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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def cross(self,p_i,p_j, max_depth = 2):
 """subtree-like swap crossover between programs p_i and p_j."""

        # only choose crossover points for out_types available in both programs
        # pdb.set_trace()
        # determine possible outttypes
        types_p_i = [t for t in [p.out_type for p in p_i]]
        types_p_j = [t for t in [p.out_type for p in p_j]]
        types = set(types_p_i).intersection(types_p_j)

        # grab subtree of p_i
        p_i_sub = [i for i,n in enumerate(p_i) if n.out_type in types]
        x_i_end = self.random_state.choice(p_i_sub)
        x_i_begin = x_i_end
        arity_sum = p_i[x_i_end].arity[p_i[x_i_end].in_type]
        # print("x_i_end:",x_i_end)
        # i = 0
        while (arity_sum > 0): #and i < 1000:
            if x_i_begin == 0:
                print("arity_sum:",arity_sum,"x_i_begin:",x_i_begin,"x_i_end:",x_i_end)
            x_i_begin -= 1
            arity_sum += p_i[x_i_begin].arity[p_i[x_i_begin].in_type]-1
        #     i += 1
        # if i == 1000:
        #     print("in variation")
        #     pdb.set_trace()

        # grab subtree of p_j with matching out_type to p_i[x_i_end]
        p_j_sub = [i for i,n in enumerate(p_j) if n.out_type == p_i[x_i_end].out_type]
        x_j_end = self.random_state.choice(p_j_sub)
        x_j_begin = x_j_end
        arity_sum = p_j[x_j_end].arity[p_j[x_j_end].in_type]
        # i = 0
        while (arity_sum > 0): #and i < 1000:
            if x_j_begin == 0:
                print("arity_sum:",arity_sum,"x_j_begin:",x_j_begin,"x_j_end:",x_j_end)
                print("p_j:",p_j)
            x_j_begin -= 1
            arity_sum += p_j[x_j_begin].arity[p_j[x_j_begin].in_type]-1
        #     i += 1
        # if i == 1000:
        #     print("in variation")
        #     pdb.set_trace()
        #swap subtrees
        tmpi = p_i[:]
        tmpj = p_j[:]
        tmpi[x_i_begin:x_i_end+1:],tmpj[x_j_begin:x_j_end+1:] = \
                        tmpj[x_j_begin:x_j_end+1:],tmpi[x_i_begin:x_i_end+1:]

        if not self.is_valid_program(p_i) or not self.is_valid_program(p_j):
            # pdb.set_trace()
            print("parent 1:",p_i,"x_i_begin:",x_i_begin,"x_i_end:",x_i_end)
            print("parent 2:",p_j,"x_j_begin:",x_j_begin,"x_j_end:",x_j_end)
            print("child 1:",tmpi)
            print("child 2:",tmpj)
            raise ValueError('Crossover produced an invalid program.')

        # size check, then assignment
        if len(tmpi) <= 2**max_depth-1:
            p_i[:] = tmpi
        if len(tmpj) <= 2**max_depth-1:
            p_j[:] = tmpj