Prop-to-bool coercion not working for `private` field definitions

Prerequisites

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• Reduce the issue to a minimal, self-contained, reproducible test case.
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Description

When assigning the fields of a structure, the private modifier can be used. When the field is Bool-valued and the private term is a decidable Prop, Lean fails to apply the coercion to Bool.

Steps to Reproduce

1. Check out the Lake project in this branch: https://github.com/datokrat/lean-bug-reproductions/tree/bugs1
2. Open Reproductions/PrivateCoercion.lean and observe that Lean reports a type mismatch.

Here's the code:

module

/--
error: Type mismatch
  instBEq._private_1
has type
  Prop
but is expected to have type
  Bool
-/
#guard_msgs in
public instance instBEq {α : Type u} [LE α] [DecidableLE α] :
    BEq α where
  beq _ _ := private True

-- works because `True` is coerced into `decide True`
public instance instBEq' {α : Type u} [LE α] [DecidableLE α] :
    BEq α where
  beq _ _ := True

Expected behavior:

Like when private is omitted, Lean should correctly insert a decide application to make a Bool out of True.

Actual behavior:

Lean fails and complains about the type mismatch.

Versions

Lean 4.23.0-nightly-2025-08-11
Target: arm64-apple-darwin23.6.0 macOS

Impact

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Start by running the minimal reproduction in Reproductions/PrivateCoercion.lean from the linked bugs1 branch and confirm the type mismatch. Trace how private field definitions are elaborated compared with the working public instance. Done means the private Bool-valued field accepts the decidable Prop through the same coercion as the non-private case, with the reproduction passing.