diff --git a/HilbertFunction.lean b/HilbertFunction.lean
index 8d9e6e8..426caea 100644
--- a/HilbertFunction.lean
+++ b/HilbertFunction.lean
@@ -48,22 +48,23 @@ macro "obviously" : tactic =>
         | fail "No, this is not obvious."))
 
 
+
+
+
+
+
 -- @Definitions (to be classified)
 section
+open GradedMonoid.GSmul
+open DirectSum
 
 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
-
+-- Make instance of M_i being an R_0-module
 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)
@@ -85,7 +86,9 @@ instance tada3 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGr
   letI := Module.compHom (⨁ j, 𝓜 j) (ofZeroRingHom 𝒜)
   exact Dfinsupp.single_injective.module (𝒜 0) (of 𝓜 i) (mylem 𝒜 𝓜 i)
 
+
 -- Definition of a Hilbert function of a graded module
+section
 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)))
@@ -95,16 +98,32 @@ noncomputable def dimensionring { A: Type _}
 
 noncomputable def dimensionmodule ( A : Type _) (M : Type _)
  [CommRing A] [AddCommGroup M] [Module A M] := krullDim (PrimeSpectrum (A ⧸ ((⊤ : Submodule A M).annihilator)) )
-
+end
 --  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
 
+
+ def Ideal.IsHomogeneous' (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)]
+  [DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)) := ∀ (i : ℤ ) ⦃r : (⨁ i, 𝒜 i)⦄, r ∈ I → DirectSum.of _ i ( r i : 𝒜 i) ∈ I 
+def HomogeneousPrime (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)):= (Ideal.IsPrime I) ∧ (Ideal.IsHomogeneous' 𝒜 I)
+def HomogeneousMax (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)):= (Ideal.IsMaximal I) ∧ (Ideal.IsHomogeneous' 𝒜 I)
+
+-- Definition of polynomail of type d 
 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
 
 end
 
 
+
+
+
+
+
+
+
+
+
 -- @[BH, 4.1.3] when d ≥ 1
 -- If M is a finite graed R-Mod of dimension d ≥ 1, then the Hilbert function H(M, n) is of polynomial type (d - 1)
 theorem hilbert_polynomial_ge1 (d : ℕ) (d1 : 1 ≤ d) (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
@@ -118,6 +137,7 @@ theorem hilbert_polynomial_ge1 (d : ℕ) (d1 : 1 ≤ d) (𝒜 : ℤ → Type _)
 
 
 
+
 -- @[BH, 4.1.3] when d = 0
 -- If M is a finite graed R-Mod of dimension zero, then the Hilbert function H(M, n) = 0 for n >> 0 
 theorem hilbert_polynomial_0 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
@@ -130,16 +150,9 @@ theorem hilbert_polynomial_0 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [
   sorry
 
 
--- @[BH, 1.5.6 (b)(ii)]
--- An associated prime of a graded R-Mod M is graded
-lemma Associated_prime_of_graded_is_graded
-(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
+
+
+
 
 -- @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 _) 
@@ -149,3 +162,20 @@ lemma Exist_chain_of_graded_submodules (𝒜 : ℤ → Type _) (𝓜 : ℤ → T
   : true := by
   sorry
 
+
+
+
+
+
+-- @[BH, 1.5.6 (b)(ii)]
+-- An associated prime of a graded R-Mod M is graded
+lemma Associated_prime_of_graded_is_graded
+(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))
+  : Ideal.IsHomogeneous' 𝒜 p := by
+  sorry
+
+
+