# Question 20445: Compose the Inner Index into the Outer Index in That Order [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-0070) 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. R0039-20445 · q-20261014-advance-20261107-r07-subsequence-0070 ## Original question and givens Use outer index k(i)=1+5i and inner index l(j)=3+2j, with nonnegative starts and positive steps. Evaluate j=20. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate). - G001 — Use outer index k(i)=1+5i and inner index l(j)=3+2j, with nonnegative starts and positive steps. Evaluate j=20. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate). ## 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-composed-index;original snapshot and mathematics retained After explicit ordered source output fields=[j, inner_index, source_index, composed_step, strictly_increasing, certificate] 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 · impliesBefore outer index k:ORIGINAL nonnegative start=1;positive integer step=5;F(q)=start+step*q for EVERY natural q>=0 After F(q)>=start>=0 and for ANY q0;strictly increasing on ALL naturals,not just the evaluated prefix Conditions: none A positive integer step preserves the order of every pair of natural indices,not merely injectivity or an arbitrary reordering.This universal mathematical law is not truncated by an evaluated-output safety range and says nothing about monotonicity of source values. Step identifiers and dependenciesRule: subseq_affine_all_natural_index_order_not_value_monotonicity · Uses: G001, S001 S003 · impliesBefore inner index l:ORIGINAL nonnegative start=3;positive integer step=2;F(q)=start+step*q for EVERY natural q>=0 After F(q)>=start>=0 and for ANY q0;strictly increasing on ALL naturals,not just the evaluated prefix Conditions: none A positive integer step preserves the order of every pair of natural indices,not merely injectivity or an arbitrary reordering.This universal mathematical law is not truncated by an evaluated-output safety range and says nothing about monotonicity of source values. Step identifiers and dependenciesRule: subseq_affine_all_natural_index_order_not_value_monotonicity · Uses: G001, S002 S004 · calculationBefore subsequence inner step times ORIGINAL j := mul(2, 20) After subsequence inner step times ORIGINAL j = 40 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 · calculationBefore subsequence ORIGINAL inner index l(j) := sum(3, 40) After subsequence ORIGINAL inner index l(j) = 43 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, S004 S006 · calculationBefore subsequence outer step times INNER result := mul(5, 43) After subsequence outer step times INNER result = 215 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 · calculationBefore subsequence ORIGINAL composed index k(l(j)) := sum(1, 215) After subsequence ORIGINAL composed index k(l(j)) = 216 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, S006 S008 · calculationBefore subsequence composed positive step := mul(5, 2) After subsequence composed positive step = 10 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 outer k(i)=1+5i;inner l(j)=3+2j;actual j=20;inner value=43 After outer(inner(j))=216;for EVERY j,composed consecutive difference=10>0;exact evaluated fields within SAFE integer range Conditions: none Apply inner to j first,then feed that index into outer,not the reverse.The composed difference is the product of positive steps for every natural j.The SAFE check concerns explicitly evaluated numeric fields,not the mathematical domain of the infinite index law. Step identifiers and dependenciesRule: subseq_composition_inner_then_outer_all_j_positive_step · Uses: G001, S008 ## Reverse checks - 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 composed positive step with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute subsequence ORIGINAL composed index k(l(j)) with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute subsequence outer step times INNER result with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute subsequence ORIGINAL inner index l(j) with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute subsequence inner step times ORIGINAL j 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 — 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 (20, 43, 216, 10, Yes, composition_remains_strictly_increasing) 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: 448301b5af448bcae515fa942ce6cd5dd5921ca7c3a262e7a94f5bd1a4c015af - candidate record SHA-256: 84586b0444a5c4a93fe7b903161e24c81b9566a83e1cb56601660967865efaeb - 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-0070.json)