Add the whole proof back since I just noticed that I had replaced them by sorried statements

CommAlg/krull.lean

@@ -377,12 +377,67 @@ lemma dim_le_one_of_pid [IsDomain R] [IsPrincipalIdealRing R] : krullDim R ≤ 1
   rw [dim_le_one_iff]
   exact Ring.DimensionLEOne.principal_ideal_ring R
 
+private lemma singleton_bot_chainHeight_one {α : Type} [Preorder α] [Bot α] : Set.chainHeight {(⊥ : α)} ≤ 1 := by
+  unfold Set.chainHeight
+  simp only [iSup_le_iff, Nat.cast_le_one]
+  intro L h
+  unfold Set.subchain at h
+  simp only [Set.mem_singleton_iff, Set.mem_setOf_eq] at h
+  rcases L with (_ | ⟨a,L⟩)
+  . simp only [List.length_nil, zero_le]
+  rcases L with (_ | ⟨b,L⟩)
+  . simp only [List.length_singleton, le_refl]
+  simp only [List.chain'_cons, List.find?, List.mem_cons, forall_eq_or_imp] at h
+  rcases h with ⟨⟨h1, _⟩,  ⟨rfl, rfl, _⟩⟩
+  exact absurd h1 (lt_irrefl _)
+
 /-- The ring of polynomials over a field has dimension one. -/
-lemma polynomial_over_field_dim_one {K : Type _} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by
+lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by
   rw [le_antisymm_iff]
   let X := @Polynomial.X K _
   constructor
-  · exact dim_le_one_of_pid
+  · unfold krullDim
+    apply @iSup_le (WithBot ℕ∞) _ _ _ _
+    intro I
+    have PIR : IsPrincipalIdealRing (Polynomial K) := by infer_instance
+    by_cases I = ⊥
+    · rw [← height_bot_iff_bot] at h
+      simp only [WithBot.coe_le_one, ge_iff_le]
+      rw [h]
+      exact bot_le
+    · push_neg at h
+      have : I.asIdeal ≠ ⊥ := by
+        by_contra a
+        have : I = ⊥ := PrimeSpectrum.ext I ⊥ a
+        contradiction
+      have maxI := IsPrime.to_maximal_ideal this
+      have sngletn : ∀P, P ∈ {J | J < I} ↔ P = ⊥ := by
+          intro P
+          constructor
+          · intro H
+            simp only [Set.mem_setOf_eq] at H
+            by_contra x
+            push_neg at x
+            have : P.asIdeal ≠ ⊥ := by
+              by_contra a
+              have : P = ⊥ := PrimeSpectrum.ext P ⊥ a
+              contradiction 
+            have maxP := IsPrime.to_maximal_ideal this
+            have IneTop := IsMaximal.ne_top maxI
+            have : P ≤ I := le_of_lt H
+            rw [←PrimeSpectrum.asIdeal_le_asIdeal] at this
+            have : P.asIdeal = I.asIdeal := Ideal.IsMaximal.eq_of_le maxP IneTop this
+            have : P = I := PrimeSpectrum.ext P I this
+            replace H : P ≠ I := ne_of_lt H
+            contradiction
+          · intro pBot
+            simp only [Set.mem_setOf_eq, pBot]
+            exact lt_of_le_of_ne bot_le h.symm
+      replace sngletn : {J | J < I} = {⊥} := Set.ext sngletn
+      unfold height
+      rw [sngletn]
+      simp only [WithBot.coe_le_one, ge_iff_le]
+      exact singleton_bot_chainHeight_one
   · 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

CommAlg/sayantan(poly_over_field).lean

@@ -22,7 +22,8 @@ private lemma singleton_bot_chainHeight_one {α : Type} [Preorder α] [Bot α] :
   exact absurd h1 (lt_irrefl _)
 
 /-- The ring of polynomials over a field has dimension one. -/
-lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by
+-- It's the exact same lemma as in krull.lean, added ' to avoid conflict
+lemma polynomial_over_field_dim_one' {K : Type} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by
   rw [le_antisymm_iff]
   let X := @Polynomial.X K _
   constructor