Research proof index

Equivalence between the global typeclass Semistable μ and interval-local semistability on the total interval. This lemma is an API bridge: it lets one freely move between the class-based semistability used in later modules and the predicate semistableI μ TotIntvl defined via StI.

semistable_of_step_cond₂ turns a strict step condition into semistability on each step. Assuming that for every intermediate z strictly between consecutive values filtration (i+1) < z < filtration i the slope strictly improves, the restricted slope Resμ ⟨(filtration (i+1), filtration i), _⟩ μ is semistable.

Semistability of a restriction vs. semistability on the quotient lattice. This lemma is the key “translation” step for coprimary filtrations: * restricting the slope μ R M to an interval (N₁, N₂) corresponds to * the induced slope on the submodule lattice of the quotient module N₂ / N₁. The statement is phrased as an equivalence between `Semistable...

Transport semistability along restriction. This theorem relates: - semistableI μ I, i.e. semistability of the interval I with respect to μ, and - Semistable (Resμ I μ), i.e. global semistability of the restricted function on the interval subtype. API note: this is a key adapter used whenever proofs switch between the “ambient interval” viewpoint a...

stable_of_step_cond₂ upgrades the previous lemma from semistability to stability. Under the same strict step condition, each restricted slope on a step interval is not only semistable but satisfies the strict inequality required for Stable.

step_cond₂_of_stable is the converse direction: stability implies the strict step condition. If each restricted slope on the step intervals is stable, then for every strict intermediate z one has the strict inequality comparing μ (filtration (i+1), z) with the step value.