testing api for StrictSeries

CommAlg/grant3.lean

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+import Mathlib.Data.Nat.Lattice
+import Mathlib.Data.ENat.Lattice
+import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
+import Mathlib.Order.CompleteLattice
+import Mathlib.Order.WithBot
+import Mathlib.Order.Height
+import CommAlg.StrictSeries
+import Mathlib.Tactic.TFAE
+
+
+noncomputable def chainLength (α : Type _) [LT α] : WithBot ℕ∞ := ⨆ (s : StrictSeries α), s.length
+
+noncomputable def height {α : Type _} [LT α] (x : α) : ℕ∞ := ⨆ s ∈ {s : StrictSeries α | s.top = x}, s.length
+
+noncomputable def krullDim (R : Type _) [CommRing R] : WithBot ℕ∞ := chainLength (PrimeSpectrum R)
+
+lemma krullDim_def (R : Type _) [CommRing R] : krullDim R = (⨆ (I : PrimeSpectrum R), height I : WithBot ℕ∞) := sorry
+lemma krullDim_def' (R : Type _) [CommRing R] : krullDim R = iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) := sorry
+
+variable {α : Type _} [LT α]
+
+lemma height_eq_chainHeight (p : α) : height p = Set.chainHeight {J : α | J < p} := sorry
+
+lemma chainLength_eq_bot : chainLength α = ⊥ ↔ IsEmpty α := by
+  sorry
+
+lemma chainLength_eq_sup_height : chainLength α = ⨆ p : α, (height p : WithBot ℕ∞) := by
+  cases' isEmpty_or_nonempty α
+  . simp [chainLength]
+  . show (⨆ (s : _), ((_ : ℕ∞) : WithBot ℕ∞)) = _
+    norm_cast
+    unfold height
+    rw [iSup_comm]
+    apply le_antisymm <;> simp [iSup_mono]
+
+#synth (IsWellOrder (WithBot ℕ∞) (·<·))
+#check le_iSup
+
+theorem lt_height_TFAE (p : α) (n : ℕ∞) :
+List.TFAE [n < height p, ∃ c : StrictSeries α, c.top = p ∧ c.length > n, ∃c : StrictSeries α, c.top = p ∧ c.length = n + 1] := by
+  cases' n using ENat.recTopCoe with n
+  . tfae_have 1 ↔ 2; simp
+    tfae_have 1 ↔ 3; simp
+    tfae_finish
+  . norm_cast
+    sorry
+    -- tfae_have 1 → 2
+    -- . sorry
+    -- tfae_have 2 → 3
+    -- . sorry
+    -- tfae_have 3 → 1
+    -- . sorry 
+    -- tfae_finish
+  
+theorem lt_height_iff {p : α} {n : ℕ∞} :
+n < height p ↔ ∃ c : StrictSeries α, c.top = p ∧ c.length = n + 1 := 
+  lt_height_TFAE p n |>.out 0 2
+
+theorem le_height_iff {p : α} {n : ℕ} :
+n ≤ height p ↔ ∃ c : StrictSeries α, c.top = p ∧ c.length = n :=
+  match n with
+  | 0 => ⟨fun _ => ⟨StrictSeries.ofElement p, by simp⟩, by simp⟩
+  | n + 1 => by
+    erw [(ENat.add_one_le_iff <| ENat.coe_ne_top _), lt_height_iff]
+    norm_cast
+  -- unfold height
+  -- constructor <;> intro h
+  -- . 
+  --   -- cases' isEmpty_or_nonempty α
+  --   -- . exfalso; exact IsEmpty.false p
+  --   by_cases case : ∃ (c : StrictSeries α) (hc : c.top = p), n ≤ c.length
+  --   . sorry
+  --   -- generalize hj : (⨆ (_), (_ : ℕ∞)) = j at h
+  --   -- have : ∃ (c : StrictSeries α) (hc : c.top = p), n ≤ c.length := by
+  --   --   induction' j using ENat.recTopCoe with j
+  --   --   . simp at hj
+  --   --     sorry
+  --   --   . norm_cast at h
+  --   --     sorry
+  --   . push_neg at case
+  --     have : 0 < n := by
+  --       let c := StrictSeries.ofList [p] (List.cons_ne_nil _ _) (List.chain'_singleton p)
+  --       specialize case c (by rw [])
+  --       simp
+  --       sorry
+  --     sorry
+  -- . rcases h with ⟨c, hc⟩
+  --   exact (hc.2 ▸ le_iSup₂ (f := fun (x : StrictSeries α) _ => (x.length : ℕ∞)) c) hc.1
+
+theorem lt_height_iff'' {p : α} {n : ℕ∞} :
+n < height p ↔ ∃ c : StrictSeries α, c.top = p ∧ c.length = n + 1 := by
+  cases' n using ENat.recTopCoe with n
+  . simp
+  . erw [←(ENat.add_one_le_iff <| ENat.coe_ne_top _), le_height_iff]
+    -- show (n + 1 : ℕ) ≤ (_ : ℕ∞) ↔ _
+    -- norm_cast
+    -- rw [le_height_iff]
+    norm_cast