Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case e'_3
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), (f (MvPolynomial.X i)).degrees ≤ d
g : MvPolynomial (Fin m) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R := eval₂Hom f.toRingHom MvPolynomial.X
hg : g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))) = ↑f
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 (σ x) = x
hg' : Surjective ⇑g
S' : ConstructibleSetData (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Finset.image
    (fun x ↦
      match x with
      | { f := fk, n := k, g := gk } =>
        { f := σ fk, n := k + n,
          g := fun s ↦
            Sum.elim (⇑σ ∘ gk) (fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i)))
              (finSumFinEquiv.symm s) })
    S
s₀ : Set (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Set.range fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i))
H : Ideal.span s₀ ≤ RingHom.ker g
p : MvPolynomial (Fin m) (MvPolynomial (Fin n) R)
hp : p ∈ RingHom.ker g
⊢ p ∈ Ideal.span s₀

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

Goal state after:
case e'_3
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), (f (MvPolynomial.X i)).degrees ≤ d
g : MvPolynomial (Fin m) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R := eval₂Hom f.toRingHom MvPolynomial.X
hg : g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))) = ↑f
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 (σ x) = x
hg' : Surjective ⇑g
S' : ConstructibleSetData (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Finset.image
    (fun x ↦
      match x with
      | { f := fk, n := k, g := gk } =>
        { f := σ fk, n := k + n,
          g := fun s ↦
            Sum.elim (⇑σ ∘ gk) (fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i)))
              (finSumFinEquiv.symm s) })
    S
s₀ : Set (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Set.range fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i))
H : Ideal.span s₀ ≤ RingHom.ker g
q₁ : MvPolynomial (Fin m) (MvPolynomial (Fin n) R)
q₂ : MvPolynomial (Fin m) R
hq₁ : q₁ ∈ Ideal.span s₀
hp : q₁ + σ q₂ ∈ RingHom.ker g
⊢ q₁ + σ q₂ ∈ Ideal.span s₀