-- 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
h_eq :
  ∀ (x : Fin n),
    (HMul.hMul (α := (Away (e.val i).leadingCoeff)[X])
          ((Polynomial.C : (a : Away (e.val i).leadingCoeff) → (Away (e.val i).leadingCoeff)[X])
            (IsLocalization.Away.invSelf (e.val i).leadingCoeff))
          (Polynomial.map (algebraMap R (Away (e.val i).leadingCoeff)) (e.val x))).degree =
      (e.val x).degree
right :
  (HMul.hMul (α := (Away (e.val i).leadingCoeff)[X])
        ((Polynomial.C : (a : Away (e.val i).leadingCoeff) → (Away (e.val i).leadingCoeff)[X])
          (IsLocalization.Away.invSelf (e.val i).leadingCoeff))
        (Polynomial.map (algebraMap R (Away (e.val i).leadingCoeff)) (e.val i))).leadingCoeff =
    1
right₁ :
  (HMul.hMul (α := (Away (e.val i).leadingCoeff)[X])
          ((Polynomial.C : (a : Away (e.val i).leadingCoeff) → (Away (e.val i).leadingCoeff)[X])
            (IsLocalization.Away.invSelf (e.val i).leadingCoeff))
          (Polynomial.map (algebraMap R (Away (e.val i).leadingCoeff)) (e.val i))).coeff
      (HMul.hMul (α := (Away (e.val i).leadingCoeff)[X])
          ((Polynomial.C : (a : Away (e.val i).leadingCoeff) → (Away (e.val i).leadingCoeff)[X])
            (IsLocalization.Away.invSelf (e.val i).leadingCoeff))
          (Polynomial.map (algebraMap R (Away (e.val i).leadingCoeff)) (e.val i))).natDegree =
    1
right₂ :
  (HMul.hMul (α := (Away (e.val i).leadingCoeff)[X])
          ((Polynomial.C : (a : Away (e.val i).leadingCoeff) → (Away (e.val i).leadingCoeff)[X])
            (IsLocalization.Away.invSelf (e.val i).leadingCoeff))
          (Polynomial.map (algebraMap R (Away (e.val i).leadingCoeff)) (e.val i))).coeff
      (e.val i).natDegree =
    1
right₃ :
  HMul.hMul (α := Away (e.val i).leadingCoeff) (IsLocalization.Away.invSelf (e.val i).leadingCoeff)
      (Polynomial.coeff (Polynomial.map (algebraMap R (Away (e.val i).leadingCoeff)) (e.val i)) (e.val i).natDegree) =
    1
right₄ :
  HMul.hMul (α := Away (e.val i).leadingCoeff) (IsLocalization.Away.invSelf (e.val i).leadingCoeff)
      ((algebraMap R (Away (e.val i).leadingCoeff) : (a : R) → Away (e.val i).leadingCoeff)
        ((e.val i).coeff (e.val i).natDegree)) =
    1
right₅ :
  HMul.hMul (α := Away (e.val i).leadingCoeff) (IsLocalization.Away.invSelf (e.val i).leadingCoeff)
      ((algebraMap R (Away (e.val i).leadingCoeff) : (a : R) → Away (e.val i).leadingCoeff) (e.val i).leadingCoeff) =
    1
right₆ :
  HMul.hMul (α := Away (e.val i).leadingCoeff)
      ((algebraMap R (Away (e.val i).leadingCoeff) : (a : R) → Away (e.val i).leadingCoeff) (e.val i).leadingCoeff)
      (IsLocalization.Away.invSelf (e.val i).leadingCoeff) =
    1
right₇ right₈ : Eq.{u + 1} (α := Away (e.val i).leadingCoeff) 1 1
⊢ (HMul.hMul (α := (Away (e.val i).leadingCoeff)[X])
      ((Polynomial.C : (a : Away (e.val i).leadingCoeff) → (Away (e.val i).leadingCoeff)[X])
        (IsLocalization.Away.invSelf (e.val i).leadingCoeff))
      (Polynomial.map (algebraMap R (Away (e.val i).leadingCoeff)) (e.val i))).Monic

Tactic:
rw [Monic.def, ← coeff_natDegree (p := _ * _), natDegree_eq_of_degree_eq (h_eq i),
          Polynomial.coeff_C_mul, Polynomial.coeff_map, coeff_natDegree, mul_comm,
          IsLocalization.Away.mul_invSelf]

Premises used: Polynomial.coeff_map, Polynomial.coeff_natDegree, Polynomial.coeff_C_mul, IsLocalization.Away.mul_invSelf, Polynomial.natDegree_eq_of_degree_eq, mul_comm, Polynomial.Monic.def