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R0021-11183 · q-20261013-advance-20261024-r07-convex-set-0008
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=-9, upper=10, 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=-9, upper=10, 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=[-9,10];arbitrary x,y in these bounds;arbitrary 0<=lambda<=1
After
lambda>=0;1-lambda>=0;x--9>=0;y--9>=0;10-x>=0;10-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=[-9,10]
After
z--9=lambda*(x--9)+(1-lambda)*(y--9)>=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=[-9,10]
After
10-z=lambda*(10-x)+(1-lambda)*(10-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 [-9,10] and all lambda in [0,1] give z in [-9,10]
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.(-9, 10, 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: ec28a7261e85b6b43246f39c48f5c4453912b0e7a057c27a17efc2f4a5e05524candidate record SHA-256: 88c4928ff1eed4e1c488461ff93c761c3e63cfe0dd765fe5a24caa8864e56abasemantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139Lean: NOT_RUN · human review: NOT_PERFORMED