from typing import List


def check_specialcase(productions_list: List[str])->bool:
    """
    >>> check_specialcase(['A->aBb','A->$','B->aAb','B->ba'])
    True
    """
    flag=0
    start_symbol= productions_list[0].split('->')[0]
    for p in productions_list:
        if p==start_symbol+'->$':
            flag=1
            break
    if flag==1:
        for p in productions_list:
            lhs=p.split('->')[0]
            rhs=p.split('->')[1]
            if lhs==start_symbol:  # means it's not a further production
                continue
            else:
                for char in rhs:
                    if char==start_symbol:
                        return True
        return False
    else:
        return False