-- 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
let let₁ let₂ :
  ∃ (T : ConstructibleSetData (MvPolynomial (Fin n) R)),
    Eq (α := Set (PrimeSpectrum (MvPolynomial (Fin n) R)))
        (Set.image (α := PrimeSpectrum (MvPolynomial (Fin m) R))
          (comap (↑f : MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)) S.toSet)
        T.toSet ∧
      ∀ C ∈ T, C.n ≤ numBound k m n d ∧ ∀ (i : Fin C.n) (j : Fin n), degreeOf j (C.g i) ≤ degBound k m n d
let₃ let₄ :
  let g := 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)),
    ∃ (T : ConstructibleSetData (MvPolynomial (Fin n) R)),
      Eq (α := Set (PrimeSpectrum (MvPolynomial (Fin n) R)))
          (Set.image (α := PrimeSpectrum (MvPolynomial (Fin m) R))
            (comap (↑f : MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)) S.toSet)
          T.toSet ∧
        ∀ C ∈ T, C.n ≤ numBound k m n d ∧ ∀ (i : Fin C.n) (j : Fin n), degreeOf j (C.g i) ≤ degBound k m n d
let₅ :
  let g := 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)),
    let 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)),
      ∃ (T : ConstructibleSetData (MvPolynomial (Fin n) R)),
        Eq (α := Set (PrimeSpectrum (MvPolynomial (Fin n) R)))
            (Set.image (α := PrimeSpectrum (MvPolynomial (Fin m) R))
              (comap (↑f : MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)) S.toSet)
            T.toSet ∧
          ∀ C ∈ T, C.n ≤ numBound k m n d ∧ ∀ (i : Fin C.n) (j : Fin n), degreeOf j (C.g i) ≤ degBound k m n d
let₆ :
  let g := 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)),
    let 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),
      ∃ (T : ConstructibleSetData (MvPolynomial (Fin n) R)),
        Eq (α := Set (PrimeSpectrum (MvPolynomial (Fin n) R)))
            (Set.image (α := PrimeSpectrum (MvPolynomial (Fin m) R))
              (comap (↑f : MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)) S.toSet)
            T.toSet ∧
          ∀ C ∈ T, C.n ≤ numBound k m n d ∧ ∀ (i : Fin C.n) (j : Fin n), degreeOf j (C.g i) ≤ degBound k m n d
let₇ let₈ let₉ :
  let g := 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)),
    let 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)),
      ∃ (T : ConstructibleSetData (MvPolynomial (Fin n) R)),
        Eq (α := Set (PrimeSpectrum (MvPolynomial (Fin n) R)))
            (Set.image (α := PrimeSpectrum (MvPolynomial (Fin m) R))
              (comap (↑f : MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)) S.toSet)
            T.toSet ∧
          ∀ C ∈ T, C.n ≤ numBound k m n d ∧ ∀ (i : Fin C.n) (j : Fin n), degreeOf j (C.g i) ≤ degBound k m n d
let₁₀ :
  let g := 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)),
    let 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)),
      let 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;
      let 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));
      ∀ (hs : zeroLocus s₀ = Set.range (comap g)),
        ∃ (T : ConstructibleSetData (MvPolynomial (Fin n) R)),
          Eq (α := Set (PrimeSpectrum (MvPolynomial (Fin n) R)))
              (Set.image (α := PrimeSpectrum (MvPolynomial (Fin m) R))
                (comap (↑f : MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)) S.toSet)
              T.toSet ∧
            ∀ C ∈ T, C.n ≤ numBound k m n d ∧ ∀ (i : Fin C.n) (j : Fin n), degreeOf j (C.g i) ≤ degBound k m n d
let₁₁ :
  let g := 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)),
    let 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)),
      let 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;
      let 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));
      ∀ (hs : zeroLocus s₀ = Set.range (comap g))
        (hg'' : ∀ (t : Set (PrimeSpectrum (MvPolynomial (Fin m) R))), comap g '' t = comap σ ⁻¹' t ∩ zeroLocus s₀),
        ∃ (T : ConstructibleSetData (MvPolynomial (Fin n) R)),
          Eq (α := Set (PrimeSpectrum (MvPolynomial (Fin n) R)))
              (Set.image (α := PrimeSpectrum (MvPolynomial (Fin m) R))
                (comap (↑f : MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)) S.toSet)
              T.toSet ∧
            ∀ C ∈ T, C.n ≤ numBound k m n d ∧ ∀ (i : Fin C.n) (j : Fin n), degreeOf j (C.g i) ≤ degBound k m n d
let₁₂ :
  let g := 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)),
    let 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)),
      let 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;
      let 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));
      ∀ (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),
        ∃ (T : ConstructibleSetData (MvPolynomial (Fin n) R)),
          Eq (α := Set (PrimeSpectrum (MvPolynomial (Fin n) R)))
              (Set.image (α := PrimeSpectrum (MvPolynomial (Fin m) R))
                (comap (↑f : MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)) S.toSet)
              T.toSet ∧
            ∀ C ∈ T, C.n ≤ numBound k m n d ∧ ∀ (i : Fin C.n) (j : Fin n), degreeOf j (C.g i) ≤ degBound k m n d
let₁₃ :
  let g := 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)),
    let 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)),
      let 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;
      let 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));
      ∀ (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) (T : ConstructibleSetData (MvPolynomial (Fin n) R))
        (hT : comap MvPolynomial.C '' S'.toSet = T.toSet)
        (hT' :
          ∀ C ∈ T,
            C.n ≤ MvPolynomialC.numBound (k + n) (fun (i : ℕ) ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) m ∧
              ∀ (i : Fin C.n),
                C.g i ∈
                  HPow.hPow (α := Submodule ℤ (MvPolynomial (Fin n) R))
                    (Submodule.restrictScalars ℤ (degreesLE R (Fin n) (Finset.val Finset.univ)))
                    (MvPolynomialC.degBound (k + n) (fun (i : ℕ) ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) m)),
        ∃ (T : ConstructibleSetData (MvPolynomial (Fin n) R)),
          Eq (α := Set (PrimeSpectrum (MvPolynomial (Fin n) R)))
              (Set.image (α := PrimeSpectrum (MvPolynomial (Fin m) R))
                (comap (↑f : MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)) S.toSet)
              T.toSet ∧
            ∀ C ∈ T, C.n ≤ numBound k m n d ∧ ∀ (i : Fin C.n) (j : Fin n), degreeOf j (C.g i) ≤ degBound k m n d
refine_1 :
  let g := 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)),
    let 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)),
      let 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;
      let 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));
      ∀ (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) (T : ConstructibleSetData (MvPolynomial (Fin n) R))
        (hT : comap MvPolynomial.C '' S'.toSet = T.toSet)
        (hT' :
          ∀ C ∈ T,
            C.n ≤ MvPolynomialC.numBound (k + n) (fun (i : ℕ) ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) m ∧
              ∀ (i : Fin C.n),
                C.g i ∈
                  HPow.hPow (α := Submodule ℤ (MvPolynomial (Fin n) R))
                    (Submodule.restrictScalars ℤ (degreesLE R (Fin n) (Finset.val Finset.univ)))
                    (MvPolynomialC.degBound (k + n) (fun (i : ℕ) ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) m)),
        Eq (α := Set (PrimeSpectrum (MvPolynomial (Fin n) R)))
          (Set.image (α := PrimeSpectrum (MvPolynomial (Fin m) R))
            (comap (↑f : MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)) S.toSet)
          T.toSet
refine_2 refine_2₁ :
  let g := 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)),
    let 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)),
      let 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;
      let 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));
      ∀ (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) (T : ConstructibleSetData (MvPolynomial (Fin n) R))
        (hT : comap MvPolynomial.C '' S'.toSet = T.toSet)
        (hT' :
          ∀ C ∈ T,
            C.n ≤ MvPolynomialC.numBound (k + n) (fun (i : ℕ) ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) m ∧
              ∀ (i : Fin C.n),
                C.g i ∈
                  HPow.hPow (α := Submodule ℤ (MvPolynomial (Fin n) R))
                    (Submodule.restrictScalars ℤ (degreesLE R (Fin n) (Finset.val Finset.univ)))
                    (MvPolynomialC.degBound (k + n) (fun (i : ℕ) ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) m))
        (C : BasicConstructibleSetData (MvPolynomial (Fin n) R)) (hCT : C ∈ T) (i : Fin C.n) (j : Fin n),
        degreeOf j (C.g i) ≤ degBound k m n d
⊢ ∃ (T : ConstructibleSetData (MvPolynomial (Fin n) R)),
    Eq (α := Set (PrimeSpectrum (MvPolynomial (Fin n) R)))
        (Set.image (α := PrimeSpectrum (MvPolynomial (Fin m) R))
          (comap (↑f : MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)) S.toSet)
        T.toSet ∧
      ∀ C ∈ T, C.n ≤ numBound k m n d ∧ ∀ (i : Fin C.n) (j : Fin n), degreeOf j (C.g i) ≤ degBound k m n d

Tactic:
classical
  let g : MvPolynomial (Fin m) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R :=
    eval₂Hom f.toRingHom X
  have hg : g.comp (algebraMap (MvPolynomial (Fin n) R) _) = f := by ext x : 2 <;> simp [g]
  let σ : MvPolynomial (Fin m) R →+* MvPolynomial (Fin m) (MvPolynomial (Fin n) R) :=
    map (algebraMap _ _)
  have hσ : g.comp σ = .id _ := by ext : 2 <;> simp [g, σ]
  have hσ' (x) : g (σ x) = x := DFunLike.congr_fun hσ x
  have hg' : Surjective g := LeftInverse.surjective hσ'
  let S' : ConstructibleSetData (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) := S.image
    fun ⟨fk, k, gk⟩ ↦ ⟨σ fk, k + n, fun s ↦ (finSumFinEquiv.symm s).elim (σ ∘ gk)
      fun i ↦ .C (.X i) - σ (f (.X i))⟩
  let s₀ : Set (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
    .range fun i ↦ .C (.X i) - σ (f (.X i))
  have hs : zeroLocus s₀ = Set.range (comap g) := by
    rw [range_comap_of_surjective _ _ hg', ← zeroLocus_span]
    congr! 2
    have H : Ideal.span s₀ ≤ RingHom.ker g := by
      simp only [Ideal.span_le, Set.range_subset_iff, SetLike.mem_coe, RingHom.mem_ker, map_sub,
        hσ', s₀]
      simp [g]
    refine H.antisymm fun p hp ↦ ?_
    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
    obtain rfl : q₂ = 0 := by simpa [hσ', show g q₁ = 0 from H hq₁] using hp
    simpa using hq₁
  have hg'' (t) : comap g '' t = comap σ ⁻¹' t ∩ zeroLocus s₀ := by
    refine Set.injOn_preimage (f := comap g) .rfl ?_ ?_ ?_
    · simp
    · simp [hs]
    · rw [Set.preimage_image_eq _ (comap_injective_of_surjective g hg'),
        Set.preimage_inter, hs, Set.preimage_range, Set.inter_univ,
        ← Set.preimage_comp, ← comap_comp, hσ]
      simp only [comap_id, Set.preimage_id']
  have hS' : comap g '' S.toSet = S'.toSet := by
    simp only [S', BasicConstructibleSetData.toSet, ConstructibleSetData.toSet, Set.image_iUnion₂,
      Finset.set_biUnion_finset_image, ← comp_def (g := finSumFinEquiv.symm), Set.range_comp,
      Equiv.range_eq_univ, Set.image_univ, Set.Sum.elim_range,
      Set.image_diff (hf := comap_injective_of_surjective g hg'), zeroLocus_union]
    simp [hg'', ← Set.inter_diff_distrib_right, Set.sdiff_inter_right_comm, s₀]
  obtain ⟨T, hT, hT'⟩ :=
    chevalley_mvPolynomialC
    (M := (degreesLE R (Fin n) Finset.univ.1).restrictScalars ℤ) (by simp) (k + n) d S'
    (Finset.forall_mem_image.mpr fun x hx ↦ (by simpa using hSn _ hx))
    (Finset.forall_mem_image.mpr fun x hx ↦ by
      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 _))
  refine ⟨T, ?_, fun C hCT ↦ ⟨(hT' C hCT).1, fun i j ↦ ?_⟩⟩
  · rwa [← hg, comap_comp, Set.image_comp, hS']
  · have := (hT' C hCT).2 i
    rw [← Submodule.restrictScalars_pow (MvPolynomialC.degBound_pos ..).ne', ← degreesLE_nsmul,
      Submodule.restrictScalars_mem, mem_degreesLE,
        Multiset.le_iff_count] at this
    simpa only [Multiset.count_nsmul, Multiset.count_univ, mul_one, ← degreeOf_def]
      using this j

Premises used: Multiset.le_iff_count, MvPolynomial.degreeOf_def, MvPolynomial.degreesLE_nsmul, LT.lt.ne', Multiset.count_nsmul, MvPolynomial.mem_degreesLE, ChevalleyThm.MvPolynomialC.degBound_pos, Submodule.restrictScalars_mem, Set.image_comp, mul_one, Multiset.count_univ, Submodule.restrictScalars_pow, PrimeSpectrum.comap_comp