diff --git a/HilbertFunction.lean b/HilbertFunction.lean
index 21715f0..4ed1a2b 100644
--- a/HilbertFunction.lean
+++ b/HilbertFunction.lean
@@ -5,6 +5,7 @@ import Mathlib.RingTheory.Ideal.AssociatedPrime
 import Mathlib.RingTheory.Artinian
 import Mathlib.Order.Height
 
+
 -- Setting for "library_search"
 set_option maxHeartbeats 0
 macro "ls" : tactic => `(tactic|library_search)
@@ -74,9 +75,8 @@ 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
+
+
 
 
 -- Definition of homogeneous ideal
@@ -91,8 +91,6 @@ def HomogeneousPrime (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [Dir
 -- Definition of homogeneous maximal ideal
 def HomogeneousMax (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)):= (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]
@@ -101,10 +99,16 @@ def HomogeneousMax (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [Direc
 end
 
 
+-- If A_0 is Artinian and local, then A is graded local
+lemma Graded_local_if_zero_component_Artinian_and_local (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)) : ∃ ( I : Ideal ((⨁ i, 𝒜 i))),(HomogeneousMax 𝒜 I) := sorry
+
+
 
 -- @[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)]
+theorem Hilbert_polynomial_ge1 (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))
@@ -118,7 +122,7 @@ 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)]
+theorem Hilbert_polynomial_0 (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))
@@ -169,7 +173,7 @@ def Component_of_graded_as_addsubgroup (𝒜 : ℤ → Type _)
   sorry
 
 
--- @ Quotient of a graded ring R by a graded ideal p is a graded R-Mod, preserving each component
+-- @Quotient of a graded ring R by a graded ideal p is a graded R-Mod, preserving each component
 instance Quotient_of_graded_is_graded
 (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
 (p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p)
@@ -177,11 +181,3 @@ instance Quotient_of_graded_is_graded
   sorry
 
 
-
--- -- @Graded submodule
--- instance Graded_submodule
--- (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
--- (p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p)
---   : DirectSum.Gmodule 𝒜 (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
---   sorry
-
diff --git a/PolyType.lean b/PolyType.lean
index 07c1be7..e69eec2 100644
--- a/PolyType.lean
+++ b/PolyType.lean
@@ -122,6 +122,11 @@ lemma Poly_constant (F : Polynomial ℚ) (c : ℚ) :
     simp
   · sorry
 
+
+-- Shifting doesn't change the polynomial type
+lemma Poly_shifting (f : ℤ → ℤ) (g : ℤ → ℤ) (hf : PolyType f d) (s : ℤ) (hfg : ∀ (n : ℤ), f (n + s) = g (n)) : PolyType g d := by
+  sorry
+
 -- PolyType 0 = constant function
 lemma PolyType_0 (f : ℤ → ℤ) : (PolyType f 0) ↔ (∃ (c : ℤ), ∃ (N : ℤ), ∀ (n : ℤ), (N ≤ n → f n = c) ∧ c ≠ 0) := by
   constructor
@@ -228,6 +233,8 @@ lemma b_to_a (f : ℤ → ℤ) (d : ℕ) : PolyType f d → (∃ (c : ℤ), ∃
   exact this1
 end
 
+
+
 -- @Additive lemma of length for a SES
 -- Given a SES 0 → A → B → C → 0, then length (A) - length (B) + length (C) = 0 
 section