```python
from typing import List


def carve_shift(start: int, end: int, span: int, rest: int, least: int) -> List[List[int]]:
    if isinstance(start, bool) or not isinstance(start, int):
        raise ValueError("start must be an integer")
    if isinstance(end, bool) or not isinstance(end, int):
        raise ValueError("end must be an integer")
    if isinstance(span, bool) or not isinstance(span, int):
        raise ValueError("span must be an integer")
    if isinstance(rest, bool) or not isinstance(rest, int):
        raise ValueError("rest must be an integer")
    if isinstance(least, bool) or not isinstance(least, int):
        raise ValueError("least must be an integer")
    if start >= end:
        raise ValueError("start must precede its end")
    if span < 1 or rest < 1:
        raise ValueError("span and rest must be at least 1")
    if least < 1 or least > span:
        raise ValueError("least must be between 1 and span")

    stretches = []
    current_start = start
    while current_start < end:
        stretch_length = min(span, end - current_start)
        stretches.append([current_start, current_start + stretch_length])
        current_start += stretch_length + rest

    return stretches
```