BT00YD

central_binom_prime_valuation_zero_of_exact_double_quotients

Alpha body-checked · checked-use disabled

An all-zero carry prefix forces the exact central valuation to zero.

Exact expanded PA statement

forall p n C v q r R s. ((~(p = 1) /\ forall frm_prime_left_bcpvzeq_prime frm_prime_right_bcpvzeq_prime. p = frm_prime_left_bcpvzeq_prime * frm_prime_right_bcpvzeq_prime -> frm_prime_left_bcpvzeq_prime = 1 \/ frm_prime_right_bcpvzeq_prime = 1)) -> (((exists bcf_lt_gap_bcpvzeq_central_out_of_range. bcf_lt_gap_bcpvzeq_central_out_of_range + S (n + n) = n) /\ C = 0) \/ ((exists bcf_le_gap_bcpvzeq_central_in_range. bcf_le_gap_bcpvzeq_central_in_range + (n) = n + n) /\ (exists bcf_row_code_code_bcpvzeq_central bcf_row_code_scale_bcpvzeq_central bcf_row_scale_code_bcpvzeq_central bcf_row_scale_scale_bcpvzeq_central bcf_row_code_bcpvzeq_central bcf_row_scale_bcpvzeq_central. ((forall bcf_row_index_bcpvzeq_central_table. (exists bcf_lt_gap_bcpvzeq_central_table_row_bound. bcf_lt_gap_bcpvzeq_central_table_row_bound + S (bcf_row_index_bcpvzeq_central_table) = S (n + n)) -> exists bcf_row_code_bcpvzeq_central_table bcf_row_scale_bcpvzeq_central_table. ((((exists bcf_height_bcpvzeq_central_table_decoded_row_code. bcf_height_bcpvzeq_central_table_decoded_row_code + S (bcf_row_code_bcpvzeq_central_table) = S ((S (bcf_row_index_bcpvzeq_central_table)) * bcf_row_code_scale_bcpvzeq_central)) /\ exists bcf_quotient_bcpvzeq_central_table_decoded_row_code. bcf_row_code_code_bcpvzeq_central = bcf_quotient_bcpvzeq_central_table_decoded_row_code * S ((S (bcf_row_index_bcpvzeq_central_table)) * bcf_row_code_scale_bcpvzeq_central) + (bcf_row_code_bcpvzeq_central_table))) /\ ((((exists bcf_height_bcpvzeq_central_table_decoded_row_scale. bcf_height_bcpvzeq_central_table_decoded_row_scale + S (bcf_row_scale_bcpvzeq_central_table) = S ((S (bcf_row_index_bcpvzeq_central_table)) * bcf_row_scale_scale_bcpvzeq_central)) /\ exists bcf_quotient_bcpvzeq_central_table_decoded_row_scale. bcf_row_scale_code_bcpvzeq_central = bcf_quotient_bcpvzeq_central_table_decoded_row_scale * S ((S (bcf_row_index_bcpvzeq_central_table)) * bcf_row_scale_scale_bcpvzeq_central) + (bcf_row_scale_bcpvzeq_central_table))) /\ ((bcf_row_index_bcpvzeq_central_table = 0 /\ (forall bcf_index_bcpvzeq_central_table_zero_row. (exists bcf_lt_gap_bcpvzeq_central_table_zero_row_bound. bcf_lt_gap_bcpvzeq_central_table_zero_row_bound + S (bcf_index_bcpvzeq_central_table_zero_row) = S (n + n)) -> exists bcf_value_bcpvzeq_central_table_zero_row. ((((exists bcf_height_bcpvzeq_central_table_zero_row_entry. bcf_height_bcpvzeq_central_table_zero_row_entry + S (bcf_value_bcpvzeq_central_table_zero_row) = S ((S (bcf_index_bcpvzeq_central_table_zero_row)) * bcf_row_scale_bcpvzeq_central_table)) /\ exists bcf_quotient_bcpvzeq_central_table_zero_row_entry. bcf_row_code_bcpvzeq_central_table = bcf_quotient_bcpvzeq_central_table_zero_row_entry * S ((S (bcf_index_bcpvzeq_central_table_zero_row)) * bcf_row_scale_bcpvzeq_central_table) + (bcf_value_bcpvzeq_central_table_zero_row))) /\ ((bcf_index_bcpvzeq_central_table_zero_row = 0 /\ bcf_value_bcpvzeq_central_table_zero_row = 1) \/ exists bcf_predecessor_bcpvzeq_central_table_zero_row. bcf_index_bcpvzeq_central_table_zero_row = S bcf_predecessor_bcpvzeq_central_table_zero_row /\ bcf_value_bcpvzeq_central_table_zero_row = 0)))) \/ exists bcf_predecessor_bcpvzeq_central_table bcf_previous_code_bcpvzeq_central_table bcf_previous_scale_bcpvzeq_central_table. bcf_row_index_bcpvzeq_central_table = S bcf_predecessor_bcpvzeq_central_table /\ ((((exists bcf_height_bcpvzeq_central_table_decoded_previous_code. bcf_height_bcpvzeq_central_table_decoded_previous_code + S (bcf_previous_code_bcpvzeq_central_table) = S ((S (bcf_predecessor_bcpvzeq_central_table)) * bcf_row_code_scale_bcpvzeq_central)) /\ exists bcf_quotient_bcpvzeq_central_table_decoded_previous_code. bcf_row_code_code_bcpvzeq_central = bcf_quotient_bcpvzeq_central_table_decoded_previous_code * S ((S (bcf_predecessor_bcpvzeq_central_table)) * bcf_row_code_scale_bcpvzeq_central) + (bcf_previous_code_bcpvzeq_central_table))) /\ ((((exists bcf_height_bcpvzeq_central_table_decoded_previous_scale. bcf_height_bcpvzeq_central_table_decoded_previous_scale + S (bcf_previous_scale_bcpvzeq_central_table) = S ((S (bcf_predecessor_bcpvzeq_central_table)) * bcf_row_scale_scale_bcpvzeq_central)) /\ exists bcf_quotient_bcpvzeq_central_table_decoded_previous_scale. bcf_row_scale_code_bcpvzeq_central = bcf_quotient_bcpvzeq_central_table_decoded_previous_scale * S ((S (bcf_predecessor_bcpvzeq_central_table)) * bcf_row_scale_scale_bcpvzeq_central) + (bcf_previous_scale_bcpvzeq_central_table))) /\ (forall bcf_index_bcpvzeq_central_table_row_step. (exists bcf_lt_gap_bcpvzeq_central_table_row_step_bound. bcf_lt_gap_bcpvzeq_central_table_row_step_bound + S (bcf_index_bcpvzeq_central_table_row_step) = S (n + n)) -> exists bcf_value_bcpvzeq_central_table_row_step. ((((exists bcf_height_bcpvzeq_central_table_row_step_entry. bcf_height_bcpvzeq_central_table_row_step_entry + S (bcf_value_bcpvzeq_central_table_row_step) = S ((S (bcf_index_bcpvzeq_central_table_row_step)) * bcf_row_scale_bcpvzeq_central_table)) /\ exists bcf_quotient_bcpvzeq_central_table_row_step_entry. bcf_row_code_bcpvzeq_central_table = bcf_quotient_bcpvzeq_central_table_row_step_entry * S ((S (bcf_index_bcpvzeq_central_table_row_step)) * bcf_row_scale_bcpvzeq_central_table) + (bcf_value_bcpvzeq_central_table_row_step))) /\ ((bcf_index_bcpvzeq_central_table_row_step = 0 /\ bcf_value_bcpvzeq_central_table_row_step = 1) \/ exists bcf_predecessor_bcpvzeq_central_table_row_step bcf_left_bcpvzeq_central_table_row_step bcf_right_bcpvzeq_central_table_row_step. bcf_index_bcpvzeq_central_table_row_step = S bcf_predecessor_bcpvzeq_central_table_row_step /\ ((((exists bcf_height_bcpvzeq_central_table_row_step_previous_left. bcf_height_bcpvzeq_central_table_row_step_previous_left + S (bcf_left_bcpvzeq_central_table_row_step) = S ((S (bcf_predecessor_bcpvzeq_central_table_row_step)) * bcf_previous_scale_bcpvzeq_central_table)) /\ exists bcf_quotient_bcpvzeq_central_table_row_step_previous_left. bcf_previous_code_bcpvzeq_central_table = bcf_quotient_bcpvzeq_central_table_row_step_previous_left * S ((S (bcf_predecessor_bcpvzeq_central_table_row_step)) * bcf_previous_scale_bcpvzeq_central_table) + (bcf_left_bcpvzeq_central_table_row_step))) /\ ((((exists bcf_height_bcpvzeq_central_table_row_step_previous_right. bcf_height_bcpvzeq_central_table_row_step_previous_right + S (bcf_right_bcpvzeq_central_table_row_step) = S ((S (S (bcf_predecessor_bcpvzeq_central_table_row_step))) * bcf_previous_scale_bcpvzeq_central_table)) /\ exists bcf_quotient_bcpvzeq_central_table_row_step_previous_right. bcf_previous_code_bcpvzeq_central_table = bcf_quotient_bcpvzeq_central_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcpvzeq_central_table_row_step))) * bcf_previous_scale_bcpvzeq_central_table) + (bcf_right_bcpvzeq_central_table_row_step))) /\ bcf_value_bcpvzeq_central_table_row_step = bcf_left_bcpvzeq_central_table_row_step + bcf_right_bcpvzeq_central_table_row_step))))))))))) /\ ((((exists bcf_height_bcpvzeq_central_decoded_row_code. bcf_height_bcpvzeq_central_decoded_row_code + S (bcf_row_code_bcpvzeq_central) = S ((S (n + n)) * bcf_row_code_scale_bcpvzeq_central)) /\ exists bcf_quotient_bcpvzeq_central_decoded_row_code. bcf_row_code_code_bcpvzeq_central = bcf_quotient_bcpvzeq_central_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_bcpvzeq_central) + (bcf_row_code_bcpvzeq_central))) /\ ((((exists bcf_height_bcpvzeq_central_decoded_row_scale. bcf_height_bcpvzeq_central_decoded_row_scale + S (bcf_row_scale_bcpvzeq_central) = S ((S (n + n)) * bcf_row_scale_scale_bcpvzeq_central)) /\ exists bcf_quotient_bcpvzeq_central_decoded_row_scale. bcf_row_scale_code_bcpvzeq_central = bcf_quotient_bcpvzeq_central_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_bcpvzeq_central) + (bcf_row_scale_bcpvzeq_central))) /\ (((exists bcf_height_bcpvzeq_central_decoded_value. bcf_height_bcpvzeq_central_decoded_value + S (C) = S ((S (n)) * bcf_row_scale_bcpvzeq_central)) /\ exists bcf_quotient_bcpvzeq_central_decoded_value. bcf_row_code_bcpvzeq_central = bcf_quotient_bcpvzeq_central_decoded_value * S ((S (n)) * bcf_row_scale_bcpvzeq_central) + (C))))))))) -> (((exists bpv_gap_bcpvzeq_valuation_exponent_bound. bpv_gap_bcpvzeq_valuation_exponent_bound + v = C) /\ (exists bpv_result_bcpvzeq_valuation_selected. ((exists ff_b_bcpvzeq_valuation_selected_power ff_c_bcpvzeq_valuation_selected_power. ((forall ff_i_bcpvzeq_valuation_selected_power_repeat. (exists ff_lt_bcpvzeq_valuation_selected_power_repeat_bound. ff_lt_bcpvzeq_valuation_selected_power_repeat_bound + S ff_i_bcpvzeq_valuation_selected_power_repeat = v) -> (((exists ff_h_bcpvzeq_valuation_selected_power_repeat_decoded. ff_h_bcpvzeq_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bcpvzeq_valuation_selected_power_repeat)) * ff_c_bcpvzeq_valuation_selected_power)) /\ exists ff_q_bcpvzeq_valuation_selected_power_repeat_decoded. ff_b_bcpvzeq_valuation_selected_power = ff_q_bcpvzeq_valuation_selected_power_repeat_decoded * S ((S (ff_i_bcpvzeq_valuation_selected_power_repeat)) * ff_c_bcpvzeq_valuation_selected_power) + (p)))) /\ (exists ff_u_bcpvzeq_valuation_selected_power_product ff_v_bcpvzeq_valuation_selected_power_product. ((((exists ff_h_bcpvzeq_valuation_selected_power_product_start. ff_h_bcpvzeq_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bcpvzeq_valuation_selected_power_product)) /\ exists ff_q_bcpvzeq_valuation_selected_power_product_start. ff_u_bcpvzeq_valuation_selected_power_product = ff_q_bcpvzeq_valuation_selected_power_product_start * S ((S (0)) * ff_v_bcpvzeq_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bcpvzeq_valuation_selected_power_product_terminal. ff_h_bcpvzeq_valuation_selected_power_product_terminal + S (bpv_result_bcpvzeq_valuation_selected) = S ((S (v)) * ff_v_bcpvzeq_valuation_selected_power_product)) /\ exists ff_q_bcpvzeq_valuation_selected_power_product_terminal. ff_u_bcpvzeq_valuation_selected_power_product = ff_q_bcpvzeq_valuation_selected_power_product_terminal * S ((S (v)) * ff_v_bcpvzeq_valuation_selected_power_product) + (bpv_result_bcpvzeq_valuation_selected))) /\ forall ff_i_bcpvzeq_valuation_selected_power_product. (exists ff_lt_bcpvzeq_valuation_selected_power_product_bound. ff_lt_bcpvzeq_valuation_selected_power_product_bound + S ff_i_bcpvzeq_valuation_selected_power_product = v) -> exists ff_p_bcpvzeq_valuation_selected_power_product ff_r_bcpvzeq_valuation_selected_power_product ff_s_bcpvzeq_valuation_selected_power_product. ((((exists ff_h_bcpvzeq_valuation_selected_power_product_factor. ff_h_bcpvzeq_valuation_selected_power_product_factor + S (ff_p_bcpvzeq_valuation_selected_power_product) = S ((S (ff_i_bcpvzeq_valuation_selected_power_product)) * ff_c_bcpvzeq_valuation_selected_power)) /\ exists ff_q_bcpvzeq_valuation_selected_power_product_factor. ff_b_bcpvzeq_valuation_selected_power = ff_q_bcpvzeq_valuation_selected_power_product_factor * S ((S (ff_i_bcpvzeq_valuation_selected_power_product)) * ff_c_bcpvzeq_valuation_selected_power) + (ff_p_bcpvzeq_valuation_selected_power_product))) /\ ((((exists ff_h_bcpvzeq_valuation_selected_power_product_partial. ff_h_bcpvzeq_valuation_selected_power_product_partial + S (ff_r_bcpvzeq_valuation_selected_power_product) = S ((S (ff_i_bcpvzeq_valuation_selected_power_product)) * ff_v_bcpvzeq_valuation_selected_power_product)) /\ exists ff_q_bcpvzeq_valuation_selected_power_product_partial. ff_u_bcpvzeq_valuation_selected_power_product = ff_q_bcpvzeq_valuation_selected_power_product_partial * S ((S (ff_i_bcpvzeq_valuation_selected_power_product)) * ff_v_bcpvzeq_valuation_selected_power_product) + (ff_r_bcpvzeq_valuation_selected_power_product))) /\ ((((exists ff_h_bcpvzeq_valuation_selected_power_product_successor. ff_h_bcpvzeq_valuation_selected_power_product_successor + S (ff_s_bcpvzeq_valuation_selected_power_product) = S ((S (S ff_i_bcpvzeq_valuation_selected_power_product)) * ff_v_bcpvzeq_valuation_selected_power_product)) /\ exists ff_q_bcpvzeq_valuation_selected_power_product_successor. ff_u_bcpvzeq_valuation_selected_power_product = ff_q_bcpvzeq_valuation_selected_power_product_successor * S ((S (S ff_i_bcpvzeq_valuation_selected_power_product)) * ff_v_bcpvzeq_valuation_selected_power_product) + (ff_s_bcpvzeq_valuation_selected_power_product))) /\ ff_s_bcpvzeq_valuation_selected_power_product = ff_r_bcpvzeq_valuation_selected_power_product * ff_p_bcpvzeq_valuation_selected_power_product)))))))) /\ (exists bpv_factor_bcpvzeq_valuation_selected_divides. C = bpv_result_bcpvzeq_valuation_selected * bpv_factor_bcpvzeq_valuation_selected_divides)))) /\ forall bpv_candidate_bcpvzeq_valuation. (exists bpv_gap_bcpvzeq_valuation_candidate_bound. bpv_gap_bcpvzeq_valuation_candidate_bound + bpv_candidate_bcpvzeq_valuation = C) -> (exists bpv_result_bcpvzeq_valuation_candidate. ((exists ff_b_bcpvzeq_valuation_candidate_power ff_c_bcpvzeq_valuation_candidate_power. ((forall ff_i_bcpvzeq_valuation_candidate_power_repeat. (exists ff_lt_bcpvzeq_valuation_candidate_power_repeat_bound. ff_lt_bcpvzeq_valuation_candidate_power_repeat_bound + S ff_i_bcpvzeq_valuation_candidate_power_repeat = bpv_candidate_bcpvzeq_valuation) -> (((exists ff_h_bcpvzeq_valuation_candidate_power_repeat_decoded. ff_h_bcpvzeq_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bcpvzeq_valuation_candidate_power_repeat)) * ff_c_bcpvzeq_valuation_candidate_power)) /\ exists ff_q_bcpvzeq_valuation_candidate_power_repeat_decoded. ff_b_bcpvzeq_valuation_candidate_power = ff_q_bcpvzeq_valuation_candidate_power_repeat_decoded * S ((S (ff_i_bcpvzeq_valuation_candidate_power_repeat)) * ff_c_bcpvzeq_valuation_candidate_power) + (p)))) /\ (exists ff_u_bcpvzeq_valuation_candidate_power_product ff_v_bcpvzeq_valuation_candidate_power_product. ((((exists ff_h_bcpvzeq_valuation_candidate_power_product_start. ff_h_bcpvzeq_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bcpvzeq_valuation_candidate_power_product)) /\ exists ff_q_bcpvzeq_valuation_candidate_power_product_start. ff_u_bcpvzeq_valuation_candidate_power_product = ff_q_bcpvzeq_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bcpvzeq_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bcpvzeq_valuation_candidate_power_product_terminal. ff_h_bcpvzeq_valuation_candidate_power_product_terminal + S (bpv_result_bcpvzeq_valuation_candidate) = S ((S (bpv_candidate_bcpvzeq_valuation)) * ff_v_bcpvzeq_valuation_candidate_power_product)) /\ exists ff_q_bcpvzeq_valuation_candidate_power_product_terminal. ff_u_bcpvzeq_valuation_candidate_power_product = ff_q_bcpvzeq_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_bcpvzeq_valuation)) * ff_v_bcpvzeq_valuation_candidate_power_product) + (bpv_result_bcpvzeq_valuation_candidate))) /\ forall ff_i_bcpvzeq_valuation_candidate_power_product. (exists ff_lt_bcpvzeq_valuation_candidate_power_product_bound. ff_lt_bcpvzeq_valuation_candidate_power_product_bound + S ff_i_bcpvzeq_valuation_candidate_power_product = bpv_candidate_bcpvzeq_valuation) -> exists ff_p_bcpvzeq_valuation_candidate_power_product ff_r_bcpvzeq_valuation_candidate_power_product ff_s_bcpvzeq_valuation_candidate_power_product. ((((exists ff_h_bcpvzeq_valuation_candidate_power_product_factor. ff_h_bcpvzeq_valuation_candidate_power_product_factor + S (ff_p_bcpvzeq_valuation_candidate_power_product) = S ((S (ff_i_bcpvzeq_valuation_candidate_power_product)) * ff_c_bcpvzeq_valuation_candidate_power)) /\ exists ff_q_bcpvzeq_valuation_candidate_power_product_factor. ff_b_bcpvzeq_valuation_candidate_power = ff_q_bcpvzeq_valuation_candidate_power_product_factor * S ((S (ff_i_bcpvzeq_valuation_candidate_power_product)) * ff_c_bcpvzeq_valuation_candidate_power) + (ff_p_bcpvzeq_valuation_candidate_power_product))) /\ ((((exists ff_h_bcpvzeq_valuation_candidate_power_product_partial. ff_h_bcpvzeq_valuation_candidate_power_product_partial + S (ff_r_bcpvzeq_valuation_candidate_power_product) = S ((S (ff_i_bcpvzeq_valuation_candidate_power_product)) * ff_v_bcpvzeq_valuation_candidate_power_product)) /\ exists ff_q_bcpvzeq_valuation_candidate_power_product_partial. ff_u_bcpvzeq_valuation_candidate_power_product = ff_q_bcpvzeq_valuation_candidate_power_product_partial * S ((S (ff_i_bcpvzeq_valuation_candidate_power_product)) * ff_v_bcpvzeq_valuation_candidate_power_product) + (ff_r_bcpvzeq_valuation_candidate_power_product))) /\ ((((exists ff_h_bcpvzeq_valuation_candidate_power_product_successor. ff_h_bcpvzeq_valuation_candidate_power_product_successor + S (ff_s_bcpvzeq_valuation_candidate_power_product) = S ((S (S ff_i_bcpvzeq_valuation_candidate_power_product)) * ff_v_bcpvzeq_valuation_candidate_power_product)) /\ exists ff_q_bcpvzeq_valuation_candidate_power_product_successor. ff_u_bcpvzeq_valuation_candidate_power_product = ff_q_bcpvzeq_valuation_candidate_power_product_successor * S ((S (S ff_i_bcpvzeq_valuation_candidate_power_product)) * ff_v_bcpvzeq_valuation_candidate_power_product) + (ff_s_bcpvzeq_valuation_candidate_power_product))) /\ ff_s_bcpvzeq_valuation_candidate_power_product = ff_r_bcpvzeq_valuation_candidate_power_product * ff_p_bcpvzeq_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_bcpvzeq_valuation_candidate_divides. C = bpv_result_bcpvzeq_valuation_candidate * bpv_factor_bcpvzeq_valuation_candidate_divides))) -> (exists bpv_gap_bcpvzeq_valuation_maximal. bpv_gap_bcpvzeq_valuation_maximal + bpv_candidate_bcpvzeq_valuation = v)) -> (exists bcf_le_gap_bcpvzeq_base. bcf_le_gap_bcpvzeq_base + (1) = p) -> (exists bpvi_b_bcpvzeq_square bpvi_c_bcpvzeq_square. ((forall bpvi_i_bcpvzeq_square. (exists bpvi_repeat_gap_bcpvzeq_square. bpvi_repeat_gap_bcpvzeq_square + S bpvi_i_bcpvzeq_square = 2) -> (((exists bpvi_h_bcpvzeq_square_repeat. bpvi_h_bcpvzeq_square_repeat + S (p) = S ((S (bpvi_i_bcpvzeq_square)) * bpvi_c_bcpvzeq_square)) /\ exists bpvi_q_bcpvzeq_square_repeat. bpvi_b_bcpvzeq_square = bpvi_q_bcpvzeq_square_repeat * S ((S (bpvi_i_bcpvzeq_square)) * bpvi_c_bcpvzeq_square) + (p)))) /\ (exists bpvi_u_bcpvzeq_square bpvi_v_bcpvzeq_square. ((((exists bpvi_h_bcpvzeq_square_start. bpvi_h_bcpvzeq_square_start + S (1) = S ((S (0)) * bpvi_v_bcpvzeq_square)) /\ exists bpvi_q_bcpvzeq_square_start. bpvi_u_bcpvzeq_square = bpvi_q_bcpvzeq_square_start * S ((S (0)) * bpvi_v_bcpvzeq_square) + (1))) /\ ((((exists bpvi_h_bcpvzeq_square_terminal. bpvi_h_bcpvzeq_square_terminal + S (s) = S ((S (2)) * bpvi_v_bcpvzeq_square)) /\ exists bpvi_q_bcpvzeq_square_terminal. bpvi_u_bcpvzeq_square = bpvi_q_bcpvzeq_square_terminal * S ((S (2)) * bpvi_v_bcpvzeq_square) + (s))) /\ forall bpvi_j_bcpvzeq_square. (exists bpvi_product_gap_bcpvzeq_square. bpvi_product_gap_bcpvzeq_square + S bpvi_j_bcpvzeq_square = 2) -> exists bpvi_factor_bcpvzeq_square bpvi_partial_bcpvzeq_square bpvi_successor_bcpvzeq_square. ((((exists bpvi_h_bcpvzeq_square_factor. bpvi_h_bcpvzeq_square_factor + S (bpvi_factor_bcpvzeq_square) = S ((S (bpvi_j_bcpvzeq_square)) * bpvi_c_bcpvzeq_square)) /\ exists bpvi_q_bcpvzeq_square_factor. bpvi_b_bcpvzeq_square = bpvi_q_bcpvzeq_square_factor * S ((S (bpvi_j_bcpvzeq_square)) * bpvi_c_bcpvzeq_square) + (bpvi_factor_bcpvzeq_square))) /\ ((((exists bpvi_h_bcpvzeq_square_partial. bpvi_h_bcpvzeq_square_partial + S (bpvi_partial_bcpvzeq_square) = S ((S (bpvi_j_bcpvzeq_square)) * bpvi_v_bcpvzeq_square)) /\ exists bpvi_q_bcpvzeq_square_partial. bpvi_u_bcpvzeq_square = bpvi_q_bcpvzeq_square_partial * S ((S (bpvi_j_bcpvzeq_square)) * bpvi_v_bcpvzeq_square) + (bpvi_partial_bcpvzeq_square))) /\ ((((exists bpvi_h_bcpvzeq_square_successor. bpvi_h_bcpvzeq_square_successor + S (bpvi_successor_bcpvzeq_square) = S ((S (S bpvi_j_bcpvzeq_square)) * bpvi_v_bcpvzeq_square)) /\ exists bpvi_q_bcpvzeq_square_successor. bpvi_u_bcpvzeq_square = bpvi_q_bcpvzeq_square_successor * S ((S (S bpvi_j_bcpvzeq_square)) * bpvi_v_bcpvzeq_square) + (bpvi_successor_bcpvzeq_square))) /\ bpvi_successor_bcpvzeq_square = bpvi_partial_bcpvzeq_square * bpvi_factor_bcpvzeq_square)))))))) -> (exists bcf_lt_gap_bcpvzeq_strict. bcf_lt_gap_bcpvzeq_strict + S (n + n) = s) -> (((n) = (p) * (q) + (r) /\ (exists bcf_lt_gap_bcpvzeq_first_bound. bcf_lt_gap_bcpvzeq_first_bound + S (r) = p))) -> (((n + n) = (p) * (q + q) + (R) /\ (exists bcf_lt_gap_bcpvzeq_double_bound. bcf_lt_gap_bcpvzeq_double_bound + S (R) = p))) -> v = 0

Structural proof guide

An all-zero carry prefix forces the exact central valuation to zero.

Direct prerequisites: central_binom_carry_bit_count, zero_or_succ, bit_count_positive_last_one, double_quotient_carry_prefix_entries_zero, beta_at_unique. The authored body proceeds by case analysis (14), intermediate claims (5), equality transport (2).

Proof neighborhood

Direct dependencies

BT00Y4 central_binom_carry_bit_count BT000Q zero_or_succ BT00Y1 bit_count_positive_last_one BT00YC double_quotient_carry_prefix_entries_zero BT0042 beta_at_unique

Direct dependents

BT00YE central_binom_prime_valuation_zero_two_thirds_range

Formal native tactic body

Dependencies are hypotheses of this body receipt. The focused endpoint audits separately check the complete empty-context certificates.

  1. 0001intro p
  2. 0002intro n
  3. 0003intro C
  4. 0004intro v
  5. 0005intro q
  6. 0006intro r
  7. 0007intro R
  8. 0008intro s
  9. 0009intro hp
  10. 0010intro hcentral
  11. 0011intro hvaluation
  12. 0012intro hbase
  13. 0013intro hsquare
  14. 0014intro hstrict
  15. 0015intro hfirst
  16. 0016intro hdouble
  17. 0017have hpackage : exists b c d e f g. (forall bls_index_bcpvzeq_left. (exists bls_gap_bcpvzeq_left_bound. bls_gap_bcpvzeq_left_bound + S (bls_index_bcpvzeq_left) = (n + n)) -> exists bls_power_bcpvzeq_left bls_quotient_bcpvzeq_left bls_remainder_bcpvzeq_left. ((exists bpvi_b_bls_bcpvzeq_left_power bpvi_c_bls_bcpvzeq_left_power. ((forall bpvi_i_bls_bcpvzeq_left_power. (exists bpvi_repeat_gap_bls_bcpvzeq_left_power. bpvi_repeat_gap_bls_bcpvzeq_left_power + S bpvi_i_bls_bcpvzeq_left_power = S bls_index_bcpvzeq_left) -> (((exists bpvi_h_bls_bcpvzeq_left_power_repeat. bpvi_h_bls_bcpvzeq_left_power_repeat + S (p) = S ((S (bpvi_i_bls_bcpvzeq_left_power)) * bpvi_c_bls_bcpvzeq_left_power)) /\ exists bpvi_q_bls_bcpvzeq_left_power_repeat. bpvi_b_bls_bcpvzeq_left_power = bpvi_q_bls_bcpvzeq_left_power_repeat * S ((S (bpvi_i_bls_bcpvzeq_left_power)) * bpvi_c_bls_bcpvzeq_left_power) + (p)))) /\ (exists bpvi_u_bls_bcpvzeq_left_power bpvi_v_bls_bcpvzeq_left_power. ((((exists bpvi_h_bls_bcpvzeq_left_power_start. bpvi_h_bls_bcpvzeq_left_power_start + S (1) = S ((S (0)) * bpvi_v_bls_bcpvzeq_left_power)) /\ exists bpvi_q_bls_bcpvzeq_left_power_start. bpvi_u_bls_bcpvzeq_left_power = bpvi_q_bls_bcpvzeq_left_power_start * S ((S (0)) * bpvi_v_bls_bcpvzeq_left_power) + (1))) /\ ((((exists bpvi_h_bls_bcpvzeq_left_power_terminal. bpvi_h_bls_bcpvzeq_left_power_terminal + S (bls_power_bcpvzeq_left) = S ((S (S bls_index_bcpvzeq_left)) * bpvi_v_bls_bcpvzeq_left_power)) /\ exists bpvi_q_bls_bcpvzeq_left_power_terminal. bpvi_u_bls_bcpvzeq_left_power = bpvi_q_bls_bcpvzeq_left_power_terminal * S ((S (S bls_index_bcpvzeq_left)) * bpvi_v_bls_bcpvzeq_left_power) + (bls_power_bcpvzeq_left))) /\ forall bpvi_j_bls_bcpvzeq_left_power. (exists bpvi_product_gap_bls_bcpvzeq_left_power. bpvi_product_gap_bls_bcpvzeq_left_power + S bpvi_j_bls_bcpvzeq_left_power = S bls_index_bcpvzeq_left) -> exists bpvi_factor_bls_bcpvzeq_left_power bpvi_partial_bls_bcpvzeq_left_power bpvi_successor_bls_bcpvzeq_left_power. ((((exists bpvi_h_bls_bcpvzeq_left_power_factor. bpvi_h_bls_bcpvzeq_left_power_factor + S (bpvi_factor_bls_bcpvzeq_left_power) = S ((S (bpvi_j_bls_bcpvzeq_left_power)) * bpvi_c_bls_bcpvzeq_left_power)) /\ exists bpvi_q_bls_bcpvzeq_left_power_factor. bpvi_b_bls_bcpvzeq_left_power = bpvi_q_bls_bcpvzeq_left_power_factor * S ((S (bpvi_j_bls_bcpvzeq_left_power)) * bpvi_c_bls_bcpvzeq_left_power) + (bpvi_factor_bls_bcpvzeq_left_power))) /\ ((((exists bpvi_h_bls_bcpvzeq_left_power_partial. bpvi_h_bls_bcpvzeq_left_power_partial + S (bpvi_partial_bls_bcpvzeq_left_power) = S ((S (bpvi_j_bls_bcpvzeq_left_power)) * bpvi_v_bls_bcpvzeq_left_power)) /\ exists bpvi_q_bls_bcpvzeq_left_power_partial. bpvi_u_bls_bcpvzeq_left_power = bpvi_q_bls_bcpvzeq_left_power_partial * S ((S (bpvi_j_bls_bcpvzeq_left_power)) * bpvi_v_bls_bcpvzeq_left_power) + (bpvi_partial_bls_bcpvzeq_left_power))) /\ ((((exists bpvi_h_bls_bcpvzeq_left_power_successor. bpvi_h_bls_bcpvzeq_left_power_successor + S (bpvi_successor_bls_bcpvzeq_left_power) = S ((S (S bpvi_j_bls_bcpvzeq_left_power)) * bpvi_v_bls_bcpvzeq_left_power)) /\ exists bpvi_q_bls_bcpvzeq_left_power_successor. bpvi_u_bls_bcpvzeq_left_power = bpvi_q_bls_bcpvzeq_left_power_successor * S ((S (S bpvi_j_bls_bcpvzeq_left_power)) * bpvi_v_bls_bcpvzeq_left_power) + (bpvi_successor_bls_bcpvzeq_left_power))) /\ bpvi_successor_bls_bcpvzeq_left_power = bpvi_partial_bls_bcpvzeq_left_power * bpvi_factor_bls_bcpvzeq_left_power)))))))) /\ ((((exists ff_h_bls_bcpvzeq_left_quotient_entry. ff_h_bls_bcpvzeq_left_quotient_entry + S (bls_quotient_bcpvzeq_left) = S ((S (bls_index_bcpvzeq_left)) * c)) /\ exists ff_q_bls_bcpvzeq_left_quotient_entry. b = ff_q_bls_bcpvzeq_left_quotient_entry * S ((S (bls_index_bcpvzeq_left)) * c) + (bls_quotient_bcpvzeq_left))) /\ ((n = bls_power_bcpvzeq_left * bls_quotient_bcpvzeq_left + bls_remainder_bcpvzeq_left /\ exists bls_remainder_gap_bcpvzeq_left_division. bls_remainder_gap_bcpvzeq_left_division + S (bls_remainder_bcpvzeq_left) = bls_power_bcpvzeq_left))))) /\ ((forall bls_index_bcpvzeq_right. (exists bls_gap_bcpvzeq_right_bound. bls_gap_bcpvzeq_right_bound + S (bls_index_bcpvzeq_right) = (n + n)) -> exists bls_power_bcpvzeq_right bls_quotient_bcpvzeq_right bls_remainder_bcpvzeq_right. ((exists bpvi_b_bls_bcpvzeq_right_power bpvi_c_bls_bcpvzeq_right_power. ((forall bpvi_i_bls_bcpvzeq_right_power. (exists bpvi_repeat_gap_bls_bcpvzeq_right_power. bpvi_repeat_gap_bls_bcpvzeq_right_power + S bpvi_i_bls_bcpvzeq_right_power = S bls_index_bcpvzeq_right) -> (((exists bpvi_h_bls_bcpvzeq_right_power_repeat. bpvi_h_bls_bcpvzeq_right_power_repeat + S (p) = S ((S (bpvi_i_bls_bcpvzeq_right_power)) * bpvi_c_bls_bcpvzeq_right_power)) /\ exists bpvi_q_bls_bcpvzeq_right_power_repeat. bpvi_b_bls_bcpvzeq_right_power = bpvi_q_bls_bcpvzeq_right_power_repeat * S ((S (bpvi_i_bls_bcpvzeq_right_power)) * bpvi_c_bls_bcpvzeq_right_power) + (p)))) /\ (exists bpvi_u_bls_bcpvzeq_right_power bpvi_v_bls_bcpvzeq_right_power. ((((exists bpvi_h_bls_bcpvzeq_right_power_start. bpvi_h_bls_bcpvzeq_right_power_start + S (1) = S ((S (0)) * bpvi_v_bls_bcpvzeq_right_power)) /\ exists bpvi_q_bls_bcpvzeq_right_power_start. bpvi_u_bls_bcpvzeq_right_power = bpvi_q_bls_bcpvzeq_right_power_start * S ((S (0)) * bpvi_v_bls_bcpvzeq_right_power) + (1))) /\ ((((exists bpvi_h_bls_bcpvzeq_right_power_terminal. bpvi_h_bls_bcpvzeq_right_power_terminal + S (bls_power_bcpvzeq_right) = S ((S (S bls_index_bcpvzeq_right)) * bpvi_v_bls_bcpvzeq_right_power)) /\ exists bpvi_q_bls_bcpvzeq_right_power_terminal. bpvi_u_bls_bcpvzeq_right_power = bpvi_q_bls_bcpvzeq_right_power_terminal * S ((S (S bls_index_bcpvzeq_right)) * bpvi_v_bls_bcpvzeq_right_power) + (bls_power_bcpvzeq_right))) /\ forall bpvi_j_bls_bcpvzeq_right_power. (exists bpvi_product_gap_bls_bcpvzeq_right_power. bpvi_product_gap_bls_bcpvzeq_right_power + S bpvi_j_bls_bcpvzeq_right_power = S bls_index_bcpvzeq_right) -> exists bpvi_factor_bls_bcpvzeq_right_power bpvi_partial_bls_bcpvzeq_right_power bpvi_successor_bls_bcpvzeq_right_power. ((((exists bpvi_h_bls_bcpvzeq_right_power_factor. bpvi_h_bls_bcpvzeq_right_power_factor + S (bpvi_factor_bls_bcpvzeq_right_power) = S ((S (bpvi_j_bls_bcpvzeq_right_power)) * bpvi_c_bls_bcpvzeq_right_power)) /\ exists bpvi_q_bls_bcpvzeq_right_power_factor. bpvi_b_bls_bcpvzeq_right_power = bpvi_q_bls_bcpvzeq_right_power_factor * S ((S (bpvi_j_bls_bcpvzeq_right_power)) * bpvi_c_bls_bcpvzeq_right_power) + (bpvi_factor_bls_bcpvzeq_right_power))) /\ ((((exists bpvi_h_bls_bcpvzeq_right_power_partial. bpvi_h_bls_bcpvzeq_right_power_partial + S (bpvi_partial_bls_bcpvzeq_right_power) = S ((S (bpvi_j_bls_bcpvzeq_right_power)) * bpvi_v_bls_bcpvzeq_right_power)) /\ exists bpvi_q_bls_bcpvzeq_right_power_partial. bpvi_u_bls_bcpvzeq_right_power = bpvi_q_bls_bcpvzeq_right_power_partial * S ((S (bpvi_j_bls_bcpvzeq_right_power)) * bpvi_v_bls_bcpvzeq_right_power) + (bpvi_partial_bls_bcpvzeq_right_power))) /\ ((((exists bpvi_h_bls_bcpvzeq_right_power_successor. bpvi_h_bls_bcpvzeq_right_power_successor + S (bpvi_successor_bls_bcpvzeq_right_power) = S ((S (S bpvi_j_bls_bcpvzeq_right_power)) * bpvi_v_bls_bcpvzeq_right_power)) /\ exists bpvi_q_bls_bcpvzeq_right_power_successor. bpvi_u_bls_bcpvzeq_right_power = bpvi_q_bls_bcpvzeq_right_power_successor * S ((S (S bpvi_j_bls_bcpvzeq_right_power)) * bpvi_v_bls_bcpvzeq_right_power) + (bpvi_successor_bls_bcpvzeq_right_power))) /\ bpvi_successor_bls_bcpvzeq_right_power = bpvi_partial_bls_bcpvzeq_right_power * bpvi_factor_bls_bcpvzeq_right_power)))))))) /\ ((((exists ff_h_bls_bcpvzeq_right_quotient_entry. ff_h_bls_bcpvzeq_right_quotient_entry + S (bls_quotient_bcpvzeq_right) = S ((S (bls_index_bcpvzeq_right)) * e)) /\ exists ff_q_bls_bcpvzeq_right_quotient_entry. d = ff_q_bls_bcpvzeq_right_quotient_entry * S ((S (bls_index_bcpvzeq_right)) * e) + (bls_quotient_bcpvzeq_right))) /\ ((n + n = bls_power_bcpvzeq_right * bls_quotient_bcpvzeq_right + bls_remainder_bcpvzeq_right /\ exists bls_remainder_gap_bcpvzeq_right_division. bls_remainder_gap_bcpvzeq_right_division + S (bls_remainder_bcpvzeq_right) = bls_power_bcpvzeq_right))))) /\ ((forall b5cc_index_bcpvzeq_carry. (exists bcf_lt_gap_bcpvzeq_carry_bound. bcf_lt_gap_bcpvzeq_carry_bound + S (b5cc_index_bcpvzeq_carry) = n + n) -> exists b5cc_left_bcpvzeq_carry b5cc_right_bcpvzeq_carry b5cc_bit_bcpvzeq_carry. (((exists fs_h_b5cc_bcpvzeq_carry_left. fs_h_b5cc_bcpvzeq_carry_left + S (b5cc_left_bcpvzeq_carry) = S ((S (b5cc_index_bcpvzeq_carry)) * c)) /\ exists fs_q_b5cc_bcpvzeq_carry_left. b = fs_q_b5cc_bcpvzeq_carry_left * S ((S (b5cc_index_bcpvzeq_carry)) * c) + (b5cc_left_bcpvzeq_carry))) /\ ((((exists fs_h_b5cc_bcpvzeq_carry_right. fs_h_b5cc_bcpvzeq_carry_right + S (b5cc_right_bcpvzeq_carry) = S ((S (b5cc_index_bcpvzeq_carry)) * e)) /\ exists fs_q_b5cc_bcpvzeq_carry_right. d = fs_q_b5cc_bcpvzeq_carry_right * S ((S (b5cc_index_bcpvzeq_carry)) * e) + (b5cc_right_bcpvzeq_carry))) /\ ((((exists fs_h_b5cc_bcpvzeq_carry_bit. fs_h_b5cc_bcpvzeq_carry_bit + S (b5cc_bit_bcpvzeq_carry) = S ((S (b5cc_index_bcpvzeq_carry)) * g)) /\ exists fs_q_b5cc_bcpvzeq_carry_bit. f = fs_q_b5cc_bcpvzeq_carry_bit * S ((S (b5cc_index_bcpvzeq_carry)) * g) + (b5cc_bit_bcpvzeq_carry))) /\ (((b5cc_bit_bcpvzeq_carry = 0 /\ b5cc_right_bcpvzeq_carry = b5cc_left_bcpvzeq_carry + b5cc_left_bcpvzeq_carry) \/ (b5cc_bit_bcpvzeq_carry = 1 /\ b5cc_right_bcpvzeq_carry = S (b5cc_left_bcpvzeq_carry + b5cc_left_bcpvzeq_carry))))))) /\ (((exists ff_u_bcpvzeq_count_sum ff_v_bcpvzeq_count_sum. ((((exists ff_h_bcpvzeq_count_sum_start. ff_h_bcpvzeq_count_sum_start + S (0) = S ((S (0)) * ff_v_bcpvzeq_count_sum)) /\ exists ff_q_bcpvzeq_count_sum_start. ff_u_bcpvzeq_count_sum = ff_q_bcpvzeq_count_sum_start * S ((S (0)) * ff_v_bcpvzeq_count_sum) + (0))) /\ ((((exists ff_h_bcpvzeq_count_sum_terminal. ff_h_bcpvzeq_count_sum_terminal + S ((v)) = S ((S ((n + n))) * ff_v_bcpvzeq_count_sum)) /\ exists ff_q_bcpvzeq_count_sum_terminal. ff_u_bcpvzeq_count_sum = ff_q_bcpvzeq_count_sum_terminal * S ((S ((n + n))) * ff_v_bcpvzeq_count_sum) + ((v)))) /\ forall ff_i_bcpvzeq_count_sum. (exists ff_lt_bcpvzeq_count_sum_bound. ff_lt_bcpvzeq_count_sum_bound + S ff_i_bcpvzeq_count_sum = (n + n)) -> exists ff_a_bcpvzeq_count_sum ff_r_bcpvzeq_count_sum ff_s_bcpvzeq_count_sum. ((((exists ff_h_bcpvzeq_count_sum_summand. ff_h_bcpvzeq_count_sum_summand + S (ff_a_bcpvzeq_count_sum) = S ((S (ff_i_bcpvzeq_count_sum)) * g)) /\ exists ff_q_bcpvzeq_count_sum_summand. f = ff_q_bcpvzeq_count_sum_summand * S ((S (ff_i_bcpvzeq_count_sum)) * g) + (ff_a_bcpvzeq_count_sum))) /\ ((((exists ff_h_bcpvzeq_count_sum_partial. ff_h_bcpvzeq_count_sum_partial + S (ff_r_bcpvzeq_count_sum) = S ((S (ff_i_bcpvzeq_count_sum)) * ff_v_bcpvzeq_count_sum)) /\ exists ff_q_bcpvzeq_count_sum_partial. ff_u_bcpvzeq_count_sum = ff_q_bcpvzeq_count_sum_partial * S ((S (ff_i_bcpvzeq_count_sum)) * ff_v_bcpvzeq_count_sum) + (ff_r_bcpvzeq_count_sum))) /\ ((((exists ff_h_bcpvzeq_count_sum_successor. ff_h_bcpvzeq_count_sum_successor + S (ff_s_bcpvzeq_count_sum) = S ((S (S ff_i_bcpvzeq_count_sum)) * ff_v_bcpvzeq_count_sum)) /\ exists ff_q_bcpvzeq_count_sum_successor. ff_u_bcpvzeq_count_sum = ff_q_bcpvzeq_count_sum_successor * S ((S (S ff_i_bcpvzeq_count_sum)) * ff_v_bcpvzeq_count_sum) + (ff_s_bcpvzeq_count_sum))) /\ ff_s_bcpvzeq_count_sum = ff_r_bcpvzeq_count_sum + ff_a_bcpvzeq_count_sum)))))) /\ (forall ff_i_bcpvzeq_count_bits. (exists ff_lt_bcpvzeq_count_bits_bound. ff_lt_bcpvzeq_count_bits_bound + S ff_i_bcpvzeq_count_bits = (n + n)) -> exists ff_bit_bcpvzeq_count_bits. ((((exists ff_h_bcpvzeq_count_bits_decoded. ff_h_bcpvzeq_count_bits_decoded + S (ff_bit_bcpvzeq_count_bits) = S ((S (ff_i_bcpvzeq_count_bits)) * g)) /\ exists ff_q_bcpvzeq_count_bits_decoded. f = ff_q_bcpvzeq_count_bits_decoded * S ((S (ff_i_bcpvzeq_count_bits)) * g) + (ff_bit_bcpvzeq_count_bits))) /\ (ff_bit_bcpvzeq_count_bits = 0 \/ ff_bit_bcpvzeq_count_bits = 1)))))))
  18. 0018specialize central_binom_carry_bit_count p
  19. 0019specialize central_binom_carry_bit_count n
  20. 0020specialize central_binom_carry_bit_count C
  21. 0021specialize central_binom_carry_bit_count v
  22. 0022apply central_binom_carry_bit_count
  23. 0023exact hp
  24. 0024exact hcentral
  25. 0025exact hvaluation
  26. 0026cases hpackage
  27. 0027cases hpackage_witness
  28. 0028cases hpackage_witness_witness
  29. 0029cases hpackage_witness_witness_witness
  30. 0030cases hpackage_witness_witness_witness_witness
  31. 0031cases hpackage_witness_witness_witness_witness_witness
  32. 0032cases hpackage_witness_witness_witness_witness_witness_witness
  33. 0033cases hpackage_witness_witness_witness_witness_witness_witness_right
  34. 0034cases hpackage_witness_witness_witness_witness_witness_witness_right_right
  35. 0035specialize zero_or_succ v
  36. 0036cases zero_or_succ
  37. 0037exact zero_or_succ_left
  38. 0038cases zero_or_succ_right
  39. 0039rewrite zero_or_succ_right_witness at hpackage_witness_witness_witness_witness_witness_witness_right_right_right
  40. 0040rewrite zero_or_succ_right_witness at hpackage_witness_witness_witness_witness_witness_witness_right_right_right
  41. 0041have hlast : exists i. (exists bcf_lt_gap_bcpvzeq_last_bound. bcf_lt_gap_bcpvzeq_last_bound + S (i) = n + n) /\ ((((exists fs_h_bcpvzeq_last_entry. fs_h_bcpvzeq_last_entry + S (1) = S ((S (i)) * x5)) /\ exists fs_q_bcpvzeq_last_entry. x4 = fs_q_bcpvzeq_last_entry * S ((S (i)) * x5) + (1))) /\ (exists bcf_le_gap_bcpvzeq_last_result. bcf_le_gap_bcpvzeq_last_result + (S x6) = S i))
  42. 0042specialize bit_count_positive_last_one x4
  43. 0043specialize bit_count_positive_last_one x5
  44. 0044specialize bit_count_positive_last_one (n + n)
  45. 0045specialize bit_count_positive_last_one x6
  46. 0046apply bit_count_positive_last_one
  47. 0047exact hpackage_witness_witness_witness_witness_witness_witness_right_right_right
  48. 0048cases hlast
  49. 0049cases hlast_witness
  50. 0050cases hlast_witness_right
  51. 0051have hentries : forall i. (exists bcf_lt_gap_bcpvzeq_zero_bound. bcf_lt_gap_bcpvzeq_zero_bound + S (i) = n + n) -> (((exists fs_h_bcpvzeq_zero_entries. fs_h_bcpvzeq_zero_entries + S (0) = S ((S (i)) * x5)) /\ exists fs_q_bcpvzeq_zero_entries. x4 = fs_q_bcpvzeq_zero_entries * S ((S (i)) * x5) + (0)))
  52. 0052specialize double_quotient_carry_prefix_entries_zero p
  53. 0053specialize double_quotient_carry_prefix_entries_zero n
  54. 0054specialize double_quotient_carry_prefix_entries_zero x
  55. 0055specialize double_quotient_carry_prefix_entries_zero x1
  56. 0056specialize double_quotient_carry_prefix_entries_zero x2
  57. 0057specialize double_quotient_carry_prefix_entries_zero x3
  58. 0058specialize double_quotient_carry_prefix_entries_zero x4
  59. 0059specialize double_quotient_carry_prefix_entries_zero x5
  60. 0060specialize double_quotient_carry_prefix_entries_zero (n + n)
  61. 0061specialize double_quotient_carry_prefix_entries_zero q
  62. 0062specialize double_quotient_carry_prefix_entries_zero r
  63. 0063specialize double_quotient_carry_prefix_entries_zero R
  64. 0064specialize double_quotient_carry_prefix_entries_zero s
  65. 0065apply double_quotient_carry_prefix_entries_zero
  66. 0066exact hbase
  67. 0067exact hsquare
  68. 0068exact hstrict
  69. 0069exact hpackage_witness_witness_witness_witness_witness_witness_left
  70. 0070exact hpackage_witness_witness_witness_witness_witness_witness_right_left
  71. 0071exact hpackage_witness_witness_witness_witness_witness_witness_right_right_left
  72. 0072exact hfirst
  73. 0073exact hdouble
  74. 0074have hzero : ((exists fs_h_bcpvzeq_zero_entry. fs_h_bcpvzeq_zero_entry + S (0) = S ((S (x7)) * x5)) /\ exists fs_q_bcpvzeq_zero_entry. x4 = fs_q_bcpvzeq_zero_entry * S ((S (x7)) * x5) + (0))
  75. 0075specialize hentries x7
  76. 0076apply hentries
  77. 0077exact hlast_witness_left
  78. 0078have hone_zero : 1 = 0
  79. 0079specialize beta_at_unique x4
  80. 0080specialize beta_at_unique x5
  81. 0081specialize beta_at_unique x7
  82. 0082specialize beta_at_unique 1
  83. 0083specialize beta_at_unique 0
  84. 0084apply beta_at_unique
  85. 0085exact hlast_witness_right_left
  86. 0086exact hzero
  87. 0087exfalso
  88. 0088apply PA1
  89. 0089exact hone_zero