# Question 20467: A Finite Increasing List Is Not an Already-Supplied Infinite Certificate [Return to catalogue page 103](https://ocme.evemisslab.com/en/questions/page/103/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261107-r07-subsequence-foundations/#q-20261014-advance-20261107-r07-subsequence-0092) 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. R0039-20467 · q-20261014-advance-20261107-r07-subsequence-0092 ## Original question and givens Supplied finite natural-index list=[16, 18, 20, 22, 24]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary). - G001 — Supplied finite natural-index list=[16, 18, 20, 22, 24]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary). ## 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-subsequence](https://ocme.evemisslab.com/en/mko/mko-subsequence/) ## Step-by-step derivation S001 · impliesBefore source operation=subseq-finite-prefix-boundary;original snapshot and mathematics retained After explicit ordered source output fields=[indices, finite_prefix_increasing, infinite_certificate_supplied, boundary] Conditions: none Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore subsequence finite prefix adjacent index1 difference := sub(18, 16) After subsequence finite prefix adjacent index1 difference = 2 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore ORIGINAL finite adjacent positions=[0, 1];actual indices=[16, 18] After exact forward difference=2;strictly increasing here=Yes Conditions: none Check every adjacent pair in original order without sorting or deduplicating to repair the prefix.Positive differences at all finite adjacent positions certify this prefix's strict increase. Step identifiers and dependenciesRule: subseq_finite_order_each_actual_adjacent_pair · Uses: G001, S002 S004 · calculationBefore subsequence finite prefix adjacent index2 difference := sub(20, 18) After subsequence finite prefix adjacent index2 difference = 2 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore ORIGINAL finite adjacent positions=[1, 2];actual indices=[18, 20] After exact forward difference=2;strictly increasing here=Yes Conditions: none Check every adjacent pair in original order without sorting or deduplicating to repair the prefix.Positive differences at all finite adjacent positions certify this prefix's strict increase. Step identifiers and dependenciesRule: subseq_finite_order_each_actual_adjacent_pair · Uses: G001, S004 S006 · calculationBefore subsequence finite prefix adjacent index3 difference := sub(22, 20) After subsequence finite prefix adjacent index3 difference = 2 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore ORIGINAL finite adjacent positions=[2, 3];actual indices=[20, 22] After exact forward difference=2;strictly increasing here=Yes Conditions: none Check every adjacent pair in original order without sorting or deduplicating to repair the prefix.Positive differences at all finite adjacent positions certify this prefix's strict increase. Step identifiers and dependenciesRule: subseq_finite_order_each_actual_adjacent_pair · Uses: G001, S006 S008 · calculationBefore subsequence finite prefix adjacent index4 difference := sub(24, 22) After subsequence finite prefix adjacent index4 difference = 2 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore ORIGINAL finite adjacent positions=[3, 4];actual indices=[22, 24] After exact forward difference=2;strictly increasing here=Yes Conditions: none Check every adjacent pair in original order without sorting or deduplicating to repair the prefix.Positive differences at all finite adjacent positions certify this prefix's strict increase. Step identifiers and dependenciesRule: subseq_finite_order_each_actual_adjacent_pair · Uses: G001, S008 S010 · impliesBefore ORIGINAL finite index list=[16, 18, 20, 22, 24];ALL adjacent strict flags=[Yes, Yes, Yes, Yes] After this finite prefix increasing=Yes;INFINITE certificate supplied by original data=No;finite_prefix_not_infinite_subsequence_certificate Conditions: none The No says the original data do not specify a universal law for all later indices.A valid finite prefix can be separately extended,for example by adding1 after its last value,but that does not determine an unspecified original continuation.If the prefix already fails,its actual adjacent violation is the obstruction. Step identifiers and dependenciesRule: subseq_prefix_No_not_nonextendability_or_sorting · Uses: G001, S009 ## Reverse checks - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 subsequence finite prefix adjacent index4 difference 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 subsequence finite prefix adjacent index3 difference 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 subsequence finite prefix adjacent index2 difference 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 subsequence finite prefix adjacent index1 difference 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 ([16, 18, 20, 22, 24], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate) 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: 1b7b57b9ad46dadc829e822e681ea7329987b7ad23fada3fd6c1b53088abe482 - candidate record SHA-256: 93b4a9c3546f20ab9eef86288459a3299ff4f82f10bc7877b7595be05539b684 - semantic review receipt SHA-256: 6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034 - question-first witness SHA-256: cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261107-r07-subsequence-0092.json)