Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: 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))
hs : zeroLocus s₀ = Set.range (comap g)
hg'' : ∀ (t : Set (PrimeSpectrum (MvPolynomial (Fin m) R))), comap g '' t = comap σ ⁻¹' t ∩ zeroLocus s₀
hS' : comap g '' S.toSet = S'.toSet
x : BasicConstructibleSetData (MvPolynomial (Fin m) R)
hx : x ∈ S
⊢ ∀
    (j :
      Fin
        (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) }).n),
    (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) }).g
        j ∈
      coeffsIn (Fin m) (Submodule.restrictScalars ℤ (degreesLE R (Fin n) Finset.univ.val)) ⊓
        Submodule.restrictScalars ℤ (degreesLE (MvPolynomial (Fin n) R) (Fin m) d)

Tactic:
intro j
      obtain ⟨j, rfl⟩ := finSumFinEquiv.surjective j
      simp only [Equiv.symm_apply_apply, Submodule.mem_inf, mem_coeffsIn,
        Submodule.restrictScalars_mem, mem_degreesLE]
      constructor
      · intro i
        obtain j | j := j
        · simp [σ, MvPolynomial.coeff_map, degrees_C]
        · simp only [MvPolynomial.algebraMap_eq, Sum.elim_inr, MvPolynomial.coeff_sub,
            MvPolynomial.coeff_C, MvPolynomial.coeff_map, σ]
          refine degrees_sub_le.trans ?_
          simp only [degrees_C, apply_ite, degrees_zero,
            Multiset.union_zero]
          split_ifs with h
          · refine (degrees_X' _).trans ?_
            simp
          · simp
      · obtain j | j := j
        · simp only [MvPolynomial.algebraMap_eq, Sum.elim_inl, comp_apply, σ]
          exact degrees_map_le.trans (hS _ hx j)
        · refine degrees_sub_le.trans ?_
          simp only [degrees_C, Multiset.zero_union]
          exact degrees_map_le.trans (hf _)

Goal state after: