```python
def shelf_fit(widths: list, shelf: int) -> int:
    if not isinstance(widths, list) or not all(isinstance(w, int) and w > 0 for w in widths):
        raise ValueError("Widths must be a list of positive integers.")
    if not isinstance(shelf, int) or shelf < 0:
        raise ValueError("Shelf width must be a non-negative integer.")

    books_on_shelf = 0
    current_width = 0

    for width in widths:
        if current_width + width <= shelf:
            books_on_shelf += 1
            current_width += width
        else:
            break

    return books_on_shelf
```