proved height_ge_iff'

CommAlg/grant.lean

@@ -17,6 +17,8 @@ import CommAlg.krull
 #check JordanHolderLattice
 
 
+section Chains
+
 variable {α : Type _} [Preorder α] (s : Set α)
 
 def finFun_to_list {n : ℕ} : (Fin n → α) → List α := by sorry
@@ -26,7 +28,6 @@ def series_to_chain : StrictSeries s → s.subchain
     ⟨ 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
@@ -40,15 +41,62 @@ lemma twoHeights : s ≠ ∅ → (some (Set.chainHeight s) : WithBot (WithTop
   -- norm_cast
   sorry
 
+end Chains
+
+section Krull
+
+variable (R : Type _) [CommRing R] (M : Type _) [AddCommGroup M] [Module R M]
+
 open Ideal
 
-lemma krullDim_le_iff' (R : Type _) [CommRing R] : 
+-- chain of primes 
+#check height 
+
+-- lemma height_ge_iff {𝔭 : PrimeSpectrum R} {n : ℕ∞} :
+--   height 𝔭 ≥ n ↔ := sorry
+
+lemma height_ge_iff' {𝔭 : PrimeSpectrum R} {n : ℕ∞} : 
+height 𝔭 > n ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀ 𝔮 ∈ c, 𝔮 < 𝔭) ∧ c.length = n + 1 := by
+  rcases n with _ | n
+  . constructor <;> intro h <;> exfalso
+    . exact (not_le.mpr h) le_top
+    . -- change ∃c, _ ∧ _ ∧ ((List.length c : ℕ∞) = ⊤ + 1) at h
+      -- rw [WithTop.top_add] at h
+      tauto
+  have (m : ℕ∞) : m > some n ↔ m ≥ some (n + 1) := by
+    symm
+    show (n + 1 ≤ m ↔ _ )
+    apply ENat.add_one_le_iff
+    exact ENat.coe_ne_top _
+  rw [this]
+  unfold Ideal.height
+  show ((↑(n + 1):ℕ∞) ≤ _) ↔ ∃c, _ ∧ _ ∧ ((_ : WithTop ℕ) = (_:ℕ∞))
+  rw [{J | J < 𝔭}.le_chainHeight_iff]
+  show (∃ c, (List.Chain' _ c ∧ ∀𝔮, 𝔮 ∈ c → 𝔮 < 𝔭) ∧ _) ↔ _
+  have h := fun (c : List (PrimeSpectrum R)) => (@WithTop.coe_eq_coe _ (List.length c) n) 
+  constructor <;> rintro ⟨c, hc⟩ <;> use c --<;> tauto--<;> exact ⟨hc.1, by tauto⟩
+  . --rw [and_assoc]
+    -- show _ ∧ _ ∧ _
+    --exact ⟨hc.1, _⟩
+    tauto
+  . change _ ∧ _ ∧ (List.length c : ℕ∞) = n + 1 at hc
+    norm_cast at hc
+    tauto
+
+lemma krullDim_le_iff' (R : Type _) [CommRing R] {n : WithBot ℕ∞} : 
   Ideal.krullDim R ≤ n ↔ (∀ c : List (PrimeSpectrum R), c.Chain' (· < ·) → c.length ≤ n + 1) := by
     sorry
 
-lemma krullDim_ge_iff' (R : Type _) [CommRing R] : 
+lemma krullDim_ge_iff' (R : Type _) [CommRing R] {n : WithBot ℕ∞} : 
   Ideal.krullDim R ≥ n ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ c.length = n + 1 := sorry
 
-
+#check (sorry : False)
+#check (sorryAx)
+#check (4 : WithBot ℕ∞)
+#check List.sum
 -- #check ((4 : ℕ∞) : WithBot (WithTop ℕ))
-#check ( (Set.chainHeight s) : WithBot (ℕ∞))
+-- #check ( (Set.chainHeight s) : WithBot (ℕ∞))
+
+variable (P : PrimeSpectrum R)
+
+#check {J | J < P}.le_chainHeight_iff (n := 4)