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Merge pull request #51 from GTBarkley/sayantan

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Develop dim_eq_dim_polynomial_add_one a bit more, with a sorried section
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Sayantan Santra 2023-06-14 12:23:19 -05:00 committed by GitHub
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18
CommAlg/sayantan(dim_eq_dim_polynomial_add_one).lean
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@ -36,11 +36,17 @@ lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
· unfold krullDim
have htPBdd : ∀ (P : PrimeSpectrum (Polynomial R)), (height P : WithBot ℕ∞) ≤ (⨆ (I : PrimeSpectrum R), ↑(height I + 1)) := by
intro P
unfold height
have : ∃ (I : PrimeSpectrum R), (height P : WithBot ℕ∞) ≤ ↑(height I + 1) := by
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
obtain ⟨I, IP⟩ := this
have : (↑(height I + 1) : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum R), ↑(height I + 1) := by
apply @le_iSup (WithBot ℕ∞) _ _ _ I
apply ge_trans this IP
have oneOut : (⨆ (I : PrimeSpectrum R), (height I : WithBot ℕ∞) + 1) ≤ (⨆ (I : PrimeSpectrum R), ↑(height I)) + 1 := by
have : ∀ P : PrimeSpectrum R, (height P : WithBot ℕ∞) + 1 ≤ (⨆ (I : PrimeSpectrum R), ↑(height I)) + 1 :=
fun P ↦ (by apply add_le_add_right (@le_iSup (WithBot ℕ∞) _ _ _ P) 1)
apply iSup_le
exact this
sorry
apply this
simp
intro P
exact ge_trans oneOut (htPBdd P)