-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
R_1 : Type u_2
inst : CommRing R_1
n : ℕ
R : Type u_2
inst_1 : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_4 : Algebra R₀ R],
    Statement R₀ R n
      { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n }
R₀ : Type u_1
inst_2 : CommRing R₀
inst_3 : Algebra R₀ R
f : R[X]
h : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus
          (Set.range
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.val) \
        zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤
        { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
            InductionObj R n }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          HPow.hPow (α := Submodule R₀ R)
            (coeffSubmodule R₀
              { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) : InductionObj R n })
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) :
                InductionObj R n }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val j (c.val j %ₘ c.val i) }
deg_bound₁ : c'.degBound ≤ c.degBound
k_1 : Fin C.n
hij : coeffSubmodule R₀ c' ≤ HPow.hPow (α := Submodule R₀ R) (coeffSubmodule R₀ c) (c.val j).degree.succ
hi' : ¬c.val i = 1
k : Fin n
h_1 : Membership.mem (γ := Finset (Fin n)) Finset.univ k
hkj : k = j
pos :
  ∀ (hle : (c.val i).degree ≤ (c.val k).degree) (hne : i ≠ k)
    (H :
      ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_4 : Algebra R₀ R],
        Statement R₀ R n
          { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val k (c.val k %ₘ c.val i) : InductionObj R n })
    (hS :
      comap Polynomial.C ''
          (zeroLocus
              (Set.range
                { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val k (c.val k %ₘ c.val i) :
                    InductionObj R n }.val) \
            zeroLocus {f}) =
        S.toSet)
    (hS' :
      ∀ C ∈ S,
        C.n ≤
            { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val k (c.val k %ₘ c.val i) :
                InductionObj R n }.degBound ∧
          ∀ (i_1 : Fin C.n),
            C.g i_1 ∈
              HPow.hPow (α := Submodule R₀ R)
                (coeffSubmodule R₀
                  { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val k (c.val k %ₘ c.val i) :
                    InductionObj R n })
                { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val k (c.val k %ₘ c.val i) :
                    InductionObj R n }.powBound),
    let c' := { val := Function.update (β := fun (a : Fin n) ↦ R[X]) c.val k (c.val k %ₘ c.val i) : InductionObj R n };
    ∀ (deg_bound₁ : c'.degBound ≤ c.degBound)
      (hij : coeffSubmodule R₀ c' ≤ HPow.hPow (α := Submodule R₀ R) (coeffSubmodule R₀ c) (c.val k).degree.succ),
      (c.val k %ₘ c.val i).degree.succ ≤ (c.val k).degree.succ
⊢ (c.val j %ₘ c.val i).degree.succ ≤ (c.val k).degree.succ

Tactic:
subst hkj