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Merge pull request #80 from GTBarkley/grant

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swapped inequalities in lt_height_iff
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GTBarkley 2023-06-15 20:54:39 -07:00 committed by GitHub
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1 changed files with 23 additions and 22 deletions
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17
CommAlg/krull.lean
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@ -112,12 +112,14 @@ lemma krullDim_eq_height [LocalRing R] : krullDim R = height (closedPoint R) :=
/-- The height of a prime `𝔭` is greater than `n` if and only if there is a chain of primes less than `𝔭`
with length `n + 1`. -/
lemma lt_height_iff' {𝔭 : PrimeSpectrum R} {n : ℕ∞} :
height 𝔭 > n ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀ 𝔮 ∈ c, 𝔮 < 𝔭) ∧ c.length = n + 1 := by
rcases n with _ | n
. constructor <;> intro h <;> exfalso
n < height 𝔭 ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀ 𝔮 ∈ c, 𝔮 < 𝔭) ∧ c.length = n + 1 := by
match n with
| ⊤ =>
constructor <;> intro h <;> exfalso
. exact (not_le.mpr h) le_top
. tauto
have (m : ℕ∞) : m > some n ↔ m ≥ some (n + 1) := by
| (n : ℕ) =>
have (m : ℕ∞) : n < m ↔ (n + 1 : ℕ∞) ≤ m := by
symm
show (n + 1 ≤ m ↔ _ )
apply ENat.add_one_le_iff
@ -135,8 +137,7 @@ height 𝔭 > n ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀
/-- Form of `lt_height_iff''` for rewriting with the height coerced to `WithBot ℕ∞`. -/
lemma lt_height_iff'' {𝔭 : PrimeSpectrum R} {n : ℕ∞} :
height 𝔭 > (n : WithBot ℕ∞) ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀ 𝔮 ∈ c, 𝔮 < 𝔭) ∧ c.length = n + 1 := by
show (_ < _) ↔ _
(n : WithBot ℕ∞) < height 𝔭 ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀ 𝔮 ∈ c, 𝔮 < 𝔭) ∧ c.length = n + 1 := by
rw [WithBot.coe_lt_coe]
exact lt_height_iff'
@ -198,7 +199,7 @@ lemma dim_le_zero_iff : krullDim R ≤ 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal
rw [hcontr] at h
exact h h𝔪.1
use 𝔪p
show (_ : WithBot ℕ∞) > (0 : ℕ∞)
show (0 : ℕ∞) < (_ : WithBot ℕ∞)
rw [lt_height_iff'']
use [I]
constructor
@ -209,7 +210,7 @@ lemma dim_le_zero_iff : krullDim R ≤ 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal
rwa [hI']
. simp
. contrapose! h
change (_ : WithBot ℕ∞) > (0 : ℕ∞) at h
change (0 : ℕ∞) < (_ : WithBot ℕ∞) at h
rw [lt_height_iff''] at h
obtain ⟨c, _, hc2, hc3⟩ := h
norm_cast at hc3