```python
def cross_up(readings: list, level: int) -> int:
    """
    Count how many times a run of readings crosses a level going upward.

    Args:
        readings (list): A list of integers representing the readings.
        level (int): The target level to cross.

    Returns:
        int: The number of crossings.
    """
    if not readings or level < 0:
        raise ValueError("Invalid input")

    count = 0
    for i in range(1, len(readings)):
        if readings[i] > level and readings[i - 1] <= level:
            count += 1

    return count
```