### Abstract

Graph lemma quantifies the defect effect of a system of word equations. That is, it provides an upper bound on the rank of the system. We formalize the proof based on the decomposition of a solution into its free basis. A direct application is an alternative proof of the fact that two noncommuting words form a code.

### License

BSD License

### History

August 17, 2023
  Updated to version v1.10.1.
August 24, 2022
  Reworked version. Added theory Glued_Codes.

### Topics

- Computer science/Automata and formal languages

### Related publications

- Development repository

### Session Combinatorics_Words_Graph_Lemma

- Glued_Codes
- Graph_Lemma

### Depends on

- Combinatorics on Words Basics

### Used by

- Binary codes that do not preserve primitivity
- Intersection of two monoids generated by two element codes

## Cite

×

Combinatorics_Words_Graph_Lemma-AFP

    @article{Combinatorics_Words_Graph_Lemma-AFP,
      author  = {Štěpán Holub and Martin Raška and Štěpán Starosta},
      title   = {Graph Lemma},
      journal = {Archive of Formal Proofs},
      month   = {May},
      year    = {2021},
      note    = {\url{https://isa-afp.org/entries/Combinatorics_Words_Graph_Lemma.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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Older releases:

- Sep 13, 2023 : Isabelle2023
- Oct 27, 2022 : Isabelle2022
- Dec 14, 2021 : Isabelle2021-1
- May 26, 2021 : Isabelle2021