feat(Algebra/Squarefree): add exists_sq_mul_squarefree

[xroblot]

Add exists_sq_mul_squarefree: every element of a unique factorization monoid is a square times a squarefree element. Also add Squarefree.dvd_of_isSquare_mul and Squarefree.associated_of_isSquare_mul, and golf the existing Nat.sq_mul_squarefree and Nat.sq_mul_squarefree_of_pos to use the new lemma.

[github-actions]

PR summary 8c4aaa4709

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ _root_.exists_sq_mul_squarefree
+ associated_of_isSquare_mul
+ dvd_of_isSquare_mul

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 8c4aaa4).

• +3 new declarations
• −0 removed declarations

+Squarefree.associated_of_isSquare_mul
+Squarefree.dvd_of_isSquare_mul
+exists_sq_mul_squarefree

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 8c4aaa4709
Reference commit 1f8806b67d

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).