distinct_odd_prime_quotient_sum_equals_rectangle_total

Exact expanded PA statement

forall p q h k tb tc qb qc ub uc cb cc Q T. p = 2 * h + 1 -> q = 2 * k + 1 -> ((~(p = 1) /\ forall frp_prime_left_outer_sum_bridge_prime_p frp_prime_right_outer_sum_bridge_prime_p. p = frp_prime_left_outer_sum_bridge_prime_p * frp_prime_right_outer_sum_bridge_prime_p -> frp_prime_left_outer_sum_bridge_prime_p = 1 \/ frp_prime_right_outer_sum_bridge_prime_p = 1)) -> ((~(q = 1) /\ forall frp_prime_left_outer_sum_bridge_prime_q frp_prime_right_outer_sum_bridge_prime_q. q = frp_prime_left_outer_sum_bridge_prime_q * frp_prime_right_outer_sum_bridge_prime_q -> frp_prime_left_outer_sum_bridge_prime_q = 1 \/ frp_prime_right_outer_sum_bridge_prime_q = 1)) -> ~(p = q) -> (forall esd_index_outer_sum_bridge_scaled esd_value_outer_sum_bridge_scaled. (exists esd_gap_outer_sum_bridge_scaled. esd_gap_outer_sum_bridge_scaled + S esd_index_outer_sum_bridge_scaled = h) -> (((exists ff_h_esd_outer_sum_bridge_scaled_decoded. ff_h_esd_outer_sum_bridge_scaled_decoded + S (esd_value_outer_sum_bridge_scaled) = S ((S (esd_index_outer_sum_bridge_scaled)) * tc)) /\ exists ff_q_esd_outer_sum_bridge_scaled_decoded. tb = ff_q_esd_outer_sum_bridge_scaled_decoded * S ((S (esd_index_outer_sum_bridge_scaled)) * tc) + (esd_value_outer_sum_bridge_scaled))) -> esd_value_outer_sum_bridge_scaled = q * (1 + esd_index_outer_sum_bridge_scaled)) -> (forall fdp_index_outer_sum_bridge_division. (exists gsp_lt_gap_outer_sum_bridge_division_index_bound. gsp_lt_gap_outer_sum_bridge_division_index_bound + S fdp_index_outer_sum_bridge_division = h) -> exists fdp_value_outer_sum_bridge_division fdp_quotient_outer_sum_bridge_division fdp_remainder_outer_sum_bridge_division. (((exists ff_h_fdp_outer_sum_bridge_division_source. ff_h_fdp_outer_sum_bridge_division_source + S (fdp_value_outer_sum_bridge_division) = S ((S (fdp_index_outer_sum_bridge_division)) * tc)) /\ exists ff_q_fdp_outer_sum_bridge_division_source. tb = ff_q_fdp_outer_sum_bridge_division_source * S ((S (fdp_index_outer_sum_bridge_division)) * tc) + (fdp_value_outer_sum_bridge_division))) /\ ((((exists ff_h_fdp_outer_sum_bridge_division_quotient_entry. ff_h_fdp_outer_sum_bridge_division_quotient_entry + S (fdp_quotient_outer_sum_bridge_division) = S ((S (fdp_index_outer_sum_bridge_division)) * qc)) /\ exists ff_q_fdp_outer_sum_bridge_division_quotient_entry. qb = ff_q_fdp_outer_sum_bridge_division_quotient_entry * S ((S (fdp_index_outer_sum_bridge_division)) * qc) + (fdp_quotient_outer_sum_bridge_division))) /\ ((((exists ff_h_fdp_outer_sum_bridge_division_remainder_entry. ff_h_fdp_outer_sum_bridge_division_remainder_entry + S (fdp_remainder_outer_sum_bridge_division) = S ((S (fdp_index_outer_sum_bridge_division)) * uc)) /\ exists ff_q_fdp_outer_sum_bridge_division_remainder_entry. ub = ff_q_fdp_outer_sum_bridge_division_remainder_entry * S ((S (fdp_index_outer_sum_bridge_division)) * uc) + (fdp_remainder_outer_sum_bridge_division))) /\ (fdp_value_outer_sum_bridge_division = p * fdp_quotient_outer_sum_bridge_division + fdp_remainder_outer_sum_bridge_division /\ (exists gsp_lt_gap_outer_sum_bridge_division_remainder_bound. gsp_lt_gap_outer_sum_bridge_division_remainder_bound + S fdp_remainder_outer_sum_bridge_division = p))))) -> (forall erc_row_outer_sum_bridge_rectangle. (exists erc_lt_gap_outer_sum_bridge_rectangle_bound. erc_lt_gap_outer_sum_bridge_rectangle_bound + S (erc_row_outer_sum_bridge_rectangle) = h) -> exists erc_count_outer_sum_bridge_rectangle. ((((exists ff_h_erc_outer_sum_bridge_rectangle_decoded. ff_h_erc_outer_sum_bridge_rectangle_decoded + S (erc_count_outer_sum_bridge_rectangle) = S ((S (erc_row_outer_sum_bridge_rectangle)) * cc)) /\ exists ff_q_erc_outer_sum_bridge_rectangle_decoded. cb = ff_q_erc_outer_sum_bridge_rectangle_decoded * S ((S (erc_row_outer_sum_bridge_rectangle)) * cc) + (erc_count_outer_sum_bridge_rectangle))) /\ (exists erc_row_code_outer_sum_bridge_rectangle_witness erc_row_scale_outer_sum_bridge_rectangle_witness. ((forall eri_column_erc_outer_sum_bridge_rectangle_witness_row. (exists eri_gap_erc_outer_sum_bridge_rectangle_witness_row_bound. eri_gap_erc_outer_sum_bridge_rectangle_witness_row_bound + S (eri_column_erc_outer_sum_bridge_rectangle_witness_row) = k) -> exists eri_bit_erc_outer_sum_bridge_rectangle_witness_row. ((((exists ff_h_eri_erc_outer_sum_bridge_rectangle_witness_row_decoded. ff_h_eri_erc_outer_sum_bridge_rectangle_witness_row_decoded + S (eri_bit_erc_outer_sum_bridge_rectangle_witness_row) = S ((S (eri_column_erc_outer_sum_bridge_rectangle_witness_row)) * erc_row_scale_outer_sum_bridge_rectangle_witness)) /\ exists ff_q_eri_erc_outer_sum_bridge_rectangle_witness_row_decoded. erc_row_code_outer_sum_bridge_rectangle_witness = ff_q_eri_erc_outer_sum_bridge_rectangle_witness_row_decoded * S ((S (eri_column_erc_outer_sum_bridge_rectangle_witness_row)) * erc_row_scale_outer_sum_bridge_rectangle_witness) + (eri_bit_erc_outer_sum_bridge_rectangle_witness_row))) /\ (((eri_bit_erc_outer_sum_bridge_rectangle_witness_row = 0 /\ ((exists eri_gap_erc_outer_sum_bridge_rectangle_witness_row_choice_left. eri_gap_erc_outer_sum_bridge_rectangle_witness_row_choice_left + S (q * S erc_row_outer_sum_bridge_rectangle) = p * S eri_column_erc_outer_sum_bridge_rectangle_witness_row) /\ ~(exists eri_gap_erc_outer_sum_bridge_rectangle_witness_row_choice_right. eri_gap_erc_outer_sum_bridge_rectangle_witness_row_choice_right + S (p * S eri_column_erc_outer_sum_bridge_rectangle_witness_row) = q * S erc_row_outer_sum_bridge_rectangle))) \/ (eri_bit_erc_outer_sum_bridge_rectangle_witness_row = 1 /\ ((exists eri_gap_erc_outer_sum_bridge_rectangle_witness_row_choice_right. eri_gap_erc_outer_sum_bridge_rectangle_witness_row_choice_right + S (p * S eri_column_erc_outer_sum_bridge_rectangle_witness_row) = q * S erc_row_outer_sum_bridge_rectangle) /\ ~(exists eri_gap_erc_outer_sum_bridge_rectangle_witness_row_choice_left. eri_gap_erc_outer_sum_bridge_rectangle_witness_row_choice_left + S (q * S erc_row_outer_sum_bridge_rectangle) = p * S eri_column_erc_outer_sum_bridge_rectangle_witness_row))))))) /\ (((exists ff_u_erc_outer_sum_bridge_rectangle_witness_count_sum ff_v_erc_outer_sum_bridge_rectangle_witness_count_sum. ((((exists ff_h_erc_outer_sum_bridge_rectangle_witness_count_sum_start. ff_h_erc_outer_sum_bridge_rectangle_witness_count_sum_start + S (0) = S ((S (0)) * ff_v_erc_outer_sum_bridge_rectangle_witness_count_sum)) /\ exists ff_q_erc_outer_sum_bridge_rectangle_witness_count_sum_start. ff_u_erc_outer_sum_bridge_rectangle_witness_count_sum = ff_q_erc_outer_sum_bridge_rectangle_witness_count_sum_start * S ((S (0)) * ff_v_erc_outer_sum_bridge_rectangle_witness_count_sum) + (0))) /\ ((((exists ff_h_erc_outer_sum_bridge_rectangle_witness_count_sum_terminal. ff_h_erc_outer_sum_bridge_rectangle_witness_count_sum_terminal + S (erc_count_outer_sum_bridge_rectangle) = S ((S (k)) * ff_v_erc_outer_sum_bridge_rectangle_witness_count_sum)) /\ exists ff_q_erc_outer_sum_bridge_rectangle_witness_count_sum_terminal. ff_u_erc_outer_sum_bridge_rectangle_witness_count_sum = ff_q_erc_outer_sum_bridge_rectangle_witness_count_sum_terminal * S ((S (k)) * ff_v_erc_outer_sum_bridge_rectangle_witness_count_sum) + (erc_count_outer_sum_bridge_rectangle))) /\ forall ff_i_erc_outer_sum_bridge_rectangle_witness_count_sum. (exists ff_lt_erc_outer_sum_bridge_rectangle_witness_count_sum_bound. ff_lt_erc_outer_sum_bridge_rectangle_witness_count_sum_bound + S ff_i_erc_outer_sum_bridge_rectangle_witness_count_sum = k) -> exists ff_a_erc_outer_sum_bridge_rectangle_witness_count_sum ff_r_erc_outer_sum_bridge_rectangle_witness_count_sum ff_s_erc_outer_sum_bridge_rectangle_witness_count_sum. ((((exists ff_h_erc_outer_sum_bridge_rectangle_witness_count_sum_summand. ff_h_erc_outer_sum_bridge_rectangle_witness_count_sum_summand + S (ff_a_erc_outer_sum_bridge_rectangle_witness_count_sum) = S ((S (ff_i_erc_outer_sum_bridge_rectangle_witness_count_sum)) * erc_row_scale_outer_sum_bridge_rectangle_witness)) /\ exists ff_q_erc_outer_sum_bridge_rectangle_witness_count_sum_summand. erc_row_code_outer_sum_bridge_rectangle_witness = ff_q_erc_outer_sum_bridge_rectangle_witness_count_sum_summand * S ((S (ff_i_erc_outer_sum_bridge_rectangle_witness_count_sum)) * erc_row_scale_outer_sum_bridge_rectangle_witness) + (ff_a_erc_outer_sum_bridge_rectangle_witness_count_sum))) /\ ((((exists ff_h_erc_outer_sum_bridge_rectangle_witness_count_sum_partial. ff_h_erc_outer_sum_bridge_rectangle_witness_count_sum_partial + S (ff_r_erc_outer_sum_bridge_rectangle_witness_count_sum) = S ((S (ff_i_erc_outer_sum_bridge_rectangle_witness_count_sum)) * ff_v_erc_outer_sum_bridge_rectangle_witness_count_sum)) /\ exists ff_q_erc_outer_sum_bridge_rectangle_witness_count_sum_partial. ff_u_erc_outer_sum_bridge_rectangle_witness_count_sum = ff_q_erc_outer_sum_bridge_rectangle_witness_count_sum_partial * S ((S (ff_i_erc_outer_sum_bridge_rectangle_witness_count_sum)) * ff_v_erc_outer_sum_bridge_rectangle_witness_count_sum) + (ff_r_erc_outer_sum_bridge_rectangle_witness_count_sum))) /\ ((((exists ff_h_erc_outer_sum_bridge_rectangle_witness_count_sum_successor. ff_h_erc_outer_sum_bridge_rectangle_witness_count_sum_successor + S (ff_s_erc_outer_sum_bridge_rectangle_witness_count_sum) = S ((S (S ff_i_erc_outer_sum_bridge_rectangle_witness_count_sum)) * ff_v_erc_outer_sum_bridge_rectangle_witness_count_sum)) /\ exists ff_q_erc_outer_sum_bridge_rectangle_witness_count_sum_successor. ff_u_erc_outer_sum_bridge_rectangle_witness_count_sum = ff_q_erc_outer_sum_bridge_rectangle_witness_count_sum_successor * S ((S (S ff_i_erc_outer_sum_bridge_rectangle_witness_count_sum)) * ff_v_erc_outer_sum_bridge_rectangle_witness_count_sum) + (ff_s_erc_outer_sum_bridge_rectangle_witness_count_sum))) /\ ff_s_erc_outer_sum_bridge_rectangle_witness_count_sum = ff_r_erc_outer_sum_bridge_rectangle_witness_count_sum + ff_a_erc_outer_sum_bridge_rectangle_witness_count_sum)))))) /\ (forall ff_i_erc_outer_sum_bridge_rectangle_witness_count_bits. (exists ff_lt_erc_outer_sum_bridge_rectangle_witness_count_bits_bound. ff_lt_erc_outer_sum_bridge_rectangle_witness_count_bits_bound + S ff_i_erc_outer_sum_bridge_rectangle_witness_count_bits = k) -> exists ff_bit_erc_outer_sum_bridge_rectangle_witness_count_bits. ((((exists ff_h_erc_outer_sum_bridge_rectangle_witness_count_bits_decoded. ff_h_erc_outer_sum_bridge_rectangle_witness_count_bits_decoded + S (ff_bit_erc_outer_sum_bridge_rectangle_witness_count_bits) = S ((S (ff_i_erc_outer_sum_bridge_rectangle_witness_count_bits)) * erc_row_scale_outer_sum_bridge_rectangle_witness)) /\ exists ff_q_erc_outer_sum_bridge_rectangle_witness_count_bits_decoded. erc_row_code_outer_sum_bridge_rectangle_witness = ff_q_erc_outer_sum_bridge_rectangle_witness_count_bits_decoded * S ((S (ff_i_erc_outer_sum_bridge_rectangle_witness_count_bits)) * erc_row_scale_outer_sum_bridge_rectangle_witness) + (ff_bit_erc_outer_sum_bridge_rectangle_witness_count_bits))) /\ (ff_bit_erc_outer_sum_bridge_rectangle_witness_count_bits = 0 \/ ff_bit_erc_outer_sum_bridge_rectangle_witness_count_bits = 1))))))))) -> (exists ff_u_outer_sum_bridge_quotient_sum ff_v_outer_sum_bridge_quotient_sum. ((((exists ff_h_outer_sum_bridge_quotient_sum_start. ff_h_outer_sum_bridge_quotient_sum_start + S (0) = S ((S (0)) * ff_v_outer_sum_bridge_quotient_sum)) /\ exists ff_q_outer_sum_bridge_quotient_sum_start. ff_u_outer_sum_bridge_quotient_sum = ff_q_outer_sum_bridge_quotient_sum_start * S ((S (0)) * ff_v_outer_sum_bridge_quotient_sum) + (0))) /\ ((((exists ff_h_outer_sum_bridge_quotient_sum_terminal. ff_h_outer_sum_bridge_quotient_sum_terminal + S (Q) = S ((S (h)) * ff_v_outer_sum_bridge_quotient_sum)) /\ exists ff_q_outer_sum_bridge_quotient_sum_terminal. ff_u_outer_sum_bridge_quotient_sum = ff_q_outer_sum_bridge_quotient_sum_terminal * S ((S (h)) * ff_v_outer_sum_bridge_quotient_sum) + (Q))) /\ forall ff_i_outer_sum_bridge_quotient_sum. (exists ff_lt_outer_sum_bridge_quotient_sum_bound. ff_lt_outer_sum_bridge_quotient_sum_bound + S ff_i_outer_sum_bridge_quotient_sum = h) -> exists ff_a_outer_sum_bridge_quotient_sum ff_r_outer_sum_bridge_quotient_sum ff_s_outer_sum_bridge_quotient_sum. ((((exists ff_h_outer_sum_bridge_quotient_sum_summand. ff_h_outer_sum_bridge_quotient_sum_summand + S (ff_a_outer_sum_bridge_quotient_sum) = S ((S (ff_i_outer_sum_bridge_quotient_sum)) * qc)) /\ exists ff_q_outer_sum_bridge_quotient_sum_summand. qb = ff_q_outer_sum_bridge_quotient_sum_summand * S ((S (ff_i_outer_sum_bridge_quotient_sum)) * qc) + (ff_a_outer_sum_bridge_quotient_sum))) /\ ((((exists ff_h_outer_sum_bridge_quotient_sum_partial. ff_h_outer_sum_bridge_quotient_sum_partial + S (ff_r_outer_sum_bridge_quotient_sum) = S ((S (ff_i_outer_sum_bridge_quotient_sum)) * ff_v_outer_sum_bridge_quotient_sum)) /\ exists ff_q_outer_sum_bridge_quotient_sum_partial. ff_u_outer_sum_bridge_quotient_sum = ff_q_outer_sum_bridge_quotient_sum_partial * S ((S (ff_i_outer_sum_bridge_quotient_sum)) * ff_v_outer_sum_bridge_quotient_sum) + (ff_r_outer_sum_bridge_quotient_sum))) /\ ((((exists ff_h_outer_sum_bridge_quotient_sum_successor. ff_h_outer_sum_bridge_quotient_sum_successor + S (ff_s_outer_sum_bridge_quotient_sum) = S ((S (S ff_i_outer_sum_bridge_quotient_sum)) * ff_v_outer_sum_bridge_quotient_sum)) /\ exists ff_q_outer_sum_bridge_quotient_sum_successor. ff_u_outer_sum_bridge_quotient_sum = ff_q_outer_sum_bridge_quotient_sum_successor * S ((S (S ff_i_outer_sum_bridge_quotient_sum)) * ff_v_outer_sum_bridge_quotient_sum) + (ff_s_outer_sum_bridge_quotient_sum))) /\ ff_s_outer_sum_bridge_quotient_sum = ff_r_outer_sum_bridge_quotient_sum + ff_a_outer_sum_bridge_quotient_sum)))))) -> (exists ff_u_outer_sum_bridge_rectangle_total ff_v_outer_sum_bridge_rectangle_total. ((((exists ff_h_outer_sum_bridge_rectangle_total_start. ff_h_outer_sum_bridge_rectangle_total_start + S (0) = S ((S (0)) * ff_v_outer_sum_bridge_rectangle_total)) /\ exists ff_q_outer_sum_bridge_rectangle_total_start. ff_u_outer_sum_bridge_rectangle_total = ff_q_outer_sum_bridge_rectangle_total_start * S ((S (0)) * ff_v_outer_sum_bridge_rectangle_total) + (0))) /\ ((((exists ff_h_outer_sum_bridge_rectangle_total_terminal. ff_h_outer_sum_bridge_rectangle_total_terminal + S (T) = S ((S (h)) * ff_v_outer_sum_bridge_rectangle_total)) /\ exists ff_q_outer_sum_bridge_rectangle_total_terminal. ff_u_outer_sum_bridge_rectangle_total = ff_q_outer_sum_bridge_rectangle_total_terminal * S ((S (h)) * ff_v_outer_sum_bridge_rectangle_total) + (T))) /\ forall ff_i_outer_sum_bridge_rectangle_total. (exists ff_lt_outer_sum_bridge_rectangle_total_bound. ff_lt_outer_sum_bridge_rectangle_total_bound + S ff_i_outer_sum_bridge_rectangle_total = h) -> exists ff_a_outer_sum_bridge_rectangle_total ff_r_outer_sum_bridge_rectangle_total ff_s_outer_sum_bridge_rectangle_total. ((((exists ff_h_outer_sum_bridge_rectangle_total_summand. ff_h_outer_sum_bridge_rectangle_total_summand + S (ff_a_outer_sum_bridge_rectangle_total) = S ((S (ff_i_outer_sum_bridge_rectangle_total)) * cc)) /\ exists ff_q_outer_sum_bridge_rectangle_total_summand. cb = ff_q_outer_sum_bridge_rectangle_total_summand * S ((S (ff_i_outer_sum_bridge_rectangle_total)) * cc) + (ff_a_outer_sum_bridge_rectangle_total))) /\ ((((exists ff_h_outer_sum_bridge_rectangle_total_partial. ff_h_outer_sum_bridge_rectangle_total_partial + S (ff_r_outer_sum_bridge_rectangle_total) = S ((S (ff_i_outer_sum_bridge_rectangle_total)) * ff_v_outer_sum_bridge_rectangle_total)) /\ exists ff_q_outer_sum_bridge_rectangle_total_partial. ff_u_outer_sum_bridge_rectangle_total = ff_q_outer_sum_bridge_rectangle_total_partial * S ((S (ff_i_outer_sum_bridge_rectangle_total)) * ff_v_outer_sum_bridge_rectangle_total) + (ff_r_outer_sum_bridge_rectangle_total))) /\ ((((exists ff_h_outer_sum_bridge_rectangle_total_successor. ff_h_outer_sum_bridge_rectangle_total_successor + S (ff_s_outer_sum_bridge_rectangle_total) = S ((S (S ff_i_outer_sum_bridge_rectangle_total)) * ff_v_outer_sum_bridge_rectangle_total)) /\ exists ff_q_outer_sum_bridge_rectangle_total_successor. ff_u_outer_sum_bridge_rectangle_total = ff_q_outer_sum_bridge_rectangle_total_successor * S ((S (S ff_i_outer_sum_bridge_rectangle_total)) * ff_v_outer_sum_bridge_rectangle_total) + (ff_s_outer_sum_bridge_rectangle_total))) /\ ff_s_outer_sum_bridge_rectangle_total = ff_r_outer_sum_bridge_rectangle_total + ff_a_outer_sum_bridge_rectangle_total)))))) -> Q = T