# Subsequence

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A subsequence selects terms by a strictly increasing natural-index map and preserves their original order.

mko-subsequence · definition · v0.10.0



## Mathematical statement

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.





## Representative formula (exact TeX)

b_j=a_{k(j)}





## Assumptions


- Index starts are nonnegative integers and index steps are positive integers.
- Arithmetic source sequences and index compositions use exact integer results.





## Symbol scope


- k — strictly increasing index map
- b — subsequence
- a — source sequence
- j — sequence index





## Intuition, definition, and misconceptions

Strict increase is stronger than arbitrary injectivity.

Composition of subsequences still preserves order.

A finite index prefix does not by itself supply a complete infinite index map.

Notation candidate: evaluate the strictly increasing index map at j before selecting term a_{k(j)}. Injectivity,a reordering or a finite increasing prefix alone does not replace that premise.

An arbitrary rearrangement or finite sample is not automatically a subsequence certificate.





## Adopted derivation guide

[An all-index subsequence certificate for affine indices](https://ocme.evemisslab.com/en/derivations/derivation-mko-subsequence-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — The source and selected sequence are distinct objects.
- [Injectivity and distinct inputs](https://ocme.evemisslab.com/en/mko/mko-injective/) — Strict increase implies injectivity, not conversely.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — The original index order must be preserved.





## Verification boundary

The symbolic certificate here is limited to affine increasing indices; other maps need their own universal certificate. The statement explicitly uses k(j); alignment with the shared abbreviated representative subscript remains under review.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add the Bolzano-Weierstrass theorem.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/source13-20261018-subsequence.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-subsequence/index.md)