chore: remove backward.privateInPublic in algebra, combinatorics and number theory

[kim-em]

This PR removes the backward.privateInPublic option in 9 files across algebra, combinatorics, number theory, order theory and set theory. In each case a private definition appeared in the statement of a public declaration, so it was already part of the public interface in all but name. The fix is to drop private from those definitions, adding docstrings where they were missing. In Algebra/EuclideanDomain/Basic the opposite direction is cleaner: the only consumer xgcdAux_P becomes private as well, matching the existing setup in Data/Int/GCD. In Combinatorics/SetFamily/FourFunctions the definition becomes Finset.collapse rather than a root-level name. In Combinatorics/SetFamily/KruskalKatona the predicate is marked @[expose], since the file's public section is not exposed and the DecidableRel instance needs to unfold it.

Files where a private lemma appeared in a public statement are handled by wrapping the use site instead, in #44032. Names are kept as they were (for example Pell.d); suggestions for better names are welcome but are not needed for this PR.

🤖 Prepared with Claude Code

[github-actions]

PR summary 2411ae2c1d

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ Finset.collapse
- collapse

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 2411ae2).

• +14 new declarations
• −1 removed declarations

-EuclideanDomain.xgcdAux_P
+Finset.UV.UsefulCompression
+Finset.UV.UsefulCompression.instDecidableRel
+Finset.collapse
+Finset.collapse.congr_simp
+FreeAlgebra.liftAux
+HomogeneousLocalization.awayMapAux
+HomogeneousLocalization.awayMapAux.congr_simp
+IsFreeGroupoid.SpanningTree.root'
+IsFreeGroupoid.symgen
+ONote.toStringAux
+Pell.d
+Pell.d.congr_simp
+SzemerediRegularity.distinctPairs
+SzemerediRegularity.distinctPairs.congr_simp

---

Decrease in strong tech debt: (relative, absolute) = (45.65, 0.36)

Current number	Change	Type (strong)
backward.privateInPublic	477	-53
backward.privateInPublic.warn	172	-43

No changes to weak technical debt.

Current commit 2411ae2c1d
Reference commit 0a6c8e0355

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.py 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).

[grunweg]

Can you take a look at the test failure, please?