Proposition 2 6
HarderNarasimhan.proposition_2_6
Plain-language statement
Proposition 2.6 (paper-facing form). For x<y<z, the statement consists of: - the unconditional monotonicity μA (x,z) ≤ μA (y,z), and - under convexity, parts (a), (b), (c) giving refined comparisons/equalities involving μA. API note: the proposition is packaged as a nested conjunction/implication structure mirroring the paper.
Exact Lean statement
lemma proposition_2_6 {ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ]
{S : Type*} [CompleteLattice S]
(μ : {p :ℒ × ℒ // p.1 < p.2} → S)
(x : ℒ) (y : ℒ) (z : ℒ)
(h : x < y ∧ y < z) :
------------
--`(2.4)`
μA μ ⟨(x, z), lt_trans h.1 h.2⟩ ≤ μA μ ⟨(y, z), h.2⟩ ∧
(
Convex μ →
--`(a)`
μA μ ⟨(x, z), lt_trans h.1 h.2⟩ ≥ (μA μ ⟨(x, y), h.1⟩ ⊓ (μA μ ⟨(y, z), h.2⟩))
∧ (
--`(b)`
(
μA μ ⟨(x, y), h.1⟩ ≥ μA μ ⟨(y, z), h.2⟩ →
μA μ ⟨(y, z), h.2⟩ = μA μ ⟨(x, z), lt_trans h.1 h.2⟩
) ∧ (
μA μ ⟨(x, y), h.1⟩ < μA μ ⟨(y, z), h.2⟩ →
μA μ ⟨(x, y), h.1⟩ ≤ μA μ ⟨(x, z), lt_trans h.1 h.2⟩ ∧
μA μ ⟨(x, z), lt_trans h.1 h.2⟩ ≤ μA μ ⟨(y, z), h.2⟩
)
) ∧ (
--`(c)`
IsComparable (μA μ ⟨(x, y), h.1⟩) (μA μ ⟨(y, z), h.2⟩) ∨
IsAttained μ ⟨(x, z), lt_trans h.1 h.2⟩ →
μA μ ⟨(y, z), h.2⟩ = μA μ ⟨(x, z), lt_trans h.1 h.2⟩ ∨
(μA μ ⟨(x, y), h.1⟩ ≤ μA μ ⟨(x, z), lt_trans h.1 h.2⟩ ∧
μA μ ⟨(x, z), lt_trans h.1 h.2⟩ < μA μ ⟨(y, z), h.2⟩)
)
)
------------Formal artifact
Lean source
lemma proposition_2_6 {ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ] {S : Type*} [CompleteLattice S] (μ : {p :ℒ × ℒ // p.1 < p.2} → S) (x : ℒ) (y : ℒ) (z : ℒ) (h : x < y ∧ y < z) :------------ --`(2.4)` μA μ ⟨(x, z), lt_trans h.1 h.2⟩ ≤ μA μ ⟨(y, z), h.2⟩ ∧ ( Convex μ → --`(a)` μA μ ⟨(x, z), lt_trans h.1 h.2⟩ ≥ (μA μ ⟨(x, y), h.1⟩ ⊓ (μA μ ⟨(y, z), h.2⟩)) ∧ ( --`(b)` ( μA μ ⟨(x, y), h.1⟩ ≥ μA μ ⟨(y, z), h.2⟩ → μA μ ⟨(y, z), h.2⟩ = μA μ ⟨(x, z), lt_trans h.1 h.2⟩ ) ∧ ( μA μ ⟨(x, y), h.1⟩ < μA μ ⟨(y, z), h.2⟩ → μA μ ⟨(x, y), h.1⟩ ≤ μA μ ⟨(x, z), lt_trans h.1 h.2⟩ ∧ μA μ ⟨(x, z), lt_trans h.1 h.2⟩ ≤ μA μ ⟨(y, z), h.2⟩ ) ) ∧ ( --`(c)` IsComparable (μA μ ⟨(x, y), h.1⟩) (μA μ ⟨(y, z), h.2⟩) ∨ IsAttained μ ⟨(x, z), lt_trans h.1 h.2⟩ → μA μ ⟨(y, z), h.2⟩ = μA μ ⟨(x, z), lt_trans h.1 h.2⟩ ∨ (μA μ ⟨(x, y), h.1⟩ ≤ μA μ ⟨(x, z), lt_trans h.1 h.2⟩ ∧ μA μ ⟨(x, z), lt_trans h.1 h.2⟩ < μA μ ⟨(y, z), h.2⟩) ) )------------:= by rw [← ConvexI_TotIntvl_iff_Convex] exact ⟨impl.prop2d6₀ μ x y z h, fun hμcvx ↦ ⟨impl.prop2d6₁I TotIntvl μ hμcvx x (in_TotIntvl x) y (in_TotIntvl y) z (in_TotIntvl z) h,⟨⟨impl.prop2d6₂I₁ TotIntvl μ hμcvx x (in_TotIntvl x) y (in_TotIntvl y) z (in_TotIntvl z) h,impl.prop2d6₂I₂ TotIntvl μ hμcvx x (in_TotIntvl x) y (in_TotIntvl y) z (in_TotIntvl z) h⟩,impl.prop2d6₃I TotIntvl μ hμcvx x (in_TotIntvl x) y (in_TotIntvl y) z (in_TotIntvl z) h⟩⟩⟩- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/Convexity/Results.lean:110-148
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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.