Merge pull request #15 from GTBarkley/grant

playing with mathlib definitions of chains

comm_alg/grant.lean

@@ -1,12 +1,61 @@
 import Mathlib.Order.KrullDimension
+import Mathlib.Order.JordanHolder
 import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
+import Mathlib.Order.Height
 
-def hello : IO Unit := do
-  IO.println "Hello, World!"
-
-#eval hello
 
 #check (p q : PrimeSpectrum _) → (p ≤ q)
 #check Preorder (PrimeSpectrum _)
 
-#check krullDim (PrimeSpectrum _)
+-- Dimension of a ring
+#check krullDim (PrimeSpectrum _)
+
+-- Length of a module
+#check krullDim (Submodule _ _)
+
+#check JordanHolderLattice
+
+
+variable {α : Type _} [Preorder α] (s : Set α)
+
+def finFun_to_list {n : ℕ} : (Fin n → α) → List α := by sorry
+
+def series_to_chain : StrictSeries s → s.subchain
+| ⟨length, toFun, strictMono⟩ => 
+    ⟨ finFun_to_list (fun x => toFun x),
+      sorry⟩
+
+
+-- there should be a coercion from WithTop ℕ to WithBot (WithTop ℕ) but it doesn't seem to work
+-- it looks like this might be because someone changed the instance from CoeCT to Coe during the port
+-- actually it looks like we can coerce to WithBot (ℕ∞) fine
+lemma twoHeights : s ≠ ∅ → (some (Set.chainHeight s) : WithBot (WithTop ℕ)) = krullDim s := by
+  intro hs
+  unfold Set.chainHeight
+  unfold krullDim
+  have hKrullSome : ∃n, krullDim s = some n := by
+    
+    sorry
+  -- norm_cast
+  sorry
+
+namespace Ideal
+noncomputable def krullDim (R : Type _) [CommRing R] := 
+  Set.chainHeight (Set.univ : Set (PrimeSpectrum R))
+
+def krullDimGE (R : Type _) [CommRing R] (n : ℕ) :=
+  ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ c.length = n + 1
+
+def krullDimLE (R : Type _) [CommRing R] (n : ℕ) :=
+  ∀ c : List (PrimeSpectrum R), c.Chain' (· < ·) → c.length ≤ n + 1
+
+end Ideal
+
+open Ideal
+
+lemma krullDim_le (R : Type _) [CommRing R] : krullDimLE R n ↔ Ideal.krullDim R ≤ n := sorry
+lemma krullDim_ge (R : Type _) [CommRing R] : krullDimGE R n ↔ Ideal.krullDim R ≥ n := sorry
+
+
+-- #check ((4 : ℕ∞) : WithBot (WithTop ℕ))
+#check ( (Set.chainHeight s) : WithBot (ℕ∞))