distinct_odd_prime_half_row_count_prefix_exists_bounded

Exact expanded PA statement

forall p q h k l. p = 2 * h + 1 -> q = 2 * k + 1 -> ((~(p = 1) /\ forall frp_prime_left_rectangle_count_prime_p frp_prime_right_rectangle_count_prime_p. p = frp_prime_left_rectangle_count_prime_p * frp_prime_right_rectangle_count_prime_p -> frp_prime_left_rectangle_count_prime_p = 1 \/ frp_prime_right_rectangle_count_prime_p = 1)) -> ((~(q = 1) /\ forall frp_prime_left_rectangle_count_prime_q frp_prime_right_rectangle_count_prime_q. q = frp_prime_left_rectangle_count_prime_q * frp_prime_right_rectangle_count_prime_q -> frp_prime_left_rectangle_count_prime_q = 1 \/ frp_prime_right_rectangle_count_prime_q = 1)) -> ~(p = q) -> (exists erc_le_gap_rectangle_count_length_bound. erc_le_gap_rectangle_count_length_bound + (l) = h) -> (exists cb cc. (forall erc_row_rectangle_count_bounded_prefix. (exists erc_lt_gap_rectangle_count_bounded_prefix_bound. erc_lt_gap_rectangle_count_bounded_prefix_bound + S (erc_row_rectangle_count_bounded_prefix) = l) -> exists erc_count_rectangle_count_bounded_prefix. ((((exists ff_h_erc_rectangle_count_bounded_prefix_decoded. ff_h_erc_rectangle_count_bounded_prefix_decoded + S (erc_count_rectangle_count_bounded_prefix) = S ((S (erc_row_rectangle_count_bounded_prefix)) * cc)) /\ exists ff_q_erc_rectangle_count_bounded_prefix_decoded. cb = ff_q_erc_rectangle_count_bounded_prefix_decoded * S ((S (erc_row_rectangle_count_bounded_prefix)) * cc) + (erc_count_rectangle_count_bounded_prefix))) /\ (exists erc_row_code_rectangle_count_bounded_prefix_witness erc_row_scale_rectangle_count_bounded_prefix_witness. ((forall eri_column_erc_rectangle_count_bounded_prefix_witness_row. (exists eri_gap_erc_rectangle_count_bounded_prefix_witness_row_bound. eri_gap_erc_rectangle_count_bounded_prefix_witness_row_bound + S (eri_column_erc_rectangle_count_bounded_prefix_witness_row) = k) -> exists eri_bit_erc_rectangle_count_bounded_prefix_witness_row. ((((exists ff_h_eri_erc_rectangle_count_bounded_prefix_witness_row_decoded. ff_h_eri_erc_rectangle_count_bounded_prefix_witness_row_decoded + S (eri_bit_erc_rectangle_count_bounded_prefix_witness_row) = S ((S (eri_column_erc_rectangle_count_bounded_prefix_witness_row)) * erc_row_scale_rectangle_count_bounded_prefix_witness)) /\ exists ff_q_eri_erc_rectangle_count_bounded_prefix_witness_row_decoded. erc_row_code_rectangle_count_bounded_prefix_witness = ff_q_eri_erc_rectangle_count_bounded_prefix_witness_row_decoded * S ((S (eri_column_erc_rectangle_count_bounded_prefix_witness_row)) * erc_row_scale_rectangle_count_bounded_prefix_witness) + (eri_bit_erc_rectangle_count_bounded_prefix_witness_row))) /\ (((eri_bit_erc_rectangle_count_bounded_prefix_witness_row = 0 /\ ((exists eri_gap_erc_rectangle_count_bounded_prefix_witness_row_choice_left. eri_gap_erc_rectangle_count_bounded_prefix_witness_row_choice_left + S (q * S erc_row_rectangle_count_bounded_prefix) = p * S eri_column_erc_rectangle_count_bounded_prefix_witness_row) /\ ~(exists eri_gap_erc_rectangle_count_bounded_prefix_witness_row_choice_right. eri_gap_erc_rectangle_count_bounded_prefix_witness_row_choice_right + S (p * S eri_column_erc_rectangle_count_bounded_prefix_witness_row) = q * S erc_row_rectangle_count_bounded_prefix))) \/ (eri_bit_erc_rectangle_count_bounded_prefix_witness_row = 1 /\ ((exists eri_gap_erc_rectangle_count_bounded_prefix_witness_row_choice_right. eri_gap_erc_rectangle_count_bounded_prefix_witness_row_choice_right + S (p * S eri_column_erc_rectangle_count_bounded_prefix_witness_row) = q * S erc_row_rectangle_count_bounded_prefix) /\ ~(exists eri_gap_erc_rectangle_count_bounded_prefix_witness_row_choice_left. eri_gap_erc_rectangle_count_bounded_prefix_witness_row_choice_left + S (q * S erc_row_rectangle_count_bounded_prefix) = p * S eri_column_erc_rectangle_count_bounded_prefix_witness_row))))))) /\ (((exists ff_u_erc_rectangle_count_bounded_prefix_witness_count_sum ff_v_erc_rectangle_count_bounded_prefix_witness_count_sum. ((((exists ff_h_erc_rectangle_count_bounded_prefix_witness_count_sum_start. ff_h_erc_rectangle_count_bounded_prefix_witness_count_sum_start + S (0) = S ((S (0)) * ff_v_erc_rectangle_count_bounded_prefix_witness_count_sum)) /\ exists ff_q_erc_rectangle_count_bounded_prefix_witness_count_sum_start. ff_u_erc_rectangle_count_bounded_prefix_witness_count_sum = ff_q_erc_rectangle_count_bounded_prefix_witness_count_sum_start * S ((S (0)) * ff_v_erc_rectangle_count_bounded_prefix_witness_count_sum) + (0))) /\ ((((exists ff_h_erc_rectangle_count_bounded_prefix_witness_count_sum_terminal. ff_h_erc_rectangle_count_bounded_prefix_witness_count_sum_terminal + S (erc_count_rectangle_count_bounded_prefix) = S ((S (k)) * ff_v_erc_rectangle_count_bounded_prefix_witness_count_sum)) /\ exists ff_q_erc_rectangle_count_bounded_prefix_witness_count_sum_terminal. ff_u_erc_rectangle_count_bounded_prefix_witness_count_sum = ff_q_erc_rectangle_count_bounded_prefix_witness_count_sum_terminal * S ((S (k)) * ff_v_erc_rectangle_count_bounded_prefix_witness_count_sum) + (erc_count_rectangle_count_bounded_prefix))) /\ forall ff_i_erc_rectangle_count_bounded_prefix_witness_count_sum. (exists ff_lt_erc_rectangle_count_bounded_prefix_witness_count_sum_bound. ff_lt_erc_rectangle_count_bounded_prefix_witness_count_sum_bound + S ff_i_erc_rectangle_count_bounded_prefix_witness_count_sum = k) -> exists ff_a_erc_rectangle_count_bounded_prefix_witness_count_sum ff_r_erc_rectangle_count_bounded_prefix_witness_count_sum ff_s_erc_rectangle_count_bounded_prefix_witness_count_sum. ((((exists ff_h_erc_rectangle_count_bounded_prefix_witness_count_sum_summand. ff_h_erc_rectangle_count_bounded_prefix_witness_count_sum_summand + S (ff_a_erc_rectangle_count_bounded_prefix_witness_count_sum) = S ((S (ff_i_erc_rectangle_count_bounded_prefix_witness_count_sum)) * erc_row_scale_rectangle_count_bounded_prefix_witness)) /\ exists ff_q_erc_rectangle_count_bounded_prefix_witness_count_sum_summand. erc_row_code_rectangle_count_bounded_prefix_witness = ff_q_erc_rectangle_count_bounded_prefix_witness_count_sum_summand * S ((S (ff_i_erc_rectangle_count_bounded_prefix_witness_count_sum)) * erc_row_scale_rectangle_count_bounded_prefix_witness) + (ff_a_erc_rectangle_count_bounded_prefix_witness_count_sum))) /\ ((((exists ff_h_erc_rectangle_count_bounded_prefix_witness_count_sum_partial. ff_h_erc_rectangle_count_bounded_prefix_witness_count_sum_partial + S (ff_r_erc_rectangle_count_bounded_prefix_witness_count_sum) = S ((S (ff_i_erc_rectangle_count_bounded_prefix_witness_count_sum)) * ff_v_erc_rectangle_count_bounded_prefix_witness_count_sum)) /\ exists ff_q_erc_rectangle_count_bounded_prefix_witness_count_sum_partial. ff_u_erc_rectangle_count_bounded_prefix_witness_count_sum = ff_q_erc_rectangle_count_bounded_prefix_witness_count_sum_partial * S ((S (ff_i_erc_rectangle_count_bounded_prefix_witness_count_sum)) * ff_v_erc_rectangle_count_bounded_prefix_witness_count_sum) + (ff_r_erc_rectangle_count_bounded_prefix_witness_count_sum))) /\ ((((exists ff_h_erc_rectangle_count_bounded_prefix_witness_count_sum_successor. ff_h_erc_rectangle_count_bounded_prefix_witness_count_sum_successor + S (ff_s_erc_rectangle_count_bounded_prefix_witness_count_sum) = S ((S (S ff_i_erc_rectangle_count_bounded_prefix_witness_count_sum)) * ff_v_erc_rectangle_count_bounded_prefix_witness_count_sum)) /\ exists ff_q_erc_rectangle_count_bounded_prefix_witness_count_sum_successor. ff_u_erc_rectangle_count_bounded_prefix_witness_count_sum = ff_q_erc_rectangle_count_bounded_prefix_witness_count_sum_successor * S ((S (S ff_i_erc_rectangle_count_bounded_prefix_witness_count_sum)) * ff_v_erc_rectangle_count_bounded_prefix_witness_count_sum) + (ff_s_erc_rectangle_count_bounded_prefix_witness_count_sum))) /\ ff_s_erc_rectangle_count_bounded_prefix_witness_count_sum = ff_r_erc_rectangle_count_bounded_prefix_witness_count_sum + ff_a_erc_rectangle_count_bounded_prefix_witness_count_sum)))))) /\ (forall ff_i_erc_rectangle_count_bounded_prefix_witness_count_bits. (exists ff_lt_erc_rectangle_count_bounded_prefix_witness_count_bits_bound. ff_lt_erc_rectangle_count_bounded_prefix_witness_count_bits_bound + S ff_i_erc_rectangle_count_bounded_prefix_witness_count_bits = k) -> exists ff_bit_erc_rectangle_count_bounded_prefix_witness_count_bits. ((((exists ff_h_erc_rectangle_count_bounded_prefix_witness_count_bits_decoded. ff_h_erc_rectangle_count_bounded_prefix_witness_count_bits_decoded + S (ff_bit_erc_rectangle_count_bounded_prefix_witness_count_bits) = S ((S (ff_i_erc_rectangle_count_bounded_prefix_witness_count_bits)) * erc_row_scale_rectangle_count_bounded_prefix_witness)) /\ exists ff_q_erc_rectangle_count_bounded_prefix_witness_count_bits_decoded. erc_row_code_rectangle_count_bounded_prefix_witness = ff_q_erc_rectangle_count_bounded_prefix_witness_count_bits_decoded * S ((S (ff_i_erc_rectangle_count_bounded_prefix_witness_count_bits)) * erc_row_scale_rectangle_count_bounded_prefix_witness) + (ff_bit_erc_rectangle_count_bounded_prefix_witness_count_bits))) /\ (ff_bit_erc_rectangle_count_bounded_prefix_witness_count_bits = 0 \/ ff_bit_erc_rectangle_count_bounded_prefix_witness_count_bits = 1))))))))))