-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
R : Type u_2
inst : CommRing R
m n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] MvPolynomial (Fin m) R
k : ℕ
d : Multiset (Fin m)
S : ConstructibleSetData (MvPolynomial (Fin m) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS : ∀ C ∈ S, ∀ (j : Fin C.n), (C.g j).degrees ≤ d
hf :
  ∀ (i : Fin n),
    degrees.{u_2, 0} (R := R) ((f : (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) R) (MvPolynomial.X i)) ≤ d
g : MvPolynomial (Fin m) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R := eval₂Hom f.toRingHom MvPolynomial.X
hg :
  Eq (α := MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)
    (g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))))
    (↑f : MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)
σ : MvPolynomial (Fin m) R →+* MvPolynomial (Fin m) (MvPolynomial (Fin n) R) :=
  MvPolynomial.map (algebraMap R (MvPolynomial (Fin n) R))
hσ : g.comp σ = RingHom.id (MvPolynomial (Fin m) R)
hσ' :
  ∀ (x : MvPolynomial (Fin m) R),
    (g : (a : MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) → MvPolynomial (Fin m) R)
        ((σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) x) =
      x
hg' :
  Surjective (α := MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) (β := MvPolynomial (Fin m) R)
    (⇑g : (a : MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) → MvPolynomial (Fin m) R)
S' : ConstructibleSetData (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Finset.image
    (fun (x : BasicConstructibleSetData (MvPolynomial (Fin m) R)) ↦
      match (motive :=
        BasicConstructibleSetData (MvPolynomial (Fin m) R) →
          BasicConstructibleSetData (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)))
        x with
      | { f := fk, n := k, g := gk } =>
        { f := (σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) fk, n := k + n,
          g := fun (s : Fin (k + n)) ↦
            Sum.elim ((⇑σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) ∘ gk)
              (fun (i : Fin n) ↦
                HSub.hSub (α := MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) (β :=
                  MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
                  ((MvPolynomial.C : (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
                      (MvPolynomial.X i) :
                    MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
                  ((σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
                      ((f : (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) R) (MvPolynomial.X i)) :
                    MvPolynomial (Fin m) (MvPolynomial (Fin n) R)))
              ((Equiv.symm finSumFinEquiv : (a : Fin (k + n)) → Fin k ⊕ Fin n) s) })
    S
s₀ : Set (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Set.range fun (i : Fin n) ↦
    HSub.hSub (α := MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) (β := MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
      ((MvPolynomial.C : (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
          (MvPolynomial.X i) :
        MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
      ((σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
          ((f : (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) R) (MvPolynomial.X i)) :
        MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
H : LE.le (α := Ideal (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))) (Ideal.span s₀) (RingHom.ker g)
p : MvPolynomial (Fin m) (MvPolynomial (Fin n) R)
hp :
  Membership.mem (γ :=
    Submodule (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)))
    (RingHom.ker g) p
h :
  ∀ (q₁ : MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) (q₂ : MvPolynomial (Fin m) R)
    (hq₁ : Membership.mem (γ := Ideal (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))) (Ideal.span s₀) q₁)
    (hp :
      Membership.mem (γ :=
        Submodule (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)))
        (RingHom.ker g)
        (HAdd.hAdd (β := MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) q₁
          ((σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) q₂ :
            MvPolynomial (Fin m) (MvPolynomial (Fin n) R)))),
    Membership.mem (γ :=
      Submodule (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)))
      (Ideal.span s₀)
      (HAdd.hAdd (β := MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) q₁
        ((σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) q₂ :
          MvPolynomial (Fin m) (MvPolynomial (Fin n) R)))
⊢ Membership.mem (γ :=
    Submodule (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)))
    (Ideal.span s₀) p

Tactic:
obtain ⟨q₁, q₂, hq₁, rfl⟩ : ∃ q₁ q₂, q₁ ∈ Ideal.span s₀ ∧ p = q₁ + σ q₂ := by
      clear hp
      obtain ⟨p, rfl⟩ := (commAlgEquiv _ _ _).surjective p
      simp_rw [← (commAlgEquiv R (Fin n) (Fin m)).symm.injective.eq_iff,
        AlgEquiv.symm_apply_apply]
      induction p using MvPolynomial.induction_on with
      | C q =>
        exact ⟨0, q, by simp, (commAlgEquiv _ _ _).injective <|
          by simp [commAlgEquiv_C, σ]⟩
      | add p q hp hq =>
        obtain ⟨q₁, q₂, hq₁, rfl⟩ := hp
        obtain ⟨q₃, q₄, hq₃, rfl⟩ := hq
        refine ⟨q₁ + q₃, q₂ + q₄, add_mem hq₁ hq₃, by simp only [map_add, add_add_add_comm]⟩
      | mul_X p i hp =>
        obtain ⟨q₁, q₂, hq₁, rfl⟩ := hp
        simp only [← (commAlgEquiv R (Fin n) (Fin m)).injective.eq_iff,
          map_mul, AlgEquiv.apply_symm_apply, commAlgEquiv_X]
        refine ⟨q₁ * .C (.X i) + σ q₂ * (.C (.X i) - σ (f (.X i))), q₂ * f (.X i), ?_, ?_⟩
        · exact add_mem (Ideal.mul_mem_right _ _ hq₁)
            (Ideal.mul_mem_left _ _ (Ideal.subset_span (Set.mem_range_self _)))
        · simp; ring

Premises used: Ideal.span, rfl