testing api for StrictSeries

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GTBarkley 2023-07-11 04:41:48 +00:00
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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 ≤ 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∞) ↔ _
-- norm_cast
-- rw [le_height_iff]
norm_cast