comm_alg/CommAlg/hilbertpolynomial.lean

import Mathlib.Order.KrullDimension
import Mathlib.Order.JordanHolder
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.Height
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Algebra.Module.GradedModule
import Mathlib.RingTheory.Ideal.AssociatedPrime
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
import Mathlib.Algebra.Module.GradedModule
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Finiteness
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.FiniteType
import Mathlib.Order.Height
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Algebra.DirectSum.Ring
import Mathlib.RingTheory.Ideal.LocalRing

-- Setting for "library_search"
set_option maxHeartbeats 0
macro "ls" : tactic => `(tactic|library_search)

-- New tactic "obviously"
macro "obviously" : tactic =>
  `(tactic| (
      first
        | dsimp; simp; done; dbg_trace "it was dsimp simp"
        | simp; done; dbg_trace "it was simp"
        | tauto; done; dbg_trace "it was tauto"
        | simp; tauto; done; dbg_trace "it was simp tauto"
        | rfl; done; dbg_trace "it was rfl"
        | norm_num; done; dbg_trace "it was norm_num"
        | /-change (@Eq ℝ _ _);-/ linarith; done; dbg_trace "it was linarith"
        -- | gcongr; done
        | ring; done; dbg_trace "it was ring"
        | trivial; done; dbg_trace "it was trivial"
        -- | nlinarith; done
        | fail "No, this is not obvious."))

-- @[BH, 1.5.6 (b)(ii)]
lemma ss (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
  [DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜] (p : associatedPrimes (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) : true := by
  sorry
-- Ideal.IsHomogeneous 𝒜 p




noncomputable def length ( A : Type _) (M : Type _)
 [CommRing A] [AddCommGroup M] [Module A M] :=  Set.chainHeight {M' : Submodule A M | M' < ⊤}


def HomogeneousPrime { A σ : Type _} [CommRing A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ℤ → σ) [GradedRing 𝒜] (I : Ideal A):= (Ideal.IsPrime I) ∧ (Ideal.IsHomogeneous 𝒜 I)
def HomogeneousMax { A σ : Type _} [CommRing A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ℤ → σ) [GradedRing 𝒜] (I : Ideal A):= (Ideal.IsMaximal I) ∧ (Ideal.IsHomogeneous 𝒜 I)

--theorem monotone_stabilizes_iff_noetherian :
-- (∀ f : ℕ →o Submodule R M, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsNoetherian R M := by
-- rw [isNoetherian_iff_wellFounded, WellFounded.monotone_chain_condition]

open GradedMonoid.GSmul
open DirectSum

instance tada1 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]  [DirectSum.GCommRing 𝒜]
  [DirectSum.Gmodule 𝒜 𝓜] (i : ℤ ) : SMul (𝒜 0) (𝓜 i)
    where smul x y := @Eq.rec ℤ (0+i) (fun a _ => 𝓜 a) (GradedMonoid.GSmul.smul x y) i (zero_add i)

lemma mylem (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]  [DirectSum.GCommRing 𝒜]
  [h : DirectSum.Gmodule 𝒜 𝓜] (i : ℤ) (a : 𝒜 0) (m : 𝓜 i) :
  of _ _ (a • m) = of _ _ a • of _ _ m := by
  refine' Eq.trans _ (Gmodule.of_smul_of 𝒜 𝓜 a m).symm
  refine' of_eq_of_gradedMonoid_eq _
  exact Sigma.ext (zero_add _).symm <| eq_rec_heq _ _

instance tada2 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]  [DirectSum.GCommRing 𝒜]
  [h : DirectSum.Gmodule 𝒜 𝓜] (i : ℤ ) : SMulWithZero (𝒜 0) (𝓜 i) := by
  letI := SMulWithZero.compHom (⨁ i, 𝓜 i) (of 𝒜 0).toZeroHom
  exact Function.Injective.smulWithZero (of 𝓜 i).toZeroHom Dfinsupp.single_injective (mylem 𝒜 𝓜 i)

instance tada3 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]  [DirectSum.GCommRing 𝒜]
  [h : DirectSum.Gmodule 𝒜 𝓜] (i : ℤ ): Module (𝒜 0) (𝓜 i) := by
  letI := Module.compHom (⨁ j, 𝓜 j) (ofZeroRingHom 𝒜)
  exact Dfinsupp.single_injective.module (𝒜 0) (of 𝓜 i) (mylem 𝒜 𝓜 i)

noncomputable def hilbert_function (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
  [DirectSum.GCommRing 𝒜]
  [DirectSum.Gmodule 𝒜 𝓜] (hilb : ℤ → ℤ) := ∀ i, hilb i = (ENat.toNat (length (𝒜 0) (𝓜 i)))

noncomputable def dimensionring { A: Type _}
 [CommRing A] := krullDim (PrimeSpectrum A)


noncomputable def dimensionmodule ( A : Type _) (M : Type _)
 [CommRing A] [AddCommGroup M] [Module A M] := krullDim (PrimeSpectrum (A ⧸ ((⊤ : Submodule A M).annihilator)) )

--  lemma graded_local (𝒜 : ℤ → Type _) [SetLike (⨁ i, 𝒜 i)] (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
--   [DirectSum.GCommRing 𝒜]
--   [DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) : ∃ ( I : Ideal ((⨁ i, 𝒜 i))),(HomogeneousMax 𝒜 I) := sorry




@[simp]
def PolyType (f : ℤ → ℤ) (d : ℕ) := ∃ Poly : Polynomial ℚ, ∃ (N : ℤ), ∀ (n : ℤ), N ≤ n → f n = Polynomial.eval (n : ℚ) Poly ∧ d = Polynomial.degree Poly

-- @[BH, 4.1.3]
theorem hilbert_polynomial (d : ℕ) (d1 : 1 ≤ d) (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) 
(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
(findim :  dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = d) (hilb : ℤ → ℤ)
 (Hhilb: hilbert_function 𝒜 𝓜 hilb)
: PolyType hilb (d - 1) := by
  sorry

-- @
lemma Graded_quotient (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)][DirectSum.GCommRing 𝒜] 
  : true := by
  sorry


-- @Existence of a chain of submodules of graded submoduels of f.g graded R-mod M
lemma Exist_chain_of_graded_submodules (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
  [DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜] (fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
  : true := by
  sorry
#print Exist_chain_of_graded_submodules
#print Finset