Research proof index

Construct a Jordan–Hölder RelSeries from an existing filtration. Given the existence instance for JordanHolderFiltration μ, we build a RelSeries for the relation JordanHolderRel μ whose head is ⊤ and whose last element is ⊥. API note: this is the RelSeries-shaped entry point corresponding to the existence instance.

Existence of a semistable RelSeries from ⊥ to ⊤ with strictly decreasing slopes. From the canonical HarderNarasimhanFiltration μ, we build a RelSeries over the relation IntervalSemistableRel μ. The step field is given by the strict-mono successor property together with piecewise_semistable. The final conjunct states the slope strictness co...

Uniqueness of the semistable RelSeries in the complete linear order case. When S is a complete linear order, Harder–Narasimhan filtrations are unique. Using impl.hHFil_of_hNSeries, any RelSeries satisfying the slope condition produces a filtration; uniqueness of filtrations then implies uniqueness of such series (up to extensional equality).

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.

Lengths of Jordan–Hölder filtrations agree under modularity. Assuming ℒ is modular and μ satisfies the standard hypotheses (including affinity), any two Jordan–Hölder filtrations for μ have the same finite length.

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.