```python
def base_amount(amount: int, unit: str, defs: dict, base: str) -> int:
    """Convert amount of unit into the base unit by unwinding definition chain."""
    if isinstance(amount, bool) or not isinstance(amount, int) or amount < 0:
        raise ValueError("amount must be a non-negative integer")
    
    if unit == base:
        return amount
    
    if unit not in defs:
        raise ValueError(f"unit '{unit}' is not defined and is not the base unit")
    
    current_unit = unit
    visited = {current_unit}
    
    while current_unit != base:
        if current_unit not in defs:
            raise ValueError(f"unit '{current_unit}' is not defined and is not the base unit")
        
        entry = defs[current_unit]
        factor = entry[0]
        finer_unit = entry[1]
        
        if not isinstance(factor, int) or isinstance(factor, bool) or factor <= 0:
            raise ValueError("factor must be a positive integer")
        
        if finer_unit in visited:
            raise ValueError("definition chain loops and cannot reach the base unit")
        
        visited.add(finer_unit)
        amount *= factor
        current_unit = finer_unit
    
    return amount
```