diff --git a/CommAlg/hilbertpolynomial.lean b/CommAlg/hilbertpolynomial.lean
new file mode 100644
index 0000000..64650dc
--- /dev/null
+++ b/CommAlg/hilbertpolynomial.lean
@@ -0,0 +1,137 @@
+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 (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]
+  [DirectSum.GCommRing π’œ] [DirectSum.Gmodule π’œ π“œ] (p : associatedPrimes (⨁ i, π’œ i) (⨁ i, π“œ i)) : true := by
+  sorry
+-- Ideal.IsHomogeneous π’œ 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 Οƒ : Type _} [CommRing A] [SetLike Οƒ A] [AddSubmonoidClass Οƒ A] (π’œ : β„€ β†’ Οƒ) [GradedRing π’œ] (I : Ideal A):= (Ideal.IsPrime I) ∧ (Ideal.IsHomogeneous π’œ I)
+def HomogeneousMax { A Οƒ : Type _} [CommRing A] [SetLike Οƒ A] [AddSubmonoidClass Οƒ A] (π’œ : β„€ β†’ Οƒ) [GradedRing π’œ] (I : Ideal A):= (Ideal.IsMaximal I) ∧ (Ideal.IsHomogeneous π’œ I)
+
+--theorem monotone_stabilizes_iff_noetherian :
+-- (βˆ€ f : β„• β†’o Submodule R M, βˆƒ n, βˆ€ m, n ≀ m β†’ f n = f m) ↔ IsNoetherian R M := by
+-- rw [isNoetherian_iff_wellFounded, WellFounded.monotone_chain_condition]
+
+open GradedMonoid.GSmul
+open DirectSum
+
+instance tada1 (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]  [DirectSum.GCommRing π’œ]
+  [DirectSum.Gmodule π’œ π“œ] (i : β„€ ) : SMul (π’œ 0) (π“œ i)
+    where smul x y := @Eq.rec β„€ (0+i) (fun a _ => π“œ a) (GradedMonoid.GSmul.smul x y) i (zero_add i)
+
+lemma mylem (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]  [DirectSum.GCommRing π’œ]
+  [h : DirectSum.Gmodule π’œ π“œ] (i : β„€) (a : π’œ 0) (m : π“œ i) :
+  of _ _ (a β€’ m) = of _ _ a β€’ of _ _ m := by
+  refine' Eq.trans _ (Gmodule.of_smul_of π’œ π“œ a m).symm
+  refine' of_eq_of_gradedMonoid_eq _
+  exact Sigma.ext (zero_add _).symm <| eq_rec_heq _ _
+
+instance tada2 (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]  [DirectSum.GCommRing π’œ]
+  [h : DirectSum.Gmodule π’œ π“œ] (i : β„€ ) : SMulWithZero (π’œ 0) (π“œ i) := by
+  letI := SMulWithZero.compHom (⨁ i, π“œ i) (of π’œ 0).toZeroHom
+  exact Function.Injective.smulWithZero (of π“œ i).toZeroHom Dfinsupp.single_injective (mylem π’œ π“œ i)
+
+instance tada3 (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]  [DirectSum.GCommRing π’œ]
+  [h : DirectSum.Gmodule π’œ π“œ] (i : β„€ ): Module (π’œ 0) (π“œ i) := by
+  letI := Module.compHom (⨁ j, π“œ j) (ofZeroRingHom π’œ)
+  exact Dfinsupp.single_injective.module (π’œ 0) (of π“œ i) (mylem π’œ π“œ i)
+
+noncomputable def hilbert_function (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]
+  [DirectSum.GCommRing π’œ]
+  [DirectSum.Gmodule π’œ π“œ] (hilb : β„€ β†’ β„€) := βˆ€ i, hilb i = (ENat.toNat (length (π’œ 0) (π“œ 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 (π’œ : β„€ β†’ Type _) [SetLike (⨁ i, π’œ i)] (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]
+--   [DirectSum.GCommRing π’œ]
+--   [DirectSum.Gmodule π’œ π“œ] (art: IsArtinianRing (π’œ 0)) (loc : LocalRing (π’œ 0)) : βˆƒ ( I : Ideal ((⨁ i, π’œ i))),(HomogeneousMax π’œ I) := sorry
+
+
+
+
+@[simp]
+def PolyType (f : β„€ β†’ β„€) (d : β„•) := βˆƒ Poly : Polynomial β„š, βˆƒ (N : β„€), βˆ€ (n : β„€), N ≀ n β†’ f n = Polynomial.eval (n : β„š) Poly ∧ d = Polynomial.degree Poly
+
+-- @[BH, 4.1.3]
+theorem hilbert_polynomial (d : β„•) (d1 : 1 ≀ d) (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]
+[DirectSum.GCommRing π’œ]
+[DirectSum.Gmodule π’œ π“œ] (art: IsArtinianRing (π’œ 0)) (loc : LocalRing (π’œ 0)) 
+(fingen : IsNoetherian (⨁ i, π’œ i) (⨁ i, π“œ i))
+(findim :  dimensionmodule (⨁ i, π’œ i) (⨁ i, π“œ i) = d) (hilb : β„€ β†’ β„€)
+ (Hhilb: hilbert_function π’œ π“œ hilb)
+: PolyType hilb (d - 1) := by
+  sorry
+
+-- @
+lemma Graded_quotient (π’œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)][DirectSum.GCommRing π’œ] 
+  : 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 (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]
+  [DirectSum.GCommRing π’œ] [DirectSum.Gmodule π’œ π“œ] (fingen : IsNoetherian (⨁ i, π’œ i) (⨁ i, π“œ i))
+  : true := by
+  sorry
+#print Exist_chain_of_graded_submodules
+#print Finset
\ No newline at end of file
diff --git a/CommAlg/krull.lean b/CommAlg/krull.lean
index 59fd048..2111469 100644
--- a/CommAlg/krull.lean
+++ b/CommAlg/krull.lean
@@ -63,7 +63,7 @@ lemma krullDim_eq_height [LocalRing R] : krullDim R = height (closedPoint R) :=
     apply height_le_of_le
     apply le_maximalIdeal
     exact I.2.1
-  . simp
+  . simp only [height_le_krullDim]
 
 #check height_le_krullDim
 --some propositions that would be nice to be able to eventually
@@ -135,7 +135,7 @@ lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by
   unfold krullDim
   simp [field_prime_height_zero]
 
-lemma isField.dim_zero {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0) : IsField D := by
+lemma domain_dim_zero.isField {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0) : IsField D := by
   by_contra x
   rw [Ring.not_isField_iff_exists_prime] at x
   obtain ⟨P, ⟨h1, primeP⟩⟩ := x
@@ -156,9 +156,9 @@ lemma isField.dim_zero {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0)
     aesop
   contradiction
 
-lemma dim_eq_zero_iff_field {D: Type _} [CommRing D] [IsDomain D] : krullDim D = 0 ↔ IsField D := by
+lemma domain_dim_eq_zero_iff_field {D: Type _} [CommRing D] [IsDomain D] : krullDim D = 0 ↔ IsField D := by
   constructor
-  Β· exact isField.dim_zero
+  Β· exact domain_dim_zero.isField
   Β· intro fieldD
     let h : Field D := IsField.toField fieldD
     exact dim_field_eq_zero
diff --git a/CommAlg/monalisa.lean b/CommAlg/monalisa.lean
new file mode 100644
index 0000000..d3a272e
--- /dev/null
+++ b/CommAlg/monalisa.lean
@@ -0,0 +1,140 @@
+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
+import Mathlib
+import Mathlib.Algebra.MonoidAlgebra.Basic
+import Mathlib.Data.Finset.Sort
+import Mathlib.Order.Height
+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
+
+
+
+
+noncomputable def length ( A : Type _) (M : Type _)
+ [CommRing A] [AddCommGroup M] [Module A M] :=  Set.chainHeight {M' : Submodule A M | M' < ⊀}
+
+
+def HomogeneousPrime { A Οƒ : Type _} [CommRing A] [SetLike Οƒ A] [AddSubmonoidClass Οƒ A] (π’œ : β„€ β†’ Οƒ) [GradedRing π’œ] (I : Ideal A):= (Ideal.IsPrime I) ∧ (Ideal.IsHomogeneous π’œ I)
+
+
+def HomogeneousMax { A Οƒ : Type _} [CommRing A] [SetLike Οƒ A] [AddSubmonoidClass Οƒ A] (π’œ : β„€ β†’ Οƒ) [GradedRing π’œ] (I : Ideal A):= (Ideal.IsMaximal I) ∧ (Ideal.IsHomogeneous π’œ I)
+
+--theorem monotone_stabilizes_iff_noetherian :
+-- (βˆ€ f : β„• β†’o Submodule R M, βˆƒ n, βˆ€ m, n ≀ m β†’ f n = f m) ↔ IsNoetherian R M := by
+-- rw [isNoetherian_iff_wellFounded, WellFounded.monotone_chain_condition]
+
+open GradedMonoid.GSmul
+
+open DirectSum
+
+instance tada1 (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]  [DirectSum.GCommRing π’œ]
+  [DirectSum.Gmodule π’œ π“œ] (i : β„€ ) : SMul (π’œ 0) (π“œ i)
+    where smul x y := @Eq.rec β„€ (0+i) (fun a _ => π“œ a) (GradedMonoid.GSmul.smul x y) i (zero_add i)
+
+lemma mylem (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]  [DirectSum.GCommRing π’œ]
+  [h : DirectSum.Gmodule π’œ π“œ] (i : β„€) (a : π’œ 0) (m : π“œ i) :
+  of _ _ (a β€’ m) = of _ _ a β€’ of _ _ m := by
+  refine' Eq.trans _ (Gmodule.of_smul_of π’œ π“œ a m).symm
+  refine' of_eq_of_gradedMonoid_eq _
+  exact Sigma.ext (zero_add _).symm <| eq_rec_heq _ _
+
+instance tada2 (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]  [DirectSum.GCommRing π’œ]
+  [h : DirectSum.Gmodule π’œ π“œ] (i : β„€ ) : SMulWithZero (π’œ 0) (π“œ i) := by
+  letI := SMulWithZero.compHom (⨁ i, π“œ i) (of π’œ 0).toZeroHom
+  exact Function.Injective.smulWithZero (of π“œ i).toZeroHom Dfinsupp.single_injective (mylem π’œ π“œ i)
+
+instance tada3 (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]  [DirectSum.GCommRing π’œ]
+  [h : DirectSum.Gmodule π’œ π“œ] (i : β„€ ): Module (π’œ 0) (π“œ i) := by
+  letI := Module.compHom (⨁ j, π“œ j) (ofZeroRingHom π’œ)
+  exact Dfinsupp.single_injective.module (π’œ 0) (of π“œ i) (mylem π’œ π“œ i)
+
+  -- (art: IsArtinianRing (π’œ 0)) (loc : LocalRing (π’œ 0))
+
+noncomputable def dummyhil_function (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]
+  [DirectSum.GCommRing π’œ]
+  [DirectSum.Gmodule π’œ π“œ] (hilb : β„€ β†’ β„•βˆž) := βˆ€ i, hilb i = (length (π’œ 0) (π“œ i))
+
+
+lemma hilbertz (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]
+  [DirectSum.GCommRing π’œ]
+  [DirectSum.Gmodule π’œ π“œ] 
+  (finlen : βˆ€ i, (length (π’œ 0) (π“œ i)) < ⊀ ) : β„€ β†’ β„€ := by
+  intro i
+  let h := dummyhil_function π’œ π“œ
+  simp  at h 
+  let n : β„€ β†’ β„• := fun i ↦ WithTop.untop _ (finlen i).ne
+  have hn : βˆ€ i, (n i : β„•βˆž) = length (π’œ 0) (π“œ i) := fun i ↦ WithTop.coe_untop _ _
+  have' := hn i
+  exact ((n i) : β„€ )
+
+
+noncomputable def hilbert_function (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]
+  [DirectSum.GCommRing π’œ]
+  [DirectSum.Gmodule π’œ π“œ] (hilb : β„€ β†’ β„€) := βˆ€ i, hilb i = (ENat.toNat (length (π’œ 0) (π“œ 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 (π’œ : β„€ β†’ Type _) [SetLike (⨁ i, π’œ i)] (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]
+--   [DirectSum.GCommRing π’œ]
+--   [DirectSum.Gmodule π’œ π“œ] (art: IsArtinianRing (π’œ 0)) (loc : LocalRing (π’œ 0)) : βˆƒ ( I : Ideal ((⨁ i, π’œ i))),(HomogeneousMax π’œ I) := sorry
+
+
+def PolyType (f : β„€ β†’ β„€) (d : β„•) := βˆƒ Poly : Polynomial β„š, βˆƒ (N : β„€), βˆ€ (n : β„€), N ≀ n β†’ f n = Polynomial.eval (n : β„š) Poly ∧ d = Polynomial.degree Poly
+
+
+
+theorem hilbert_polynomial (π’œ : β„€ β†’ Type _) (π“œ : β„€ β†’ Type _) [βˆ€ i, AddCommGroup (π’œ i)] [βˆ€ i, AddCommGroup (π“œ i)]
+[DirectSum.GCommRing π’œ]
+[DirectSum.Gmodule π’œ π“œ] (art: IsArtinianRing (π’œ 0)) (loc : LocalRing (π’œ 0)) (fingen : IsNoetherian (⨁ i, π’œ i) (⨁ i, π“œ i))
+(findim : βˆƒ d : β„• , dimensionmodule (⨁ i, π’œ i) (⨁ i, π“œ i) = d):True := sorry
+
+-- Semiring A]
+
+-- variable [SetLike Οƒ A]
\ No newline at end of file
diff --git a/CommAlg/sameer(artinian-rings).lean b/CommAlg/sameer(artinian-rings).lean
index e6bb667..417cc96 100644
--- a/CommAlg/sameer(artinian-rings).lean
+++ b/CommAlg/sameer(artinian-rings).lean
@@ -6,7 +6,9 @@ import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
 import Mathlib.RingTheory.DedekindDomain.DVR
 
 
-lemma FieldisArtinian (R : Type _) [CommRing R] (IsField : ):= by sorry
+lemma FieldisArtinian (R : Type _) [CommRing R] (h: IsField R) : 
+  IsArtinianRing R := by sorry
+
 
 
 lemma ArtinianDomainIsField (R : Type _) [CommRing R] [IsDomain R]
@@ -47,8 +49,7 @@ lemma isArtinianRing_of_quotient_of_artinian (R : Type _) [CommRing R]
 lemma IsPrimeMaximal (R : Type _) [CommRing R] (P : Ideal R) 
 (IsArt : IsArtinianRing R) (isPrime : Ideal.IsPrime P) : Ideal.IsMaximal P := 
 by 
--- if R is Artinian and P is prime then R/P is Integral Domain 
--- which is Artinian Domain 
+-- if R is Artinian and P is prime then R/P is Artinian Domain 
 -- R⧸P is a field by the above lemma
 -- P is maximal
 
@@ -56,13 +57,13 @@ by
   have artRP : IsArtinianRing (R⧸P) := by
     exact isArtinianRing_of_quotient_of_artinian R P IsArt
   
-
+  have artRPField : IsField (R⧸P) := by
+    exact ArtinianDomainIsField (R⧸P) artRP
+  
+  have h := Ideal.Quotient.maximal_of_isField P artRPField
+  exact h
+  
 -- Then R/I is Artinian 
  --  have' : IsArtinianRing R ∧ Ideal.IsPrime I β†’ IsDomain (Rβ§ΈI) := by
 
 -- R⧸I.IsArtinian → monotone_stabilizes_iff_artinian.R⧸I
-
-
-
-
--- Use Stacks project proof since it's broken into lemmas
diff --git a/CommAlg/sayantan(dim_eq_dim_polynomial_add_one).lean b/CommAlg/sayantan(dim_eq_dim_polynomial_add_one).lean
index a9269a2..089c4b5 100644
--- a/CommAlg/sayantan(dim_eq_dim_polynomial_add_one).lean
+++ b/CommAlg/sayantan(dim_eq_dim_polynomial_add_one).lean
@@ -22,11 +22,7 @@ noncomputable instance : CompleteLattice (WithBot (β„•βˆž)) := WithBot.WithTop.c
 lemma dim_le_dim_polynomial_add_one [Nontrivial R] :
   krullDim R + 1 ≀ krullDim (Polynomial R) := sorry -- Others are working on it
 
--- private lemma sum_succ_of_succ_sum {ΞΉ : Type} (a : β„•βˆž) [inst : Nonempty ΞΉ] :
---       (⨆ (x : ΞΉ), a + 1) = (⨆ (x : ΞΉ), a) + 1 := by
---   have : a + 1 = (⨆ (x : ΞΉ), a) + 1 := by rw [ciSup_const]
---   have : a + 1 = (⨆ (x : ΞΉ), a + 1) := Eq.symm ciSup_const
---   simp
+lemma height_le_of_le {I J : PrimeSpectrum R} (I_le_J : I ≀ J) : height I ≀ height J := sorry -- Already done in main file
 
 lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
   krullDim R + 1 = krullDim (Polynomial R) := by
@@ -34,13 +30,33 @@ lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
   constructor
   Β· exact dim_le_dim_polynomial_add_one
   Β· unfold krullDim
-    have htPBdd : βˆ€ (P : PrimeSpectrum (Polynomial R)), (height P: WithBot β„•βˆž) ≀ (⨆ (I : PrimeSpectrum R), ↑(height I + 1)) := by
+    have htPBdd : βˆ€ (P : PrimeSpectrum (Polynomial R)), (height P : WithBot β„•βˆž) ≀ (⨆ (I : PrimeSpectrum R), ↑(height I + 1)) := by
       intro P
-      unfold height
-      sorry
-    have : (⨆ (I : PrimeSpectrum R), ↑(height I) + 1) ≀ (⨆ (I : PrimeSpectrum R), ↑(height I)) + 1 := by
-      have : βˆ€ P : PrimeSpectrum R, ↑(height P) + 1 ≀ (⨆ (I : PrimeSpectrum R), ↑(height I)) + 1 :=
-        fun _ ↦ add_le_add_right (le_iSup height _) 1
+      have : βˆƒ (I : PrimeSpectrum R), (height P : WithBot β„•βˆž) ≀ ↑(height I + 1) := by
+        have : βˆƒ M, Ideal.IsMaximal M ∧ P.asIdeal ≀ M := by
+          apply exists_le_maximal
+          apply IsPrime.ne_top
+          apply P.IsPrime
+        obtain ⟨M, maxM, PleM⟩ := this
+        let P' : PrimeSpectrum (Polynomial R) := PrimeSpectrum.mk M (IsMaximal.isPrime maxM)
+        have PleP' : P ≀ P' := PleM
+        have : height P ≀ height P' := height_le_of_le PleP'
+        simp only [WithBot.coe_le_coe]
+        have : βˆƒ (I : PrimeSpectrum R), height P' ≀ height I + 1 := by
+
+          sorry
+        obtain ⟨I, h⟩ := this
+        use I
+        exact ge_trans h this
+      obtain ⟨I, IP⟩ := this
+      have : (↑(height I + 1) : WithBot β„•βˆž) ≀ ⨆ (I : PrimeSpectrum R), ↑(height I + 1) := by
+        apply @le_iSup (WithBot β„•βˆž) _ _ _ I
+      exact ge_trans this IP
+    have oneOut : (⨆ (I : PrimeSpectrum R), (height I : WithBot β„•βˆž) + 1) ≀ (⨆ (I : PrimeSpectrum R), ↑(height I)) + 1 := by
+      have : βˆ€ P : PrimeSpectrum R, (height P : WithBot β„•βˆž) + 1 ≀ (⨆ (I : PrimeSpectrum R), ↑(height I)) + 1 :=
+        fun P ↦ (by apply add_le_add_right (@le_iSup (WithBot β„•βˆž) _ _ _ P) 1)
       apply iSup_le
-      exact this
-    sorry
\ No newline at end of file
+      apply this
+    simp only [iSup_le_iff]
+    intro P
+    exact ge_trans oneOut (htPBdd P)