feat(Data/List): decidability of HasPeriod and periods of repeated lists

[theebayuser]

Hello, this PR extends Mathlib/Data/List/PeriodicityLemma.lean with:

• a Decidable (List.HasPeriod w p) instance (it is a prefix test), and
• two directions that relate periods to repetition:
  • hasPeriod_flatten_replicate — (replicate n l).flatten has period l.length
  • eq_flatten_replicate_of_hasPeriod — a word with period p and length r * p is the r-fold repetition of its length-p prefix.

This characterizes r-th powers of words by a period plus a length constraint, which a future follow-up PR that I plan to work on will use.

AI disclosure: This code was mostly generated by Claude Fable 5 and Opus 4.8 as part of my larger project on combinatorics on words, which I then reviewed and revised to contribute to mathlib.

[github-actions]

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[github-actions]

PR summary 728c687d4b

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ eq_flatten_replicate_of_hasPeriod
+ hasPeriod_flatten_replicate
+ instance [DecidableEq α] (w : List α) (p : ℕ) : Decidable (w.HasPeriod p)
+ length_flatten_replicate

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

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

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

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 728c687).

• +4 new declarations
• −0 removed declarations

+List.eq_flatten_replicate_of_hasPeriod
+List.hasPeriod_flatten_replicate
+List.instDecidableHasPeriodOfDecidableEq
+List.length_flatten_replicate

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 728c687d4b
Reference commit abb22825db

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/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).

[theebayuser]

LLM-generated