central_binom_prime_valuation_zero_above_third_quotient

Exact expanded PA statement

forall p n C v q r. ((~(p = 1) /\ forall frm_prime_left_bcpvzatq_prime frm_prime_right_bcpvzatq_prime. p = frm_prime_left_bcpvzatq_prime * frm_prime_right_bcpvzatq_prime -> frm_prime_left_bcpvzatq_prime = 1 \/ frm_prime_right_bcpvzatq_prime = 1)) -> (exists bcf_lt_gap_bcpvzatq_positive. bcf_lt_gap_bcpvzatq_positive + S (2) = n) -> (((n + n) = (3) * (q) + (r) /\ (exists bcf_lt_gap_bcpvzatq_division_bound. bcf_lt_gap_bcpvzatq_division_bound + S (r) = 3))) -> (exists bcf_lt_gap_bcpvzatq_above. bcf_lt_gap_bcpvzatq_above + S (q) = p) -> (exists bcf_le_gap_bcpvzatq_bound. bcf_le_gap_bcpvzatq_bound + (p) = n) -> (((exists bcf_lt_gap_bcpvzatq_central_out_of_range. bcf_lt_gap_bcpvzatq_central_out_of_range + S (n + n) = n) /\ C = 0) \/ ((exists bcf_le_gap_bcpvzatq_central_in_range. bcf_le_gap_bcpvzatq_central_in_range + (n) = n + n) /\ (exists bcf_row_code_code_bcpvzatq_central bcf_row_code_scale_bcpvzatq_central bcf_row_scale_code_bcpvzatq_central bcf_row_scale_scale_bcpvzatq_central bcf_row_code_bcpvzatq_central bcf_row_scale_bcpvzatq_central. ((forall bcf_row_index_bcpvzatq_central_table. (exists bcf_lt_gap_bcpvzatq_central_table_row_bound. bcf_lt_gap_bcpvzatq_central_table_row_bound + S (bcf_row_index_bcpvzatq_central_table) = S (n + n)) -> exists bcf_row_code_bcpvzatq_central_table bcf_row_scale_bcpvzatq_central_table. ((((exists bcf_height_bcpvzatq_central_table_decoded_row_code. bcf_height_bcpvzatq_central_table_decoded_row_code + S (bcf_row_code_bcpvzatq_central_table) = S ((S (bcf_row_index_bcpvzatq_central_table)) * bcf_row_code_scale_bcpvzatq_central)) /\ exists bcf_quotient_bcpvzatq_central_table_decoded_row_code. bcf_row_code_code_bcpvzatq_central = bcf_quotient_bcpvzatq_central_table_decoded_row_code * S ((S (bcf_row_index_bcpvzatq_central_table)) * bcf_row_code_scale_bcpvzatq_central) + (bcf_row_code_bcpvzatq_central_table))) /\ ((((exists bcf_height_bcpvzatq_central_table_decoded_row_scale. bcf_height_bcpvzatq_central_table_decoded_row_scale + S (bcf_row_scale_bcpvzatq_central_table) = S ((S (bcf_row_index_bcpvzatq_central_table)) * bcf_row_scale_scale_bcpvzatq_central)) /\ exists bcf_quotient_bcpvzatq_central_table_decoded_row_scale. bcf_row_scale_code_bcpvzatq_central = bcf_quotient_bcpvzatq_central_table_decoded_row_scale * S ((S (bcf_row_index_bcpvzatq_central_table)) * bcf_row_scale_scale_bcpvzatq_central) + (bcf_row_scale_bcpvzatq_central_table))) /\ ((bcf_row_index_bcpvzatq_central_table = 0 /\ (forall bcf_index_bcpvzatq_central_table_zero_row. (exists bcf_lt_gap_bcpvzatq_central_table_zero_row_bound. bcf_lt_gap_bcpvzatq_central_table_zero_row_bound + S (bcf_index_bcpvzatq_central_table_zero_row) = S (n + n)) -> exists bcf_value_bcpvzatq_central_table_zero_row. ((((exists bcf_height_bcpvzatq_central_table_zero_row_entry. bcf_height_bcpvzatq_central_table_zero_row_entry + S (bcf_value_bcpvzatq_central_table_zero_row) = S ((S (bcf_index_bcpvzatq_central_table_zero_row)) * bcf_row_scale_bcpvzatq_central_table)) /\ exists bcf_quotient_bcpvzatq_central_table_zero_row_entry. bcf_row_code_bcpvzatq_central_table = bcf_quotient_bcpvzatq_central_table_zero_row_entry * S ((S (bcf_index_bcpvzatq_central_table_zero_row)) * bcf_row_scale_bcpvzatq_central_table) + (bcf_value_bcpvzatq_central_table_zero_row))) /\ ((bcf_index_bcpvzatq_central_table_zero_row = 0 /\ bcf_value_bcpvzatq_central_table_zero_row = 1) \/ exists bcf_predecessor_bcpvzatq_central_table_zero_row. bcf_index_bcpvzatq_central_table_zero_row = S bcf_predecessor_bcpvzatq_central_table_zero_row /\ bcf_value_bcpvzatq_central_table_zero_row = 0)))) \/ exists bcf_predecessor_bcpvzatq_central_table bcf_previous_code_bcpvzatq_central_table bcf_previous_scale_bcpvzatq_central_table. bcf_row_index_bcpvzatq_central_table = S bcf_predecessor_bcpvzatq_central_table /\ ((((exists bcf_height_bcpvzatq_central_table_decoded_previous_code. bcf_height_bcpvzatq_central_table_decoded_previous_code + S (bcf_previous_code_bcpvzatq_central_table) = S ((S (bcf_predecessor_bcpvzatq_central_table)) * bcf_row_code_scale_bcpvzatq_central)) /\ exists bcf_quotient_bcpvzatq_central_table_decoded_previous_code. bcf_row_code_code_bcpvzatq_central = bcf_quotient_bcpvzatq_central_table_decoded_previous_code * S ((S (bcf_predecessor_bcpvzatq_central_table)) * bcf_row_code_scale_bcpvzatq_central) + (bcf_previous_code_bcpvzatq_central_table))) /\ ((((exists bcf_height_bcpvzatq_central_table_decoded_previous_scale. bcf_height_bcpvzatq_central_table_decoded_previous_scale + S (bcf_previous_scale_bcpvzatq_central_table) = S ((S (bcf_predecessor_bcpvzatq_central_table)) * bcf_row_scale_scale_bcpvzatq_central)) /\ exists bcf_quotient_bcpvzatq_central_table_decoded_previous_scale. bcf_row_scale_code_bcpvzatq_central = bcf_quotient_bcpvzatq_central_table_decoded_previous_scale * S ((S (bcf_predecessor_bcpvzatq_central_table)) * bcf_row_scale_scale_bcpvzatq_central) + (bcf_previous_scale_bcpvzatq_central_table))) /\ (forall bcf_index_bcpvzatq_central_table_row_step. (exists bcf_lt_gap_bcpvzatq_central_table_row_step_bound. bcf_lt_gap_bcpvzatq_central_table_row_step_bound + S (bcf_index_bcpvzatq_central_table_row_step) = S (n + n)) -> exists bcf_value_bcpvzatq_central_table_row_step. ((((exists bcf_height_bcpvzatq_central_table_row_step_entry. bcf_height_bcpvzatq_central_table_row_step_entry + S (bcf_value_bcpvzatq_central_table_row_step) = S ((S (bcf_index_bcpvzatq_central_table_row_step)) * bcf_row_scale_bcpvzatq_central_table)) /\ exists bcf_quotient_bcpvzatq_central_table_row_step_entry. bcf_row_code_bcpvzatq_central_table = bcf_quotient_bcpvzatq_central_table_row_step_entry * S ((S (bcf_index_bcpvzatq_central_table_row_step)) * bcf_row_scale_bcpvzatq_central_table) + (bcf_value_bcpvzatq_central_table_row_step))) /\ ((bcf_index_bcpvzatq_central_table_row_step = 0 /\ bcf_value_bcpvzatq_central_table_row_step = 1) \/ exists bcf_predecessor_bcpvzatq_central_table_row_step bcf_left_bcpvzatq_central_table_row_step bcf_right_bcpvzatq_central_table_row_step. bcf_index_bcpvzatq_central_table_row_step = S bcf_predecessor_bcpvzatq_central_table_row_step /\ ((((exists bcf_height_bcpvzatq_central_table_row_step_previous_left. bcf_height_bcpvzatq_central_table_row_step_previous_left + S (bcf_left_bcpvzatq_central_table_row_step) = S ((S (bcf_predecessor_bcpvzatq_central_table_row_step)) * bcf_previous_scale_bcpvzatq_central_table)) /\ exists bcf_quotient_bcpvzatq_central_table_row_step_previous_left. bcf_previous_code_bcpvzatq_central_table = bcf_quotient_bcpvzatq_central_table_row_step_previous_left * S ((S (bcf_predecessor_bcpvzatq_central_table_row_step)) * bcf_previous_scale_bcpvzatq_central_table) + (bcf_left_bcpvzatq_central_table_row_step))) /\ ((((exists bcf_height_bcpvzatq_central_table_row_step_previous_right. bcf_height_bcpvzatq_central_table_row_step_previous_right + S (bcf_right_bcpvzatq_central_table_row_step) = S ((S (S (bcf_predecessor_bcpvzatq_central_table_row_step))) * bcf_previous_scale_bcpvzatq_central_table)) /\ exists bcf_quotient_bcpvzatq_central_table_row_step_previous_right. bcf_previous_code_bcpvzatq_central_table = bcf_quotient_bcpvzatq_central_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcpvzatq_central_table_row_step))) * bcf_previous_scale_bcpvzatq_central_table) + (bcf_right_bcpvzatq_central_table_row_step))) /\ bcf_value_bcpvzatq_central_table_row_step = bcf_left_bcpvzatq_central_table_row_step + bcf_right_bcpvzatq_central_table_row_step))))))))))) /\ ((((exists bcf_height_bcpvzatq_central_decoded_row_code. bcf_height_bcpvzatq_central_decoded_row_code + S (bcf_row_code_bcpvzatq_central) = S ((S (n + n)) * bcf_row_code_scale_bcpvzatq_central)) /\ exists bcf_quotient_bcpvzatq_central_decoded_row_code. bcf_row_code_code_bcpvzatq_central = bcf_quotient_bcpvzatq_central_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_bcpvzatq_central) + (bcf_row_code_bcpvzatq_central))) /\ ((((exists bcf_height_bcpvzatq_central_decoded_row_scale. bcf_height_bcpvzatq_central_decoded_row_scale + S (bcf_row_scale_bcpvzatq_central) = S ((S (n + n)) * bcf_row_scale_scale_bcpvzatq_central)) /\ exists bcf_quotient_bcpvzatq_central_decoded_row_scale. bcf_row_scale_code_bcpvzatq_central = bcf_quotient_bcpvzatq_central_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_bcpvzatq_central) + (bcf_row_scale_bcpvzatq_central))) /\ (((exists bcf_height_bcpvzatq_central_decoded_value. bcf_height_bcpvzatq_central_decoded_value + S (C) = S ((S (n)) * bcf_row_scale_bcpvzatq_central)) /\ exists bcf_quotient_bcpvzatq_central_decoded_value. bcf_row_code_bcpvzatq_central = bcf_quotient_bcpvzatq_central_decoded_value * S ((S (n)) * bcf_row_scale_bcpvzatq_central) + (C))))))))) -> (((exists bpv_gap_bcpvzatq_valuation_exponent_bound. bpv_gap_bcpvzatq_valuation_exponent_bound + v = C) /\ (exists bpv_result_bcpvzatq_valuation_selected. ((exists ff_b_bcpvzatq_valuation_selected_power ff_c_bcpvzatq_valuation_selected_power. ((forall ff_i_bcpvzatq_valuation_selected_power_repeat. (exists ff_lt_bcpvzatq_valuation_selected_power_repeat_bound. ff_lt_bcpvzatq_valuation_selected_power_repeat_bound + S ff_i_bcpvzatq_valuation_selected_power_repeat = v) -> (((exists ff_h_bcpvzatq_valuation_selected_power_repeat_decoded. ff_h_bcpvzatq_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bcpvzatq_valuation_selected_power_repeat)) * ff_c_bcpvzatq_valuation_selected_power)) /\ exists ff_q_bcpvzatq_valuation_selected_power_repeat_decoded. ff_b_bcpvzatq_valuation_selected_power = ff_q_bcpvzatq_valuation_selected_power_repeat_decoded * S ((S (ff_i_bcpvzatq_valuation_selected_power_repeat)) * ff_c_bcpvzatq_valuation_selected_power) + (p)))) /\ (exists ff_u_bcpvzatq_valuation_selected_power_product ff_v_bcpvzatq_valuation_selected_power_product. ((((exists ff_h_bcpvzatq_valuation_selected_power_product_start. ff_h_bcpvzatq_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bcpvzatq_valuation_selected_power_product)) /\ exists ff_q_bcpvzatq_valuation_selected_power_product_start. ff_u_bcpvzatq_valuation_selected_power_product = ff_q_bcpvzatq_valuation_selected_power_product_start * S ((S (0)) * ff_v_bcpvzatq_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bcpvzatq_valuation_selected_power_product_terminal. ff_h_bcpvzatq_valuation_selected_power_product_terminal + S (bpv_result_bcpvzatq_valuation_selected) = S ((S (v)) * ff_v_bcpvzatq_valuation_selected_power_product)) /\ exists ff_q_bcpvzatq_valuation_selected_power_product_terminal. ff_u_bcpvzatq_valuation_selected_power_product = ff_q_bcpvzatq_valuation_selected_power_product_terminal * S ((S (v)) * ff_v_bcpvzatq_valuation_selected_power_product) + (bpv_result_bcpvzatq_valuation_selected))) /\ forall ff_i_bcpvzatq_valuation_selected_power_product. (exists ff_lt_bcpvzatq_valuation_selected_power_product_bound. ff_lt_bcpvzatq_valuation_selected_power_product_bound + S ff_i_bcpvzatq_valuation_selected_power_product = v) -> exists ff_p_bcpvzatq_valuation_selected_power_product ff_r_bcpvzatq_valuation_selected_power_product ff_s_bcpvzatq_valuation_selected_power_product. ((((exists ff_h_bcpvzatq_valuation_selected_power_product_factor. ff_h_bcpvzatq_valuation_selected_power_product_factor + S (ff_p_bcpvzatq_valuation_selected_power_product) = S ((S (ff_i_bcpvzatq_valuation_selected_power_product)) * ff_c_bcpvzatq_valuation_selected_power)) /\ exists ff_q_bcpvzatq_valuation_selected_power_product_factor. ff_b_bcpvzatq_valuation_selected_power = ff_q_bcpvzatq_valuation_selected_power_product_factor * S ((S (ff_i_bcpvzatq_valuation_selected_power_product)) * ff_c_bcpvzatq_valuation_selected_power) + (ff_p_bcpvzatq_valuation_selected_power_product))) /\ ((((exists ff_h_bcpvzatq_valuation_selected_power_product_partial. ff_h_bcpvzatq_valuation_selected_power_product_partial + S (ff_r_bcpvzatq_valuation_selected_power_product) = S ((S (ff_i_bcpvzatq_valuation_selected_power_product)) * ff_v_bcpvzatq_valuation_selected_power_product)) /\ exists ff_q_bcpvzatq_valuation_selected_power_product_partial. ff_u_bcpvzatq_valuation_selected_power_product = ff_q_bcpvzatq_valuation_selected_power_product_partial * S ((S (ff_i_bcpvzatq_valuation_selected_power_product)) * ff_v_bcpvzatq_valuation_selected_power_product) + (ff_r_bcpvzatq_valuation_selected_power_product))) /\ ((((exists ff_h_bcpvzatq_valuation_selected_power_product_successor. ff_h_bcpvzatq_valuation_selected_power_product_successor + S (ff_s_bcpvzatq_valuation_selected_power_product) = S ((S (S ff_i_bcpvzatq_valuation_selected_power_product)) * ff_v_bcpvzatq_valuation_selected_power_product)) /\ exists ff_q_bcpvzatq_valuation_selected_power_product_successor. ff_u_bcpvzatq_valuation_selected_power_product = ff_q_bcpvzatq_valuation_selected_power_product_successor * S ((S (S ff_i_bcpvzatq_valuation_selected_power_product)) * ff_v_bcpvzatq_valuation_selected_power_product) + (ff_s_bcpvzatq_valuation_selected_power_product))) /\ ff_s_bcpvzatq_valuation_selected_power_product = ff_r_bcpvzatq_valuation_selected_power_product * ff_p_bcpvzatq_valuation_selected_power_product)))))))) /\ (exists bpv_factor_bcpvzatq_valuation_selected_divides. C = bpv_result_bcpvzatq_valuation_selected * bpv_factor_bcpvzatq_valuation_selected_divides)))) /\ forall bpv_candidate_bcpvzatq_valuation. (exists bpv_gap_bcpvzatq_valuation_candidate_bound. bpv_gap_bcpvzatq_valuation_candidate_bound + bpv_candidate_bcpvzatq_valuation = C) -> (exists bpv_result_bcpvzatq_valuation_candidate. ((exists ff_b_bcpvzatq_valuation_candidate_power ff_c_bcpvzatq_valuation_candidate_power. ((forall ff_i_bcpvzatq_valuation_candidate_power_repeat. (exists ff_lt_bcpvzatq_valuation_candidate_power_repeat_bound. ff_lt_bcpvzatq_valuation_candidate_power_repeat_bound + S ff_i_bcpvzatq_valuation_candidate_power_repeat = bpv_candidate_bcpvzatq_valuation) -> (((exists ff_h_bcpvzatq_valuation_candidate_power_repeat_decoded. ff_h_bcpvzatq_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bcpvzatq_valuation_candidate_power_repeat)) * ff_c_bcpvzatq_valuation_candidate_power)) /\ exists ff_q_bcpvzatq_valuation_candidate_power_repeat_decoded. ff_b_bcpvzatq_valuation_candidate_power = ff_q_bcpvzatq_valuation_candidate_power_repeat_decoded * S ((S (ff_i_bcpvzatq_valuation_candidate_power_repeat)) * ff_c_bcpvzatq_valuation_candidate_power) + (p)))) /\ (exists ff_u_bcpvzatq_valuation_candidate_power_product ff_v_bcpvzatq_valuation_candidate_power_product. ((((exists ff_h_bcpvzatq_valuation_candidate_power_product_start. ff_h_bcpvzatq_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bcpvzatq_valuation_candidate_power_product)) /\ exists ff_q_bcpvzatq_valuation_candidate_power_product_start. ff_u_bcpvzatq_valuation_candidate_power_product = ff_q_bcpvzatq_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bcpvzatq_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bcpvzatq_valuation_candidate_power_product_terminal. ff_h_bcpvzatq_valuation_candidate_power_product_terminal + S (bpv_result_bcpvzatq_valuation_candidate) = S ((S (bpv_candidate_bcpvzatq_valuation)) * ff_v_bcpvzatq_valuation_candidate_power_product)) /\ exists ff_q_bcpvzatq_valuation_candidate_power_product_terminal. ff_u_bcpvzatq_valuation_candidate_power_product = ff_q_bcpvzatq_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_bcpvzatq_valuation)) * ff_v_bcpvzatq_valuation_candidate_power_product) + (bpv_result_bcpvzatq_valuation_candidate))) /\ forall ff_i_bcpvzatq_valuation_candidate_power_product. (exists ff_lt_bcpvzatq_valuation_candidate_power_product_bound. ff_lt_bcpvzatq_valuation_candidate_power_product_bound + S ff_i_bcpvzatq_valuation_candidate_power_product = bpv_candidate_bcpvzatq_valuation) -> exists ff_p_bcpvzatq_valuation_candidate_power_product ff_r_bcpvzatq_valuation_candidate_power_product ff_s_bcpvzatq_valuation_candidate_power_product. ((((exists ff_h_bcpvzatq_valuation_candidate_power_product_factor. ff_h_bcpvzatq_valuation_candidate_power_product_factor + S (ff_p_bcpvzatq_valuation_candidate_power_product) = S ((S (ff_i_bcpvzatq_valuation_candidate_power_product)) * ff_c_bcpvzatq_valuation_candidate_power)) /\ exists ff_q_bcpvzatq_valuation_candidate_power_product_factor. ff_b_bcpvzatq_valuation_candidate_power = ff_q_bcpvzatq_valuation_candidate_power_product_factor * S ((S (ff_i_bcpvzatq_valuation_candidate_power_product)) * ff_c_bcpvzatq_valuation_candidate_power) + (ff_p_bcpvzatq_valuation_candidate_power_product))) /\ ((((exists ff_h_bcpvzatq_valuation_candidate_power_product_partial. ff_h_bcpvzatq_valuation_candidate_power_product_partial + S (ff_r_bcpvzatq_valuation_candidate_power_product) = S ((S (ff_i_bcpvzatq_valuation_candidate_power_product)) * ff_v_bcpvzatq_valuation_candidate_power_product)) /\ exists ff_q_bcpvzatq_valuation_candidate_power_product_partial. ff_u_bcpvzatq_valuation_candidate_power_product = ff_q_bcpvzatq_valuation_candidate_power_product_partial * S ((S (ff_i_bcpvzatq_valuation_candidate_power_product)) * ff_v_bcpvzatq_valuation_candidate_power_product) + (ff_r_bcpvzatq_valuation_candidate_power_product))) /\ ((((exists ff_h_bcpvzatq_valuation_candidate_power_product_successor. ff_h_bcpvzatq_valuation_candidate_power_product_successor + S (ff_s_bcpvzatq_valuation_candidate_power_product) = S ((S (S ff_i_bcpvzatq_valuation_candidate_power_product)) * ff_v_bcpvzatq_valuation_candidate_power_product)) /\ exists ff_q_bcpvzatq_valuation_candidate_power_product_successor. ff_u_bcpvzatq_valuation_candidate_power_product = ff_q_bcpvzatq_valuation_candidate_power_product_successor * S ((S (S ff_i_bcpvzatq_valuation_candidate_power_product)) * ff_v_bcpvzatq_valuation_candidate_power_product) + (ff_s_bcpvzatq_valuation_candidate_power_product))) /\ ff_s_bcpvzatq_valuation_candidate_power_product = ff_r_bcpvzatq_valuation_candidate_power_product * ff_p_bcpvzatq_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_bcpvzatq_valuation_candidate_divides. C = bpv_result_bcpvzatq_valuation_candidate * bpv_factor_bcpvzatq_valuation_candidate_divides))) -> (exists bpv_gap_bcpvzatq_valuation_maximal. bpv_gap_bcpvzatq_valuation_maximal + bpv_candidate_bcpvzatq_valuation = v)) -> v = 0