bit_count_functional

Exact expanded PA statement

forall b c l n m. (((exists ff_u_l_sum ff_v_l_sum. ((((exists ff_h_l_sum_start. ff_h_l_sum_start + S (0) = S ((S (0)) * ff_v_l_sum)) /\ exists ff_q_l_sum_start. ff_u_l_sum = ff_q_l_sum_start * S ((S (0)) * ff_v_l_sum) + (0))) /\ ((((exists ff_h_l_sum_terminal. ff_h_l_sum_terminal + S (n) = S ((S (l)) * ff_v_l_sum)) /\ exists ff_q_l_sum_terminal. ff_u_l_sum = ff_q_l_sum_terminal * S ((S (l)) * ff_v_l_sum) + (n))) /\ forall ff_i_l_sum. (exists ff_lt_l_sum_bound. ff_lt_l_sum_bound + S ff_i_l_sum = l) -> exists ff_a_l_sum ff_r_l_sum ff_s_l_sum. ((((exists ff_h_l_sum_summand. ff_h_l_sum_summand + S (ff_a_l_sum) = S ((S (ff_i_l_sum)) * c)) /\ exists ff_q_l_sum_summand. b = ff_q_l_sum_summand * S ((S (ff_i_l_sum)) * c) + (ff_a_l_sum))) /\ ((((exists ff_h_l_sum_partial. ff_h_l_sum_partial + S (ff_r_l_sum) = S ((S (ff_i_l_sum)) * ff_v_l_sum)) /\ exists ff_q_l_sum_partial. ff_u_l_sum = ff_q_l_sum_partial * S ((S (ff_i_l_sum)) * ff_v_l_sum) + (ff_r_l_sum))) /\ ((((exists ff_h_l_sum_successor. ff_h_l_sum_successor + S (ff_s_l_sum) = S ((S (S ff_i_l_sum)) * ff_v_l_sum)) /\ exists ff_q_l_sum_successor. ff_u_l_sum = ff_q_l_sum_successor * S ((S (S ff_i_l_sum)) * ff_v_l_sum) + (ff_s_l_sum))) /\ ff_s_l_sum = ff_r_l_sum + ff_a_l_sum)))))) /\ (forall ff_i_l_bits. (exists ff_lt_l_bits_bound. ff_lt_l_bits_bound + S ff_i_l_bits = l) -> exists ff_bit_l_bits. ((((exists ff_h_l_bits_decoded. ff_h_l_bits_decoded + S (ff_bit_l_bits) = S ((S (ff_i_l_bits)) * c)) /\ exists ff_q_l_bits_decoded. b = ff_q_l_bits_decoded * S ((S (ff_i_l_bits)) * c) + (ff_bit_l_bits))) /\ (ff_bit_l_bits = 0 \/ ff_bit_l_bits = 1))))) -> (((exists ff_u_r_sum ff_v_r_sum. ((((exists ff_h_r_sum_start. ff_h_r_sum_start + S (0) = S ((S (0)) * ff_v_r_sum)) /\ exists ff_q_r_sum_start. ff_u_r_sum = ff_q_r_sum_start * S ((S (0)) * ff_v_r_sum) + (0))) /\ ((((exists ff_h_r_sum_terminal. ff_h_r_sum_terminal + S (m) = S ((S (l)) * ff_v_r_sum)) /\ exists ff_q_r_sum_terminal. ff_u_r_sum = ff_q_r_sum_terminal * S ((S (l)) * ff_v_r_sum) + (m))) /\ forall ff_i_r_sum. (exists ff_lt_r_sum_bound. ff_lt_r_sum_bound + S ff_i_r_sum = l) -> exists ff_a_r_sum ff_r_r_sum ff_s_r_sum. ((((exists ff_h_r_sum_summand. ff_h_r_sum_summand + S (ff_a_r_sum) = S ((S (ff_i_r_sum)) * c)) /\ exists ff_q_r_sum_summand. b = ff_q_r_sum_summand * S ((S (ff_i_r_sum)) * c) + (ff_a_r_sum))) /\ ((((exists ff_h_r_sum_partial. ff_h_r_sum_partial + S (ff_r_r_sum) = S ((S (ff_i_r_sum)) * ff_v_r_sum)) /\ exists ff_q_r_sum_partial. ff_u_r_sum = ff_q_r_sum_partial * S ((S (ff_i_r_sum)) * ff_v_r_sum) + (ff_r_r_sum))) /\ ((((exists ff_h_r_sum_successor. ff_h_r_sum_successor + S (ff_s_r_sum) = S ((S (S ff_i_r_sum)) * ff_v_r_sum)) /\ exists ff_q_r_sum_successor. ff_u_r_sum = ff_q_r_sum_successor * S ((S (S ff_i_r_sum)) * ff_v_r_sum) + (ff_s_r_sum))) /\ ff_s_r_sum = ff_r_r_sum + ff_a_r_sum)))))) /\ (forall ff_i_r_bits. (exists ff_lt_r_bits_bound. ff_lt_r_bits_bound + S ff_i_r_bits = l) -> exists ff_bit_r_bits. ((((exists ff_h_r_bits_decoded. ff_h_r_bits_decoded + S (ff_bit_r_bits) = S ((S (ff_i_r_bits)) * c)) /\ exists ff_q_r_bits_decoded. b = ff_q_r_bits_decoded * S ((S (ff_i_r_bits)) * c) + (ff_bit_r_bits))) /\ (ff_bit_r_bits = 0 \/ ff_bit_r_bits = 1))))) -> n = m