import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.Tactic.NormNum
import OAI.NumberTheory.PiExponent.LocalAlgebra.RegularHilbertStep

namespace OAI

namespace PiExponentJets.W64

attribute [local instance] MvPolynomial.gradedAlgebra

variable {k σ : Type*} [Field k] [Finite σ]

noncomputable def sectionHilbertSeries (I : Ideal (MvPolynomial σ k)) : PowerSeries ℤ :=
  PowerSeries.mk (fun n => (Module.finrank k (quotientSection I n) : ℤ))

omit [Finite σ] in
@[simp] theorem coeff_sectionHilbertSeries (I : Ideal (MvPolynomial σ k)) (n : ℕ) :
    PowerSeries.coeff n (sectionHilbertSeries I) =
      (Module.finrank k (quotientSection I n) : ℤ) :=
  PowerSeries.coeff_mk _ _

theorem sectionHilbertSeries_regular_step (I : Ideal (MvPolynomial σ k))
    (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k))
    {d : ℕ} (f : MvPolynomial σ k) (hf : f.IsHomogeneous d)
    (hreg : IsRightRegular (Ideal.Quotient.mk I f)) :
    sectionHilbertSeries (Ideal.span {f} ⊔ I) =
      (1 - PowerSeries.X ^ d) * sectionHilbertSeries I := by
  ext n
  rw [sub_mul, one_mul, map_sub, PowerSeries.coeff_X_pow_mul']
  simp only [coeff_sectionHilbertSeries]
  by_cases hdn : d ≤ n
  · rw [ite_eq_left hdn]
    have h := regular_hilbert_step I hI f hf hreg (n-d)
    rw [Nat.sub_add_cancel hdn] at h
    apply eq_sub_of_add_eq
    exact_mod_cast h
  · rw [ite_eq_right hdn, sub_zero]
    exact congrArg (fun m : ℕ => (m : ℤ))
      (hilbert_step_of_degree_lt I hI f hf (Nat.lt_of_not_ge hdn))

end PiExponentJets.W64

end OAI