Research proof index

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.

Restriction commutes with the “right-anchored infimum” construction μmin from Basic.lean. This is the dual statement to μmax_res_intvl.

Let $f:\mathbb R\to\mathbb R$. If the same function, viewed as taking values in $\mathbb C$, has derivative $u\in\mathbb C$ at a real point $z$, then $u$ is real and equals the ordinary real derivative of $f$ at $z$.