Μmax res intvl
HarderNarasimhan.μmax_res_intvl
Plain-language statement
Restriction commutes with the “left-anchored supremum” construction μmax from Basic.lean. Mathematically, taking μmax inside an interval is the same as taking μmax in ℒ after forgetting the interval subtype.
Exact Lean statement
lemma μmax_res_intvl {ℒ : Type*} [Nontrivial ℒ] [PartialOrder ℒ] [BoundedOrder ℒ]
{I : {p : ℒ × ℒ // p.1 < p.2}}
{S : Type*} [CompleteLattice S]
{μ : {p :ℒ × ℒ // p.1 < p.2} → S}
{J : {p :(Interval I) × (Interval I) // p.1 < p.2}} :
------------
μmax (Resμ I μ) J = μmax μ ⟨(J.val.1.val,J.val.2.val),lt_lt⟩
------------Formal artifact
Lean source
lemma μmax_res_intvl {ℒ : Type*} [Nontrivial ℒ] [PartialOrder ℒ] [BoundedOrder ℒ]{I : {p : ℒ × ℒ // p.1 < p.2}}{S : Type*} [CompleteLattice S]{μ : {p :ℒ × ℒ // p.1 < p.2} → S}{J : {p :(Interval I) × (Interval I) // p.1 < p.2}} :------------μmax (Resμ I μ) J = μmax μ ⟨(J.val.1.val,J.val.2.val),lt_lt⟩------------:= by unfold μmax simp only [μ_res_intvl, ne_eq] congr ext x constructor · rintro ⟨u,hu1,hu2⟩ use u.val use ⟨hu1.1,fun hc ↦ hu1.right (Subtype.coe_inj.1 hc)⟩ · rintro ⟨u,hu1,hu2⟩ use ⟨u,le_trans ((J.val).1.prop.1) hu1.1.1 ,le_trans hu1.1.2 ((J.val).2.prop.2)⟩ rw [← hu2] simp only [exists_prop, and_true] exact ⟨hu1.1,fun hc ↦ hu1.right (Subtype.coe_inj.2 hc)⟩- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/Interval.lean:202-224
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
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.