```python
def base_amount(amount: int, unit: str, defs: dict, base: str) -> int:
    """
    Convert amount of unit into the base unit by unwinding one definition at a time.

    :param amount: The amount of the unit to convert.
    :param unit: The name of the unit to convert from.
    :param defs: A dictionary where each key is a unit name and the value is a tuple [factor, finer].
    :param base: The name of the base unit.
    :return: The converted amount in the base unit.
    """
    
    if amount < 0 or not isinstance(amount, int):
        raise ValueError("Amount must be a non-negative whole number.")
    
    if unit not in defs:
        raise ValueError(f"Unit '{unit}' is not defined.")

    def convert_unit(amount, unit):
        factor, finer = defs[unit]
        
        if factor <= 0 or not isinstance(factor, int):
            raise ValueError(f"Factor for unit '{unit}' must be a positive whole number.")
        
        # If the unit is the base unit, return the amount as it is.
        elif unit == base:
            return amount
        
        # Otherwise, convert to the finer unit and then back to the base.
        else:
            finer_amount = convert_unit(amount * factor, finer)
            base_amount = convert_unit(finer_amount // factor, finer)
            return base_amount + (amount % factor) * factor

    return convert_unit(amount, unit)
```