Handler for `deriving Inhabited` on private structure introduces public auxiliary definition

[wupr]

Prerequisites

• Check that your issue is not already filed:
  https://github.com/leanprover/lean4/issues
• Reduce the issue to a minimal, self-contained, reproducible test case.
  Avoid dependencies to Mathlib or Batteries.
• Test your test case against the latest nightly release, for example on
  https://live.lean-lang.org/#project=lean-nightly
  (You can also use the settings there to switch to “Lean nightly”)

Description

In a module, deriving Inhabited for a private structure produces a private instance, but the auxiliary definition holding the default value is not private. This is a regression in v4.31.0, likely introduced by the new handler for deriving Inhabited in #9815.

This bug causes problems downstream. For example, the linter in Batteries would complain that the (public) auxiliary definition doesn't have a doc string.

Context

I raised this on Zulip: #general > Missing doc string for `deriving Inhabited`.

Steps to Reproduce

Build the following file:

module

private structure A where
deriving Inhabited

/--
info: @[instance_reducible] private def instInhabitedA : Inhabited A :=
{ default := instInhabitedA.default }
-/
#guard_msgs in
#print instInhabitedA

/--
info: private def instInhabitedA.default : A :=
{ }
-/
#guard_msgs in
#print instInhabitedA.default

Expected behavior: Build succeeds with no messages.

Actual behavior: Build fails with error message:

❌️ Docstring on `#guard_msgs` does not match generated message:

- info: private def instInhabitedA.default : A :=
+ info: def instInhabitedA.default : A :=
  { }

Versions

Lean 4.33.0-nightly-2026-07-09
Target: x86_64-unknown-linux-gnu

Additional Information

None.

Impact

Add 👍 to issues you consider important. If others are impacted by this issue, please ask them to add 👍 to it.

[wupr]

I attempted a fix on my fork: https://github.com/wupr/lean4/tree/deriving-inhabited-private-instance

It seems to work, but I'm not sure if it's the best fix.