An all-index subsequence certificate for affine indices

Goal and scope

Construct a subsequence of a_n=3n-2 using k(j)=1+2j, then take a further subsequence.

Nonnegative integer indices and complete affine maps, not finite prefixes or arbitrary rearrangements.

Givens and symbols

• algebra
• sets
• definition_here

• G1 — a_n=3n-2 for n≥0; k(j)=1+2j; a further index is ell(i)=2+3i for i≥0.

• k,\ell,m — index maps
• b_j,c_i — selected terms

Rules and sources

• mko_definition · cited — The map k:N→N must be strictly increasing, not merely injective, and b_j=a_{k(j)}. This slice uses affine k(j)=s+d*j with d≥1 as a symbolic all-j certificate. A finite increasing prefix is not an already-supplied infinite map; arbitrary rearrangements and finite samples are not automatic certificates. · mko-subsequence
• algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.
• sets · prerequisite — Compute unions, intersections, complements, inclusion and preimages element by element.
• definition_here · prerequisite — A subsequence b_j=a_{k(j)} requires k strictly increasing on all nonnegative integers; index composition evaluates the inner map first.

Main derivation

j\in\mathbb N_0

k(j)=1+2j\in\mathbb N_0

j_2>j_1\ge0

k(j_2)-k(j_1)=2(j_2-j_1)>0

b_j=a_{k(j)}

b_j=3(1+2j)-2=1+6j

b_j=a_{k(j)},\quad k\text{ strictly increasing on }\mathbb N_0

(b_j)\text{ is an infinite subsequence of }(a_n)

i_2>i_1\ge0

\ell(i_2)-\ell(i_1)=3(i_2-i_1)>0

m(i)=k(\ell(i))

m(i)=1+2(2+3i)=5+6i

c_i=b_{\ell(i)}=a_{m(i)}

c_i=13+18i

r(0)=1,\ r(1)=0,\ r(j)=j\ (j\ge2)

r\text{ injective but not increasing};\quad(a_{r(j)})\text{ is not certified as a subsequence}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S04 — r(1)<r(0) is an explicit failure, not merely an inconclusive finite sample.
• S07 · premise_audit · S03, S06 — The equality checks values, while the positive step separately checks order.
• S06 · inverse_equivalence · G1, S05 — The defining composition applies ell then k. These particular affine laws accidentally commute numerically, but arbitrary index maps cannot be interchanged.
• S05 · premise_audit · G1 — Its start 2 is nonnegative and step 3 positive.
• S04 · premise_audit · S01, S02, S03 — A finite prefix alone would not supply this total function.
• S03 · inverse_equivalence · G1, S02 — The substituted subscript of a is exactly k(j).
• S02 · premise_audit · G1, S01 — Strict increase implies injectivity, but injectivity alone is not sufficient for an index map to certify a subsequence.
• S01 · premise_audit · G1 — The definition is not just a finite list of indices.

Error controls and negative controls

Index first

Evaluate k(j) before a and retain the parentheses.

Infinite certificate

Use the positive-step difference to certify all indices, not a finite table.

• An injective index map always gives a subsequence. → reject — The swapping injection in S08 is not increasing.
• The original index of c_i is k(i). → reject — The correct index is k(ell(i))=5+6i, whereas k(i)=1+2i; already at i=0 they are five and one.
• Ten increasing indices prove an infinite subsequence. → reject — A map and increasing certificate on all of N_0 are still required.

Result and limitations

b_j=1+6j and c_i=13+18i, both certified by complete strictly increasing index maps.

Verification and source binding

1 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 723f86ede56e20fd17954f55eaf9ac1bb183cabb027071619eab37816d8ea520
• original source SHA-256: b3f9513bca7f0ae4d56cfc0a6777cfda19771c065785d1014d30696c2f912b93
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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