Research proof index

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.

Restriction commutes with μA, the infimum over right-endpoints of μmax values. This lemma is a key “locality” principle: computations of μA can be performed on subintervals.

Restriction commutes with μB, the supremum over left-endpoints of μmin values. This is the μB-analogue of μA_res_intvl.

Restriction commutes with the “left-anchored supremum” construction μmax from Basic.lean. Mathematically, taking μmax inside an interval is the same as taking μmax in ℒ after forgetting the interval subtype.