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change : Improved singleton_chainHeight_le_one to singleton_chainHeight_one

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Sayantan Santra 2023-06-17 15:13:53 -07:00
parent 3b9a47ac6f
commit 9f0ed8cee3
Signed by: SinTan1729
GPG key ID: EB3E68BFBA25C85F
1 changed files with 22 additions and 14 deletions
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CommAlg/krull.lean
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@ -356,19 +356,27 @@ 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_chainHeight_le_one {α : Type _} {x : α} [Preorder α] : Set.chainHeight {x} ≤ 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 _)
/-- Singleton sets have chainHeight 1 -/
lemma singleton_chainHeight_one {α : Type _} {x : α} [Preorder α] : Set.chainHeight {x} = 1 := by
have le : Set.chainHeight {x} ≤ 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 _)
suffices : Set.chainHeight {x} > 0
· change _ < _ at this
rw [←ENat.one_le_iff_pos] at this
apply le_antisymm <;> trivial
by_contra x
simp only [gt_iff_lt, not_lt, nonpos_iff_eq_zero, Set.chainHeight_eq_zero_iff, Set.singleton_ne_empty] at x
/-- 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
@ -416,7 +424,7 @@ lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullD
unfold height
rw [sngletn]
simp only [WithBot.coe_le_one, ge_iff_le]
exact singleton_chainHeight_le_one
exact le_of_eq singleton_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