# Question 2674: Repeated Inputs versus a Single-Valued Pair Table [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-0074) 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. R0004-02674 · q-20261013-advance-20261021-r01-mathematical-representation-0074 ## 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. Pairs=[[0, 11], [1, 12], [1, 12], [2, 41]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table. - 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. Pairs=[[0, 11], [1, 12], [1, 12], [2, 41]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table. ## 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 input=0; distinct outputs=[11] After one output=Yes Conditions: none Different outputs for the same input violate a function table;duplicate identical pairs are removed and are not a conflict. Step identifiers and dependenciesRule: single_valued_fiber_audit · Uses: G001, S001 S003 · impliesBefore input=1; distinct outputs=[12] After one output=Yes Conditions: none Different outputs for the same input violate a function table;duplicate identical pairs are removed and are not a conflict. Step identifiers and dependenciesRule: single_valued_fiber_audit · Uses: G001, S002 S004 · impliesBefore input=2; distinct outputs=[41] After one output=Yes Conditions: none Different outputs for the same input violate a function table;duplicate identical pairs are removed and are not a conflict. Step identifiers and dependenciesRule: single_valued_fiber_audit · Uses: G001, S003 S005 · impliesBefore [] After function on listed domain=Yes Conditions: none Judge single-valuedness on the listed domain;do not invent a larger domain requiring additional values. Step identifiers and dependenciesRule: listed_domain_function_verdict · Uses: G001, S004 ## Reverse checks - 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 — 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 ([[0, 11], [1, 12], [2, 41]], [0, 1, 2], [], 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: 823aaa46a24524e61f97803f5e5f00673032adb405af605382369f4b7bdafadc - candidate record SHA-256: af4a123da4e09660d268e2cb5c83b8d652875a52447f509a76b5dc8705908956 - 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-0074.json)