# Question 5720: Closed Comparison Truths and Signed Witnesses [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-r03-proposition-foundations/#q-20261008-advance-20261012-r03-proposition-0020) 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. R0010-05720 · q-20261008-advance-20261012-r03-proposition-0020 ## Original question and givens In a classical integer context, ordered pairs=[[7, 0], [-18, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values). - G001 — In a classical integer context, ordered pairs=[[7, 0], [-18, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values). ## 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-proposition](https://ocme.evemisslab.com/en/mko/mko-proposition/) ## Step-by-step derivation S001 · impliesBefore classical integer formulas with explicit binding/data After a truth verdict is not a delivered general proof object Conditions: none A false closed statement is still a proposition.These finite substitutions and comparison witnesses do not promote a truth label to a general formal proof or completeness of arbitrary inference rules. Step identifiers and dependenciesRule: proposition_truth_and_proof_scope · Uses: G001 S002 · calculationBefore integer comparison difference0 := sub(7, 0) After integer comparison difference0 = 7 Conditions: none Compute a−b from the ordered pair,retaining zero and either sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore row=0;a=7;b=0;a−b=7 After statement=a≤b;truth=False Conditions: none Equality requires difference0 and ≤ requires a nonpositive difference;the signed difference supplies the exact comparison witness,not a candidate report. Step identifiers and dependenciesRule: closed_integer_statement_row · Uses: G001, S002 S004 · calculationBefore integer comparison difference1 := sub(-18, 2) After integer comparison difference1 = -20 Conditions: none Compute a−b from the ordered pair,retaining zero and either sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore row=1;a=-18;b=2;a−b=-20 After statement=a≤b;truth=True Conditions: none Equality requires difference0 and ≤ requires a nonpositive difference;the signed difference supplies the exact comparison witness,not a candidate report. Step identifiers and dependenciesRule: closed_integer_statement_row · Uses: G001, S004 S006 · calculationBefore integer comparison difference2 := sub(0, 0) After integer comparison difference2 = 0 Conditions: none Compute a−b from the ordered pair,retaining zero and either sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore row=2;a=0;b=0;a−b=0 After statement=a≤b;truth=True Conditions: none Equality requires difference0 and ≤ requires a nonpositive difference;the signed difference supplies the exact comparison witness,not a candidate report. Step identifiers and dependenciesRule: closed_integer_statement_row · Uses: G001, S006 ## Reverse checks - 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 integer comparison difference2 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 integer comparison difference1 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 integer comparison difference0 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 ([False, True, True], [7, -20, 0]) 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: bc168b01d0787990646b07360d02f41f0e6d9b2006938a8f7d7bc89dd31d595b - candidate record SHA-256: e07f978055f65d186922c44c2532f7e815a522ffa911082edc73e83afd0ce506 - 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-r03-proposition-0020.json)