`mut_mutex_lock` in closures doesn't consider `Fn` bound

Summary

When a &mut Mutex<T> is given and used in a closure, the lint fails to consider the type constraints on the given closure, and wrongfully suggests get_mut which would require a FnMut bound instead of Fn, creating code that fails to compile.

Lint Name

mut_mutex_lock

Reproducer

I tried this code:

use std::sync::Mutex;

fn _closure(_: impl Fn()) {}
fn _func(mutex: &mut Mutex<()>) {
    _closure(|| {
        let _guard = mutex.lock();
    })
}

I saw this happen:

warning: calling `&mut Mutex::lock` unnecessarily locks an exclusive (mutable) reference
 --> src/lib.rs:6:28
  |
6 |         let _guard = mutex.lock();
  |                            ^^^^ help: change this to: `get_mut`
  |
  = help: for further information visit https://rust-lang.github.io/rust-clippy/master/index.html#mut_mutex_lock
  = note: `#[warn(clippy::mut_mutex_lock)]` on by default

I expected to see this happen:
No lint

Version

rustc 1.81.0 (eeb90cda1 2024-09-04)
binary: rustc
commit-hash: eeb90cda1969383f56a2637cbd3037bdf598841c
commit-date: 2024-09-04
host: x86_64-unknown-linux-gnu
release: 1.81.0
LLVM version: 18.1.7

Additional Labels

No response

Start by locating the implementation and tests for the mut_mutex_lock lint, then run the Rust reproducer from the issue. Check how closure trait bounds are considered before suggesting get_mut. Done means an Fn closure using &mut Mutex is not linted when the suggestion would fail to compile.