four_pow_lt_mul_central_binom

Proof neighborhood

Direct dependencies

Direct dependents

Formal native tactic body

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

1. 0001have hzero_lt_four : exists bcf_lt_gap_bfplcb_zero_lt_four. bcf_lt_gap_bfplcb_zero_lt_four + S (0) = 4
2. 0002exists 3
3. 0003norm_num
4. 0004have hone_lt_four : exists bcf_lt_gap_bfplcb_one_lt_four. bcf_lt_gap_bfplcb_one_lt_four + S (1) = 4
5. 0005exists 2
6. 0006norm_num
7. 0007have htwo_lt_four : exists bcf_lt_gap_bfplcb_two_lt_four. bcf_lt_gap_bfplcb_two_lt_four + S (2) = 4
8. 0008exists 1
9. 0009norm_num
10. 0010have hthree_lt_four : exists bcf_lt_gap_bfplcb_three_lt_four. bcf_lt_gap_bfplcb_three_lt_four + S (3) = 4
11. 0011exists 0
12. 0012norm_num
13. 0013have hpackage : (forall n. exists z. (((exists bcf_lt_gap_bcb4we_exists_out_of_range. bcf_lt_gap_bcb4we_exists_out_of_range + S (n + n) = n) /\ z = 0) \/ ((exists bcf_le_gap_bcb4we_exists_in_range. bcf_le_gap_bcb4we_exists_in_range + (n) = n + n) /\ (exists bcf_row_code_code_bcb4we_exists bcf_row_code_scale_bcb4we_exists bcf_row_scale_code_bcb4we_exists bcf_row_scale_scale_bcb4we_exists bcf_row_code_bcb4we_exists bcf_row_scale_bcb4we_exists. ((forall bcf_row_index_bcb4we_exists_table. (exists bcf_lt_gap_bcb4we_exists_table_row_bound. bcf_lt_gap_bcb4we_exists_table_row_bound + S (bcf_row_index_bcb4we_exists_table) = S (n + n)) -> exists bcf_row_code_bcb4we_exists_table bcf_row_scale_bcb4we_exists_table. ((((exists bcf_height_bcb4we_exists_table_decoded_row_code. bcf_height_bcb4we_exists_table_decoded_row_code + S (bcf_row_code_bcb4we_exists_table) = S ((S (bcf_row_index_bcb4we_exists_table)) * bcf_row_code_scale_bcb4we_exists)) /\ exists bcf_quotient_bcb4we_exists_table_decoded_row_code. bcf_row_code_code_bcb4we_exists = bcf_quotient_bcb4we_exists_table_decoded_row_code * S ((S (bcf_row_index_bcb4we_exists_table)) * bcf_row_code_scale_bcb4we_exists) + (bcf_row_code_bcb4we_exists_table))) /\ ((((exists bcf_height_bcb4we_exists_table_decoded_row_scale. bcf_height_bcb4we_exists_table_decoded_row_scale + S (bcf_row_scale_bcb4we_exists_table) = S ((S (bcf_row_index_bcb4we_exists_table)) * bcf_row_scale_scale_bcb4we_exists)) /\ exists bcf_quotient_bcb4we_exists_table_decoded_row_scale. bcf_row_scale_code_bcb4we_exists = bcf_quotient_bcb4we_exists_table_decoded_row_scale * S ((S (bcf_row_index_bcb4we_exists_table)) * bcf_row_scale_scale_bcb4we_exists) + (bcf_row_scale_bcb4we_exists_table))) /\ ((bcf_row_index_bcb4we_exists_table = 0 /\ (forall bcf_index_bcb4we_exists_table_zero_row. (exists bcf_lt_gap_bcb4we_exists_table_zero_row_bound. bcf_lt_gap_bcb4we_exists_table_zero_row_bound + S (bcf_index_bcb4we_exists_table_zero_row) = S (n + n)) -> exists bcf_value_bcb4we_exists_table_zero_row. ((((exists bcf_height_bcb4we_exists_table_zero_row_entry. bcf_height_bcb4we_exists_table_zero_row_entry + S (bcf_value_bcb4we_exists_table_zero_row) = S ((S (bcf_index_bcb4we_exists_table_zero_row)) * bcf_row_scale_bcb4we_exists_table)) /\ exists bcf_quotient_bcb4we_exists_table_zero_row_entry. bcf_row_code_bcb4we_exists_table = bcf_quotient_bcb4we_exists_table_zero_row_entry * S ((S (bcf_index_bcb4we_exists_table_zero_row)) * bcf_row_scale_bcb4we_exists_table) + (bcf_value_bcb4we_exists_table_zero_row))) /\ ((bcf_index_bcb4we_exists_table_zero_row = 0 /\ bcf_value_bcb4we_exists_table_zero_row = 1) \/ exists bcf_predecessor_bcb4we_exists_table_zero_row. bcf_index_bcb4we_exists_table_zero_row = S bcf_predecessor_bcb4we_exists_table_zero_row /\ bcf_value_bcb4we_exists_table_zero_row = 0)))) \/ exists bcf_predecessor_bcb4we_exists_table bcf_previous_code_bcb4we_exists_table bcf_previous_scale_bcb4we_exists_table. bcf_row_index_bcb4we_exists_table = S bcf_predecessor_bcb4we_exists_table /\ ((((exists bcf_height_bcb4we_exists_table_decoded_previous_code. bcf_height_bcb4we_exists_table_decoded_previous_code + S (bcf_previous_code_bcb4we_exists_table) = S ((S (bcf_predecessor_bcb4we_exists_table)) * bcf_row_code_scale_bcb4we_exists)) /\ exists bcf_quotient_bcb4we_exists_table_decoded_previous_code. bcf_row_code_code_bcb4we_exists = bcf_quotient_bcb4we_exists_table_decoded_previous_code * S ((S (bcf_predecessor_bcb4we_exists_table)) * bcf_row_code_scale_bcb4we_exists) + (bcf_previous_code_bcb4we_exists_table))) /\ ((((exists bcf_height_bcb4we_exists_table_decoded_previous_scale. bcf_height_bcb4we_exists_table_decoded_previous_scale + S (bcf_previous_scale_bcb4we_exists_table) = S ((S (bcf_predecessor_bcb4we_exists_table)) * bcf_row_scale_scale_bcb4we_exists)) /\ exists bcf_quotient_bcb4we_exists_table_decoded_previous_scale. bcf_row_scale_code_bcb4we_exists = bcf_quotient_bcb4we_exists_table_decoded_previous_scale * S ((S (bcf_predecessor_bcb4we_exists_table)) * bcf_row_scale_scale_bcb4we_exists) + (bcf_previous_scale_bcb4we_exists_table))) /\ (forall bcf_index_bcb4we_exists_table_row_step. (exists bcf_lt_gap_bcb4we_exists_table_row_step_bound. bcf_lt_gap_bcb4we_exists_table_row_step_bound + S (bcf_index_bcb4we_exists_table_row_step) = S (n + n)) -> exists bcf_value_bcb4we_exists_table_row_step. ((((exists bcf_height_bcb4we_exists_table_row_step_entry. bcf_height_bcb4we_exists_table_row_step_entry + S (bcf_value_bcb4we_exists_table_row_step) = S ((S (bcf_index_bcb4we_exists_table_row_step)) * bcf_row_scale_bcb4we_exists_table)) /\ exists bcf_quotient_bcb4we_exists_table_row_step_entry. bcf_row_code_bcb4we_exists_table = bcf_quotient_bcb4we_exists_table_row_step_entry * S ((S (bcf_index_bcb4we_exists_table_row_step)) * bcf_row_scale_bcb4we_exists_table) + (bcf_value_bcb4we_exists_table_row_step))) /\ ((bcf_index_bcb4we_exists_table_row_step = 0 /\ bcf_value_bcb4we_exists_table_row_step = 1) \/ exists bcf_predecessor_bcb4we_exists_table_row_step bcf_left_bcb4we_exists_table_row_step bcf_right_bcb4we_exists_table_row_step. bcf_index_bcb4we_exists_table_row_step = S bcf_predecessor_bcb4we_exists_table_row_step /\ ((((exists bcf_height_bcb4we_exists_table_row_step_previous_left. bcf_height_bcb4we_exists_table_row_step_previous_left + S (bcf_left_bcb4we_exists_table_row_step) = S ((S (bcf_predecessor_bcb4we_exists_table_row_step)) * bcf_previous_scale_bcb4we_exists_table)) /\ exists bcf_quotient_bcb4we_exists_table_row_step_previous_left. bcf_previous_code_bcb4we_exists_table = bcf_quotient_bcb4we_exists_table_row_step_previous_left * S ((S (bcf_predecessor_bcb4we_exists_table_row_step)) * bcf_previous_scale_bcb4we_exists_table) + (bcf_left_bcb4we_exists_table_row_step))) /\ ((((exists bcf_height_bcb4we_exists_table_row_step_previous_right. bcf_height_bcb4we_exists_table_row_step_previous_right + S (bcf_right_bcb4we_exists_table_row_step) = S ((S (S (bcf_predecessor_bcb4we_exists_table_row_step))) * bcf_previous_scale_bcb4we_exists_table)) /\ exists bcf_quotient_bcb4we_exists_table_row_step_previous_right. bcf_previous_code_bcb4we_exists_table = bcf_quotient_bcb4we_exists_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcb4we_exists_table_row_step))) * bcf_previous_scale_bcb4we_exists_table) + (bcf_right_bcb4we_exists_table_row_step))) /\ bcf_value_bcb4we_exists_table_row_step = bcf_left_bcb4we_exists_table_row_step + bcf_right_bcb4we_exists_table_row_step))))))))))) /\ ((((exists bcf_height_bcb4we_exists_decoded_row_code. bcf_height_bcb4we_exists_decoded_row_code + S (bcf_row_code_bcb4we_exists) = S ((S (n + n)) * bcf_row_code_scale_bcb4we_exists)) /\ exists bcf_quotient_bcb4we_exists_decoded_row_code. bcf_row_code_code_bcb4we_exists = bcf_quotient_bcb4we_exists_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_bcb4we_exists) + (bcf_row_code_bcb4we_exists))) /\ ((((exists bcf_height_bcb4we_exists_decoded_row_scale. bcf_height_bcb4we_exists_decoded_row_scale + S (bcf_row_scale_bcb4we_exists) = S ((S (n + n)) * bcf_row_scale_scale_bcb4we_exists)) /\ exists bcf_quotient_bcb4we_exists_decoded_row_scale. bcf_row_scale_code_bcb4we_exists = bcf_quotient_bcb4we_exists_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_bcb4we_exists) + (bcf_row_scale_bcb4we_exists))) /\ (((exists bcf_height_bcb4we_exists_decoded_value. bcf_height_bcb4we_exists_decoded_value + S (z) = S ((S (n)) * bcf_row_scale_bcb4we_exists)) /\ exists bcf_quotient_bcb4we_exists_decoded_value. bcf_row_code_bcb4we_exists = bcf_quotient_bcb4we_exists_decoded_value * S ((S (n)) * bcf_row_scale_bcb4we_exists) + (z)))))))))) /\ (forall p c. (exists pa_b_bfplcb4_power pa_c_bfplcb4_power. ((forall pa_i_bfplcb4_power_repeat. (exists pa_lt_bfplcb4_power_repeat_bound. pa_lt_bfplcb4_power_repeat_bound + S pa_i_bfplcb4_power_repeat = 4) -> (((exists pa_h_bfplcb4_power_repeat_decoded. pa_h_bfplcb4_power_repeat_decoded + S (4) = S ((S (pa_i_bfplcb4_power_repeat)) * pa_c_bfplcb4_power)) /\ exists pa_q_bfplcb4_power_repeat_decoded. pa_b_bfplcb4_power = pa_q_bfplcb4_power_repeat_decoded * S ((S (pa_i_bfplcb4_power_repeat)) * pa_c_bfplcb4_power) + (4)))) /\ (exists pa_u_bfplcb4_power_product pa_v_bfplcb4_power_product. ((((exists pa_h_bfplcb4_power_product_start. pa_h_bfplcb4_power_product_start + S (1) = S ((S (0)) * pa_v_bfplcb4_power_product)) /\ exists pa_q_bfplcb4_power_product_start. pa_u_bfplcb4_power_product = pa_q_bfplcb4_power_product_start * S ((S (0)) * pa_v_bfplcb4_power_product) + (1))) /\ ((((exists pa_h_bfplcb4_power_product_terminal. pa_h_bfplcb4_power_product_terminal + S (p) = S ((S (4)) * pa_v_bfplcb4_power_product)) /\ exists pa_q_bfplcb4_power_product_terminal. pa_u_bfplcb4_power_product = pa_q_bfplcb4_power_product_terminal * S ((S (4)) * pa_v_bfplcb4_power_product) + (p))) /\ forall pa_i_bfplcb4_power_product. (exists pa_lt_bfplcb4_power_product_bound. pa_lt_bfplcb4_power_product_bound + S pa_i_bfplcb4_power_product = 4) -> exists pa_p_bfplcb4_power_product pa_r_bfplcb4_power_product pa_s_bfplcb4_power_product. ((((exists pa_h_bfplcb4_power_product_factor. pa_h_bfplcb4_power_product_factor + S (pa_p_bfplcb4_power_product) = S ((S (pa_i_bfplcb4_power_product)) * pa_c_bfplcb4_power)) /\ exists pa_q_bfplcb4_power_product_factor. pa_b_bfplcb4_power = pa_q_bfplcb4_power_product_factor * S ((S (pa_i_bfplcb4_power_product)) * pa_c_bfplcb4_power) + (pa_p_bfplcb4_power_product))) /\ ((((exists pa_h_bfplcb4_power_product_partial. pa_h_bfplcb4_power_product_partial + S (pa_r_bfplcb4_power_product) = S ((S (pa_i_bfplcb4_power_product)) * pa_v_bfplcb4_power_product)) /\ exists pa_q_bfplcb4_power_product_partial. pa_u_bfplcb4_power_product = pa_q_bfplcb4_power_product_partial * S ((S (pa_i_bfplcb4_power_product)) * pa_v_bfplcb4_power_product) + (pa_r_bfplcb4_power_product))) /\ ((((exists pa_h_bfplcb4_power_product_successor. pa_h_bfplcb4_power_product_successor + S (pa_s_bfplcb4_power_product) = S ((S (S pa_i_bfplcb4_power_product)) * pa_v_bfplcb4_power_product)) /\ exists pa_q_bfplcb4_power_product_successor. pa_u_bfplcb4_power_product = pa_q_bfplcb4_power_product_successor * S ((S (S pa_i_bfplcb4_power_product)) * pa_v_bfplcb4_power_product) + (pa_s_bfplcb4_power_product))) /\ pa_s_bfplcb4_power_product = pa_r_bfplcb4_power_product * pa_p_bfplcb4_power_product)))))))) -> (((exists bcf_lt_gap_bfplcb4_central_out_of_range. bcf_lt_gap_bfplcb4_central_out_of_range + S (4 + 4) = 4) /\ c = 0) \/ ((exists bcf_le_gap_bfplcb4_central_in_range. bcf_le_gap_bfplcb4_central_in_range + (4) = 4 + 4) /\ (exists bcf_row_code_code_bfplcb4_central bcf_row_code_scale_bfplcb4_central bcf_row_scale_code_bfplcb4_central bcf_row_scale_scale_bfplcb4_central bcf_row_code_bfplcb4_central bcf_row_scale_bfplcb4_central. ((forall bcf_row_index_bfplcb4_central_table. (exists bcf_lt_gap_bfplcb4_central_table_row_bound. bcf_lt_gap_bfplcb4_central_table_row_bound + S (bcf_row_index_bfplcb4_central_table) = S (4 + 4)) -> exists bcf_row_code_bfplcb4_central_table bcf_row_scale_bfplcb4_central_table. ((((exists bcf_height_bfplcb4_central_table_decoded_row_code. bcf_height_bfplcb4_central_table_decoded_row_code + S (bcf_row_code_bfplcb4_central_table) = S ((S (bcf_row_index_bfplcb4_central_table)) * bcf_row_code_scale_bfplcb4_central)) /\ exists bcf_quotient_bfplcb4_central_table_decoded_row_code. bcf_row_code_code_bfplcb4_central = bcf_quotient_bfplcb4_central_table_decoded_row_code * S ((S (bcf_row_index_bfplcb4_central_table)) * bcf_row_code_scale_bfplcb4_central) + (bcf_row_code_bfplcb4_central_table))) /\ ((((exists bcf_height_bfplcb4_central_table_decoded_row_scale. bcf_height_bfplcb4_central_table_decoded_row_scale + S (bcf_row_scale_bfplcb4_central_table) = S ((S (bcf_row_index_bfplcb4_central_table)) * bcf_row_scale_scale_bfplcb4_central)) /\ exists bcf_quotient_bfplcb4_central_table_decoded_row_scale. bcf_row_scale_code_bfplcb4_central = bcf_quotient_bfplcb4_central_table_decoded_row_scale * S ((S (bcf_row_index_bfplcb4_central_table)) * bcf_row_scale_scale_bfplcb4_central) + (bcf_row_scale_bfplcb4_central_table))) /\ ((bcf_row_index_bfplcb4_central_table = 0 /\ (forall bcf_index_bfplcb4_central_table_zero_row. (exists bcf_lt_gap_bfplcb4_central_table_zero_row_bound. bcf_lt_gap_bfplcb4_central_table_zero_row_bound + S (bcf_index_bfplcb4_central_table_zero_row) = S (4 + 4)) -> exists bcf_value_bfplcb4_central_table_zero_row. ((((exists bcf_height_bfplcb4_central_table_zero_row_entry. bcf_height_bfplcb4_central_table_zero_row_entry + S (bcf_value_bfplcb4_central_table_zero_row) = S ((S (bcf_index_bfplcb4_central_table_zero_row)) * bcf_row_scale_bfplcb4_central_table)) /\ exists bcf_quotient_bfplcb4_central_table_zero_row_entry. bcf_row_code_bfplcb4_central_table = bcf_quotient_bfplcb4_central_table_zero_row_entry * S ((S (bcf_index_bfplcb4_central_table_zero_row)) * bcf_row_scale_bfplcb4_central_table) + (bcf_value_bfplcb4_central_table_zero_row))) /\ ((bcf_index_bfplcb4_central_table_zero_row = 0 /\ bcf_value_bfplcb4_central_table_zero_row = 1) \/ exists bcf_predecessor_bfplcb4_central_table_zero_row. bcf_index_bfplcb4_central_table_zero_row = S bcf_predecessor_bfplcb4_central_table_zero_row /\ bcf_value_bfplcb4_central_table_zero_row = 0)))) \/ exists bcf_predecessor_bfplcb4_central_table bcf_previous_code_bfplcb4_central_table bcf_previous_scale_bfplcb4_central_table. bcf_row_index_bfplcb4_central_table = S bcf_predecessor_bfplcb4_central_table /\ ((((exists bcf_height_bfplcb4_central_table_decoded_previous_code. bcf_height_bfplcb4_central_table_decoded_previous_code + S (bcf_previous_code_bfplcb4_central_table) = S ((S (bcf_predecessor_bfplcb4_central_table)) * bcf_row_code_scale_bfplcb4_central)) /\ exists bcf_quotient_bfplcb4_central_table_decoded_previous_code. bcf_row_code_code_bfplcb4_central = bcf_quotient_bfplcb4_central_table_decoded_previous_code * S ((S (bcf_predecessor_bfplcb4_central_table)) * bcf_row_code_scale_bfplcb4_central) + (bcf_previous_code_bfplcb4_central_table))) /\ ((((exists bcf_height_bfplcb4_central_table_decoded_previous_scale. bcf_height_bfplcb4_central_table_decoded_previous_scale + S (bcf_previous_scale_bfplcb4_central_table) = S ((S (bcf_predecessor_bfplcb4_central_table)) * bcf_row_scale_scale_bfplcb4_central)) /\ exists bcf_quotient_bfplcb4_central_table_decoded_previous_scale. bcf_row_scale_code_bfplcb4_central = bcf_quotient_bfplcb4_central_table_decoded_previous_scale * S ((S (bcf_predecessor_bfplcb4_central_table)) * bcf_row_scale_scale_bfplcb4_central) + (bcf_previous_scale_bfplcb4_central_table))) /\ (forall bcf_index_bfplcb4_central_table_row_step. (exists bcf_lt_gap_bfplcb4_central_table_row_step_bound. bcf_lt_gap_bfplcb4_central_table_row_step_bound + S (bcf_index_bfplcb4_central_table_row_step) = S (4 + 4)) -> exists bcf_value_bfplcb4_central_table_row_step. ((((exists bcf_height_bfplcb4_central_table_row_step_entry. bcf_height_bfplcb4_central_table_row_step_entry + S (bcf_value_bfplcb4_central_table_row_step) = S ((S (bcf_index_bfplcb4_central_table_row_step)) * bcf_row_scale_bfplcb4_central_table)) /\ exists bcf_quotient_bfplcb4_central_table_row_step_entry. bcf_row_code_bfplcb4_central_table = bcf_quotient_bfplcb4_central_table_row_step_entry * S ((S (bcf_index_bfplcb4_central_table_row_step)) * bcf_row_scale_bfplcb4_central_table) + (bcf_value_bfplcb4_central_table_row_step))) /\ ((bcf_index_bfplcb4_central_table_row_step = 0 /\ bcf_value_bfplcb4_central_table_row_step = 1) \/ exists bcf_predecessor_bfplcb4_central_table_row_step bcf_left_bfplcb4_central_table_row_step bcf_right_bfplcb4_central_table_row_step. bcf_index_bfplcb4_central_table_row_step = S bcf_predecessor_bfplcb4_central_table_row_step /\ ((((exists bcf_height_bfplcb4_central_table_row_step_previous_left. bcf_height_bfplcb4_central_table_row_step_previous_left + S (bcf_left_bfplcb4_central_table_row_step) = S ((S (bcf_predecessor_bfplcb4_central_table_row_step)) * bcf_previous_scale_bfplcb4_central_table)) /\ exists bcf_quotient_bfplcb4_central_table_row_step_previous_left. bcf_previous_code_bfplcb4_central_table = bcf_quotient_bfplcb4_central_table_row_step_previous_left * S ((S (bcf_predecessor_bfplcb4_central_table_row_step)) * bcf_previous_scale_bfplcb4_central_table) + (bcf_left_bfplcb4_central_table_row_step))) /\ ((((exists bcf_height_bfplcb4_central_table_row_step_previous_right. bcf_height_bfplcb4_central_table_row_step_previous_right + S (bcf_right_bfplcb4_central_table_row_step) = S ((S (S (bcf_predecessor_bfplcb4_central_table_row_step))) * bcf_previous_scale_bfplcb4_central_table)) /\ exists bcf_quotient_bfplcb4_central_table_row_step_previous_right. bcf_previous_code_bfplcb4_central_table = bcf_quotient_bfplcb4_central_table_row_step_previous_right * S ((S (S (bcf_predecessor_bfplcb4_central_table_row_step))) * bcf_previous_scale_bfplcb4_central_table) + (bcf_right_bfplcb4_central_table_row_step))) /\ bcf_value_bfplcb4_central_table_row_step = bcf_left_bfplcb4_central_table_row_step + bcf_right_bfplcb4_central_table_row_step))))))))))) /\ ((((exists bcf_height_bfplcb4_central_decoded_row_code. bcf_height_bfplcb4_central_decoded_row_code + S (bcf_row_code_bfplcb4_central) = S ((S (4 + 4)) * bcf_row_code_scale_bfplcb4_central)) /\ exists bcf_quotient_bfplcb4_central_decoded_row_code. bcf_row_code_code_bfplcb4_central = bcf_quotient_bfplcb4_central_decoded_row_code * S ((S (4 + 4)) * bcf_row_code_scale_bfplcb4_central) + (bcf_row_code_bfplcb4_central))) /\ ((((exists bcf_height_bfplcb4_central_decoded_row_scale. bcf_height_bfplcb4_central_decoded_row_scale + S (bcf_row_scale_bfplcb4_central) = S ((S (4 + 4)) * bcf_row_scale_scale_bfplcb4_central)) /\ exists bcf_quotient_bfplcb4_central_decoded_row_scale. bcf_row_scale_code_bfplcb4_central = bcf_quotient_bfplcb4_central_decoded_row_scale * S ((S (4 + 4)) * bcf_row_scale_scale_bfplcb4_central) + (bcf_row_scale_bfplcb4_central))) /\ (((exists bcf_height_bfplcb4_central_decoded_value. bcf_height_bfplcb4_central_decoded_value + S (c) = S ((S (4)) * bcf_row_scale_bfplcb4_central)) /\ exists bcf_quotient_bfplcb4_central_decoded_value. bcf_row_code_bfplcb4_central = bcf_quotient_bfplcb4_central_decoded_value * S ((S (4)) * bcf_row_scale_bfplcb4_central) + (c))))))))) -> (exists bcf_lt_gap_bfplcb4_result. bcf_lt_gap_bfplcb4_result + S (p) = 4 * c))
14. 0014apply four_pow_central_seed_package
15. 0015exact central_binom_succ_recurrence
16. 0016cases hpackage
17. 0017induction n
18. 0018intro p
19. 0019intro c
20. 0020intro hbound
21. 0021intro hpower
22. 0022intro hcentral
23. 0023exfalso
24. 0024specialize lt_not_le 0
25. 0025specialize lt_not_le 4
26. 0026apply lt_not_le
27. 0027exact hzero_lt_four
28. 0028exact hbound
29. 0029induction n
30. 0030intro p
31. 0031intro c
32. 0032intro hbound
33. 0033intro hpower
34. 0034intro hcentral
35. 0035exfalso
36. 0036specialize lt_not_le 1
37. 0037specialize lt_not_le 4
38. 0038apply lt_not_le
39. 0039exact hone_lt_four
40. 0040exact hbound
41. 0041induction n
42. 0042intro p
43. 0043intro c
44. 0044intro hbound
45. 0045intro hpower
46. 0046intro hcentral
47. 0047exfalso
48. 0048specialize lt_not_le 2
49. 0049specialize lt_not_le 4
50. 0050apply lt_not_le
51. 0051exact htwo_lt_four
52. 0052exact hbound
53. 0053induction n
54. 0054intro p
55. 0055intro c
56. 0056intro hbound
57. 0057intro hpower
58. 0058intro hcentral
59. 0059exfalso
60. 0060specialize lt_not_le 3
61. 0061specialize lt_not_le 4
62. 0062apply lt_not_le
63. 0063exact hthree_lt_four
64. 0064exact hbound
65. 0065induction n
66. 0066intro p
67. 0067intro c
68. 0068intro hbound
69. 0069intro hpower
70. 0070intro hcentral
71. 0071apply hpackage_right
72. 0072exact hpower
73. 0073exact hcentral
74. 0074intro p
75. 0075intro c
76. 0076intro hbound
77. 0077intro hpower
78. 0078intro hcentral
79. 0079have hpower_step : exists r. (exists pa_b_bfplcb_predecessor_power pa_c_bfplcb_predecessor_power. ((forall pa_i_bfplcb_predecessor_power_repeat. (exists pa_lt_bfplcb_predecessor_power_repeat_bound. pa_lt_bfplcb_predecessor_power_repeat_bound + S pa_i_bfplcb_predecessor_power_repeat = S (S (S (S n)))) -> (((exists pa_h_bfplcb_predecessor_power_repeat_decoded. pa_h_bfplcb_predecessor_power_repeat_decoded + S (4) = S ((S (pa_i_bfplcb_predecessor_power_repeat)) * pa_c_bfplcb_predecessor_power)) /\ exists pa_q_bfplcb_predecessor_power_repeat_decoded. pa_b_bfplcb_predecessor_power = pa_q_bfplcb_predecessor_power_repeat_decoded * S ((S (pa_i_bfplcb_predecessor_power_repeat)) * pa_c_bfplcb_predecessor_power) + (4)))) /\ (exists pa_u_bfplcb_predecessor_power_product pa_v_bfplcb_predecessor_power_product. ((((exists pa_h_bfplcb_predecessor_power_product_start. pa_h_bfplcb_predecessor_power_product_start + S (1) = S ((S (0)) * pa_v_bfplcb_predecessor_power_product)) /\ exists pa_q_bfplcb_predecessor_power_product_start. pa_u_bfplcb_predecessor_power_product = pa_q_bfplcb_predecessor_power_product_start * S ((S (0)) * pa_v_bfplcb_predecessor_power_product) + (1))) /\ ((((exists pa_h_bfplcb_predecessor_power_product_terminal. pa_h_bfplcb_predecessor_power_product_terminal + S (r) = S ((S (S (S (S (S n))))) * pa_v_bfplcb_predecessor_power_product)) /\ exists pa_q_bfplcb_predecessor_power_product_terminal. pa_u_bfplcb_predecessor_power_product = pa_q_bfplcb_predecessor_power_product_terminal * S ((S (S (S (S (S n))))) * pa_v_bfplcb_predecessor_power_product) + (r))) /\ forall pa_i_bfplcb_predecessor_power_product. (exists pa_lt_bfplcb_predecessor_power_product_bound. pa_lt_bfplcb_predecessor_power_product_bound + S pa_i_bfplcb_predecessor_power_product = S (S (S (S n)))) -> exists pa_p_bfplcb_predecessor_power_product pa_r_bfplcb_predecessor_power_product pa_s_bfplcb_predecessor_power_product. ((((exists pa_h_bfplcb_predecessor_power_product_factor. pa_h_bfplcb_predecessor_power_product_factor + S (pa_p_bfplcb_predecessor_power_product) = S ((S (pa_i_bfplcb_predecessor_power_product)) * pa_c_bfplcb_predecessor_power)) /\ exists pa_q_bfplcb_predecessor_power_product_factor. pa_b_bfplcb_predecessor_power = pa_q_bfplcb_predecessor_power_product_factor * S ((S (pa_i_bfplcb_predecessor_power_product)) * pa_c_bfplcb_predecessor_power) + (pa_p_bfplcb_predecessor_power_product))) /\ ((((exists pa_h_bfplcb_predecessor_power_product_partial. pa_h_bfplcb_predecessor_power_product_partial + S (pa_r_bfplcb_predecessor_power_product) = S ((S (pa_i_bfplcb_predecessor_power_product)) * pa_v_bfplcb_predecessor_power_product)) /\ exists pa_q_bfplcb_predecessor_power_product_partial. pa_u_bfplcb_predecessor_power_product = pa_q_bfplcb_predecessor_power_product_partial * S ((S (pa_i_bfplcb_predecessor_power_product)) * pa_v_bfplcb_predecessor_power_product) + (pa_r_bfplcb_predecessor_power_product))) /\ ((((exists pa_h_bfplcb_predecessor_power_product_successor. pa_h_bfplcb_predecessor_power_product_successor + S (pa_s_bfplcb_predecessor_power_product) = S ((S (S pa_i_bfplcb_predecessor_power_product)) * pa_v_bfplcb_predecessor_power_product)) /\ exists pa_q_bfplcb_predecessor_power_product_successor. pa_u_bfplcb_predecessor_power_product = pa_q_bfplcb_predecessor_power_product_successor * S ((S (S pa_i_bfplcb_predecessor_power_product)) * pa_v_bfplcb_predecessor_power_product) + (pa_s_bfplcb_predecessor_power_product))) /\ pa_s_bfplcb_predecessor_power_product = pa_r_bfplcb_predecessor_power_product * pa_p_bfplcb_predecessor_power_product)))))))) /\ p = r * 4
80. 0080apply pow_successor_decompose
81. 0081refl
82. 0082exact hpower
83. 0083cases hpower_step
84. 0084cases hpower_step_witness
85. 0085have hpredecessor_exists : exists a. (((exists bcf_lt_gap_bfplcb_predecessor_central_out_of_range. bcf_lt_gap_bfplcb_predecessor_central_out_of_range + S (S S S S n + S S S S n) = S S S S n) /\ a = 0) \/ ((exists bcf_le_gap_bfplcb_predecessor_central_in_range. bcf_le_gap_bfplcb_predecessor_central_in_range + (S S S S n) = S S S S n + S S S S n) /\ (exists bcf_row_code_code_bfplcb_predecessor_central bcf_row_code_scale_bfplcb_predecessor_central bcf_row_scale_code_bfplcb_predecessor_central bcf_row_scale_scale_bfplcb_predecessor_central bcf_row_code_bfplcb_predecessor_central bcf_row_scale_bfplcb_predecessor_central. ((forall bcf_row_index_bfplcb_predecessor_central_table. (exists bcf_lt_gap_bfplcb_predecessor_central_table_row_bound. bcf_lt_gap_bfplcb_predecessor_central_table_row_bound + S (bcf_row_index_bfplcb_predecessor_central_table) = S (S S S S n + S S S S n)) -> exists bcf_row_code_bfplcb_predecessor_central_table bcf_row_scale_bfplcb_predecessor_central_table. ((((exists bcf_height_bfplcb_predecessor_central_table_decoded_row_code. bcf_height_bfplcb_predecessor_central_table_decoded_row_code + S (bcf_row_code_bfplcb_predecessor_central_table) = S ((S (bcf_row_index_bfplcb_predecessor_central_table)) * bcf_row_code_scale_bfplcb_predecessor_central)) /\ exists bcf_quotient_bfplcb_predecessor_central_table_decoded_row_code. bcf_row_code_code_bfplcb_predecessor_central = bcf_quotient_bfplcb_predecessor_central_table_decoded_row_code * S ((S (bcf_row_index_bfplcb_predecessor_central_table)) * bcf_row_code_scale_bfplcb_predecessor_central) + (bcf_row_code_bfplcb_predecessor_central_table))) /\ ((((exists bcf_height_bfplcb_predecessor_central_table_decoded_row_scale. bcf_height_bfplcb_predecessor_central_table_decoded_row_scale + S (bcf_row_scale_bfplcb_predecessor_central_table) = S ((S (bcf_row_index_bfplcb_predecessor_central_table)) * bcf_row_scale_scale_bfplcb_predecessor_central)) /\ exists bcf_quotient_bfplcb_predecessor_central_table_decoded_row_scale. bcf_row_scale_code_bfplcb_predecessor_central = bcf_quotient_bfplcb_predecessor_central_table_decoded_row_scale * S ((S (bcf_row_index_bfplcb_predecessor_central_table)) * bcf_row_scale_scale_bfplcb_predecessor_central) + (bcf_row_scale_bfplcb_predecessor_central_table))) /\ ((bcf_row_index_bfplcb_predecessor_central_table = 0 /\ (forall bcf_index_bfplcb_predecessor_central_table_zero_row. (exists bcf_lt_gap_bfplcb_predecessor_central_table_zero_row_bound. bcf_lt_gap_bfplcb_predecessor_central_table_zero_row_bound + S (bcf_index_bfplcb_predecessor_central_table_zero_row) = S (S S S S n + S S S S n)) -> exists bcf_value_bfplcb_predecessor_central_table_zero_row. ((((exists bcf_height_bfplcb_predecessor_central_table_zero_row_entry. bcf_height_bfplcb_predecessor_central_table_zero_row_entry + S (bcf_value_bfplcb_predecessor_central_table_zero_row) = S ((S (bcf_index_bfplcb_predecessor_central_table_zero_row)) * bcf_row_scale_bfplcb_predecessor_central_table)) /\ exists bcf_quotient_bfplcb_predecessor_central_table_zero_row_entry. bcf_row_code_bfplcb_predecessor_central_table = bcf_quotient_bfplcb_predecessor_central_table_zero_row_entry * S ((S (bcf_index_bfplcb_predecessor_central_table_zero_row)) * bcf_row_scale_bfplcb_predecessor_central_table) + (bcf_value_bfplcb_predecessor_central_table_zero_row))) /\ ((bcf_index_bfplcb_predecessor_central_table_zero_row = 0 /\ bcf_value_bfplcb_predecessor_central_table_zero_row = 1) \/ exists bcf_predecessor_bfplcb_predecessor_central_table_zero_row. bcf_index_bfplcb_predecessor_central_table_zero_row = S bcf_predecessor_bfplcb_predecessor_central_table_zero_row /\ bcf_value_bfplcb_predecessor_central_table_zero_row = 0)))) \/ exists bcf_predecessor_bfplcb_predecessor_central_table bcf_previous_code_bfplcb_predecessor_central_table bcf_previous_scale_bfplcb_predecessor_central_table. bcf_row_index_bfplcb_predecessor_central_table = S bcf_predecessor_bfplcb_predecessor_central_table /\ ((((exists bcf_height_bfplcb_predecessor_central_table_decoded_previous_code. bcf_height_bfplcb_predecessor_central_table_decoded_previous_code + S (bcf_previous_code_bfplcb_predecessor_central_table) = S ((S (bcf_predecessor_bfplcb_predecessor_central_table)) * bcf_row_code_scale_bfplcb_predecessor_central)) /\ exists bcf_quotient_bfplcb_predecessor_central_table_decoded_previous_code. bcf_row_code_code_bfplcb_predecessor_central = bcf_quotient_bfplcb_predecessor_central_table_decoded_previous_code * S ((S (bcf_predecessor_bfplcb_predecessor_central_table)) * bcf_row_code_scale_bfplcb_predecessor_central) + (bcf_previous_code_bfplcb_predecessor_central_table))) /\ ((((exists bcf_height_bfplcb_predecessor_central_table_decoded_previous_scale. bcf_height_bfplcb_predecessor_central_table_decoded_previous_scale + S (bcf_previous_scale_bfplcb_predecessor_central_table) = S ((S (bcf_predecessor_bfplcb_predecessor_central_table)) * bcf_row_scale_scale_bfplcb_predecessor_central)) /\ exists bcf_quotient_bfplcb_predecessor_central_table_decoded_previous_scale. bcf_row_scale_code_bfplcb_predecessor_central = bcf_quotient_bfplcb_predecessor_central_table_decoded_previous_scale * S ((S (bcf_predecessor_bfplcb_predecessor_central_table)) * bcf_row_scale_scale_bfplcb_predecessor_central) + (bcf_previous_scale_bfplcb_predecessor_central_table))) /\ (forall bcf_index_bfplcb_predecessor_central_table_row_step. (exists bcf_lt_gap_bfplcb_predecessor_central_table_row_step_bound. bcf_lt_gap_bfplcb_predecessor_central_table_row_step_bound + S (bcf_index_bfplcb_predecessor_central_table_row_step) = S (S S S S n + S S S S n)) -> exists bcf_value_bfplcb_predecessor_central_table_row_step. ((((exists bcf_height_bfplcb_predecessor_central_table_row_step_entry. bcf_height_bfplcb_predecessor_central_table_row_step_entry + S (bcf_value_bfplcb_predecessor_central_table_row_step) = S ((S (bcf_index_bfplcb_predecessor_central_table_row_step)) * bcf_row_scale_bfplcb_predecessor_central_table)) /\ exists bcf_quotient_bfplcb_predecessor_central_table_row_step_entry. bcf_row_code_bfplcb_predecessor_central_table = bcf_quotient_bfplcb_predecessor_central_table_row_step_entry * S ((S (bcf_index_bfplcb_predecessor_central_table_row_step)) * bcf_row_scale_bfplcb_predecessor_central_table) + (bcf_value_bfplcb_predecessor_central_table_row_step))) /\ ((bcf_index_bfplcb_predecessor_central_table_row_step = 0 /\ bcf_value_bfplcb_predecessor_central_table_row_step = 1) \/ exists bcf_predecessor_bfplcb_predecessor_central_table_row_step bcf_left_bfplcb_predecessor_central_table_row_step bcf_right_bfplcb_predecessor_central_table_row_step. bcf_index_bfplcb_predecessor_central_table_row_step = S bcf_predecessor_bfplcb_predecessor_central_table_row_step /\ ((((exists bcf_height_bfplcb_predecessor_central_table_row_step_previous_left. bcf_height_bfplcb_predecessor_central_table_row_step_previous_left + S (bcf_left_bfplcb_predecessor_central_table_row_step) = S ((S (bcf_predecessor_bfplcb_predecessor_central_table_row_step)) * bcf_previous_scale_bfplcb_predecessor_central_table)) /\ exists bcf_quotient_bfplcb_predecessor_central_table_row_step_previous_left. bcf_previous_code_bfplcb_predecessor_central_table = bcf_quotient_bfplcb_predecessor_central_table_row_step_previous_left * S ((S (bcf_predecessor_bfplcb_predecessor_central_table_row_step)) * bcf_previous_scale_bfplcb_predecessor_central_table) + (bcf_left_bfplcb_predecessor_central_table_row_step))) /\ ((((exists bcf_height_bfplcb_predecessor_central_table_row_step_previous_right. bcf_height_bfplcb_predecessor_central_table_row_step_previous_right + S (bcf_right_bfplcb_predecessor_central_table_row_step) = S ((S (S (bcf_predecessor_bfplcb_predecessor_central_table_row_step))) * bcf_previous_scale_bfplcb_predecessor_central_table)) /\ exists bcf_quotient_bfplcb_predecessor_central_table_row_step_previous_right. bcf_previous_code_bfplcb_predecessor_central_table = bcf_quotient_bfplcb_predecessor_central_table_row_step_previous_right * S ((S (S (bcf_predecessor_bfplcb_predecessor_central_table_row_step))) * bcf_previous_scale_bfplcb_predecessor_central_table) + (bcf_right_bfplcb_predecessor_central_table_row_step))) /\ bcf_value_bfplcb_predecessor_central_table_row_step = bcf_left_bfplcb_predecessor_central_table_row_step + bcf_right_bfplcb_predecessor_central_table_row_step))))))))))) /\ ((((exists bcf_height_bfplcb_predecessor_central_decoded_row_code. bcf_height_bfplcb_predecessor_central_decoded_row_code + S (bcf_row_code_bfplcb_predecessor_central) = S ((S (S S S S n + S S S S n)) * bcf_row_code_scale_bfplcb_predecessor_central)) /\ exists bcf_quotient_bfplcb_predecessor_central_decoded_row_code. bcf_row_code_code_bfplcb_predecessor_central = bcf_quotient_bfplcb_predecessor_central_decoded_row_code * S ((S (S S S S n + S S S S n)) * bcf_row_code_scale_bfplcb_predecessor_central) + (bcf_row_code_bfplcb_predecessor_central))) /\ ((((exists bcf_height_bfplcb_predecessor_central_decoded_row_scale. bcf_height_bfplcb_predecessor_central_decoded_row_scale + S (bcf_row_scale_bfplcb_predecessor_central) = S ((S (S S S S n + S S S S n)) * bcf_row_scale_scale_bfplcb_predecessor_central)) /\ exists bcf_quotient_bfplcb_predecessor_central_decoded_row_scale. bcf_row_scale_code_bfplcb_predecessor_central = bcf_quotient_bfplcb_predecessor_central_decoded_row_scale * S ((S (S S S S n + S S S S n)) * bcf_row_scale_scale_bfplcb_predecessor_central) + (bcf_row_scale_bfplcb_predecessor_central))) /\ (((exists bcf_height_bfplcb_predecessor_central_decoded_value. bcf_height_bfplcb_predecessor_central_decoded_value + S (a) = S ((S (S S S S n)) * bcf_row_scale_bfplcb_predecessor_central)) /\ exists bcf_quotient_bfplcb_predecessor_central_decoded_value. bcf_row_code_bfplcb_predecessor_central = bcf_quotient_bfplcb_predecessor_central_decoded_value * S ((S (S S S S n)) * bcf_row_scale_bfplcb_predecessor_central) + (a)))))))))
86. 0086apply hpackage_left
87. 0087cases hpredecessor_exists
88. 0088have hpredecessor_bound : exists bcf_le_gap_bfplcb_predecessor_bound. bcf_le_gap_bfplcb_predecessor_bound + (4) = S (S (S (S n)))
89. 0089exists n
90. 0090simp
91. 0091have hstrict : exists bcf_lt_gap_bfplcb_predecessor_result. bcf_lt_gap_bfplcb_predecessor_result + S (x) = S (S (S (S n))) * x1
92. 0092apply IH4
93. 0093exact hpredecessor_bound
94. 0094exact hpower_step_witness_left
95. 0095exact hpredecessor_exists_witness
96. 0096have hrecurrence : S (S (S (S (S n)))) * c = (2 * S (S (S (S (S n))) + S (S (S (S n))))) * x1
97. 0097apply central_binom_succ_recurrence
98. 0098exact hpredecessor_exists_witness
99. 0099exact hcentral
100. 0100have hstep : exists bcf_lt_gap_bfplcb_successor_result. bcf_lt_gap_bfplcb_successor_result + S (x * 4) = S (S (S (S (S n)))) * c
101. 0101apply four_power_central_recurrence_step
102. 0102exact hstrict
103. 0103exact hrecurrence
104. 0104rewrite hpower_step_witness_right
105. 0105exact hstep