Plain-language statement
Lemma 2.4 (paper-facing form). Assuming global convexity of μ, this provides the two inequalities labelled (2.2) and (2.3) in the file, packaged as a conjunction. API note: the proof reduces to the interval-local lemmas in HarderNarasimhan.Convexity.Impl by using the equivalence ConvexI TotIntvl μ ↔ Convex μ.
Exact Lean statement
lemma lemma_2_4 {ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ]
{S : Type*} [CompleteLattice S]
(μ : {p :ℒ × ℒ // p.1 < p.2} → S) (hμcvx : Convex μ)
(x : ℒ) (w : ℒ) (hxw : ¬ x ≤ w)
(u : ℒ) (t : ℒ)
(huxw : u ≤ x ⊓ w) (hxwt : x ⊔ w ≤ t) :
------------
(
--`(2.2)`
μA μ ⟨(u, x), lt_of_le_of_lt huxw (inf_lt_left.2 hxw)⟩ ≤
μmax μ ⟨(x ⊓ w, x), inf_lt_left.2 hxw⟩ ∧
μmax μ ⟨(x ⊓ w, x), inf_lt_left.2 hxw⟩ ≤
μmax μ ⟨(w, t), lt_of_le_of_lt' hxwt <| right_lt_sup.2 hxw⟩
) ∧
--`(2.3)`
μA μ ⟨(u, x), lt_of_le_of_lt huxw (inf_lt_left.2 hxw)⟩ ≤ μA μ ⟨(w, x ⊔ w), right_lt_sup.2 hxw⟩
------------Formal artifact
Lean source
lemma lemma_2_4 {ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ] {S : Type*} [CompleteLattice S] (μ : {p :ℒ × ℒ // p.1 < p.2} → S) (hμcvx : Convex μ) (x : ℒ) (w : ℒ) (hxw : ¬ x ≤ w) (u : ℒ) (t : ℒ) (huxw : u ≤ x ⊓ w) (hxwt : x ⊔ w ≤ t) :------------ ( --`(2.2)` μA μ ⟨(u, x), lt_of_le_of_lt huxw (inf_lt_left.2 hxw)⟩ ≤ μmax μ ⟨(x ⊓ w, x), inf_lt_left.2 hxw⟩ ∧ μmax μ ⟨(x ⊓ w, x), inf_lt_left.2 hxw⟩ ≤ μmax μ ⟨(w, t), lt_of_le_of_lt' hxwt <| right_lt_sup.2 hxw⟩ ) ∧ --`(2.3)` μA μ ⟨(u, x), lt_of_le_of_lt huxw (inf_lt_left.2 hxw)⟩ ≤ μA μ ⟨(w, x ⊔ w), right_lt_sup.2 hxw⟩------------ := by apply (ConvexI_TotIntvl_iff_Convex _).2 at hμcvx exact ⟨⟨impl.lem2d4₁ μ x w hxw u huxw,impl.lem2d4₂I TotIntvl μ hμcvx x (in_TotIntvl x) w (in_TotIntvl w) hxw t hxwt⟩,impl.lem2d4₃I TotIntvl μ hμcvx x (in_TotIntvl x) w (in_TotIntvl w) hxw u huxw⟩- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/Convexity/Results.lean:50-71
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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.
Associated Primes ker mk Linear Map subset
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_ker_mkLinearMap_subset
Plain-language statement
One-sided inclusion for associated primes of the kernel of the localization map. If p ∈ associatedPrimes R (ker (mkLinearMap S M)), then p is an associated prime of M and p is not disjoint from the multiplicative set S. This is one direction of associatedPrimes_ker_mkLinearMap_eq.
Source project: Harder-Narasimhan
Person-level attribution pending.
Associated Primes localized Module subset disjoint
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_localizedModule_subset_disjoint
Plain-language statement
Associated primes of a localized module are disjoint from the multiplicative set. More precisely, if p ∈ associatedPrimes R (LocalizedModule S M), then p.carrier ∩ S = ∅. This is a standard fact in commutative algebra: an element of S becomes a unit after localization, so no associated prime of the localized module can contain an element of S.
Source project: Harder-Narasimhan
Person-level attribution pending.