HNFil μA pseudo strict anti
HarderNarasimhan.impl.HNFil_μA_pseudo_strict_anti
Project documentation
Strict decrease condition on consecutive μA-slopes for HNFil. This is the analogue of “HN slopes are strictly decreasing”, phrased as the non-comparability statement ¬ μA(i,i+1) ≤ μA(i+1,i+2). The proof is an application of the internal obstruction lemma prop3d7₂.
Exact Lean statement
lemma HNFil_μA_pseudo_strict_anti {ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ]
[WellFoundedGT ℒ] {S : Type*} [CompleteLattice S]
(μ : {p :ℒ × ℒ // p.1 < p.2} → S)
[hμ : μA_DescendingChainCondition μ] [hμcvx : ConvexI TotIntvl μ] [h : μ_Admissible μ] :
∀ i : ℕ, (hi : i + 1 < Nat.find (HNFil_of_fin_len μ)) →
¬ μA μ ⟨(HNFil μ i, HNFil μ (i+1)),
HNFil_is_strict_mono μ i (Nat.find_min (HNFil_of_fin_len μ) (Nat.lt_of_succ_lt hi))⟩ ≤
μA μ ⟨(HNFil μ (i+1), HNFil μ (i+2)),
HNFil_is_strict_mono μ (i + 1) (Nat.find_min (HNFil_of_fin_len μ) hi)⟩Formal artifact
Lean source
lemma HNFil_μA_pseudo_strict_anti {ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ][WellFoundedGT ℒ] {S : Type*} [CompleteLattice S](μ : {p :ℒ × ℒ // p.1 < p.2} → S)[hμ : μA_DescendingChainCondition μ] [hμcvx : ConvexI TotIntvl μ] [h : μ_Admissible μ] :∀ i : ℕ, (hi : i + 1 < Nat.find (HNFil_of_fin_len μ)) → ¬ μA μ ⟨(HNFil μ i, HNFil μ (i+1)), HNFil_is_strict_mono μ i (Nat.find_min (HNFil_of_fin_len μ) (Nat.lt_of_succ_lt hi))⟩ ≤ μA μ ⟨(HNFil μ (i+1), HNFil μ (i+2)), HNFil_is_strict_mono μ (i + 1) (Nat.find_min (HNFil_of_fin_len μ) hi)⟩ := by intro i hj refine impl.prop3d7₂ μ ⟨(HNFil μ i,⊤),lt_top_iff_ne_top.2 <| Nat.find_min (HNFil_of_fin_len μ) <| lt_trans (lt_add_one i) hj⟩ (Convex_of_Convex_large TotIntvl ⟨(HNFil μ i,⊤),lt_top_iff_ne_top.2 <| Nat.find_min (HNFil_of_fin_len μ) <| lt_trans (lt_add_one i) hj⟩ ⟨bot_le,le_top⟩ μ hμcvx) (HNFil μ (i + 1)) (HNFil_prop_of_def μ i <| Nat.find_min (HNFil_of_fin_len μ) <| lt_trans (lt_add_one i) hj).1 (HNFil μ (i + 1 + 1)) (?_) ?_ exact ⟨le_of_lt <| lt_trans (HNFil_is_strict_mono μ i <| Nat.find_min (HNFil_of_fin_len μ) <| lt_trans (lt_add_one i) hj) <| HNFil_is_strict_mono μ (i + 1) <| Nat.find_min (HNFil_of_fin_len μ) <| hj,le_top⟩- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/Filtration/Impl.lean:210-227
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Related declarations
Associated Primes ker mk Linear Map eq
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_ker_mkLinearMap_eq
Project documentation
Associated primes of the kernel of the localization map. This identifies the associated primes of ker (LocalizedModule.mkLinearMap S M) : Submodule R M with the associated primes of M that do meet the multiplicative set S. Equivalently, these are the associated primes of M after removing those disjoint from S. This lemma is used as the “kernel...
Source project: Harder-Narasimhan
Person-level attribution pending.
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HarderNarasimhan.CommutativeAlgebra.associatedPrimes_ker_mkLinearMap_subset
Plain-language statement
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Person-level attribution pending.
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HarderNarasimhan.CommutativeAlgebra.associatedPrimes_localizedModule_subset_disjoint
Plain-language statement
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Source project: Harder-Narasimhan
Person-level attribution pending.