
                      (a : R[X]) → (R ⧸ Ideal.span {c})[X]) ∘
                    e.val } →
            P R e
v : DegreeType n
H_IH : ∀ y < v, ∀ {R : Type u} [inst : CommRing R] (e : InductionObj R n) (hv : y = e.degree), P R e
R : Type u
inst : CommRing R
e : InductionObj R n
hv : v = e.degree
h : Nontrivial R
he0 : ∃ (x : Fin n), ¬e.val x = 0
H : ¬∃ (i : Fin n), (e.val i).Monic ∧ ∀ (j : Fin n), e.val j ≠ 0 → (e.val i).degree ≤ (e.val j).degree
i : Fin n
hi : e.val i ≠ 0
i_min : ∀ (j : Fin n), e.val j ≠ 0 → (e.val i).degree ≤ (e.val j).degree
refine_1 :
  Function.comp (β := (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {(e.val i).leadingCoeff}))[X])
      Polynomial.degree
      (val
        {
          val :=
            (⇑(mapRingHom
                    (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {(e.val i).leadingCoeff})))) :
                (a : R[X]) → (R ⧸ Ideal.span {(e.val i).leadingCoeff})[X]) ∘
              e.val :
          InductionObj (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {(e.val i).leadingCoeff})) n }) ≤
    Polynomial.degree ∘ e.val
refine_2 :
  Function.comp (β := (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {(e.val i).leadingCoeff}))[X])
      Polynomial.degree
      (val
        {
          val :=
            (⇑(mapRingHom
                    (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {(e.val i).leadingCoeff})))) :
                (a : R[X]) → (R ⧸ Ideal.span {(e.val i).leadingCoeff})[X]) ∘
              e.val :
          InductionObj (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {(e.val i).leadingCoeff})) n }) ≠
    Polynomial.degree ∘ e.val
⊢ InductionObj.degree (R := HasQuotient.Quotient R (B := Ideal R) (Ideal.span {(e.val i).leadingCoeff}))
      {
        val :=
          (⇑(mapRingHom (algebraMap R (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {(e.val i).leadingCoeff})))) :
              (a : R[X]) → (R ⧸ Ideal.span {(e.val i).leadingCoeff})[X]) ∘
            e.val :
        InductionObj (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {(e.val i).leadingCoeff})) n } <
    e.degree

Tactic:
refine .left _ _ (lt_of_le_of_ne ?_ ?_)