Merge pull request #103 from GTBarkley/monalisa4

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CommAlg/final_hil_pol.lean

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+import Mathlib.Order.KrullDimension
+import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
+import Mathlib.Algebra.Module.GradedModule
+import Mathlib.RingTheory.Ideal.AssociatedPrime
+import Mathlib.RingTheory.Artinian
+import Mathlib.Order.Height
+import Mathlib.RingTheory.Ideal.Quotient
+import Mathlib.RingTheory.SimpleModule
+import Mathlib.Algebra.Module.LinearMap
+import Mathlib.Algebra.Field.Defs
+import CommAlg.krull
+
+
+
+-- 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
+        | aesop; done; dbg_trace "it was aesop"
+        | fail "No, this is not obvious."))
+
+
+
+open GradedMonoid.GSmul
+open DirectSum
+
+
+
+-- @Definitions (to be classified)
+section
+
+-- 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
+
+
+-- 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)
+
+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)
+
+
+-- Definition of a Hilbert function of a graded module
+section
+
+noncomputable def length ( A : Type _) (M : Type _)
+ [CommRing A] [AddCommGroup M] [Module A M] :=  Set.chainHeight {M' : Submodule A M | M' < ⊤}
+
+noncomputable def dimensionmodule ( A : Type _) (M : Type _)
+ [CommRing A] [AddCommGroup M] [Module A M] := Ideal.krullDim  (A ⧸ ((⊤ : Submodule A M).annihilator))  
+
+
+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)))  
+
+
+lemma lengthfield ( k : Type _) [Field k] : length (k) (k) = 1 := by
+sorry
+
+
+lemma equaldim ( A : Type _) [CommRing A] (I : Ideal A): dimensionmodule (A) (A ⧸ I) = Ideal.krullDim (A ⧸ I) := by
+sorry
+
+lemma dim_iso ( A : Type _) (M : Type _) (N : Type _) [CommRing A] [AddCommGroup M] [Module A M] [AddCommGroup N] [Module A N] (h : Nonempty (M →ₗ[A] N)) : dimensionmodule A M = dimensionmodule A N := by
+sorry
+
+
+end
+
+
+
+-- Definition of homogeneous ideal
+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
+
+-- Definition of homogeneous prime ideal
+def HomogeneousPrime (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] (I : Ideal (⨁ i, 𝒜 i)):= (Ideal.IsPrime I) ∧ (Ideal.IsHomogeneous' 𝒜 I)
+
+-- 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]
+
+
+instance {𝒜 : ℤ → Type _} [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] :
+    Algebra (𝒜 0) (⨁ i, 𝒜 i) :=
+  Algebra.ofModule'
+  (by
+    intro r x
+    sorry)
+  (by
+    intro r x
+    sorry)
+
+
+
+class StandardGraded (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜] : Prop where
+  gen_in_first_piece :
+    Algebra.adjoin (𝒜 0) (DirectSum.of _ 1 : 𝒜 1 →+ ⨁ i, 𝒜 i).range = (⊤ : Subalgebra (𝒜 0) (⨁ i, 𝒜 i))
+
+
+-- Each component of a graded ring is an additive subgroup
+def Component_of_graded_as_addsubgroup (𝒜 : ℤ → Type _) 
+[∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
+(p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p) (i : ℤ) : AddSubgroup (𝒜 i) := by
+  sorry
+
+
+def graded_ring_morphism (𝒜 : ℤ → Type _) (ℬ : ℤ → Type _)
+[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (ℬ i)]
+[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing ℬ] (f : (⨁ i, 𝒜 i) →+* (⨁ i, ℬ i)) := ∀ i, ∀ (r : 𝒜 i), ∀ j, (j ≠ i → f (DirectSum.of _ i r) j = 0)
+
+structure GradedLinearMap (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) (𝓝 : ℤ → Type _)
+    [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
+    [DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜] [DirectSum.Gmodule 𝒜 𝓝]
+    extends LinearMap (RingHom.id (⨁ i, 𝒜 i)) (⨁ i, 𝓜 i) (⨁ i, 𝓝 i) where
+  respects_grading (i : ℤ) (r : 𝓜 i) (j : ℤ) : j ≠ i → toFun (DirectSum.of _ i r) j = 0
+
+/-- `𝓜 →ᵍₗ[𝒜] 𝓝` denotes the type of graded `𝒜`-linear maps from `𝓜` to `𝓝`. -/
+notation:25 𝓜 " →ᵍₗ[" 𝒜:25 "] " 𝓝:0 => GradedLinearMap 𝒜 𝓜 𝓝
+
+structure GradedLinearEquiv (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) (𝓝 : ℤ → Type _)
+    [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
+    [DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜][DirectSum.Gmodule 𝒜 𝓝] 
+    extends (⨁ i, 𝓜 i) ≃ (⨁ i, 𝓝 i), 𝓜 →ᵍₗ[𝒜] 𝓝
+
+/-- `𝓜 ≃ᵍₗ[𝒜] 𝓝` denotes the type of graded `𝒜`-linear isomorphisms from `(⨁ i, 𝓜 i)` to `(⨁ i, 𝓝 i)`. -/
+notation:25 𝓜 " ≃ᵍₗ[" 𝒜:25 "] " 𝓝:0 => GradedLinearEquiv 𝒜 𝓜 𝓝
+
+def graded_ring_isomorphism (𝒜 : ℤ → Type _) (𝓑 : ℤ → Type _)
+[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓑 i)]
+[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝓑]
+(f : (⨁ i, 𝒜 i) →+*  (⨁ i, 𝓑 i))
+:= (graded_ring_morphism 𝒜 𝓑 f) ∧ (Function.Bijective f)
+
+def graded_ring_isomorphic (𝒜 : ℤ → Type _) (𝓑 : ℤ → Type _)
+[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓑 i)]
+[DirectSum.GCommRing 𝒜] [DirectSum.GCommRing 𝓑] := ∃ (f : (⨁ i, 𝒜 i) →+*  (⨁ i, 𝓑 i)), graded_ring_isomorphism 𝒜 𝓑 f
+
+-- def graded_submodule
+--     (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) (𝓝 : ℤ → Type _)
+--     [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
+--     [DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜][DirectSum.Gmodule 𝒜 𝓝]
+--     (h (⨁ i, 𝓝 i) : Submodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) :
+--     Prop :=
+--   ∃ (piece : Submodule (𝒜 0) (𝓜 i)), piece = 𝓝 i
+
+
+end
+
+class DirectSum.GalgebrA
+  (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
+  (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝓜]
+  extends DirectSum.Gmodule 𝒜 𝓜
+
+-- def graded_algebra_morphism (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
+--   (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝓜] [DirectSum.GalgebrA 𝒜 𝓜]
+--   (𝓝 : ℤ → Type _) [∀ i, AddCommGroup (𝓝 i)] [DirectSum.GCommRing 𝓝] [DirectSum.GalgebrA 𝒜 𝓝] 
+--   (f : (⨁ i, 𝓜 i) →  (⨁ i, 𝓝 i)) := (graded_ring_morphism 𝓜 𝓝 f) ∧ (GradedLinearMap 𝒜 𝓜 𝓝 toFun)  
+
+
+
+-- @Quotient of a graded ring R by a graded ideal p is a graded R-alg, preserving each component
+
+instance Quotient_of_graded_gradedring
+    (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
+    (p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p) :
+    DirectSum.GCommRing (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
+  sorry
+
+instance Quotient_of_graded_is_gradedalg
+    (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
+    (p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p) :
+    DirectSum.GalgebrA 𝒜 (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)) := by
+  sorry
+
+section
+variable (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜]
+    [LocalRing (𝒜 0)] (m : LocalRing.maximalIdeal (𝒜 0)) 
+
+-- check if `Pi.Single` or something writes this more elegantly
+def GradedOneComponent (i : ℤ) : Type _ := ite (i = 0) (𝒜 0 ⧸ LocalRing.maximalIdeal (𝒜 0)) PUnit 
+
+instance (i : ℤ) : AddMonoid (GradedOneComponent 𝒜 i) := by
+  unfold GradedOneComponent
+  sorry -- split into 0 and nonzero cases and then `inferInstance`
+
+instance : DirectSum.Gmodule 𝒜 (GradedOneComponent 𝒜) := by sorry
+
+
+
+lemma Graded_local [StandardGraded 𝒜] (I : Ideal (⨁ i, 𝒜 i)) (hp : (HomogeneousMax 𝒜 I)) [∀ i, Module (𝒜 0) ((𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 I hp.2 i))] (art: IsArtinianRing (𝒜 0)) : (∀ (i : ℤ ), (i ≠ 0 → Nonempty (((𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 I hp.2 i)) →ₗ[𝒜 0] (𝒜 i))) )  := by
+  sorry
+
+end
+
+lemma Quotient_of_graded_ringiso (𝒜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [DirectSum.GCommRing 𝒜](p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p)
+-- (hm : 𝓜 = (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)))
+: Nonempty (((⨁ i, (𝒜 i))⧸p) →ₗ[(⨁ i, 𝒜 i)] (⨁ i, (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i))) := by
+  sorry
+
+def Is.Graded_local (𝒜 : ℤ → Type _)
+[∀ i, AddCommGroup (𝒜 i)][DirectSum.GCommRing 𝒜] := ∃! ( I : Ideal ((⨁ i, 𝒜 i))),(HomogeneousMax 𝒜 I)
+
+lemma hilfun_eq (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) (𝓝 : ℤ → Type _)
+[∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [∀ i, AddCommGroup (𝓝 i)]
+[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜][DirectSum.Gmodule 𝒜 𝓝] (iso : GradedLinearEquiv 𝒜 𝓜 𝓝)(hilbm : ℤ → ℤ) (Hhilbm: hilbert_function 𝒜 𝓜 hilbm) (hilbn : ℤ → ℤ) (Hhilbn: hilbert_function 𝒜 𝓝 hilbn) : ∀ (n : ℤ), hilbm n = hilbn n := by
+sorry
+
+-- If A_0 is Artinian and local, then A is graded local
+
+
+
+-- @Existence of a chain of submodules of graded submoduels of a 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))
+  : ∃ (c : List (Submodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))), c.Chain' (· < ·) ∧ ∀ M ∈ c, Ture := 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) ∧ ((∃ (i : ℤ ), ∃ (x :  𝒜 i), p = (Submodule.span (⨁ i, 𝒜 i) {DirectSum.of _ i x}).annihilator)) := by
+  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_d_ge_1 (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] [StandardGraded 𝒜] (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
+
+
+-- (reduced version) [BH, 4.1.3] when d ≥ 1
+-- If M is a finite graed R-Mod of dimension d ≥ 1, and M = R⧸ 𝓅 for a graded prime ideal 𝓅, then the Hilbert function H(M, n) is of polynomial type (d - 1)
+theorem Hilbert_polynomial_d_ge_1_reduced 
+(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] [StandardGraded 𝒜] (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)
+(p : Ideal (⨁ i, 𝒜 i)) (hp : Ideal.IsHomogeneous' 𝒜 p)
+(hm : 𝓜 = (fun i => (𝒜 i)⧸(Component_of_graded_as_addsubgroup 𝒜 p hp i)))
+: PolyType hilb (d - 1) := by
+  sorry
+
+
+-- @[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_d_0 (𝒜 : ℤ → Type _) (𝓜 : ℤ → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
+[DirectSum.GCommRing 𝒜]
+[DirectSum.Gmodule 𝒜 𝓜] [StandardGraded 𝒜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) 
+(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
+(findim :  dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = 0)
+(hilb : ℤ → ℤ) (Hhilb : hilbert_function 𝒜 𝓜 hilb)
+: (∃ (N : ℤ), ∀ (n : ℤ), n ≥ N → hilb n = 0) := by
+  sorry