```python
def carve_shift(start: int, end: int, span: int, rest: int, least: int) -> list:
    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("shift 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

    while current < end:
        if end - current < span:
            span = end - current

        stretch_end = min(current + span, end)
        stretches.append((current, stretch_end))

        current += rest

    return stretches
```