# Question 8212: Closed-Interval Intersection Including Touching Endpoints [Return to catalogue page 42](https://ocme.evemisslab.com/en/questions/page/42/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261017-r04-order-relation-foundations/#q-20261012-advance-20261017-r04-order-relation-0037) 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. R0015-08212 · q-20261012-advance-20261017-r04-order-relation-0037 ## Original question and givens Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-1/2, 5/2] and [2/2, 8/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set). - G001 — Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-1/2, 5/2] and [2/2, 8/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set). ## 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-order-relation](https://ocme.evemisslab.com/en/mko/mko-order-relation/) ## Step-by-step derivation S001 · impliesBefore usual total order on R;all original denominators positive After closed endpoints included;negative scaling reverses;zero collapses Conditions: none These exact rationals use the usual real total order,not complex numbers or an arbitrary partial order.Original denominators must be positive;normalizing a negative input denominator does not waive that contract. Step identifiers and dependenciesRule: real_rational_order_scope · Uses: G001 S002 · calculationBefore lower bounds left cross := mul(-1, 1) After lower bounds left cross = -1 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore lower bounds right cross := mul(1, 2) After lower bounds right cross = 2 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore lower bounds cross residual := sub(-1, 2) After lower bounds cross residual = -3 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore lower bounds:fractions=[-1/2, 1/1] After cross residual=-3;comparison=< Conditions: none Reduced denominators are positive,so the cross-difference sign determines order.Retain equality and use original indices separately to break equal-value sorting ties. Step identifiers and dependenciesRule: rational_positive_denominator_comparison · Uses: G001, S004 S006 · calculationBefore upper bounds left cross := mul(5, 1) After upper bounds left cross = 5 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore upper bounds right cross := mul(4, 2) After upper bounds right cross = 8 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore upper bounds cross residual := sub(5, 8) After upper bounds cross residual = -3 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore upper bounds:fractions=[5/2, 4/1] After cross residual=-3;comparison=< Conditions: none Reduced denominators are positive,so the cross-difference sign determines order.Retain equality and use original indices separately to break equal-value sorting ties. Step identifiers and dependenciesRule: rational_positive_denominator_comparison · Uses: G001, S008 S010 · calculationBefore intersection inner bounds left cross := mul(1, 2) After intersection inner bounds left cross = 2 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore intersection inner bounds right cross := mul(5, 1) After intersection inner bounds right cross = 5 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore intersection inner bounds cross residual := sub(2, 5) After intersection inner bounds cross residual = -3 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore intersection inner bounds:fractions=[1/1, 5/2] After cross residual=-3;comparison=< Conditions: none Reduced denominators are positive,so the cross-difference sign determines order.Retain equality and use original indices separately to break equal-value sorting ties. Step identifiers and dependenciesRule: rational_positive_denominator_comparison · Uses: G001, S012 S014 · impliesBefore intervals=[[-1/2, 5/2], [1/1, 4/1]] After max lower=1/1;min upper=5/2;nonempty=Yes Conditions: none Satisfying both lower bounds means using their maximum;both upper bounds means their minimum.Equal inner bounds still include that single point. Step identifiers and dependenciesRule: closed_interval_intersection_guard · Uses: G001, S013 ## Reverse checks - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute intersection inner bounds cross residual with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute intersection inner bounds right cross with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute intersection inner bounds left cross with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute upper bounds cross residual with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute upper bounds right cross with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute upper bounds left cross with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute lower bounds cross residual with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute lower bounds right cross with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute lower bounds left cross with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([-1/2, 5/2], [1/1, 4/1], 1/1, 5/2, Yes, [1/1, 5/2]) 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: b48b95a224a74d0d5464dc02d40ba0abd5fe0f74f41d9c97706957931c7105c0 - candidate record SHA-256: aa0d2b2c585085bfd0a525735a419434b08bdecbb2de4f0dde70d3d61e587909 - semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18 - question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261017-r04-order-relation-0037.json)