Made some progress on dim_eq_dim_polynomial_add_one

CommAlg/sayantan(dim_eq_dim_polynomial_add_one).lean

@@ -0,0 +1,46 @@
+import Mathlib.RingTheory.Ideal.Basic
+import Mathlib.Order.Height
+import Mathlib.RingTheory.PrincipalIdealDomain
+import Mathlib.RingTheory.DedekindDomain.Basic
+import Mathlib.RingTheory.Ideal.Quotient
+import Mathlib.RingTheory.Localization.AtPrime
+import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
+import Mathlib.Order.ConditionallyCompleteLattice.Basic
+
+namespace Ideal
+
+variable {R : Type _} [CommRing R] (I : PrimeSpectrum R)
+noncomputable def height : ℕ∞ := Set.chainHeight {J : PrimeSpectrum R | J < I}
+noncomputable def krullDim (R : Type) [CommRing R] : WithBot ℕ∞ := ⨆ (I : PrimeSpectrum R), height I
+
+lemma height_def : height I = Set.chainHeight {J : PrimeSpectrum R | J < I} := rfl
+lemma krullDim_def (R : Type) [CommRing R] : krullDim R = (⨆ (I : PrimeSpectrum R), height I : WithBot ℕ∞) := rfl
+lemma krullDim_def' (R : Type) [CommRing R] : krullDim R = iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) := rfl
+
+noncomputable instance : CompleteLattice (WithBot (ℕ∞)) := WithBot.WithTop.completeLattice
+
+lemma dim_le_dim_polynomial_add_one [Nontrivial R] :
+  krullDim R + 1 ≤ krullDim (Polynomial R) := sorry -- Others are working on it
+
+-- private lemma sum_succ_of_succ_sum {ι : Type} (a : ℕ∞) [inst : Nonempty ι] :
+--       (⨆ (x : ι), a + 1) = (⨆ (x : ι), a) + 1 := by
+--   have : a + 1 = (⨆ (x : ι), a) + 1 := by rw [ciSup_const]
+--   have : a + 1 = (⨆ (x : ι), a + 1) := Eq.symm ciSup_const
+--   simp
+
+lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
+  krullDim R + 1 = krullDim (Polynomial R) := by
+  rw [le_antisymm_iff]
+  constructor
+  · exact dim_le_dim_polynomial_add_one
+  · unfold krullDim
+    have htPBdd : ∀ (P : PrimeSpectrum (Polynomial R)), (height P: WithBot ℕ∞) ≤ (⨆ (I : PrimeSpectrum R), ↑(height I + 1)) := by
+      intro P
+      unfold height
+      sorry
+    have : (⨆ (I : PrimeSpectrum R), ↑(height I) + 1) ≤ (⨆ (I : PrimeSpectrum R), ↑(height I)) + 1 := by
+      have : ∀ P : PrimeSpectrum R, ↑(height P) + 1 ≤ (⨆ (I : PrimeSpectrum R), ↑(height I)) + 1 :=
+        fun _ ↦ add_le_add_right (le_iSup height _) 1
+      apply iSup_le
+      exact this
+    sorry