Plain-language statement
Remark 2.5 (part 2): idempotence of μmax. The statement μmax μ I = μmax (μmax μ) I says that applying μmax twice does not change the result. Convexity is used to relate the two suprema.
Exact Lean statement
lemma rmk2d5₂
(I : {p : ℒ × ℒ // p.1 < p.2})
(μ : {p :ℒ × ℒ // p.1 < p.2} → S) (hμcvx : ConvexI I μ) :
μmax μ I = μmax (μmax μ) IFormal artifact
Lean source
lemma rmk2d5₂ (I : {p : ℒ × ℒ // p.1 < p.2}) (μ : {p :ℒ × ℒ // p.1 < p.2} → S) (hμcvx : ConvexI I μ) : μmax μ I = μmax (μmax μ) I := by apply eq_of_le_of_ge · apply le_sSup use I.val.2, ⟨⟨le_of_lt I.prop, le_refl I.val.2⟩, ne_of_lt I.prop⟩ · apply sSup_le simp only [ne_eq, Set.mem_setOf_eq, forall_exists_index] intro b v hv res rw [← res] have h : μmax μ ⟨(v ⊓ I.val.1, v), inf_lt_left.2 (not_le_of_gt (lt_of_le_of_ne hv.1.1 hv.2))⟩ ≤ μmax μ ⟨(I.val.1, I.val.2), lt_of_le_of_lt' ((sup_eq_left.2 hv.1.1).symm ▸ hv.1.2) <| right_lt_sup.2 <| not_le_of_gt <| lt_of_le_of_ne hv.1.1 hv.2⟩ := lem2d4₂I I μ hμcvx v hv.1 I.val.1 ⟨le_rfl, le_of_lt I.prop⟩ (not_le_of_gt <| lt_of_le_of_ne hv.1.1 hv.2) I.val.2 <| (sup_eq_left.2 hv.1.1).symm ▸ hv.1.2 simp only [inf_eq_right.2 hv.1.1] at h exact h- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/Convexity/Impl.lean:233-251
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.