Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case hP₂
R : Type u_2
inst✝ : CommRing R
n : ℕ
⊢ ∀ (R : Type u_2) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),
    (e.val i).Monic →
      (∀ (j : Fin n), j ≠ i → e.val j = 0) →
        ∀ {R₀ : Type u_1} [inst_1 : CommRing R₀] [inst_2 : Algebra R₀ R], Statement R₀ R n e

pp: case hP₃
R : Type u_2
inst✝ : CommRing R
n : ℕ
⊢ ∀ (R : Type u_2) [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 →
          (∀ {R₀ : Type u_1} [inst_1 : CommRing R₀] [inst_2 : Algebra R₀ R],
              Statement R₀ R n { val := Function.update e.val j (e.val j %ₘ e.val i) }) →
            ∀ {R₀ : Type u_1} [inst_1 : CommRing R₀] [inst_2 : Algebra R₀ R], Statement R₀ R n e

pp: case hP₄
R : Type u_2
inst✝ : CommRing R
n : ℕ
⊢ ∀ (R : Type u_2) [inst : CommRing R] (c : R) (i : Fin n) (e : InductionObj R n),
    c = (e.val i).leadingCoeff →
      c ≠ 0 →
        (∀ {R₀ : Type u_1} [inst_1 : CommRing R₀] [inst_2 : Algebra R₀ (Away c)],
            Statement R₀ (Away c) n
              { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom (algebraMap R (Away c))) ∘ e.val }) →
          (∀ {R₀ : Type u_1} [inst_1 : CommRing R₀] [inst_2 : Algebra R₀ (R ⧸ Ideal.span {c})],
              Statement R₀ (R ⧸ Ideal.span {c}) n
                { val := ⇑(mapRingHom (algebraMap R (R ⧸ Ideal.span {c}))) ∘ e.val }) →
            ∀ {R₀ : Type u_1} [inst_1 : CommRing R₀] [inst_2 : Algebra R₀ R], Statement R₀ R n e

Tactic:
· intro R _ g i hi hi_min _ R₀ _ f
    let M := R[X] ⧸ Ideal.span {g.1 i}
    have : Module.Free R M := .of_basis (AdjoinRoot.powerBasis' hi).basis
    have : Module.Finite R M := .of_basis (AdjoinRoot.powerBasis' hi).basis
    refine ⟨(Finset.range (Module.finrank R M)).image
      fun j ↦ ⟨(Algebra.lmul R M (Ideal.Quotient.mk _ f)).charpoly.coeff j, 0, 0⟩, ?_, ?_⟩
    · ext x
      have : zeroLocus (Set.range g.val) = zeroLocus {g.1 i} := by
        rw [Set.range_eq_iUnion, zeroLocus_iUnion]
        refine (Set.iInter_subset _ _).antisymm (Set.subset_iInter fun j ↦ ?_)
        by_cases hij : i = j
        · subst hij; rfl
        · rw [hi_min j (.symm hij), zeroLocus_singleton_zero]; exact Set.subset_univ _
      rw [this, ← Polynomial.algebraMap_eq, mem_image_comap_zeroLocus_sdiff,
        IsScalarTower.algebraMap_apply R[X] M, isNilpotent_tensor_residueField_iff]
      simp [BasicConstructibleSetData.toSet, ConstructibleSetData.toSet, Set.subset_def, M]
    · simp

Goal state after:
case hP₃
R : Type u_2
inst✝ : CommRing R
n : ℕ
⊢ ∀ (R : Type u_2) [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 →
          (∀ {R₀ : Type u_1} [inst_1 : CommRing R₀] [inst_2 : Algebra R₀ R],
              Statement R₀ R n { val := Function.update e.val j (e.val j %ₘ e.val i) }) →
            ∀ {R₀ : Type u_1} [inst_1 : CommRing R₀] [inst_2 : Algebra R₀ R], Statement R₀ R n e

case hP₄
R : Type u_2
inst✝ : CommRing R
n : ℕ
⊢ ∀ (R : Type u_2) [inst : CommRing R] (c : R) (i : Fin n) (e : InductionObj R n),
    c = (e.val i).leadingCoeff →
      c ≠ 0 →
        (∀ {R₀ : Type u_1} [inst_1 : CommRing R₀] [inst_2 : Algebra R₀ (Away c)],
            Statement R₀ (Away c) n
              { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom (algebraMap R (Away c))) ∘ e.val }) →
          (∀ {R₀ : Type u_1} [inst_1 : CommRing R₀] [inst_2 : Algebra R₀ (R ⧸ Ideal.span {c})],
              Statement R₀ (R ⧸ Ideal.span {c}) n
                { val := ⇑(mapRingHom (algebraMap R (R ⧸ Ideal.span {c}))) ∘ e.val }) →
            ∀ {R₀ : Type u_1} [inst_1 : CommRing R₀] [inst_2 : Algebra R₀ R], Statement R₀ R n e