feat(RingTheory): add nilpotence lemmas for zero sums

[sankalpsthakur]

In a commutative semiring, adds:

• sq_eq_sq_of_add_eq_zero: if x + y = 0 then x^2 = y^2
• IsNilpotent.of_add_eq_zero_left: if x + y = 0 and y is nilpotent, so is x
• isNilpotent_iff_of_add_eq_zero: under x + y = 0, x is nilpotent iff y is

Motivated by #10539.

Validation

Built locally: lake build Mathlib.RingTheory.Nilpotent.Basic — succeeds.

Closes #10539

AI/LLM disclosure

AI coding tools were used to help draft this change and PR description. I reviewed the complete change, understand the reasoning, and verified the build locally before submitting.

[github-actions]

Welcome new contributor!

Thank you for contributing to Mathlib! If you haven't done so already, please review our contribution guidelines, as well as the style guide and naming conventions. In particular, we kindly remind contributors that we have guidelines regarding the use of AI when making pull requests.

We use a review queue to manage reviews. If your PR does not appear there, it is probably because it is not successfully building (i.e., it doesn't have a green checkmark), has the awaiting-author tag, or another reason described in the Lifecycle of a PR. The review dashboard has a dedicated webpage which shows whether your PR is on the review queue, and (if not), why.

If you haven't already done so, please come to https://leanprover.zulipchat.com/, introduce yourself, and mention your new PR.

Thank you again for joining our community.

[github-actions]

PR summary 6ec4388c57

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ IsNilpotent.of_add_eq_zero_left
+ isNilpotent_iff_of_add_eq_zero
+ sq_eq_sq_of_add_eq_zero

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 6ec4388).

• +3 new declarations
• −0 removed declarations

+IsNilpotent.of_add_eq_zero_left
+isNilpotent_iff_of_add_eq_zero
+sq_eq_sq_of_add_eq_zero

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 6ec4388c57
Reference commit 54df8a3f96

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).