-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
R : Type u_2
inst : CommRing R
n_1 n : ℕ
IH :
  ∀ {M : Submodule ℤ R} (hM : 1 ∈ M) (k : ℕ) (d : Multiset (Fin n)) (S : ConstructibleSetData (MvPolynomial (Fin n) R))
    (hSn : ∀ C ∈ S, C.n ≤ k)
    (hS :
      ∀ C ∈ S,
        ∀ (j : Fin C.n),
          Membership.mem (γ := Submodule ℤ (MvPolynomial (Fin n) R))
            (coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) d)) (C.g j)),
    ∃ (T : ConstructibleSetData R),
      comap MvPolynomial.C '' S.toSet = T.toSet ∧
        ∀ C ∈ T,
          C.n ≤ numBound k (fun (i : ℕ) ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n ∧
            ∀ (i : Fin C.n), C.g i ∈ M ^ degBound k (fun (i : ℕ) ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n
M : Submodule ℤ R
hM : 1 ∈ M
k : ℕ
d : Multiset (Fin (n + 1))
S : ConstructibleSetData (MvPolynomial (Fin (n + 1)) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS :
  ∀ C ∈ S,
    ∀ (j : Fin C.n),
      Membership.mem (γ := Submodule ℤ (MvPolynomial (Fin (n + 1)) R))
        (coeffsIn (Fin (n + 1)) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin (n + 1)) d)) (C.g j)
e : MvPolynomial (Fin (n + 1)) R ≃ₐ[R] (MvPolynomial (Fin n) R)[X] := MvPolynomial.finSuccEquiv R n
S' : ConstructibleSetData (MvPolynomial (Fin n) R)[X] := ConstructibleSetData.map e.toRingEquiv.toRingHom S
C : BasicConstructibleSetData (MvPolynomial (Fin (n + 1)) R)
hxS : C ∈ S
h :
  ∀
    (j :
      Fin
        (BasicConstructibleSetData.n (R := (MvPolynomial (Fin n) R)[X])
          ({ f := (e : (a : MvPolynomial (Fin (n + 1)) R) → (MvPolynomial (Fin n) R)[X]) C.f, n := C.n,
              g := (⇑e : (a : MvPolynomial (Fin (n + 1)) R) → (MvPolynomial (Fin n) R)[X]) ∘ C.g } :
            BasicConstructibleSetData (MvPolynomial (Fin n) R)[X])))
    (hj :
      Membership.mem (γ :=
        Finset
          (Fin
            (BasicConstructibleSetData.n (R := (MvPolynomial (Fin n) R)[X])
              ({ f := (e : (a : MvPolynomial (Fin (n + 1)) R) → (MvPolynomial (Fin n) R)[X]) C.f, n := C.n,
                  g := (⇑e : (a : MvPolynomial (Fin (n + 1)) R) → (MvPolynomial (Fin n) R)[X]) ∘ C.g } :
                BasicConstructibleSetData (MvPolynomial (Fin n) R)[X]))))
        Finset.univ j),
    (degree.{u_2} (R := MvPolynomial (Fin n) R)
          (BasicConstructibleSetData.g
            { f := (e : (a : MvPolynomial (Fin (n + 1)) R) → (MvPolynomial (Fin n) R)[X]) C.f, n := C.n,
              g := (⇑e : (a : MvPolynomial (Fin (n + 1)) R) → (MvPolynomial (Fin n) R)[X]) ∘ C.g :
              BasicConstructibleSetData (MvPolynomial (Fin n) R)[X] }
            j)).succ ≤
      1 + Multiset.count 0 d
⊢ LE.le (α := ℕ)
    (∑ i :
      Fin
        { f := (e : (a : MvPolynomial (Fin (n + 1)) R) → (MvPolynomial (Fin n) R)[X]) C.f, n := C.n,
            g := (⇑e : (a : MvPolynomial (Fin (n + 1)) R) → (MvPolynomial (Fin n) R)[X]) ∘ C.g }.n,
      (degree.{u_2} (R := MvPolynomial (Fin n) R)
          (BasicConstructibleSetData.g
            { f := (e : (a : MvPolynomial (Fin (n + 1)) R) → (MvPolynomial (Fin n) R)[X]) C.f, n := C.n,
              g := (⇑e : (a : MvPolynomial (Fin (n + 1)) R) → (MvPolynomial (Fin n) R)[X]) ∘ C.g }
            i)).succ)
    (∑ i : Fin C.n, (1 + Multiset.count 0 d))

Tactic:
gcongr with j hj