feat(RingTheory): coprimality of a sum and difference

[sankalpsthakur]

Adds Int.isCoprime_add_sub: if m, n are coprime and m + n is odd (equivalently, m, n have opposite parity), then m + n and m - n are coprime.

Motivated by #37366.

Validation

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

Closes #37366

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]

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[github-actions]

PR summary d06498125a

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ isCoprime_add_sub

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 d064981).

• +1 new declarations
• −0 removed declarations

+Int.isCoprime_add_sub

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit d06498125a
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).