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change: Refactoring

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Sayantan Santra 2023-06-12 16:25:02 -07:00
parent 6496bc43d9
commit a41873ac1b
Signed by: SinTan1729
GPG key ID: EB3E68BFBA25C85F
1 changed files with 9 additions and 19 deletions
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CommAlg/sayantan.lean
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@ -9,18 +9,8 @@ import Mathlib.Order.ConditionallyCompleteLattice.Basic
-- import Mathlib.Data.ENat.Lattice -- import Mathlib.Data.ENat.Lattice
-- import Mathlib.Order.OrderIsoNat -- import Mathlib.Order.OrderIsoNat
-- import Mathlib.Tactic.TFAE -- import Mathlib.Tactic.TFAE
namespace Ideal namespace Ideal
-- def foo : List Nat := [1, 2, 3, 4, 5]
-- #check List.Chain'
-- example : List.Chain' (· < ·) foo := by
-- repeat { constructor; norm_num }
example (x : Nat) : List.Chain' (· < ·) [x] := by example (x : Nat) : List.Chain' (· < ·) [x] := by
constructor constructor
@ -36,10 +26,8 @@ lemma height_def : height I = Set.chainHeight {J : PrimeSpectrum R | J < I} := r
lemma krullDim_def (R : Type) [CommRing R] : krullDim R = (⨆ (I : PrimeSpectrum R), height I : WithBot ℕ∞) := rfl lemma krullDim_def (R : Type) [CommRing R] : krullDim R = (⨆ (I : PrimeSpectrum R), height I : WithBot ℕ∞) := rfl
lemma krullDim_def' (R : Type) [CommRing R] : krullDim R = iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) := rfl lemma krullDim_def' (R : Type) [CommRing R] : krullDim R = iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) := rfl
variable {K : Type _} [Field K] @[simp]
lemma field_prime_bot {K: Type _} [Field K] (P : Ideal K) : IsPrime P ↔ P = ⊥ := by
lemma dim_field_eq_zero : krullDim K = 0 := by
have prime_bot (P : Ideal K) : IsPrime P ↔ P = ⊥ := by
constructor constructor
· intro primeP · intro primeP
obtain T := eq_bot_or_top P obtain T := eq_bot_or_top P
@ -48,9 +36,8 @@ lemma dim_field_eq_zero : krullDim K = 0 := by
· intro botP · intro botP
rw [botP] rw [botP]
exact bot_prime exact bot_prime
unfold krullDim
have height_zero : ∀ P : PrimeSpectrum K, height P = 0 := by lemma field_prime_height_zero {K: Type _} [Field K] (P : PrimeSpectrum K) : height P = 0 := by
intro P
unfold height unfold height
simp simp
by_contra spec by_contra spec
@ -59,9 +46,12 @@ lemma dim_field_eq_zero : krullDim K = 0 := by
obtain ⟨J, JlP : J < P⟩ := spec obtain ⟨J, JlP : J < P⟩ := spec
have P0 : IsPrime P.asIdeal := P.IsPrime have P0 : IsPrime P.asIdeal := P.IsPrime
have J0 : IsPrime J.asIdeal := J.IsPrime have J0 : IsPrime J.asIdeal := J.IsPrime
rw [prime_bot] at P0 J0 rw [field_prime_bot] at P0 J0
have : J.asIdeal = P.asIdeal := Eq.trans J0 (Eq.symm P0) have : J.asIdeal = P.asIdeal := Eq.trans J0 (Eq.symm P0)
have JeqP : J = P := PrimeSpectrum.ext J P this have JeqP : J = P := PrimeSpectrum.ext J P this
have JneqP : J ≠ P := ne_of_lt JlP have JneqP : J ≠ P := ne_of_lt JlP
contradiction contradiction
simp [height_zero]
lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by
unfold krullDim
simp [field_prime_height_zero]