# Question 5672: Finite Predicate Extension and Aligned Truths [Return to catalogue page 29](https://ocme.evemisslab.com/en/questions/page/29/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261012-r04-set-membership-foundations/#q-20261008-advance-20261012-r04-set-membership-0097) 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. R0010-05672 · q-20261008-advance-20261012-r04-set-membership-0097 ## Original question and givens Finite integer set U=[-21, -1, 0, 1, 21]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U). - G001 — Finite integer set U=[-21, -1, 0, 1, 21]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U). ## 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-set-membership](https://ocme.evemisslab.com/en/mko/mko-set-membership/) ## Step-by-step derivation S001 · impliesBefore declared finite type or bounded integer universe After membership truth is distinct from eligibility to ask the question Conditions: none Being outside the type is not the same as false membership.Check only the declared finite type or integer set,not all set theory or unrestricted comprehension. Step identifiers and dependenciesRule: finite_membership_scope · Uses: G001 S002 · calculationBefore predicate difference at-21 := sub(-21, 0) After predicate difference at-21 = -21 Conditions: none Compute x minus the parameter as an exact integer comparison witness. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore x=-21;predicate=ge 0 After False Conditions: none Evaluate every element in normalized U order;predicate success does not expand U. Step identifiers and dependenciesRule: finite_membership_predicate_row · Uses: G001, S002 S004 · calculationBefore predicate difference at-1 := sub(-1, 0) After predicate difference at-1 = -1 Conditions: none Compute x minus the parameter as an exact integer comparison witness. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore x=-1;predicate=ge 0 After False Conditions: none Evaluate every element in normalized U order;predicate success does not expand U. Step identifiers and dependenciesRule: finite_membership_predicate_row · Uses: G001, S004 S006 · calculationBefore predicate difference at0 := sub(0, 0) After predicate difference at0 = 0 Conditions: none Compute x minus the parameter as an exact integer comparison witness. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore x=0;predicate=ge 0 After True Conditions: none Evaluate every element in normalized U order;predicate success does not expand U. Step identifiers and dependenciesRule: finite_membership_predicate_row · Uses: G001, S006 S008 · calculationBefore predicate difference at1 := sub(1, 0) After predicate difference at1 = 1 Conditions: none Compute x minus the parameter as an exact integer comparison witness. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore x=1;predicate=ge 0 After True Conditions: none Evaluate every element in normalized U order;predicate success does not expand U. Step identifiers and dependenciesRule: finite_membership_predicate_row · Uses: G001, S008 S010 · calculationBefore predicate difference at21 := sub(21, 0) After predicate difference at21 = 21 Conditions: none Compute x minus the parameter as an exact integer comparison witness. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore x=21;predicate=ge 0 After True Conditions: none Evaluate every element in normalized U order;predicate success does not expand U. Step identifiers and dependenciesRule: finite_membership_predicate_row · Uses: G001, S010 S012 · impliesBefore normalized U=[-21, -1, 0, 1, 21];truths=[False, False, True, True, True] After A={x∈U:P(x)}=[0, 1, 21] Conditions: none The full extension contains exactly the successful elements of the supplied U;the truth list remains aligned with all sorted U-elements.Empty U gives two empty lists. Step identifiers and dependenciesRule: bounded_predicate_extension · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute predicate difference at21 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 predicate difference at1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute predicate difference at0 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 predicate difference at-1 with exact rational arithmetic, not rounded decimals. - 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 — Recompute predicate difference at-21 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 ([0, 1, 21], [False, False, True, True, True]) 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: 3173f7c1d32f768f7420e6bf3fc32608909db7ee27c57e582222856d145b3462 - candidate record SHA-256: 1cd939dbfe7ebc10271d484a7eff9ece7f608865e68e4ece52bae4b9a0e1da26 - semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f - question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r04-set-membership-0097.json)