feat(Computability): add Kleene's algorithm to prove regular languages have regex matching

[yisiox]

This PR adds a proof that every regular language has some regular expression matching it.

This was achieved by the following constructions and proofs of their correctness

• Define toSingleεNFA to transform any epsilon-NFA to an equivalent one with only a single start and accept state (and the type is ExtendedState)
• Define a bijection between ExtendedState and Fin (FinEnum.card (ExtendedState σ)) to index the states
• Define pathRegex which implements Kleene's algorithm
• Define toRegex which takes an epsilon-NFA, applies toSingleεNFA on it, and computes the corresponding regex using pathRegex

The proofs of correctness chain together to yield (toRegex M).matches' = M.accepts which asserts the direction of Kleene's theorem required.

---

As in #mathlib4 > Regular languages: the review queue @ 💬, this overlaps with #15654, which uses GNFA instead but has been inactive.

[github-actions]

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[github-actions]

PR summary 592989bdba

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Computability.EpsilonNFA	719	729	+10 (+1.39%)

Import changes for all files

Files	Import difference
Mathlib.Computability.EpsilonNFA	10

---

Declarations diff

+ ExtendedState
+ ExtendedState.equivSum
+ IsPath.from_accept
+ IsPath.state_accept
+ IsPath.toSingleεNFA_lift_extendedState
+ IsRestrictedMatch
+ IsRestrictedMatch.mono
+ IsRestrictedPath
+ accepts_toRegex
+ accepts_toSingleεNFA
+ directRegex
+ e
+ instance [FinEnum σ] : FinEnum (ExtendedState σ)
+ instance {M : εNFA α σ}
+ isRestrictedMatch_iff_exists_isPath
+ isRestrictedMatch_iff_exists_isRestrictedPath
+ isRestrictedMatch_nil
+ isRestrictedMatch_of_mem_pathRegex
+ isRestrictedMatch_star
+ isRestrictedPath_iff_isPath
+ matches'_sum_map
+ mem_matches'_directRegex
+ mem_matches'_mul_star_mul
+ mem_matches'_star
+ mem_matches'_star_concat
+ mem_pathRegex_iff_isRestrictedMatch
+ mem_pathRegex_of_isRestrictedMatch
+ pathRegex
+ pathRegex_mono
+ pathRegex_trans
+ toRegex
+ toSingleεNFA

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[YaelDillies]

Are you sure you don't want to contribute this to cslib instead? I am by no means a computability expert, and you will probably experience much faster review there than here.

[YaelDillies]

Is this not the same as

Suggested change

	(L.foldl (fun acc a => acc + f a) acc).matches' =
	(acc + (L.map f).sum).matches' =

? Let's use existing primitives where possible. Eg this shows that probably we want lemmas about (a + b).matches' and (L.map f).sum.matches'

[yisiox]

I was struggling with representing the base case of Kleene's algorithm $R^0_{ij}=\bigcup_{a:q_j\in \delta(q_i,a)}a$ and had the awkward foldl in directRegex, hence this lemma about foldl and matches'. I changed it now to sum and map as suggested and only a lemma about (L.map f).sum.matches' is needed since there is no accumulator anymore. Hopefully it's less awkward.

[yisiox]

Are you sure you don't want to contribute this to cslib instead? I am by no means a computability expert, and you will probably experience much faster review there than here.

This came about because I needed algebraic stuff for Mazurkiewicz trace theory and thus relied on mathlib. I thought it would be appropriate to add this to mathlib as at some point I realised I needed Kleene's theorem.

[yisiox]

-awaiting-author

[YaelDillies]

Interesting, thank you for letting me know. However, are you aware that cslib depends on mathlib? It therefore looks to me like cslib is a perfect fit for your contribution

[yisiox]

I see, thanks for the suggestion! I looked at cslib and noticed that cslib has their own definition of automata, only imports mathlib computability to define IsRegular, and has no definition for regular expressions. Perhaps some translations are needed to get this suitable for cslib, and will require importing regular expressions from mathlib too. I definitely want to support the cslib ecosystem though and will see if I can reach out to someone from the cslib community for their thoughts on the matter.

But still mathlib computability has definitions of automata, regular languages, and regular expressions. Kleene's theorem seems like a natural completion of the API.

[YaelDillies]

still mathlib computability has definitions of automata, regular languages, and regular expressions. Kleene's theorem seems like a natural completion of the API.

These were added to mathlib before cslib existed. It is IMO reasonable to move these to cslib now