Looking at ** rt_mutex_waiter_less () ** per,

kernel/kernel/locking/rtmutex.c

static inline int
rt_mutex_waiter_less(struct rt_mutex_waiter *left,
		     struct rt_mutex_waiter *right)
{
	if (left->prio < right->prio)
		return 1;

	/*
	 * If both waiters have dl_prio(), we check the deadlines of the
	 * associated tasks.
	 * If left waiter has a dl_prio(), and we didn't return 1 above,
	 * then right waiter has a dl_prio() too.
	 */
	if (dl_prio(left->prio))
		return dl_time_before(left->deadline, right->deadline);

	return 0;
}

Per ** rt_mutex_waiter_equal () ** is

kernel/kernel/locking/rtmutex.c

static inline int
rt_mutex_waiter_equal(struct rt_mutex_waiter *left,
		      struct rt_mutex_waiter *right)
{
	if (left->prio != right->prio)
		return 0;

	/*
	 * If both waiters have dl_prio(), we check the deadlines of the
	 * associated tasks.
	 * If left waiter has a dl_prio(), and we didn't return 0 above,
	 * then right waiter has a dl_prio() too.
	 */
	if (dl_prio(left->prio))
		return left->deadline == right->deadline;

	return 1;
}

You can see how the real-time tasks are sorted by priority. I will omit the details, but I have implemented these comparison functions so that tasks are sorted in order of priority in rbtree. Well, I just got an explanation article about rbtree, so let me introduce it. -Essence of red-black tree --Qiita

The ** rt_mutex_adjust_prio_chain () ** that appears in the documentation is long. If you know that

kernel/kernel/locking/rtmutex.c

/*
 * Adjust the priority chain. Also used for deadlock detection.
 * Decreases task's usage by one - may thus free the task.
 *
 * @task:	the task owning the mutex (owner) for which a chain walk is
 *		probably needed
 * @chwalk:	do we have to carry out deadlock detection?
 * @orig_lock:	the mutex (can be NULL if we are walking the chain to recheck
 *		things for a task that has just got its priority adjusted, and
 *		is waiting on a mutex)
 * @next_lock:	the mutex on which the owner of @orig_lock was blocked before
 *		we dropped its pi_lock. Is never dereferenced, only used for
 *		comparison to detect lock chain changes.
 * @orig_waiter: rt_mutex_waiter struct for the task that has just donated
 *		its priority to the mutex owner (can be NULL in the case
 *		depicted above or if the top waiter is gone away and we are
 *		actually deboosting the owner)
 * @top_task:	the current top waiter
 *
 * Returns 0 or -EDEADLK.
 *
 * Chain walk basics and protection scope
 *
 * [R] refcount on task
 * [P] task->pi_lock held
 * [L] rtmutex->wait_lock held
 *
 * Step	Description				Protected by
 *	function arguments:
 *	@task					[R]
 *	@orig_lock if != NULL			@top_task is blocked on it
 *	@next_lock				Unprotected. Cannot be
 *						dereferenced. Only used for
 *						comparison.
 *	@orig_waiter if != NULL			@top_task is blocked on it
 *	@top_task				current, or in case of proxy
 *						locking protected by calling
 *						code
 *	again:
 *	  loop_sanity_check();
 *	retry:
 * [1]	  lock(task->pi_lock);			[R] acquire [P]
 * [2]	  waiter = task->pi_blocked_on;		[P]
 * [3]	  check_exit_conditions_1();		[P]
 * [4]	  lock = waiter->lock;			[P]
 * [5]	  if (!try_lock(lock->wait_lock)) {	[P] try to acquire [L]
 *	    unlock(task->pi_lock);		release [P]
 *	    goto retry;
 *	  }
 * [6]	  check_exit_conditions_2();		[P] + [L]
 * [7]	  requeue_lock_waiter(lock, waiter);	[P] + [L]
 * [8]	  unlock(task->pi_lock);		release [P]
 *	  put_task_struct(task);		release [R]
 * [9]	  check_exit_conditions_3();		[L]
 * [10]	  task = owner(lock);			[L]
 *	  get_task_struct(task);		[L] acquire [R]
 *	  lock(task->pi_lock);			[L] acquire [P]
 * [11]	  requeue_pi_waiter(tsk, waiters(lock));[P] + [L]
 * [12]	  check_exit_conditions_4();		[P] + [L]
 * [13]	  unlock(task->pi_lock);		release [P]
 *	  unlock(lock->wait_lock);		release [L]
 *	  goto again;
 */
static int rt_mutex_adjust_prio_chain(struct task_struct *task,
				      enum rtmutex_chainwalk chwalk,
				      struct rt_mutex *orig_lock,
				      struct rt_mutex *next_lock,
				      struct rt_mutex_waiter *orig_waiter,
				      struct task_struct *top_task)
{
	struct rt_mutex_waiter *waiter, *top_waiter = orig_waiter;
	struct rt_mutex_waiter *prerequeue_top_waiter;
	int ret = 0, depth = 0;
	struct rt_mutex *lock;
	bool detect_deadlock;
	bool requeue = true;

	detect_deadlock = rt_mutex_cond_detect_deadlock(orig_waiter, chwalk);

	/*
	 * The (de)boosting is a step by step approach with a lot of
	 * pitfalls. We want this to be preemptible and we want hold a
	 * maximum of two locks per step. So we have to check
	 * carefully whether things change under us.
	 */
 again:
	/*
	 * We limit the lock chain length for each invocation.
	 */
	if (++depth > max_lock_depth) {
		static int prev_max;

		/*
		 * Print this only once. If the admin changes the limit,
		 * print a new message when reaching the limit again.
		 */
		if (prev_max != max_lock_depth) {
			prev_max = max_lock_depth;
			printk(KERN_WARNING "Maximum lock depth %d reached "
			       "task: %s (%d)\n", max_lock_depth,
			       top_task->comm, task_pid_nr(top_task));
		}
		put_task_struct(task);

		return -EDEADLK;
	}

	/*
	 * We are fully preemptible here and only hold the refcount on
	 * @task. So everything can have changed under us since the
	 * caller or our own code below (goto retry/again) dropped all
	 * locks.
	 */
 retry:
	/*
	 * [1] Task cannot go away as we did a get_task() before !
	 */
	raw_spin_lock_irq(&task->pi_lock);

	/*
	 * [2] Get the waiter on which @task is blocked on.
	 */
	waiter = task->pi_blocked_on;

	/*
	 * [3] check_exit_conditions_1() protected by task->pi_lock.
	 */

	/*
	 * Check whether the end of the boosting chain has been
	 * reached or the state of the chain has changed while we
	 * dropped the locks.
	 */
	if (!waiter)
		goto out_unlock_pi;

	/*
	 * Check the orig_waiter state. After we dropped the locks,
	 * the previous owner of the lock might have released the lock.
	 */
	if (orig_waiter && !rt_mutex_owner(orig_lock))
		goto out_unlock_pi;

	/*
	 * We dropped all locks after taking a refcount on @task, so
	 * the task might have moved on in the lock chain or even left
	 * the chain completely and blocks now on an unrelated lock or
	 * on @orig_lock.
	 *
	 * We stored the lock on which @task was blocked in @next_lock,
	 * so we can detect the chain change.
	 */
	if (next_lock != waiter->lock)
		goto out_unlock_pi;

	/*
	 * Drop out, when the task has no waiters. Note,
	 * top_waiter can be NULL, when we are in the deboosting
	 * mode!
	 */
	if (top_waiter) {
		if (!task_has_pi_waiters(task))
			goto out_unlock_pi;
		/*
		 * If deadlock detection is off, we stop here if we
		 * are not the top pi waiter of the task. If deadlock
		 * detection is enabled we continue, but stop the
		 * requeueing in the chain walk.
		 */
		if (top_waiter != task_top_pi_waiter(task)) {
			if (!detect_deadlock)
				goto out_unlock_pi;
			else
				requeue = false;
		}
	}

	/*
	 * If the waiter priority is the same as the task priority
	 * then there is no further priority adjustment necessary.  If
	 * deadlock detection is off, we stop the chain walk. If its
	 * enabled we continue, but stop the requeueing in the chain
	 * walk.
	 */
	if (rt_mutex_waiter_equal(waiter, task_to_waiter(task))) {
		if (!detect_deadlock)
			goto out_unlock_pi;
		else
			requeue = false;
	}

	/*
	 * [4] Get the next lock
	 */
	lock = waiter->lock;
	/*
	 * [5] We need to trylock here as we are holding task->pi_lock,
	 * which is the reverse lock order versus the other rtmutex
	 * operations.
	 */
	if (!raw_spin_trylock(&lock->wait_lock)) {
		raw_spin_unlock_irq(&task->pi_lock);
		cpu_relax();
		goto retry;
	}

	/*
	 * [6] check_exit_conditions_2() protected by task->pi_lock and
	 * lock->wait_lock.
	 *
	 * Deadlock detection. If the lock is the same as the original
	 * lock which caused us to walk the lock chain or if the
	 * current lock is owned by the task which initiated the chain
	 * walk, we detected a deadlock.
	 */
	if (lock == orig_lock || rt_mutex_owner(lock) == top_task) {
		debug_rt_mutex_deadlock(chwalk, orig_waiter, lock);
		raw_spin_unlock(&lock->wait_lock);
		ret = -EDEADLK;
		goto out_unlock_pi;
	}

	/*
	 * If we just follow the lock chain for deadlock detection, no
	 * need to do all the requeue operations. To avoid a truckload
	 * of conditionals around the various places below, just do the
	 * minimum chain walk checks.
	 */
	if (!requeue) {
		/*
		 * No requeue[7] here. Just release @task [8]
		 */
		raw_spin_unlock(&task->pi_lock);
		put_task_struct(task);

		/*
		 * [9] check_exit_conditions_3 protected by lock->wait_lock.
		 * If there is no owner of the lock, end of chain.
		 */
		if (!rt_mutex_owner(lock)) {
			raw_spin_unlock_irq(&lock->wait_lock);
			return 0;
		}

		/* [10] Grab the next task, i.e. owner of @lock */
		task = get_task_struct(rt_mutex_owner(lock));
		raw_spin_lock(&task->pi_lock);

		/*
		 * No requeue [11] here. We just do deadlock detection.
		 *
		 * [12] Store whether owner is blocked
		 * itself. Decision is made after dropping the locks
		 */
		next_lock = task_blocked_on_lock(task);
		/*
		 * Get the top waiter for the next iteration
		 */
		top_waiter = rt_mutex_top_waiter(lock);

		/* [13] Drop locks */
		raw_spin_unlock(&task->pi_lock);
		raw_spin_unlock_irq(&lock->wait_lock);

		/* If owner is not blocked, end of chain. */
		if (!next_lock)
			goto out_put_task;
		goto again;
	}

	/*
	 * Store the current top waiter before doing the requeue
	 * operation on @lock. We need it for the boost/deboost
	 * decision below.
	 */
	prerequeue_top_waiter = rt_mutex_top_waiter(lock);

	/* [7] Requeue the waiter in the lock waiter tree. */
	rt_mutex_dequeue(lock, waiter);

	/*
	 * Update the waiter prio fields now that we're dequeued.
	 *
	 * These values can have changed through either:
	 *
	 *   sys_sched_set_scheduler() / sys_sched_setattr()
	 *
	 * or
	 *
	 *   DL CBS enforcement advancing the effective deadline.
	 *
	 * Even though pi_waiters also uses these fields, and that tree is only
	 * updated in [11], we can do this here, since we hold [L], which
	 * serializes all pi_waiters access and rb_erase() does not care about
	 * the values of the node being removed.
	 */
	waiter->prio = task->prio;
	waiter->deadline = task->dl.deadline;

	rt_mutex_enqueue(lock, waiter);

	/* [8] Release the task */
	raw_spin_unlock(&task->pi_lock);
	put_task_struct(task);

	/*
	 * [9] check_exit_conditions_3 protected by lock->wait_lock.
	 *
	 * We must abort the chain walk if there is no lock owner even
	 * in the dead lock detection case, as we have nothing to
	 * follow here. This is the end of the chain we are walking.
	 */
	if (!rt_mutex_owner(lock)) {
		/*
		 * If the requeue [7] above changed the top waiter,
		 * then we need to wake the new top waiter up to try
		 * to get the lock.
		 */
		if (prerequeue_top_waiter != rt_mutex_top_waiter(lock))
			wake_up_process(rt_mutex_top_waiter(lock)->task);
		raw_spin_unlock_irq(&lock->wait_lock);
		return 0;
	}

	/* [10] Grab the next task, i.e. the owner of @lock */
	task = get_task_struct(rt_mutex_owner(lock));
	raw_spin_lock(&task->pi_lock);

	/* [11] requeue the pi waiters if necessary */
	if (waiter == rt_mutex_top_waiter(lock)) {
		/*
		 * The waiter became the new top (highest priority)
		 * waiter on the lock. Replace the previous top waiter
		 * in the owner tasks pi waiters tree with this waiter
		 * and adjust the priority of the owner.
		 */
		rt_mutex_dequeue_pi(task, prerequeue_top_waiter);
		rt_mutex_enqueue_pi(task, waiter);
		rt_mutex_adjust_prio(task);

	} else if (prerequeue_top_waiter == waiter) {
		/*
		 * The waiter was the top waiter on the lock, but is
		 * no longer the top prority waiter. Replace waiter in
		 * the owner tasks pi waiters tree with the new top
		 * (highest priority) waiter and adjust the priority
		 * of the owner.
		 * The new top waiter is stored in @waiter so that
		 * @waiter == @top_waiter evaluates to true below and
		 * we continue to deboost the rest of the chain.
		 */
		rt_mutex_dequeue_pi(task, waiter);
		waiter = rt_mutex_top_waiter(lock);
		rt_mutex_enqueue_pi(task, waiter);
		rt_mutex_adjust_prio(task);
	} else {
		/*
		 * Nothing changed. No need to do any priority
		 * adjustment.
		 */
	}

	/*
	 * [12] check_exit_conditions_4() protected by task->pi_lock
	 * and lock->wait_lock. The actual decisions are made after we
	 * dropped the locks.
	 *
	 * Check whether the task which owns the current lock is pi
	 * blocked itself. If yes we store a pointer to the lock for
	 * the lock chain change detection above. After we dropped
	 * task->pi_lock next_lock cannot be dereferenced anymore.
	 */
	next_lock = task_blocked_on_lock(task);
	/*
	 * Store the top waiter of @lock for the end of chain walk
	 * decision below.
	 */
	top_waiter = rt_mutex_top_waiter(lock);

	/* [13] Drop the locks */
	raw_spin_unlock(&task->pi_lock);
	raw_spin_unlock_irq(&lock->wait_lock);

	/*
	 * Make the actual exit decisions [12], based on the stored
	 * values.
	 *
	 * We reached the end of the lock chain. Stop right here. No
	 * point to go back just to figure that out.
	 */
	if (!next_lock)
		goto out_put_task;

	/*
	 * If the current waiter is not the top waiter on the lock,
	 * then we can stop the chain walk here if we are not in full
	 * deadlock detection mode.
	 */
	if (!detect_deadlock && waiter != top_waiter)
		goto out_put_task;

	goto again;

 out_unlock_pi:
	raw_spin_unlock_irq(&task->pi_lock);
 out_put_task:
	put_task_struct(task);

	return ret;
}

It can be seen that priority inheritance is being implemented with difficulty while covering various conditions. Also, ** rt_mutex_setprio () ** has different implementations when CONFIG_RT_MUTEXES is enabled. It's also long, but if you put it all in the same way,

kernel/sched/core.c

#ifdef CONFIG_RT_MUTEXES

static inline int __rt_effective_prio(struct task_struct *pi_task, int prio)
{
	if (pi_task)
		prio = min(prio, pi_task->prio);

	return prio;
}

static inline int rt_effective_prio(struct task_struct *p, int prio)
{
	struct task_struct *pi_task = rt_mutex_get_top_task(p);

	return __rt_effective_prio(pi_task, prio);
}

/*
 * rt_mutex_setprio - set the current priority of a task
 * @p: task to boost
 * @pi_task: donor task
 *
 * This function changes the 'effective' priority of a task. It does
 * not touch ->normal_prio like __setscheduler().
 *
 * Used by the rt_mutex code to implement priority inheritance
 * logic. Call site only calls if the priority of the task changed.
 */
void rt_mutex_setprio(struct task_struct *p, struct task_struct *pi_task)
{
	int prio, oldprio, queued, running, queue_flag =
		DEQUEUE_SAVE | DEQUEUE_MOVE | DEQUEUE_NOCLOCK;
	const struct sched_class *prev_class;
	struct rq_flags rf;
	struct rq *rq;

	/* XXX used to be waiter->prio, not waiter->task->prio */
	prio = __rt_effective_prio(pi_task, p->normal_prio);

	/*
	 * If nothing changed; bail early.
	 */
	if (p->pi_top_task == pi_task && prio == p->prio && !dl_prio(prio))
		return;

	rq = __task_rq_lock(p, &rf);
	update_rq_clock(rq);
	/*
	 * Set under pi_lock && rq->lock, such that the value can be used under
	 * either lock.
	 *
	 * Note that there is loads of tricky to make this pointer cache work
	 * right. rt_mutex_slowunlock()+rt_mutex_postunlock() work together to
	 * ensure a task is de-boosted (pi_task is set to NULL) before the
	 * task is allowed to run again (and can exit). This ensures the pointer
	 * points to a blocked task -- which guaratees the task is present.
	 */
	p->pi_top_task = pi_task;

	/*
	 * For FIFO/RR we only need to set prio, if that matches we're done.
	 */
	if (prio == p->prio && !dl_prio(prio))
		goto out_unlock;

	/*
	 * Idle task boosting is a nono in general. There is one
	 * exception, when PREEMPT_RT and NOHZ is active:
	 *
	 * The idle task calls get_next_timer_interrupt() and holds
	 * the timer wheel base->lock on the CPU and another CPU wants
	 * to access the timer (probably to cancel it). We can safely
	 * ignore the boosting request, as the idle CPU runs this code
	 * with interrupts disabled and will complete the lock
	 * protected section without being interrupted. So there is no
	 * real need to boost.
	 */
	if (unlikely(p == rq->idle)) {
		WARN_ON(p != rq->curr);
		WARN_ON(p->pi_blocked_on);
		goto out_unlock;
	}

	trace_sched_pi_setprio(p, pi_task);
	oldprio = p->prio;

	if (oldprio == prio)
		queue_flag &= ~DEQUEUE_MOVE;

	prev_class = p->sched_class;
	queued = task_on_rq_queued(p);
	running = task_current(rq, p);
	if (queued)
		dequeue_task(rq, p, queue_flag);
	if (running)
		put_prev_task(rq, p);

	/*
	 * Boosting condition are:
	 * 1. -rt task is running and holds mutex A
	 *      --> -dl task blocks on mutex A
	 *
	 * 2. -dl task is running and holds mutex A
	 *      --> -dl task blocks on mutex A and could preempt the
	 *          running task
	 */
	if (dl_prio(prio)) {
		if (!dl_prio(p->normal_prio) ||
		    (pi_task && dl_entity_preempt(&pi_task->dl, &p->dl))) {
			p->dl.dl_boosted = 1;
			queue_flag |= ENQUEUE_REPLENISH;
		} else
			p->dl.dl_boosted = 0;
		p->sched_class = &dl_sched_class;
	} else if (rt_prio(prio)) {
		if (dl_prio(oldprio))
			p->dl.dl_boosted = 0;
		if (oldprio < prio)
			queue_flag |= ENQUEUE_HEAD;
		p->sched_class = &rt_sched_class;
	} else {
		if (dl_prio(oldprio))
			p->dl.dl_boosted = 0;
		if (rt_prio(oldprio))
			p->rt.timeout = 0;
		p->sched_class = &fair_sched_class;
	}

	p->prio = prio;

	if (queued)
		enqueue_task(rq, p, queue_flag);
	if (running)
		set_next_task(rq, p);

	check_class_changed(rq, p, prev_class, oldprio);
out_unlock:
	/* Avoid rq from going away on us: */
	preempt_disable();
	__task_rq_unlock(rq, &rf);

	balance_callback(rq);
	preempt_enable();
}。

You can see how the task priority is actually changed and rescheduled.