comm_alg/CommAlg/grant.lean

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import Mathlib.Order.KrullDimension
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import Mathlib.Order.JordanHolder
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
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import Mathlib.Order.Height
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#check (p q : PrimeSpectrum _) → (p ≤ q)
#check Preorder (PrimeSpectrum _)
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-- 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 : } : (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⟩
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-- there should be a coercion from WithTop to WithBot (WithTop ) 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
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lemma twoHeights : s ≠ ∅ → (some (Set.chainHeight s) : WithBot (WithTop )) = krullDim s := by
intro hs
unfold Set.chainHeight
unfold krullDim
have hKrullSome : ∃n, krullDim s = some n := by
sorry
-- norm_cast
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sorry
namespace Ideal
noncomputable def krullDim (R : Type _) [CommRing R] :=
Set.chainHeight (Set.univ : Set (PrimeSpectrum R))
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def krullDimGE (R : Type _) [CommRing R] (n : ) :=
∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ c.length = n + 1
def krullDimLE (R : Type _) [CommRing R] (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 ))
#check ( (Set.chainHeight s) : WithBot (ℕ∞))