# Pumping lemma for regular languages

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

Every regular language has a positive pumping length such that each sufficiently long member word has a split that remains in the language for every pump count.

mko-pumping-lemma-regular-languages · theorem · v0.10.0



## Mathematical statement

Keep the quantifier order: there exists p, for every w in L with |w|≥p, there exist x,y,z, for every i≥0, with w=xyz, |xy|≤p, |y|>0 and xy^i z in L. Positive regular families use a repeated-state loop in a complete DFA run and a symbolic invariant for every i, not a few pump tests. For L={a^n b^n}, choose w=a^p b^p and address every legal split: y consists only of a symbols, so i=0 leaves the language. One decomposition, bounded words or finite checks do not prove regularity. The adversary chooses the word before the prover chooses the split and the adversary chooses i after that.





## Representative formula (exact TeX)

P(L,p,w)=b





## Assumptions


- p is a positive integer and word/split indices are exact.
- A positive witness uses a complete DFA state loop and an invariant for all pump counts.
- A negative adversary addresses every split satisfying |xy|≤p and |y|>0.





## Symbol scope


- P — regular-pumping operator
- L — formal language
- p — pumping length
- w — sufficiently long member word
- b — pumping-property statement





## Intuition, definition, and misconceptions

Check the whole quantifier order and split constraints first.

A DFA loop supports every i, not merely finitely tested pump counts.

The a^p b^p adversary addresses all legal splits.

One pumpable split does not prove regularity, and one unfavorable preselected split does not prove nonregularity.





## Adopted derivation guide

[DFA-loop quantifiers and the a^n b^n contradiction](https://ocme.evemisslab.com/en/derivations/derivation-mko-pumping-lemma-regular-languages-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Regular language](https://ocme.evemisslab.com/en/mko/mko-regular-language/) — Specify the applicable class.
- [Deterministic finite automaton](https://ocme.evemisslab.com/en/mko/mko-finite-automaton/) — Supply the repeated-state loop.
- [Mathematical proof](https://ocme.evemisslab.com/en/mko/mko-proof/) — Use contradiction and quantifier opposition.
- [Pigeonhole principle](https://ocme.evemisslab.com/en/mko/mko-pigeonhole-principle/) — Obtain a repeated state among p+1 positions.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — Retain the adversarial choice order.
- [Mathematical induction](https://ocme.evemisslab.com/en/mko/mko-mathematical-induction/) — Retain the run invariant for every pump count.





## Verification boundary

The pumping property is necessary, not sufficient, for regularity. Fixed-p split tables do not alone replace the argument for every proposed pumping length.

not_formalized

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





## Next obligation

Add Myhill–Nerode and context-free pumping boundaries.





## 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/pumping-lemma-regular-languages.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-pumping-lemma-regular-languages/index.md)