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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 10:22:14 -07:00
parent 2737f8bba6
commit 39c06f39fd
Signed by: SinTan1729
GPG key ID: EB3E68BFBA25C85F
1 changed files with 14 additions and 8 deletions
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CommAlg/sayantan(dim_eq_dim_polynomial_add_one).lean
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@ -34,13 +34,19 @@ lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
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
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)