# Question 5452: Finite Coverage and Next Uncovered Index [Return to catalogue page 28](https://ocme.evemisslab.com/en/questions/page/28/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261012-r01-mathematical-induction-foundations/#q-20261008-advance-20261012-r01-mathematical-induction-0077) 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. R0009-05452 · q-20261008-advance-20261012-r01-mathematical-induction-0077 ## Original question and givens N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(1000) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index). - G001 — N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(1000) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index). ## 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-induction](https://ocme.evemisslab.com/en/mko/mko-mathematical-induction/) ## Step-by-step derivation S001 · impliesBefore natural indices start at0;finite induction protocol After index bookkeeping and finite values are not proofs of all P(n) Conditions: none Natural indices do not require nonnegative sequence values.Checking a finite range or a successor index does not supply proposition truth or an induction proof over all naturals. Step identifiers and dependenciesRule: finite_induction_claim_boundary · Uses: G001 S002 · calculationBefore covered inclusive index count := sum(1000, 1) After covered inclusive index count = 1001 Conditions: none P(0) through P(through) includes both endpoints,so there are through+1 indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore covered indices0,...,1000 After 1000 adjacent steps;next uncovered target=1001 Conditions: none The adjacent-step count equals the final index and the next target is through+1.This counts declared coverage,not independent verification of each P. Step identifiers and dependenciesRule: finite_range_obligation_counts · Uses: G001, S002 ## Reverse checks - 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 covered inclusive index count 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 (1001, 1000, 1001) 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: 5d2cda00f61e422eec3964d3658c1f6c12f1e9b7984b5061aba94fb634407c6a - candidate record SHA-256: 6e270588ac86243384077cbed0479d511adc97f9af2e4c8f3b0336e0db8ea3eb - semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56 - question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r01-mathematical-induction-0077.json)