Question 20450: Compose the Inner Index into the Outer Index in That Order

Original question and givens

Use outer index k(i)=0+5i and inner index l(j)=0+1j, with nonnegative starts and positive steps. Evaluate j=25. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

• G001 — Use outer index k(i)=0+5i and inner index l(j)=0+1j, with nonnegative starts and positive steps. Evaluate j=25. 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

Step-by-step derivation

source operation=subseq-composed-index;original snapshot and mathematics retained

explicit ordered source output fields=[j, inner_index, source_index, composed_step, strictly_increasing, certificate]

outer index k:ORIGINAL nonnegative start=0;positive integer step=5;F(q)=start+step*q for EVERY natural q>=0

F(q)>=start>=0 and for ANY q<r,F(r)-F(q)=(5)*(r-q)>0;strictly increasing on ALL naturals,not just the evaluated prefix

inner index l:ORIGINAL nonnegative start=0;positive integer step=1;F(q)=start+step*q for EVERY natural q>=0

F(q)>=start>=0 and for ANY q<r,F(r)-F(q)=(1)*(r-q)>0;strictly increasing on ALL naturals,not just the evaluated prefix

subsequence inner step times ORIGINAL j := mul(1, 25)

subsequence inner step times ORIGINAL j = 25

subsequence ORIGINAL inner index l(j) := sum(0, 25)

subsequence ORIGINAL inner index l(j) = 25

subsequence outer step times INNER result := mul(5, 25)

subsequence outer step times INNER result = 125

subsequence ORIGINAL composed index k(l(j)) := sum(0, 125)

subsequence ORIGINAL composed index k(l(j)) = 125

subsequence composed positive step := mul(5, 1)

subsequence composed positive step = 5

ORIGINAL outer k(i)=0+5i;inner l(j)=0+1j;actual j=25;inner value=25

outer(inner(j))=125;for EVERY j,composed consecutive difference=5>0;exact evaluated fields within SAFE integer range

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

(25, 25, 125, 5, 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.