# An all-index subsequence certificate for affine indices [← Guide index](https://ocme.evemisslab.com/en/derivations/) · [Original MKO](https://ocme.evemisslab.com/en/mko/mko-subsequence/) This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented. derivation-mko-subsequence-01 · ocme-new142-reader-adoption-20261021 ## 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 S01 · impliesBefore j\in\mathbb N_0 After k(j)=1+2j\in\mathbb N_0 Conditions: none Rule: algebra · Uses: G1 Leaf basis: algebra k maps every nonnegative integer to a legal source index. S02 · impliesBefore j_2>j_1\ge0 After k(j_2)-k(j_1)=2(j_2-j_1)>0 Conditions: none Rule: algebra · Uses: G1, S01 Leaf basis: algebra Prove strict increase for any ordered pair of indices. S03 · equivalentBefore b_j=a_{k(j)} After b_j=3(1+2j)-2=1+6j Conditions: none Rule: algebra · Uses: G1, S02 Leaf basis: algebra Evaluate k(j) before substituting into the source sequence; k is an index map, not a term value. S04 · definitionBefore b_j=a_{k(j)},\quad k\text{ strictly increasing on }\mathbb N_0 After (b_j)\text{ is an infinite subsequence of }(a_n) Conditions: none Rule: definition_here · Uses: S01, S02, S03 Leaf basis: definition_here The complete infinite increasing map and term formula together establish the subsequence definition. S05 · impliesBefore i_2>i_1\ge0 After \ell(i_2)-\ell(i_1)=3(i_2-i_1)>0 Conditions: none Rule: algebra · Uses: G1 Leaf basis: algebra The second index map is likewise nonnegative and strictly increasing. S06 · equivalentBefore m(i)=k(\ell(i)) After m(i)=1+2(2+3i)=5+6i Conditions: none Rule: algebra · Uses: G1, S05 Leaf basis: algebra The composed index remains a nonnegative affine map. S07 · impliesBefore c_i=b_{\ell(i)}=a_{m(i)} After c_i=13+18i Conditions: none Rule: algebra · Uses: S03, S06 Leaf basis: algebra Both selection routes give 13+18i; the positive step 6 of m certifies a subsequence of the original sequence. S08 · impliesBefore r(0)=1,\ r(1)=0,\ r(j)=j\ (j\ge2) After r\text{ injective but not increasing};\quad(a_{r(j)})\text{ is not certified as a subsequence} Conditions: none Rule: sets · Uses: G1, S04 Leaf basis: sets Swapping the first two indices gives an infinite injective counterexample, distinguishing injectivity from order preservation. ## Substeps and lemmas No substeps in this guide. ## Reverse audits - S08 · premise_audit · G1, S04 — r(1)