Research proof index

Semistability of the initial segment and the “no improvement to the right” property. If x ∈ St μ, then: 1. the restriction of μ to the interval (⊥, x) is semistable, and 2. for any y > x, the μA-slope on (⊥, x) is not dominated by the slope on (x, y). This packages the two internal statements impl.prop3d7₁ and impl.prop3d7₂. API note:...

Totality/maximality consequences and the slope decomposition formula. Assuming convexity and an additional admissibility hypothesis (either totality of ≤ on S, or an attainment condition on μ), the internal results show: 1. St μ is totally ordered and, under DCC, admits a greatest element; and 2. for any stable breakpoint x and any y > x, the...

Proposition 4.16: a TFAE package relating the three equalities for μmin/μmax on TotIntvl, and (under chain conditions) equivalence with Nash equilibrium.

Remark 2.5 (paper-facing form). Under convexity, this states: - μmax μ is convex, and - μmax is idempotent and leaves μA unchanged on every interval. API note: the second component is universally quantified over intervals to facilitate rewriting in later files.