Question 11831: Qualify a Declared Nonempty Interval before Using Jensen Semantics

Original question and givens

The interval record is [lower,upper]=[-7, 8]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.

• G001 — The interval record is [lower,upper]=[-7, 8]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-convex-function

Step-by-step derivation

declared interval record=[-7, 8]

ordered_nonempty=Yes;nondegenerate=Yes

arbitrary x,y in [-7,8];0<=lambda<=1;z=lambda*x+(1-lambda)*y

z--7=lambda*(x--7)+(1-lambda)*(y--7)>=0;8-z=lambda*(8-x)+(1-lambda)*(8-y)>=0

Reverse checks

• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([-7, 8], Yes, Yes, all segment points remain in the interval)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.