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R0021-11198 · q-20261013-advance-20261024-r07-convex-set-0023
Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-24, upper=25, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).
G001 — Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-24, upper=25, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).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.
Before
interval:closed bounds=[-24,25];arbitrary x,y in these bounds;arbitrary 0<=lambda<=1
After
lambda>=0;1-lambda>=0;x--24>=0;y--24>=0;25-x>=0;25-y>=0
Conditions: none
Choose arbitrary interval members and any permitted lambda.All four endpoint slacks are nonnegative;equal bounds are valid.
Rule: universal_interval_arbitrary_members · Uses: G001
Before
z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-24,25]
After
z--24=lambda*(x--24)+(1-lambda)*(y--24)>=0
Conditions: none
Use the weights'unit sum to expand z minus the lower bound as a sum of nonnegative products.This is not sampling.
Rule: universal_interval_lower_identity · Uses: G001, S001
Before
z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-24,25]
After
25-z=lambda*(25-x)+(1-lambda)*(25-y)>=0
Conditions: none
Likewise the upper bound minus z is nonnegative,so both bounding inequalities hold.
Rule: universal_interval_upper_identity · Uses: G001, S002
Before
interval:both symbolic slack identities hold for every chosen x,y,lambda
After
all x,y in [-24,25] and all lambda in [0,1] give z in [-24,25]
Conditions: none
No special restriction was added to the arbitrary endpoints or lambda.Discharge their universal quantifiers to obtain convexity of the whole closed interval.
Rule: universal_interval_quantifier_discharge · Uses: G001, S003
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(-24, 25, all x,y in interval and all lambda in [0,1], Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 2312f444c972a04c882ac1fca3048edfcacb8f60beeb57d5ac2cefe619b17380candidate record SHA-256: fe6606b71a615034fce036c77aa44d07477d6a4219c64e4b600d803be506f38dsemantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139Lean: NOT_RUN · human review: NOT_PERFORMED