Ok, I'm stupid.
Here's a function that checks if a matrix respects the Lunar Condition.
bool lunar_check() {
    for(int i = 0; i != N-1;++i) {
        int count = 0;
        for(int j = 0; j != M;++j) {
            count += (matrix[j][i]==0&&matrix[j][i+1]==1);
        }
        if(count>K) return false;
    }
    return true;
}
I hope that nobody else's have problems reading the statement. ;-;