One side of poly_over_field

CommAlg/sayantan(poly_over_field).lean

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+import CommAlg.krull
+import Mathlib.RingTheory.Ideal.Operations
+import Mathlib.RingTheory.FiniteType
+import Mathlib.Order.Height
+import Mathlib.RingTheory.PrincipalIdealDomain
+import Mathlib.RingTheory.DedekindDomain.Basic
+import Mathlib.RingTheory.Ideal.Quotient
+import Mathlib.RingTheory.Ideal.MinimalPrime
+import Mathlib.RingTheory.Localization.AtPrime
+import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
+import Mathlib.Order.ConditionallyCompleteLattice.Basic
+
+namespace Ideal
+
+lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by
+  -- unfold krullDim
+  rw [le_antisymm_iff]
+  constructor
+  · sorry
+  · suffices : ∃I : PrimeSpectrum (Polynomial K), 1 ≤ (height I : WithBot ℕ∞)
+    · obtain ⟨I, h⟩ := this
+      have :  (height I : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum (Polynomial K)), ↑(height I) := by
+        apply @le_iSup (WithBot ℕ∞) _ _ _ I
+      exact le_trans h this
+    have primeX : Prime Polynomial.X := @Polynomial.prime_X K _ _
+    let X := @Polynomial.X K _
+    have : IsPrime (span {X}) := by
+      refine Iff.mpr (span_singleton_prime ?hp) primeX
+      exact Polynomial.X_ne_zero
+    let P := PrimeSpectrum.mk (span {X}) this
+    unfold height
+    use P
+    have : ⊥ ∈ {J | J < P} := by
+      simp only [Set.mem_setOf_eq]
+      rw [bot_lt_iff_ne_bot]
+      suffices : P.asIdeal ≠ ⊥
+      · by_contra x
+        rw [PrimeSpectrum.ext_iff] at x
+        contradiction
+      by_contra x
+      simp only [span_singleton_eq_bot] at x
+      have := @Polynomial.X_ne_zero K _ _
+      contradiction
+    have : {J | J < P}.Nonempty := Set.nonempty_of_mem this
+    rwa [←Set.one_le_chainHeight_iff, ←WithBot.one_le_coe] at this