# Question 2618: A Rule-to-Table Bridge on an Explicit Finite Domain [Return to catalogue page 14](https://ocme.evemisslab.com/en/questions/page/14/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261021-r01-mathematical-representation-foundations/#q-20261013-advance-20261021-r01-mathematical-representation-0018) 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. R0004-02618 · q-20261013-advance-20261021-r01-mathematical-representation-0018 ## Original question and givens Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=0, b=5, inputs=[0, 1, 3]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row). - G001 — Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=0, b=5, inputs=[0, 1, 3]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row). ## 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-mathematical-representation](https://ocme.evemisslab.com/en/mko/mko-mathematical-representation/) ## Step-by-step derivation S001 · impliesBefore declared symbolic rule or complete finite table After keep the stated domain and representation distinct Conditions: none Use only the declared rule or table;agreement on a finite table alone does not establish function equality on an unlisted domain. Step identifiers and dependenciesRule: representation_scope · Uses: G001 S002 · impliesBefore [0, 1, 3] After [0, 1, 3] Conditions: none Deduplicate and numerically order the domain;repeating an input creates no additional function value. Step identifiers and dependenciesRule: distinct_numeric_domain · Uses: G001, S001 S003 · calculationBefore x=0 ax := mul(0, 0) After x=0 ax = 0 Conditions: none Evaluate a·x first;do not replace the rule by a(x+b). Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore x=0 ax+b := sum(0, 5) After x=0 ax+b = 5 Conditions: none Then add the intercept b,fully substituting this input. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore x=0; ax+b=5 After [0, 5] Conditions: none Each input is paired with its calculated output;the ordered pair lists input before output. Step identifiers and dependenciesRule: representation_table_row · Uses: G001, S004 S006 · calculationBefore x=1 ax := mul(0, 1) After x=1 ax = 0 Conditions: none Evaluate a·x first;do not replace the rule by a(x+b). Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore x=1 ax+b := sum(0, 5) After x=1 ax+b = 5 Conditions: none Then add the intercept b,fully substituting this input. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore x=1; ax+b=5 After [1, 5] Conditions: none Each input is paired with its calculated output;the ordered pair lists input before output. Step identifiers and dependenciesRule: representation_table_row · Uses: G001, S007 S009 · calculationBefore x=3 ax := mul(0, 3) After x=3 ax = 0 Conditions: none Evaluate a·x first;do not replace the rule by a(x+b). Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore x=3 ax+b := sum(0, 5) After x=3 ax+b = 5 Conditions: none Then add the intercept b,fully substituting this input. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore x=3; ax+b=5 After [3, 5] Conditions: none Each input is paired with its calculated output;the ordered pair lists input before output. Step identifiers and dependenciesRule: representation_table_row · Uses: G001, S010 ## Reverse checks - 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 x=3 ax+b with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute x=3 ax with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute x=1 ax+b with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute x=1 ax 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 x=0 ax+b with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute x=0 ax with exact rational arithmetic, not rounded decimals. - 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 (0, 5, [0, 1, 3], [[0, 5], [1, 5], [3, 5]]) 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: 56ae3a020676d2ea83cc3951931de5d7a5c6f6d75aea20cd3feb6f3817006703 - candidate record SHA-256: 44a78c3e8ba0b7ff713c8b039c70b0fe1b5cff7a26499c369ea250f392a57116 - semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56 - question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261021-r01-mathematical-representation-0018.json)