# Question 11189: A Universal Closed-Interval Convexity Certificate

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0021-11189 · q-20261013-advance-20261024-r07-convex-set-0014



## Original question and givens

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=-15, upper=16, 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=-15, upper=16, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).





## 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-set](https://ocme.evemisslab.com/en/mko/mko-convex-set/)





## Step-by-step derivation

S001 · impliesBefore

interval:closed bounds=[-15,16];arbitrary x,y in these bounds;arbitrary 0<=lambda<=1

After

lambda>=0;1-lambda>=0;x--15>=0;y--15>=0;16-x>=0;16-y>=0

Conditions: none

Choose arbitrary interval members and any permitted lambda.All four endpoint slacks are nonnegative;equal bounds are valid.

Step identifiers and dependenciesRule: universal_interval_arbitrary_members · Uses: G001





S002 · impliesBefore

z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-15,16]

After

z--15=lambda*(x--15)+(1-lambda)*(y--15)>=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.

Step identifiers and dependenciesRule: universal_interval_lower_identity · Uses: G001, S001





S003 · impliesBefore

z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-15,16]

After

16-z=lambda*(16-x)+(1-lambda)*(16-y)>=0

Conditions: none

Likewise the upper bound minus z is nonnegative,so both bounding inequalities hold.

Step identifiers and dependenciesRule: universal_interval_upper_identity · Uses: G001, S002





S004 · impliesBefore

interval:both symbolic slack identities hold for every chosen x,y,lambda

After

all x,y in [-15,16] and all lambda in [0,1] give z in [-15,16]

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.

Step identifiers and dependenciesRule: universal_interval_quantifier_discharge · Uses: G001, S003









## Reverse checks


- 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.





## Result

(-15, 16, 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.





## 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 9dfe2069ea0d577100c412874edcace30bc0f4ecea70e4f10c48c101b47e1247
- candidate record SHA-256: 9c80a86c9452d2332c88b7fd412ca5b1ef8ed1e24c712603245e9369a3a5cc3b
- semantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258
- question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261024-r07-convex-set-0014.json)