Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case left
R : Type u_2
inst✝ : CommRing R
n✝ n : ℕ
IH :
  ∀ {M : Submodule ℤ R},
    1 ∈ M →
      ∀ (k : ℕ) (d : Multiset (Fin n)) (S : ConstructibleSetData (MvPolynomial (Fin n) R)),
        (∀ C ∈ S, C.n ≤ k) →
          (∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) d)) →
            ∃ T,
              comap MvPolynomial.C '' S.toSet = T.toSet ∧
                ∀ C ∈ T,
                  C.n ≤ numBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n ∧
                    ∀ (i : Fin C.n), C.g i ∈ M ^ degBound k (fun i ↦ 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), C.g j ∈ coeffsIn (Fin (n + 1)) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin (n + 1)) d✝)
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
hS' : S'.degBound ≤ k✝ * (1 + Multiset.count 0 d✝)
B : Multiset (Fin n) := Finsupp.toMultiset (Finsupp.comapDomain Fin.succ (Multiset.toFinsupp d✝) ⋯)
k : ℕ
C : BasicConstructibleSetData (MvPolynomial (Fin (n + 1)) R)
hxS : C ∈ S
j : Fin { f := (MvPolynomial.finSuccEquiv R n) C.f, n := C.n, g := ⇑(MvPolynomial.finSuccEquiv R n) ∘ C.g }.n
d : Fin n →₀ ℕ
⊢ MvPolynomial.coeff d (((MvPolynomial.finSuccEquiv R n) (C.g j)).coeff k) ∈ M

Tactic:
simp only [finSuccEquiv_coeff_coeff]

Goal state after:
case left
R : Type u_2
inst✝ : CommRing R
n✝ n : ℕ
IH :
  ∀ {M : Submodule ℤ R},
    1 ∈ M →
      ∀ (k : ℕ) (d : Multiset (Fin n)) (S : ConstructibleSetData (MvPolynomial (Fin n) R)),
        (∀ C ∈ S, C.n ≤ k) →
          (∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) d)) →
            ∃ T,
              comap MvPolynomial.C '' S.toSet = T.toSet ∧
                ∀ C ∈ T,
                  C.n ≤ numBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n ∧
                    ∀ (i : Fin C.n), C.g i ∈ M ^ degBound k (fun i ↦ 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), C.g j ∈ coeffsIn (Fin (n + 1)) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin (n + 1)) d✝)
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
hS' : S'.degBound ≤ k✝ * (1 + Multiset.count 0 d✝)
B : Multiset (Fin n) := Finsupp.toMultiset (Finsupp.comapDomain Fin.succ (Multiset.toFinsupp d✝) ⋯)
k : ℕ
C : BasicConstructibleSetData (MvPolynomial (Fin (n + 1)) R)
hxS : C ∈ S
j : Fin { f := (MvPolynomial.finSuccEquiv R n) C.f, n := C.n, g := ⇑(MvPolynomial.finSuccEquiv R n) ∘ C.g }.n
d : Fin n →₀ ℕ
⊢ MvPolynomial.coeff (Finsupp.cons k d) (C.g j) ∈ M