BT0114 — central_binom_le_of_no_bertrand

Exact expanded PA statement

forall n s q r C A B. (forall bpr_prime_candidate_b5cblonbp_exclusion. ((exists bpr_gap_b5cblonbp_exclusion_lower. bpr_gap_b5cblonbp_exclusion_lower + S (n) = bpr_prime_candidate_b5cblonbp_exclusion) /\ (exists bpr_le_gap_b5cblonbp_exclusion_upper. bpr_le_gap_b5cblonbp_exclusion_upper + (bpr_prime_candidate_b5cblonbp_exclusion) = (n + n))) -> ~((~(bpr_prime_candidate_b5cblonbp_exclusion = 1) /\ forall bpr_left_b5cblonbp_exclusion_prime bpr_right_b5cblonbp_exclusion_prime. bpr_prime_candidate_b5cblonbp_exclusion = bpr_left_b5cblonbp_exclusion_prime * bpr_right_b5cblonbp_exclusion_prime -> bpr_left_b5cblonbp_exclusion_prime = 1 \/ bpr_right_b5cblonbp_exclusion_prime = 1))) -> (exists bcf_lt_gap_b5cblonbp_positive. bcf_lt_gap_b5cblonbp_positive + S (2) = n) -> (((exists bcs_sqrt_lower_gap_b5cblonbp_floor. bcs_sqrt_lower_gap_b5cblonbp_floor + (s) * (s) = (n + n)) /\ exists bcs_sqrt_upper_gap_b5cblonbp_floor. bcs_sqrt_upper_gap_b5cblonbp_floor + S (n + n) = S (s) * S (s))) -> (((n + n) = (3) * (q) + (r) /\ (exists bcf_lt_gap_b5cblonbp_division_bound. bcf_lt_gap_b5cblonbp_division_bound + S (r) = 3))) -> (((exists bcf_lt_gap_b5cblonbp_central_out_of_range. bcf_lt_gap_b5cblonbp_central_out_of_range + S (n + n) = n) /\ C = 0) \/ ((exists bcf_le_gap_b5cblonbp_central_in_range. bcf_le_gap_b5cblonbp_central_in_range + (n) = n + n) /\ (exists bcf_row_code_code_b5cblonbp_central bcf_row_code_scale_b5cblonbp_central bcf_row_scale_code_b5cblonbp_central bcf_row_scale_scale_b5cblonbp_central bcf_row_code_b5cblonbp_central bcf_row_scale_b5cblonbp_central. ((forall bcf_row_index_b5cblonbp_central_table. (exists bcf_lt_gap_b5cblonbp_central_table_row_bound. bcf_lt_gap_b5cblonbp_central_table_row_bound + S (bcf_row_index_b5cblonbp_central_table) = S (n + n)) -> exists bcf_row_code_b5cblonbp_central_table bcf_row_scale_b5cblonbp_central_table. ((((exists bcf_height_b5cblonbp_central_table_decoded_row_code. bcf_height_b5cblonbp_central_table_decoded_row_code + S (bcf_row_code_b5cblonbp_central_table) = S ((S (bcf_row_index_b5cblonbp_central_table)) * bcf_row_code_scale_b5cblonbp_central)) /\ exists bcf_quotient_b5cblonbp_central_table_decoded_row_code. bcf_row_code_code_b5cblonbp_central = bcf_quotient_b5cblonbp_central_table_decoded_row_code * S ((S (bcf_row_index_b5cblonbp_central_table)) * bcf_row_code_scale_b5cblonbp_central) + (bcf_row_code_b5cblonbp_central_table))) /\ ((((exists bcf_height_b5cblonbp_central_table_decoded_row_scale. bcf_height_b5cblonbp_central_table_decoded_row_scale + S (bcf_row_scale_b5cblonbp_central_table) = S ((S (bcf_row_index_b5cblonbp_central_table)) * bcf_row_scale_scale_b5cblonbp_central)) /\ exists bcf_quotient_b5cblonbp_central_table_decoded_row_scale. bcf_row_scale_code_b5cblonbp_central = bcf_quotient_b5cblonbp_central_table_decoded_row_scale * S ((S (bcf_row_index_b5cblonbp_central_table)) * bcf_row_scale_scale_b5cblonbp_central) + (bcf_row_scale_b5cblonbp_central_table))) /\ ((bcf_row_index_b5cblonbp_central_table = 0 /\ (forall bcf_index_b5cblonbp_central_table_zero_row. (exists bcf_lt_gap_b5cblonbp_central_table_zero_row_bound. bcf_lt_gap_b5cblonbp_central_table_zero_row_bound + S (bcf_index_b5cblonbp_central_table_zero_row) = S (n + n)) -> exists bcf_value_b5cblonbp_central_table_zero_row. ((((exists bcf_height_b5cblonbp_central_table_zero_row_entry. bcf_height_b5cblonbp_central_table_zero_row_entry + S (bcf_value_b5cblonbp_central_table_zero_row) = S ((S (bcf_index_b5cblonbp_central_table_zero_row)) * bcf_row_scale_b5cblonbp_central_table)) /\ exists bcf_quotient_b5cblonbp_central_table_zero_row_entry. bcf_row_code_b5cblonbp_central_table = bcf_quotient_b5cblonbp_central_table_zero_row_entry * S ((S (bcf_index_b5cblonbp_central_table_zero_row)) * bcf_row_scale_b5cblonbp_central_table) + (bcf_value_b5cblonbp_central_table_zero_row))) /\ ((bcf_index_b5cblonbp_central_table_zero_row = 0 /\ bcf_value_b5cblonbp_central_table_zero_row = 1) \/ exists bcf_predecessor_b5cblonbp_central_table_zero_row. bcf_index_b5cblonbp_central_table_zero_row = S bcf_predecessor_b5cblonbp_central_table_zero_row /\ bcf_value_b5cblonbp_central_table_zero_row = 0)))) \/ exists bcf_predecessor_b5cblonbp_central_table bcf_previous_code_b5cblonbp_central_table bcf_previous_scale_b5cblonbp_central_table. bcf_row_index_b5cblonbp_central_table = S bcf_predecessor_b5cblonbp_central_table /\ ((((exists bcf_height_b5cblonbp_central_table_decoded_previous_code. bcf_height_b5cblonbp_central_table_decoded_previous_code + S (bcf_previous_code_b5cblonbp_central_table) = S ((S (bcf_predecessor_b5cblonbp_central_table)) * bcf_row_code_scale_b5cblonbp_central)) /\ exists bcf_quotient_b5cblonbp_central_table_decoded_previous_code. bcf_row_code_code_b5cblonbp_central = bcf_quotient_b5cblonbp_central_table_decoded_previous_code * S ((S (bcf_predecessor_b5cblonbp_central_table)) * bcf_row_code_scale_b5cblonbp_central) + (bcf_previous_code_b5cblonbp_central_table))) /\ ((((exists bcf_height_b5cblonbp_central_table_decoded_previous_scale. bcf_height_b5cblonbp_central_table_decoded_previous_scale + S (bcf_previous_scale_b5cblonbp_central_table) = S ((S (bcf_predecessor_b5cblonbp_central_table)) * bcf_row_scale_scale_b5cblonbp_central)) /\ exists bcf_quotient_b5cblonbp_central_table_decoded_previous_scale. bcf_row_scale_code_b5cblonbp_central = bcf_quotient_b5cblonbp_central_table_decoded_previous_scale * S ((S (bcf_predecessor_b5cblonbp_central_table)) * bcf_row_scale_scale_b5cblonbp_central) + (bcf_previous_scale_b5cblonbp_central_table))) /\ (forall bcf_index_b5cblonbp_central_table_row_step. (exists bcf_lt_gap_b5cblonbp_central_table_row_step_bound. bcf_lt_gap_b5cblonbp_central_table_row_step_bound + S (bcf_index_b5cblonbp_central_table_row_step) = S (n + n)) -> exists bcf_value_b5cblonbp_central_table_row_step. ((((exists bcf_height_b5cblonbp_central_table_row_step_entry. bcf_height_b5cblonbp_central_table_row_step_entry + S (bcf_value_b5cblonbp_central_table_row_step) = S ((S (bcf_index_b5cblonbp_central_table_row_step)) * bcf_row_scale_b5cblonbp_central_table)) /\ exists bcf_quotient_b5cblonbp_central_table_row_step_entry. bcf_row_code_b5cblonbp_central_table = bcf_quotient_b5cblonbp_central_table_row_step_entry * S ((S (bcf_index_b5cblonbp_central_table_row_step)) * bcf_row_scale_b5cblonbp_central_table) + (bcf_value_b5cblonbp_central_table_row_step))) /\ ((bcf_index_b5cblonbp_central_table_row_step = 0 /\ bcf_value_b5cblonbp_central_table_row_step = 1) \/ exists bcf_predecessor_b5cblonbp_central_table_row_step bcf_left_b5cblonbp_central_table_row_step bcf_right_b5cblonbp_central_table_row_step. bcf_index_b5cblonbp_central_table_row_step = S bcf_predecessor_b5cblonbp_central_table_row_step /\ ((((exists bcf_height_b5cblonbp_central_table_row_step_previous_left. bcf_height_b5cblonbp_central_table_row_step_previous_left + S (bcf_left_b5cblonbp_central_table_row_step) = S ((S (bcf_predecessor_b5cblonbp_central_table_row_step)) * bcf_previous_scale_b5cblonbp_central_table)) /\ exists bcf_quotient_b5cblonbp_central_table_row_step_previous_left. bcf_previous_code_b5cblonbp_central_table = bcf_quotient_b5cblonbp_central_table_row_step_previous_left * S ((S (bcf_predecessor_b5cblonbp_central_table_row_step)) * bcf_previous_scale_b5cblonbp_central_table) + (bcf_left_b5cblonbp_central_table_row_step))) /\ ((((exists bcf_height_b5cblonbp_central_table_row_step_previous_right. bcf_height_b5cblonbp_central_table_row_step_previous_right + S (bcf_right_b5cblonbp_central_table_row_step) = S ((S (S (bcf_predecessor_b5cblonbp_central_table_row_step))) * bcf_previous_scale_b5cblonbp_central_table)) /\ exists bcf_quotient_b5cblonbp_central_table_row_step_previous_right. bcf_previous_code_b5cblonbp_central_table = bcf_quotient_b5cblonbp_central_table_row_step_previous_right * S ((S (S (bcf_predecessor_b5cblonbp_central_table_row_step))) * bcf_previous_scale_b5cblonbp_central_table) + (bcf_right_b5cblonbp_central_table_row_step))) /\ bcf_value_b5cblonbp_central_table_row_step = bcf_left_b5cblonbp_central_table_row_step + bcf_right_b5cblonbp_central_table_row_step))))))))))) /\ ((((exists bcf_height_b5cblonbp_central_decoded_row_code. bcf_height_b5cblonbp_central_decoded_row_code + S (bcf_row_code_b5cblonbp_central) = S ((S (n + n)) * bcf_row_code_scale_b5cblonbp_central)) /\ exists bcf_quotient_b5cblonbp_central_decoded_row_code. bcf_row_code_code_b5cblonbp_central = bcf_quotient_b5cblonbp_central_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_b5cblonbp_central) + (bcf_row_code_b5cblonbp_central))) /\ ((((exists bcf_height_b5cblonbp_central_decoded_row_scale. bcf_height_b5cblonbp_central_decoded_row_scale + S (bcf_row_scale_b5cblonbp_central) = S ((S (n + n)) * bcf_row_scale_scale_b5cblonbp_central)) /\ exists bcf_quotient_b5cblonbp_central_decoded_row_scale. bcf_row_scale_code_b5cblonbp_central = bcf_quotient_b5cblonbp_central_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_b5cblonbp_central) + (bcf_row_scale_b5cblonbp_central))) /\ (((exists bcf_height_b5cblonbp_central_decoded_value. bcf_height_b5cblonbp_central_decoded_value + S (C) = S ((S (n)) * bcf_row_scale_b5cblonbp_central)) /\ exists bcf_quotient_b5cblonbp_central_decoded_value. bcf_row_code_b5cblonbp_central = bcf_quotient_b5cblonbp_central_decoded_value * S ((S (n)) * bcf_row_scale_b5cblonbp_central) + (C))))))))) -> (exists bpvi_b_b5cblonbp_power_a bpvi_c_b5cblonbp_power_a. ((forall bpvi_i_b5cblonbp_power_a. (exists bpvi_repeat_gap_b5cblonbp_power_a. bpvi_repeat_gap_b5cblonbp_power_a + S bpvi_i_b5cblonbp_power_a = s) -> (((exists bpvi_h_b5cblonbp_power_a_repeat. bpvi_h_b5cblonbp_power_a_repeat + S (n + n) = S ((S (bpvi_i_b5cblonbp_power_a)) * bpvi_c_b5cblonbp_power_a)) /\ exists bpvi_q_b5cblonbp_power_a_repeat. bpvi_b_b5cblonbp_power_a = bpvi_q_b5cblonbp_power_a_repeat * S ((S (bpvi_i_b5cblonbp_power_a)) * bpvi_c_b5cblonbp_power_a) + (n + n)))) /\ (exists bpvi_u_b5cblonbp_power_a bpvi_v_b5cblonbp_power_a. ((((exists bpvi_h_b5cblonbp_power_a_start. bpvi_h_b5cblonbp_power_a_start + S (1) = S ((S (0)) * bpvi_v_b5cblonbp_power_a)) /\ exists bpvi_q_b5cblonbp_power_a_start. bpvi_u_b5cblonbp_power_a = bpvi_q_b5cblonbp_power_a_start * S ((S (0)) * bpvi_v_b5cblonbp_power_a) + (1))) /\ ((((exists bpvi_h_b5cblonbp_power_a_terminal. bpvi_h_b5cblonbp_power_a_terminal + S (A) = S ((S (s)) * bpvi_v_b5cblonbp_power_a)) /\ exists bpvi_q_b5cblonbp_power_a_terminal. bpvi_u_b5cblonbp_power_a = bpvi_q_b5cblonbp_power_a_terminal * S ((S (s)) * bpvi_v_b5cblonbp_power_a) + (A))) /\ forall bpvi_j_b5cblonbp_power_a. (exists bpvi_product_gap_b5cblonbp_power_a. bpvi_product_gap_b5cblonbp_power_a + S bpvi_j_b5cblonbp_power_a = s) -> exists bpvi_factor_b5cblonbp_power_a bpvi_partial_b5cblonbp_power_a bpvi_successor_b5cblonbp_power_a. ((((exists bpvi_h_b5cblonbp_power_a_factor. bpvi_h_b5cblonbp_power_a_factor + S (bpvi_factor_b5cblonbp_power_a) = S ((S (bpvi_j_b5cblonbp_power_a)) * bpvi_c_b5cblonbp_power_a)) /\ exists bpvi_q_b5cblonbp_power_a_factor. bpvi_b_b5cblonbp_power_a = bpvi_q_b5cblonbp_power_a_factor * S ((S (bpvi_j_b5cblonbp_power_a)) * bpvi_c_b5cblonbp_power_a) + (bpvi_factor_b5cblonbp_power_a))) /\ ((((exists bpvi_h_b5cblonbp_power_a_partial. bpvi_h_b5cblonbp_power_a_partial + S (bpvi_partial_b5cblonbp_power_a) = S ((S (bpvi_j_b5cblonbp_power_a)) * bpvi_v_b5cblonbp_power_a)) /\ exists bpvi_q_b5cblonbp_power_a_partial. bpvi_u_b5cblonbp_power_a = bpvi_q_b5cblonbp_power_a_partial * S ((S (bpvi_j_b5cblonbp_power_a)) * bpvi_v_b5cblonbp_power_a) + (bpvi_partial_b5cblonbp_power_a))) /\ ((((exists bpvi_h_b5cblonbp_power_a_successor. bpvi_h_b5cblonbp_power_a_successor + S (bpvi_successor_b5cblonbp_power_a) = S ((S (S bpvi_j_b5cblonbp_power_a)) * bpvi_v_b5cblonbp_power_a)) /\ exists bpvi_q_b5cblonbp_power_a_successor. bpvi_u_b5cblonbp_power_a = bpvi_q_b5cblonbp_power_a_successor * S ((S (S bpvi_j_b5cblonbp_power_a)) * bpvi_v_b5cblonbp_power_a) + (bpvi_successor_b5cblonbp_power_a))) /\ bpvi_successor_b5cblonbp_power_a = bpvi_partial_b5cblonbp_power_a * bpvi_factor_b5cblonbp_power_a)))))))) -> (exists bpvi_b_b5cblonbp_power_b bpvi_c_b5cblonbp_power_b. ((forall bpvi_i_b5cblonbp_power_b. (exists bpvi_repeat_gap_b5cblonbp_power_b. bpvi_repeat_gap_b5cblonbp_power_b + S bpvi_i_b5cblonbp_power_b = q) -> (((exists bpvi_h_b5cblonbp_power_b_repeat. bpvi_h_b5cblonbp_power_b_repeat + S (4) = S ((S (bpvi_i_b5cblonbp_power_b)) * bpvi_c_b5cblonbp_power_b)) /\ exists bpvi_q_b5cblonbp_power_b_repeat. bpvi_b_b5cblonbp_power_b = bpvi_q_b5cblonbp_power_b_repeat * S ((S (bpvi_i_b5cblonbp_power_b)) * bpvi_c_b5cblonbp_power_b) + (4)))) /\ (exists bpvi_u_b5cblonbp_power_b bpvi_v_b5cblonbp_power_b. ((((exists bpvi_h_b5cblonbp_power_b_start. bpvi_h_b5cblonbp_power_b_start + S (1) = S ((S (0)) * bpvi_v_b5cblonbp_power_b)) /\ exists bpvi_q_b5cblonbp_power_b_start. bpvi_u_b5cblonbp_power_b = bpvi_q_b5cblonbp_power_b_start * S ((S (0)) * bpvi_v_b5cblonbp_power_b) + (1))) /\ ((((exists bpvi_h_b5cblonbp_power_b_terminal. bpvi_h_b5cblonbp_power_b_terminal + S (B) = S ((S (q)) * bpvi_v_b5cblonbp_power_b)) /\ exists bpvi_q_b5cblonbp_power_b_terminal. bpvi_u_b5cblonbp_power_b = bpvi_q_b5cblonbp_power_b_terminal * S ((S (q)) * bpvi_v_b5cblonbp_power_b) + (B))) /\ forall bpvi_j_b5cblonbp_power_b. (exists bpvi_product_gap_b5cblonbp_power_b. bpvi_product_gap_b5cblonbp_power_b + S bpvi_j_b5cblonbp_power_b = q) -> exists bpvi_factor_b5cblonbp_power_b bpvi_partial_b5cblonbp_power_b bpvi_successor_b5cblonbp_power_b. ((((exists bpvi_h_b5cblonbp_power_b_factor. bpvi_h_b5cblonbp_power_b_factor + S (bpvi_factor_b5cblonbp_power_b) = S ((S (bpvi_j_b5cblonbp_power_b)) * bpvi_c_b5cblonbp_power_b)) /\ exists bpvi_q_b5cblonbp_power_b_factor. bpvi_b_b5cblonbp_power_b = bpvi_q_b5cblonbp_power_b_factor * S ((S (bpvi_j_b5cblonbp_power_b)) * bpvi_c_b5cblonbp_power_b) + (bpvi_factor_b5cblonbp_power_b))) /\ ((((exists bpvi_h_b5cblonbp_power_b_partial. bpvi_h_b5cblonbp_power_b_partial + S (bpvi_partial_b5cblonbp_power_b) = S ((S (bpvi_j_b5cblonbp_power_b)) * bpvi_v_b5cblonbp_power_b)) /\ exists bpvi_q_b5cblonbp_power_b_partial. bpvi_u_b5cblonbp_power_b = bpvi_q_b5cblonbp_power_b_partial * S ((S (bpvi_j_b5cblonbp_power_b)) * bpvi_v_b5cblonbp_power_b) + (bpvi_partial_b5cblonbp_power_b))) /\ ((((exists bpvi_h_b5cblonbp_power_b_successor. bpvi_h_b5cblonbp_power_b_successor + S (bpvi_successor_b5cblonbp_power_b) = S ((S (S bpvi_j_b5cblonbp_power_b)) * bpvi_v_b5cblonbp_power_b)) /\ exists bpvi_q_b5cblonbp_power_b_successor. bpvi_u_b5cblonbp_power_b = bpvi_q_b5cblonbp_power_b_successor * S ((S (S bpvi_j_b5cblonbp_power_b)) * bpvi_v_b5cblonbp_power_b) + (bpvi_successor_b5cblonbp_power_b))) /\ bpvi_successor_b5cblonbp_power_b = bpvi_partial_b5cblonbp_power_b * bpvi_factor_b5cblonbp_power_b)))))))) -> (exists bcf_le_gap_b5cblonbp_result. bcf_le_gap_b5cblonbp_result + (C) = A * B)

Structural proof guide

No Bertrand prime forces the reviewed central-binomial upper bound.

Direct prerequisites: mul_le_mul, floor_third_double_gap_package, central_binom_prime_contribution_product_exists, no_bertrand_small_contribution_product_le_power, no_bertrand_middle_contribution_interval_le_four_pow, central_binom_factorization_small. The authored body proceeds by case analysis (9), intermediate claims (6), equality transport (2).

Proof neighborhood

Direct dependencies

BT00PV mul_le_mul BT010J floor_third_double_gap_package BT0107 central_binom_prime_contribution_product_exists BT010Y no_bertrand_small_contribution_product_le_power BT0111 no_bertrand_middle_contribution_interval_le_four_pow BT0113 central_binom_factorization_small

Direct dependents

BT0115 bertrand_eventually_closed_upper

Formal native tactic body

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

0001intro n
0002intro s
0003intro q
0004intro r
0005intro C
0006intro A
0007intro B
0008intro hexclusion
0009intro hpositive
0010intro hfloor
0011intro hdivision
0012intro hcentral
0013intro hpower_a
0014intro hpower_b
0015have hgaps : exists g h. s + g = q /\ q + h = n + n
0016specialize floor_third_double_gap_package n
0017specialize floor_third_double_gap_package s
0018specialize floor_third_double_gap_package q
0019specialize floor_third_double_gap_package r
0020apply floor_third_double_gap_package
0021exact hpositive
0022exact hfloor
0023exact hdivision
0024cases hgaps
0025cases hgaps_witness
0026cases hgaps_witness_witness
0027have hproduct : exists z. (exists bpr_product_code_b5cblonbp_product bpr_product_scale_b5cblonbp_product. ((forall bpr_prefix_index_b5cblonbp_product_prefix. (exists bpr_gap_b5cblonbp_product_prefix_bound. bpr_gap_b5cblonbp_product_prefix_bound + S (bpr_prefix_index_b5cblonbp_product_prefix) = n + n) -> exists bpr_prefix_value_b5cblonbp_product_prefix. ((((exists bpr_height_b5cblonbp_product_prefix_decoded. bpr_height_b5cblonbp_product_prefix_decoded + S (bpr_prefix_value_b5cblonbp_product_prefix) = S ((S (bpr_prefix_index_b5cblonbp_product_prefix)) * bpr_product_scale_b5cblonbp_product)) /\ exists bpr_quotient_b5cblonbp_product_prefix_decoded. bpr_product_code_b5cblonbp_product = bpr_quotient_b5cblonbp_product_prefix_decoded * S ((S (bpr_prefix_index_b5cblonbp_product_prefix)) * bpr_product_scale_b5cblonbp_product) + (bpr_prefix_value_b5cblonbp_product_prefix))) /\ (((((~(S (bpr_prefix_index_b5cblonbp_product_prefix) = 1) /\ forall bpr_left_b5cblonbp_product_prefix_choice_prime bpr_right_b5cblonbp_product_prefix_choice_prime. S (bpr_prefix_index_b5cblonbp_product_prefix) = bpr_left_b5cblonbp_product_prefix_choice_prime * bpr_right_b5cblonbp_product_prefix_choice_prime -> bpr_left_b5cblonbp_product_prefix_choice_prime = 1 \/ bpr_right_b5cblonbp_product_prefix_choice_prime = 1)) /\ exists bpr_choice_exponent_b5cblonbp_product_prefix_choice. ((((exists bpr_le_gap_b5cblonbp_product_prefix_choice_valuation_selected_bound. bpr_le_gap_b5cblonbp_product_prefix_choice_valuation_selected_bound + (bpr_choice_exponent_b5cblonbp_product_prefix_choice) = (C)) /\ (exists bpr_power_value_b5cblonbp_product_prefix_choice_valuation_selected. ((exists bpr_power_code_b5cblonbp_product_prefix_choice_valuation_selected_power bpr_power_scale_b5cblonbp_product_prefix_choice_valuation_selected_power. ((forall bpr_power_index_b5cblonbp_product_prefix_choice_valuation_selected_power. (exists bpr_gap_b5cblonbp_product_prefix_choice_valuation_selected_power_repeat_bound. bpr_gap_b5cblonbp_product_prefix_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_b5cblonbp_product_prefix_choice_valuation_selected_power) = bpr_choice_exponent_b5cblonbp_product_prefix_choice) -> (((exists bpr_height_b5cblonbp_product_prefix_choice_valuation_selected_power_repeat_entry. bpr_height_b5cblonbp_product_prefix_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_b5cblonbp_product_prefix)) = S ((S (bpr_power_index_b5cblonbp_product_prefix_choice_valuation_selected_power)) * bpr_power_scale_b5cblonbp_product_prefix_choice_valuation_selected_power)) /\ exists bpr_quotient_b5cblonbp_product_prefix_choice_valuation_selected_power_repeat_entry. bpr_power_code_b5cblonbp_product_prefix_choice_valuation_selected_power = bpr_quotient_b5cblonbp_product_prefix_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_b5cblonbp_product_prefix_choice_valuation_selected_power)) * bpr_power_scale_b5cblonbp_product_prefix_choice_valuation_selected_power) + (S (bpr_prefix_index_b5cblonbp_product_prefix))))) /\ (exists ff_u_b5cblonbp_product_prefix_choice_valuation_selected_power_product ff_v_b5cblonbp_product_prefix_choice_valuation_selected_power_product. ((((exists ff_h_b5cblonbp_product_prefix_choice_valuation_selected_power_product_start. ff_h_b5cblonbp_product_prefix_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_b5cblonbp_product_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5cblonbp_product_prefix_choice_valuation_selected_power_product_start. ff_u_b5cblonbp_product_prefix_choice_valuation_selected_power_product = ff_q_b5cblonbp_product_prefix_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_b5cblonbp_product_prefix_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_b5cblonbp_product_prefix_choice_valuation_selected_power_product_terminal. ff_h_b5cblonbp_product_prefix_choice_valuation_selected_power_product_terminal + S (bpr_power_value_b5cblonbp_product_prefix_choice_valuation_selected) = S ((S (bpr_choice_exponent_b5cblonbp_product_prefix_choice)) * ff_v_b5cblonbp_product_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5cblonbp_product_prefix_choice_valuation_selected_power_product_terminal. ff_u_b5cblonbp_product_prefix_choice_valuation_selected_power_product = ff_q_b5cblonbp_product_prefix_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_b5cblonbp_product_prefix_choice)) * ff_v_b5cblonbp_product_prefix_choice_valuation_selected_power_product) + (bpr_power_value_b5cblonbp_product_prefix_choice_valuation_selected))) /\ forall ff_i_b5cblonbp_product_prefix_choice_valuation_selected_power_product. (exists ff_lt_b5cblonbp_product_prefix_choice_valuation_selected_power_product_bound. ff_lt_b5cblonbp_product_prefix_choice_valuation_selected_power_product_bound + S ff_i_b5cblonbp_product_prefix_choice_valuation_selected_power_product = bpr_choice_exponent_b5cblonbp_product_prefix_choice) -> exists ff_p_b5cblonbp_product_prefix_choice_valuation_selected_power_product ff_r_b5cblonbp_product_prefix_choice_valuation_selected_power_product ff_s_b5cblonbp_product_prefix_choice_valuation_selected_power_product. ((((exists ff_h_b5cblonbp_product_prefix_choice_valuation_selected_power_product_factor. ff_h_b5cblonbp_product_prefix_choice_valuation_selected_power_product_factor + S (ff_p_b5cblonbp_product_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_b5cblonbp_product_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_b5cblonbp_product_prefix_choice_valuation_selected_power)) /\ exists ff_q_b5cblonbp_product_prefix_choice_valuation_selected_power_product_factor. bpr_power_code_b5cblonbp_product_prefix_choice_valuation_selected_power = ff_q_b5cblonbp_product_prefix_choice_valuation_selected_power_product_factor * S ((S (ff_i_b5cblonbp_product_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_b5cblonbp_product_prefix_choice_valuation_selected_power) + (ff_p_b5cblonbp_product_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_b5cblonbp_product_prefix_choice_valuation_selected_power_product_partial. ff_h_b5cblonbp_product_prefix_choice_valuation_selected_power_product_partial + S (ff_r_b5cblonbp_product_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_b5cblonbp_product_prefix_choice_valuation_selected_power_product)) * ff_v_b5cblonbp_product_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5cblonbp_product_prefix_choice_valuation_selected_power_product_partial. ff_u_b5cblonbp_product_prefix_choice_valuation_selected_power_product = ff_q_b5cblonbp_product_prefix_choice_valuation_selected_power_product_partial * S ((S (ff_i_b5cblonbp_product_prefix_choice_valuation_selected_power_product)) * ff_v_b5cblonbp_product_prefix_choice_valuation_selected_power_product) + (ff_r_b5cblonbp_product_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_b5cblonbp_product_prefix_choice_valuation_selected_power_product_successor. ff_h_b5cblonbp_product_prefix_choice_valuation_selected_power_product_successor + S (ff_s_b5cblonbp_product_prefix_choice_valuation_selected_power_product) = S ((S (S ff_i_b5cblonbp_product_prefix_choice_valuation_selected_power_product)) * ff_v_b5cblonbp_product_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5cblonbp_product_prefix_choice_valuation_selected_power_product_successor. ff_u_b5cblonbp_product_prefix_choice_valuation_selected_power_product = ff_q_b5cblonbp_product_prefix_choice_valuation_selected_power_product_successor * S ((S (S ff_i_b5cblonbp_product_prefix_choice_valuation_selected_power_product)) * ff_v_b5cblonbp_product_prefix_choice_valuation_selected_power_product) + (ff_s_b5cblonbp_product_prefix_choice_valuation_selected_power_product))) /\ ff_s_b5cblonbp_product_prefix_choice_valuation_selected_power_product = ff_r_b5cblonbp_product_prefix_choice_valuation_selected_power_product * ff_p_b5cblonbp_product_prefix_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_b5cblonbp_product_prefix_choice_valuation_selected_divides. C = (bpr_power_value_b5cblonbp_product_prefix_choice_valuation_selected) * bpr_divides_quotient_b5cblonbp_product_prefix_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_b5cblonbp_product_prefix_choice_valuation. (exists bpr_le_gap_b5cblonbp_product_prefix_choice_valuation_candidate_bound. bpr_le_gap_b5cblonbp_product_prefix_choice_valuation_candidate_bound + (bpr_valuation_candidate_b5cblonbp_product_prefix_choice_valuation) = (C)) -> (exists bpr_power_value_b5cblonbp_product_prefix_choice_valuation_candidate. ((exists bpr_power_code_b5cblonbp_product_prefix_choice_valuation_candidate_power bpr_power_scale_b5cblonbp_product_prefix_choice_valuation_candidate_power. ((forall bpr_power_index_b5cblonbp_product_prefix_choice_valuation_candidate_power. (exists bpr_gap_b5cblonbp_product_prefix_choice_valuation_candidate_power_repeat_bound. bpr_gap_b5cblonbp_product_prefix_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_b5cblonbp_product_prefix_choice_valuation_candidate_power) = bpr_valuation_candidate_b5cblonbp_product_prefix_choice_valuation) -> (((exists bpr_height_b5cblonbp_product_prefix_choice_valuation_candidate_power_repeat_entry. bpr_height_b5cblonbp_product_prefix_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_b5cblonbp_product_prefix)) = S ((S (bpr_power_index_b5cblonbp_product_prefix_choice_valuation_candidate_power)) * bpr_power_scale_b5cblonbp_product_prefix_choice_valuation_candidate_power)) /\ exists bpr_quotient_b5cblonbp_product_prefix_choice_valuation_candidate_power_repeat_entry. bpr_power_code_b5cblonbp_product_prefix_choice_valuation_candidate_power = bpr_quotient_b5cblonbp_product_prefix_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_b5cblonbp_product_prefix_choice_valuation_candidate_power)) * bpr_power_scale_b5cblonbp_product_prefix_choice_valuation_candidate_power) + (S (bpr_prefix_index_b5cblonbp_product_prefix))))) /\ (exists ff_u_b5cblonbp_product_prefix_choice_valuation_candidate_power_product ff_v_b5cblonbp_product_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_start. ff_h_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_b5cblonbp_product_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_start. ff_u_b5cblonbp_product_prefix_choice_valuation_candidate_power_product = ff_q_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_b5cblonbp_product_prefix_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_terminal. ff_h_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_b5cblonbp_product_prefix_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_b5cblonbp_product_prefix_choice_valuation)) * ff_v_b5cblonbp_product_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_terminal. ff_u_b5cblonbp_product_prefix_choice_valuation_candidate_power_product = ff_q_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_b5cblonbp_product_prefix_choice_valuation)) * ff_v_b5cblonbp_product_prefix_choice_valuation_candidate_power_product) + (bpr_power_value_b5cblonbp_product_prefix_choice_valuation_candidate))) /\ forall ff_i_b5cblonbp_product_prefix_choice_valuation_candidate_power_product. (exists ff_lt_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_bound. ff_lt_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_bound + S ff_i_b5cblonbp_product_prefix_choice_valuation_candidate_power_product = bpr_valuation_candidate_b5cblonbp_product_prefix_choice_valuation) -> exists ff_p_b5cblonbp_product_prefix_choice_valuation_candidate_power_product ff_r_b5cblonbp_product_prefix_choice_valuation_candidate_power_product ff_s_b5cblonbp_product_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_factor. ff_h_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_factor + S (ff_p_b5cblonbp_product_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_b5cblonbp_product_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_b5cblonbp_product_prefix_choice_valuation_candidate_power)) /\ exists ff_q_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_factor. bpr_power_code_b5cblonbp_product_prefix_choice_valuation_candidate_power = ff_q_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_factor * S ((S (ff_i_b5cblonbp_product_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_b5cblonbp_product_prefix_choice_valuation_candidate_power) + (ff_p_b5cblonbp_product_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_partial. ff_h_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_partial + S (ff_r_b5cblonbp_product_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_b5cblonbp_product_prefix_choice_valuation_candidate_power_product)) * ff_v_b5cblonbp_product_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_partial. ff_u_b5cblonbp_product_prefix_choice_valuation_candidate_power_product = ff_q_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_partial * S ((S (ff_i_b5cblonbp_product_prefix_choice_valuation_candidate_power_product)) * ff_v_b5cblonbp_product_prefix_choice_valuation_candidate_power_product) + (ff_r_b5cblonbp_product_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_successor. ff_h_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_successor + S (ff_s_b5cblonbp_product_prefix_choice_valuation_candidate_power_product) = S ((S (S ff_i_b5cblonbp_product_prefix_choice_valuation_candidate_power_product)) * ff_v_b5cblonbp_product_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_successor. ff_u_b5cblonbp_product_prefix_choice_valuation_candidate_power_product = ff_q_b5cblonbp_product_prefix_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_b5cblonbp_product_prefix_choice_valuation_candidate_power_product)) * ff_v_b5cblonbp_product_prefix_choice_valuation_candidate_power_product) + (ff_s_b5cblonbp_product_prefix_choice_valuation_candidate_power_product))) /\ ff_s_b5cblonbp_product_prefix_choice_valuation_candidate_power_product = ff_r_b5cblonbp_product_prefix_choice_valuation_candidate_power_product * ff_p_b5cblonbp_product_prefix_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_b5cblonbp_product_prefix_choice_valuation_candidate_divides. C = (bpr_power_value_b5cblonbp_product_prefix_choice_valuation_candidate) * bpr_divides_quotient_b5cblonbp_product_prefix_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_b5cblonbp_product_prefix_choice_valuation_candidate_below. bpr_le_gap_b5cblonbp_product_prefix_choice_valuation_candidate_below + (bpr_valuation_candidate_b5cblonbp_product_prefix_choice_valuation) = (bpr_choice_exponent_b5cblonbp_product_prefix_choice))) /\ (exists bpr_power_code_b5cblonbp_product_prefix_choice_power bpr_power_scale_b5cblonbp_product_prefix_choice_power. ((forall bpr_power_index_b5cblonbp_product_prefix_choice_power. (exists bpr_gap_b5cblonbp_product_prefix_choice_power_repeat_bound. bpr_gap_b5cblonbp_product_prefix_choice_power_repeat_bound + S (bpr_power_index_b5cblonbp_product_prefix_choice_power) = bpr_choice_exponent_b5cblonbp_product_prefix_choice) -> (((exists bpr_height_b5cblonbp_product_prefix_choice_power_repeat_entry. bpr_height_b5cblonbp_product_prefix_choice_power_repeat_entry + S (S (bpr_prefix_index_b5cblonbp_product_prefix)) = S ((S (bpr_power_index_b5cblonbp_product_prefix_choice_power)) * bpr_power_scale_b5cblonbp_product_prefix_choice_power)) /\ exists bpr_quotient_b5cblonbp_product_prefix_choice_power_repeat_entry. bpr_power_code_b5cblonbp_product_prefix_choice_power = bpr_quotient_b5cblonbp_product_prefix_choice_power_repeat_entry * S ((S (bpr_power_index_b5cblonbp_product_prefix_choice_power)) * bpr_power_scale_b5cblonbp_product_prefix_choice_power) + (S (bpr_prefix_index_b5cblonbp_product_prefix))))) /\ (exists ff_u_b5cblonbp_product_prefix_choice_power_product ff_v_b5cblonbp_product_prefix_choice_power_product. ((((exists ff_h_b5cblonbp_product_prefix_choice_power_product_start. ff_h_b5cblonbp_product_prefix_choice_power_product_start + S (1) = S ((S (0)) * ff_v_b5cblonbp_product_prefix_choice_power_product)) /\ exists ff_q_b5cblonbp_product_prefix_choice_power_product_start. ff_u_b5cblonbp_product_prefix_choice_power_product = ff_q_b5cblonbp_product_prefix_choice_power_product_start * S ((S (0)) * ff_v_b5cblonbp_product_prefix_choice_power_product) + (1))) /\ ((((exists ff_h_b5cblonbp_product_prefix_choice_power_product_terminal. ff_h_b5cblonbp_product_prefix_choice_power_product_terminal + S (bpr_prefix_value_b5cblonbp_product_prefix) = S ((S (bpr_choice_exponent_b5cblonbp_product_prefix_choice)) * ff_v_b5cblonbp_product_prefix_choice_power_product)) /\ exists ff_q_b5cblonbp_product_prefix_choice_power_product_terminal. ff_u_b5cblonbp_product_prefix_choice_power_product = ff_q_b5cblonbp_product_prefix_choice_power_product_terminal * S ((S (bpr_choice_exponent_b5cblonbp_product_prefix_choice)) * ff_v_b5cblonbp_product_prefix_choice_power_product) + (bpr_prefix_value_b5cblonbp_product_prefix))) /\ forall ff_i_b5cblonbp_product_prefix_choice_power_product. (exists ff_lt_b5cblonbp_product_prefix_choice_power_product_bound. ff_lt_b5cblonbp_product_prefix_choice_power_product_bound + S ff_i_b5cblonbp_product_prefix_choice_power_product = bpr_choice_exponent_b5cblonbp_product_prefix_choice) -> exists ff_p_b5cblonbp_product_prefix_choice_power_product ff_r_b5cblonbp_product_prefix_choice_power_product ff_s_b5cblonbp_product_prefix_choice_power_product. ((((exists ff_h_b5cblonbp_product_prefix_choice_power_product_factor. ff_h_b5cblonbp_product_prefix_choice_power_product_factor + S (ff_p_b5cblonbp_product_prefix_choice_power_product) = S ((S (ff_i_b5cblonbp_product_prefix_choice_power_product)) * bpr_power_scale_b5cblonbp_product_prefix_choice_power)) /\ exists ff_q_b5cblonbp_product_prefix_choice_power_product_factor. bpr_power_code_b5cblonbp_product_prefix_choice_power = ff_q_b5cblonbp_product_prefix_choice_power_product_factor * S ((S (ff_i_b5cblonbp_product_prefix_choice_power_product)) * bpr_power_scale_b5cblonbp_product_prefix_choice_power) + (ff_p_b5cblonbp_product_prefix_choice_power_product))) /\ ((((exists ff_h_b5cblonbp_product_prefix_choice_power_product_partial. ff_h_b5cblonbp_product_prefix_choice_power_product_partial + S (ff_r_b5cblonbp_product_prefix_choice_power_product) = S ((S (ff_i_b5cblonbp_product_prefix_choice_power_product)) * ff_v_b5cblonbp_product_prefix_choice_power_product)) /\ exists ff_q_b5cblonbp_product_prefix_choice_power_product_partial. ff_u_b5cblonbp_product_prefix_choice_power_product = ff_q_b5cblonbp_product_prefix_choice_power_product_partial * S ((S (ff_i_b5cblonbp_product_prefix_choice_power_product)) * ff_v_b5cblonbp_product_prefix_choice_power_product) + (ff_r_b5cblonbp_product_prefix_choice_power_product))) /\ ((((exists ff_h_b5cblonbp_product_prefix_choice_power_product_successor. ff_h_b5cblonbp_product_prefix_choice_power_product_successor + S (ff_s_b5cblonbp_product_prefix_choice_power_product) = S ((S (S ff_i_b5cblonbp_product_prefix_choice_power_product)) * ff_v_b5cblonbp_product_prefix_choice_power_product)) /\ exists ff_q_b5cblonbp_product_prefix_choice_power_product_successor. ff_u_b5cblonbp_product_prefix_choice_power_product = ff_q_b5cblonbp_product_prefix_choice_power_product_successor * S ((S (S ff_i_b5cblonbp_product_prefix_choice_power_product)) * ff_v_b5cblonbp_product_prefix_choice_power_product) + (ff_s_b5cblonbp_product_prefix_choice_power_product))) /\ ff_s_b5cblonbp_product_prefix_choice_power_product = ff_r_b5cblonbp_product_prefix_choice_power_product * ff_p_b5cblonbp_product_prefix_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_b5cblonbp_product_prefix) = 1) /\ forall bpr_left_b5cblonbp_product_prefix_choice_prime bpr_right_b5cblonbp_product_prefix_choice_prime. S (bpr_prefix_index_b5cblonbp_product_prefix) = bpr_left_b5cblonbp_product_prefix_choice_prime * bpr_right_b5cblonbp_product_prefix_choice_prime -> bpr_left_b5cblonbp_product_prefix_choice_prime = 1 \/ bpr_right_b5cblonbp_product_prefix_choice_prime = 1)) /\ bpr_prefix_value_b5cblonbp_product_prefix = 1))))) /\ (exists ff_u_b5cblonbp_product_product ff_v_b5cblonbp_product_product. ((((exists ff_h_b5cblonbp_product_product_start. ff_h_b5cblonbp_product_product_start + S (1) = S ((S (0)) * ff_v_b5cblonbp_product_product)) /\ exists ff_q_b5cblonbp_product_product_start. ff_u_b5cblonbp_product_product = ff_q_b5cblonbp_product_product_start * S ((S (0)) * ff_v_b5cblonbp_product_product) + (1))) /\ ((((exists ff_h_b5cblonbp_product_product_terminal. ff_h_b5cblonbp_product_product_terminal + S (z) = S ((S (n + n)) * ff_v_b5cblonbp_product_product)) /\ exists ff_q_b5cblonbp_product_product_terminal. ff_u_b5cblonbp_product_product = ff_q_b5cblonbp_product_product_terminal * S ((S (n + n)) * ff_v_b5cblonbp_product_product) + (z))) /\ forall ff_i_b5cblonbp_product_product. (exists ff_lt_b5cblonbp_product_product_bound. ff_lt_b5cblonbp_product_product_bound + S ff_i_b5cblonbp_product_product = n + n) -> exists ff_p_b5cblonbp_product_product ff_r_b5cblonbp_product_product ff_s_b5cblonbp_product_product. ((((exists ff_h_b5cblonbp_product_product_factor. ff_h_b5cblonbp_product_product_factor + S (ff_p_b5cblonbp_product_product) = S ((S (ff_i_b5cblonbp_product_product)) * bpr_product_scale_b5cblonbp_product)) /\ exists ff_q_b5cblonbp_product_product_factor. bpr_product_code_b5cblonbp_product = ff_q_b5cblonbp_product_product_factor * S ((S (ff_i_b5cblonbp_product_product)) * bpr_product_scale_b5cblonbp_product) + (ff_p_b5cblonbp_product_product))) /\ ((((exists ff_h_b5cblonbp_product_product_partial. ff_h_b5cblonbp_product_product_partial + S (ff_r_b5cblonbp_product_product) = S ((S (ff_i_b5cblonbp_product_product)) * ff_v_b5cblonbp_product_product)) /\ exists ff_q_b5cblonbp_product_product_partial. ff_u_b5cblonbp_product_product = ff_q_b5cblonbp_product_product_partial * S ((S (ff_i_b5cblonbp_product_product)) * ff_v_b5cblonbp_product_product) + (ff_r_b5cblonbp_product_product))) /\ ((((exists ff_h_b5cblonbp_product_product_successor. ff_h_b5cblonbp_product_product_successor + S (ff_s_b5cblonbp_product_product) = S ((S (S ff_i_b5cblonbp_product_product)) * ff_v_b5cblonbp_product_product)) /\ exists ff_q_b5cblonbp_product_product_successor. ff_u_b5cblonbp_product_product = ff_q_b5cblonbp_product_product_successor * S ((S (S ff_i_b5cblonbp_product_product)) * ff_v_b5cblonbp_product_product) + (ff_s_b5cblonbp_product_product))) /\ ff_s_b5cblonbp_product_product = ff_r_b5cblonbp_product_product * ff_p_b5cblonbp_product_product)))))))) /\ C = z
0028specialize central_binom_prime_contribution_product_exists n
0029specialize central_binom_prime_contribution_product_exists C
0030apply central_binom_prime_contribution_product_exists
0031exact hcentral
0032cases hproduct
0033cases hproduct_witness
0034have hfactorization : exists u v. (exists bpr_product_code_b5cblonbp_small bpr_product_scale_b5cblonbp_small. ((forall bpr_prefix_index_b5cblonbp_small_prefix. (exists bpr_gap_b5cblonbp_small_prefix_bound. bpr_gap_b5cblonbp_small_prefix_bound + S (bpr_prefix_index_b5cblonbp_small_prefix) = s) -> exists bpr_prefix_value_b5cblonbp_small_prefix. ((((exists bpr_height_b5cblonbp_small_prefix_decoded. bpr_height_b5cblonbp_small_prefix_decoded + S (bpr_prefix_value_b5cblonbp_small_prefix) = S ((S (bpr_prefix_index_b5cblonbp_small_prefix)) * bpr_product_scale_b5cblonbp_small)) /\ exists bpr_quotient_b5cblonbp_small_prefix_decoded. bpr_product_code_b5cblonbp_small = bpr_quotient_b5cblonbp_small_prefix_decoded * S ((S (bpr_prefix_index_b5cblonbp_small_prefix)) * bpr_product_scale_b5cblonbp_small) + (bpr_prefix_value_b5cblonbp_small_prefix))) /\ (((((~(S (bpr_prefix_index_b5cblonbp_small_prefix) = 1) /\ forall bpr_left_b5cblonbp_small_prefix_choice_prime bpr_right_b5cblonbp_small_prefix_choice_prime. S (bpr_prefix_index_b5cblonbp_small_prefix) = bpr_left_b5cblonbp_small_prefix_choice_prime * bpr_right_b5cblonbp_small_prefix_choice_prime -> bpr_left_b5cblonbp_small_prefix_choice_prime = 1 \/ bpr_right_b5cblonbp_small_prefix_choice_prime = 1)) /\ exists bpr_choice_exponent_b5cblonbp_small_prefix_choice. ((((exists bpr_le_gap_b5cblonbp_small_prefix_choice_valuation_selected_bound. bpr_le_gap_b5cblonbp_small_prefix_choice_valuation_selected_bound + (bpr_choice_exponent_b5cblonbp_small_prefix_choice) = (C)) /\ (exists bpr_power_value_b5cblonbp_small_prefix_choice_valuation_selected. ((exists bpr_power_code_b5cblonbp_small_prefix_choice_valuation_selected_power bpr_power_scale_b5cblonbp_small_prefix_choice_valuation_selected_power. ((forall bpr_power_index_b5cblonbp_small_prefix_choice_valuation_selected_power. (exists bpr_gap_b5cblonbp_small_prefix_choice_valuation_selected_power_repeat_bound. bpr_gap_b5cblonbp_small_prefix_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_b5cblonbp_small_prefix_choice_valuation_selected_power) = bpr_choice_exponent_b5cblonbp_small_prefix_choice) -> (((exists bpr_height_b5cblonbp_small_prefix_choice_valuation_selected_power_repeat_entry. bpr_height_b5cblonbp_small_prefix_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_b5cblonbp_small_prefix)) = S ((S (bpr_power_index_b5cblonbp_small_prefix_choice_valuation_selected_power)) * bpr_power_scale_b5cblonbp_small_prefix_choice_valuation_selected_power)) /\ exists bpr_quotient_b5cblonbp_small_prefix_choice_valuation_selected_power_repeat_entry. bpr_power_code_b5cblonbp_small_prefix_choice_valuation_selected_power = bpr_quotient_b5cblonbp_small_prefix_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_b5cblonbp_small_prefix_choice_valuation_selected_power)) * bpr_power_scale_b5cblonbp_small_prefix_choice_valuation_selected_power) + (S (bpr_prefix_index_b5cblonbp_small_prefix))))) /\ (exists ff_u_b5cblonbp_small_prefix_choice_valuation_selected_power_product ff_v_b5cblonbp_small_prefix_choice_valuation_selected_power_product. ((((exists ff_h_b5cblonbp_small_prefix_choice_valuation_selected_power_product_start. ff_h_b5cblonbp_small_prefix_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_b5cblonbp_small_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5cblonbp_small_prefix_choice_valuation_selected_power_product_start. ff_u_b5cblonbp_small_prefix_choice_valuation_selected_power_product = ff_q_b5cblonbp_small_prefix_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_b5cblonbp_small_prefix_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_b5cblonbp_small_prefix_choice_valuation_selected_power_product_terminal. ff_h_b5cblonbp_small_prefix_choice_valuation_selected_power_product_terminal + S (bpr_power_value_b5cblonbp_small_prefix_choice_valuation_selected) = S ((S (bpr_choice_exponent_b5cblonbp_small_prefix_choice)) * ff_v_b5cblonbp_small_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5cblonbp_small_prefix_choice_valuation_selected_power_product_terminal. ff_u_b5cblonbp_small_prefix_choice_valuation_selected_power_product = ff_q_b5cblonbp_small_prefix_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_b5cblonbp_small_prefix_choice)) * ff_v_b5cblonbp_small_prefix_choice_valuation_selected_power_product) + (bpr_power_value_b5cblonbp_small_prefix_choice_valuation_selected))) /\ forall ff_i_b5cblonbp_small_prefix_choice_valuation_selected_power_product. (exists ff_lt_b5cblonbp_small_prefix_choice_valuation_selected_power_product_bound. ff_lt_b5cblonbp_small_prefix_choice_valuation_selected_power_product_bound + S ff_i_b5cblonbp_small_prefix_choice_valuation_selected_power_product = bpr_choice_exponent_b5cblonbp_small_prefix_choice) -> exists ff_p_b5cblonbp_small_prefix_choice_valuation_selected_power_product ff_r_b5cblonbp_small_prefix_choice_valuation_selected_power_product ff_s_b5cblonbp_small_prefix_choice_valuation_selected_power_product. ((((exists ff_h_b5cblonbp_small_prefix_choice_valuation_selected_power_product_factor. ff_h_b5cblonbp_small_prefix_choice_valuation_selected_power_product_factor + S (ff_p_b5cblonbp_small_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_b5cblonbp_small_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_b5cblonbp_small_prefix_choice_valuation_selected_power)) /\ exists ff_q_b5cblonbp_small_prefix_choice_valuation_selected_power_product_factor. bpr_power_code_b5cblonbp_small_prefix_choice_valuation_selected_power = ff_q_b5cblonbp_small_prefix_choice_valuation_selected_power_product_factor * S ((S (ff_i_b5cblonbp_small_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_b5cblonbp_small_prefix_choice_valuation_selected_power) + (ff_p_b5cblonbp_small_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_b5cblonbp_small_prefix_choice_valuation_selected_power_product_partial. ff_h_b5cblonbp_small_prefix_choice_valuation_selected_power_product_partial + S (ff_r_b5cblonbp_small_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_b5cblonbp_small_prefix_choice_valuation_selected_power_product)) * ff_v_b5cblonbp_small_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5cblonbp_small_prefix_choice_valuation_selected_power_product_partial. ff_u_b5cblonbp_small_prefix_choice_valuation_selected_power_product = ff_q_b5cblonbp_small_prefix_choice_valuation_selected_power_product_partial * S ((S (ff_i_b5cblonbp_small_prefix_choice_valuation_selected_power_product)) * ff_v_b5cblonbp_small_prefix_choice_valuation_selected_power_product) + (ff_r_b5cblonbp_small_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_b5cblonbp_small_prefix_choice_valuation_selected_power_product_successor. ff_h_b5cblonbp_small_prefix_choice_valuation_selected_power_product_successor + S (ff_s_b5cblonbp_small_prefix_choice_valuation_selected_power_product) = S ((S (S ff_i_b5cblonbp_small_prefix_choice_valuation_selected_power_product)) * ff_v_b5cblonbp_small_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5cblonbp_small_prefix_choice_valuation_selected_power_product_successor. ff_u_b5cblonbp_small_prefix_choice_valuation_selected_power_product = ff_q_b5cblonbp_small_prefix_choice_valuation_selected_power_product_successor * S ((S (S ff_i_b5cblonbp_small_prefix_choice_valuation_selected_power_product)) * ff_v_b5cblonbp_small_prefix_choice_valuation_selected_power_product) + (ff_s_b5cblonbp_small_prefix_choice_valuation_selected_power_product))) /\ ff_s_b5cblonbp_small_prefix_choice_valuation_selected_power_product = ff_r_b5cblonbp_small_prefix_choice_valuation_selected_power_product * ff_p_b5cblonbp_small_prefix_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_b5cblonbp_small_prefix_choice_valuation_selected_divides. C = (bpr_power_value_b5cblonbp_small_prefix_choice_valuation_selected) * bpr_divides_quotient_b5cblonbp_small_prefix_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_b5cblonbp_small_prefix_choice_valuation. (exists bpr_le_gap_b5cblonbp_small_prefix_choice_valuation_candidate_bound. bpr_le_gap_b5cblonbp_small_prefix_choice_valuation_candidate_bound + (bpr_valuation_candidate_b5cblonbp_small_prefix_choice_valuation) = (C)) -> (exists bpr_power_value_b5cblonbp_small_prefix_choice_valuation_candidate. ((exists bpr_power_code_b5cblonbp_small_prefix_choice_valuation_candidate_power bpr_power_scale_b5cblonbp_small_prefix_choice_valuation_candidate_power. ((forall bpr_power_index_b5cblonbp_small_prefix_choice_valuation_candidate_power. (exists bpr_gap_b5cblonbp_small_prefix_choice_valuation_candidate_power_repeat_bound. bpr_gap_b5cblonbp_small_prefix_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_b5cblonbp_small_prefix_choice_valuation_candidate_power) = bpr_valuation_candidate_b5cblonbp_small_prefix_choice_valuation) -> (((exists bpr_height_b5cblonbp_small_prefix_choice_valuation_candidate_power_repeat_entry. bpr_height_b5cblonbp_small_prefix_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_b5cblonbp_small_prefix)) = S ((S (bpr_power_index_b5cblonbp_small_prefix_choice_valuation_candidate_power)) * bpr_power_scale_b5cblonbp_small_prefix_choice_valuation_candidate_power)) /\ exists bpr_quotient_b5cblonbp_small_prefix_choice_valuation_candidate_power_repeat_entry. bpr_power_code_b5cblonbp_small_prefix_choice_valuation_candidate_power = bpr_quotient_b5cblonbp_small_prefix_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_b5cblonbp_small_prefix_choice_valuation_candidate_power)) * bpr_power_scale_b5cblonbp_small_prefix_choice_valuation_candidate_power) + (S (bpr_prefix_index_b5cblonbp_small_prefix))))) /\ (exists ff_u_b5cblonbp_small_prefix_choice_valuation_candidate_power_product ff_v_b5cblonbp_small_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_start. ff_h_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_b5cblonbp_small_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_start. ff_u_b5cblonbp_small_prefix_choice_valuation_candidate_power_product = ff_q_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_b5cblonbp_small_prefix_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_terminal. ff_h_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_b5cblonbp_small_prefix_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_b5cblonbp_small_prefix_choice_valuation)) * ff_v_b5cblonbp_small_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_terminal. ff_u_b5cblonbp_small_prefix_choice_valuation_candidate_power_product = ff_q_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_b5cblonbp_small_prefix_choice_valuation)) * ff_v_b5cblonbp_small_prefix_choice_valuation_candidate_power_product) + (bpr_power_value_b5cblonbp_small_prefix_choice_valuation_candidate))) /\ forall ff_i_b5cblonbp_small_prefix_choice_valuation_candidate_power_product. (exists ff_lt_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_bound. ff_lt_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_bound + S ff_i_b5cblonbp_small_prefix_choice_valuation_candidate_power_product = bpr_valuation_candidate_b5cblonbp_small_prefix_choice_valuation) -> exists ff_p_b5cblonbp_small_prefix_choice_valuation_candidate_power_product ff_r_b5cblonbp_small_prefix_choice_valuation_candidate_power_product ff_s_b5cblonbp_small_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_factor. ff_h_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_factor + S (ff_p_b5cblonbp_small_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_b5cblonbp_small_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_b5cblonbp_small_prefix_choice_valuation_candidate_power)) /\ exists ff_q_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_factor. bpr_power_code_b5cblonbp_small_prefix_choice_valuation_candidate_power = ff_q_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_factor * S ((S (ff_i_b5cblonbp_small_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_b5cblonbp_small_prefix_choice_valuation_candidate_power) + (ff_p_b5cblonbp_small_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_partial. ff_h_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_partial + S (ff_r_b5cblonbp_small_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_b5cblonbp_small_prefix_choice_valuation_candidate_power_product)) * ff_v_b5cblonbp_small_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_partial. ff_u_b5cblonbp_small_prefix_choice_valuation_candidate_power_product = ff_q_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_partial * S ((S (ff_i_b5cblonbp_small_prefix_choice_valuation_candidate_power_product)) * ff_v_b5cblonbp_small_prefix_choice_valuation_candidate_power_product) + (ff_r_b5cblonbp_small_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_successor. ff_h_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_successor + S (ff_s_b5cblonbp_small_prefix_choice_valuation_candidate_power_product) = S ((S (S ff_i_b5cblonbp_small_prefix_choice_valuation_candidate_power_product)) * ff_v_b5cblonbp_small_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_successor. ff_u_b5cblonbp_small_prefix_choice_valuation_candidate_power_product = ff_q_b5cblonbp_small_prefix_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_b5cblonbp_small_prefix_choice_valuation_candidate_power_product)) * ff_v_b5cblonbp_small_prefix_choice_valuation_candidate_power_product) + (ff_s_b5cblonbp_small_prefix_choice_valuation_candidate_power_product))) /\ ff_s_b5cblonbp_small_prefix_choice_valuation_candidate_power_product = ff_r_b5cblonbp_small_prefix_choice_valuation_candidate_power_product * ff_p_b5cblonbp_small_prefix_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_b5cblonbp_small_prefix_choice_valuation_candidate_divides. C = (bpr_power_value_b5cblonbp_small_prefix_choice_valuation_candidate) * bpr_divides_quotient_b5cblonbp_small_prefix_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_b5cblonbp_small_prefix_choice_valuation_candidate_below. bpr_le_gap_b5cblonbp_small_prefix_choice_valuation_candidate_below + (bpr_valuation_candidate_b5cblonbp_small_prefix_choice_valuation) = (bpr_choice_exponent_b5cblonbp_small_prefix_choice))) /\ (exists bpr_power_code_b5cblonbp_small_prefix_choice_power bpr_power_scale_b5cblonbp_small_prefix_choice_power. ((forall bpr_power_index_b5cblonbp_small_prefix_choice_power. (exists bpr_gap_b5cblonbp_small_prefix_choice_power_repeat_bound. bpr_gap_b5cblonbp_small_prefix_choice_power_repeat_bound + S (bpr_power_index_b5cblonbp_small_prefix_choice_power) = bpr_choice_exponent_b5cblonbp_small_prefix_choice) -> (((exists bpr_height_b5cblonbp_small_prefix_choice_power_repeat_entry. bpr_height_b5cblonbp_small_prefix_choice_power_repeat_entry + S (S (bpr_prefix_index_b5cblonbp_small_prefix)) = S ((S (bpr_power_index_b5cblonbp_small_prefix_choice_power)) * bpr_power_scale_b5cblonbp_small_prefix_choice_power)) /\ exists bpr_quotient_b5cblonbp_small_prefix_choice_power_repeat_entry. bpr_power_code_b5cblonbp_small_prefix_choice_power = bpr_quotient_b5cblonbp_small_prefix_choice_power_repeat_entry * S ((S (bpr_power_index_b5cblonbp_small_prefix_choice_power)) * bpr_power_scale_b5cblonbp_small_prefix_choice_power) + (S (bpr_prefix_index_b5cblonbp_small_prefix))))) /\ (exists ff_u_b5cblonbp_small_prefix_choice_power_product ff_v_b5cblonbp_small_prefix_choice_power_product. ((((exists ff_h_b5cblonbp_small_prefix_choice_power_product_start. ff_h_b5cblonbp_small_prefix_choice_power_product_start + S (1) = S ((S (0)) * ff_v_b5cblonbp_small_prefix_choice_power_product)) /\ exists ff_q_b5cblonbp_small_prefix_choice_power_product_start. ff_u_b5cblonbp_small_prefix_choice_power_product = ff_q_b5cblonbp_small_prefix_choice_power_product_start * S ((S (0)) * ff_v_b5cblonbp_small_prefix_choice_power_product) + (1))) /\ ((((exists ff_h_b5cblonbp_small_prefix_choice_power_product_terminal. ff_h_b5cblonbp_small_prefix_choice_power_product_terminal + S (bpr_prefix_value_b5cblonbp_small_prefix) = S ((S (bpr_choice_exponent_b5cblonbp_small_prefix_choice)) * ff_v_b5cblonbp_small_prefix_choice_power_product)) /\ exists ff_q_b5cblonbp_small_prefix_choice_power_product_terminal. ff_u_b5cblonbp_small_prefix_choice_power_product = ff_q_b5cblonbp_small_prefix_choice_power_product_terminal * S ((S (bpr_choice_exponent_b5cblonbp_small_prefix_choice)) * ff_v_b5cblonbp_small_prefix_choice_power_product) + (bpr_prefix_value_b5cblonbp_small_prefix))) /\ forall ff_i_b5cblonbp_small_prefix_choice_power_product. (exists ff_lt_b5cblonbp_small_prefix_choice_power_product_bound. ff_lt_b5cblonbp_small_prefix_choice_power_product_bound + S ff_i_b5cblonbp_small_prefix_choice_power_product = bpr_choice_exponent_b5cblonbp_small_prefix_choice) -> exists ff_p_b5cblonbp_small_prefix_choice_power_product ff_r_b5cblonbp_small_prefix_choice_power_product ff_s_b5cblonbp_small_prefix_choice_power_product. ((((exists ff_h_b5cblonbp_small_prefix_choice_power_product_factor. ff_h_b5cblonbp_small_prefix_choice_power_product_factor + S (ff_p_b5cblonbp_small_prefix_choice_power_product) = S ((S (ff_i_b5cblonbp_small_prefix_choice_power_product)) * bpr_power_scale_b5cblonbp_small_prefix_choice_power)) /\ exists ff_q_b5cblonbp_small_prefix_choice_power_product_factor. bpr_power_code_b5cblonbp_small_prefix_choice_power = ff_q_b5cblonbp_small_prefix_choice_power_product_factor * S ((S (ff_i_b5cblonbp_small_prefix_choice_power_product)) * bpr_power_scale_b5cblonbp_small_prefix_choice_power) + (ff_p_b5cblonbp_small_prefix_choice_power_product))) /\ ((((exists ff_h_b5cblonbp_small_prefix_choice_power_product_partial. ff_h_b5cblonbp_small_prefix_choice_power_product_partial + S (ff_r_b5cblonbp_small_prefix_choice_power_product) = S ((S (ff_i_b5cblonbp_small_prefix_choice_power_product)) * ff_v_b5cblonbp_small_prefix_choice_power_product)) /\ exists ff_q_b5cblonbp_small_prefix_choice_power_product_partial. ff_u_b5cblonbp_small_prefix_choice_power_product = ff_q_b5cblonbp_small_prefix_choice_power_product_partial * S ((S (ff_i_b5cblonbp_small_prefix_choice_power_product)) * ff_v_b5cblonbp_small_prefix_choice_power_product) + (ff_r_b5cblonbp_small_prefix_choice_power_product))) /\ ((((exists ff_h_b5cblonbp_small_prefix_choice_power_product_successor. ff_h_b5cblonbp_small_prefix_choice_power_product_successor + S (ff_s_b5cblonbp_small_prefix_choice_power_product) = S ((S (S ff_i_b5cblonbp_small_prefix_choice_power_product)) * ff_v_b5cblonbp_small_prefix_choice_power_product)) /\ exists ff_q_b5cblonbp_small_prefix_choice_power_product_successor. ff_u_b5cblonbp_small_prefix_choice_power_product = ff_q_b5cblonbp_small_prefix_choice_power_product_successor * S ((S (S ff_i_b5cblonbp_small_prefix_choice_power_product)) * ff_v_b5cblonbp_small_prefix_choice_power_product) + (ff_s_b5cblonbp_small_prefix_choice_power_product))) /\ ff_s_b5cblonbp_small_prefix_choice_power_product = ff_r_b5cblonbp_small_prefix_choice_power_product * ff_p_b5cblonbp_small_prefix_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_b5cblonbp_small_prefix) = 1) /\ forall bpr_left_b5cblonbp_small_prefix_choice_prime bpr_right_b5cblonbp_small_prefix_choice_prime. S (bpr_prefix_index_b5cblonbp_small_prefix) = bpr_left_b5cblonbp_small_prefix_choice_prime * bpr_right_b5cblonbp_small_prefix_choice_prime -> bpr_left_b5cblonbp_small_prefix_choice_prime = 1 \/ bpr_right_b5cblonbp_small_prefix_choice_prime = 1)) /\ bpr_prefix_value_b5cblonbp_small_prefix = 1))))) /\ (exists ff_u_b5cblonbp_small_product ff_v_b5cblonbp_small_product. ((((exists ff_h_b5cblonbp_small_product_start. ff_h_b5cblonbp_small_product_start + S (1) = S ((S (0)) * ff_v_b5cblonbp_small_product)) /\ exists ff_q_b5cblonbp_small_product_start. ff_u_b5cblonbp_small_product = ff_q_b5cblonbp_small_product_start * S ((S (0)) * ff_v_b5cblonbp_small_product) + (1))) /\ ((((exists ff_h_b5cblonbp_small_product_terminal. ff_h_b5cblonbp_small_product_terminal + S (u) = S ((S (s)) * ff_v_b5cblonbp_small_product)) /\ exists ff_q_b5cblonbp_small_product_terminal. ff_u_b5cblonbp_small_product = ff_q_b5cblonbp_small_product_terminal * S ((S (s)) * ff_v_b5cblonbp_small_product) + (u))) /\ forall ff_i_b5cblonbp_small_product. (exists ff_lt_b5cblonbp_small_product_bound. ff_lt_b5cblonbp_small_product_bound + S ff_i_b5cblonbp_small_product = s) -> exists ff_p_b5cblonbp_small_product ff_r_b5cblonbp_small_product ff_s_b5cblonbp_small_product. ((((exists ff_h_b5cblonbp_small_product_factor. ff_h_b5cblonbp_small_product_factor + S (ff_p_b5cblonbp_small_product) = S ((S (ff_i_b5cblonbp_small_product)) * bpr_product_scale_b5cblonbp_small)) /\ exists ff_q_b5cblonbp_small_product_factor. bpr_product_code_b5cblonbp_small = ff_q_b5cblonbp_small_product_factor * S ((S (ff_i_b5cblonbp_small_product)) * bpr_product_scale_b5cblonbp_small) + (ff_p_b5cblonbp_small_product))) /\ ((((exists ff_h_b5cblonbp_small_product_partial. ff_h_b5cblonbp_small_product_partial + S (ff_r_b5cblonbp_small_product) = S ((S (ff_i_b5cblonbp_small_product)) * ff_v_b5cblonbp_small_product)) /\ exists ff_q_b5cblonbp_small_product_partial. ff_u_b5cblonbp_small_product = ff_q_b5cblonbp_small_product_partial * S ((S (ff_i_b5cblonbp_small_product)) * ff_v_b5cblonbp_small_product) + (ff_r_b5cblonbp_small_product))) /\ ((((exists ff_h_b5cblonbp_small_product_successor. ff_h_b5cblonbp_small_product_successor + S (ff_s_b5cblonbp_small_product) = S ((S (S ff_i_b5cblonbp_small_product)) * ff_v_b5cblonbp_small_product)) /\ exists ff_q_b5cblonbp_small_product_successor. ff_u_b5cblonbp_small_product = ff_q_b5cblonbp_small_product_successor * S ((S (S ff_i_b5cblonbp_small_product)) * ff_v_b5cblonbp_small_product) + (ff_s_b5cblonbp_small_product))) /\ ff_s_b5cblonbp_small_product = ff_r_b5cblonbp_small_product * ff_p_b5cblonbp_small_product)))))))) /\ ((exists bpr_code_b5cblonbp_middle bpr_scale_b5cblonbp_middle. ((forall bpr_index_b5cblonbp_middle_prefix. (exists bpr_gap_b5cblonbp_middle_prefix_bound. bpr_gap_b5cblonbp_middle_prefix_bound + S (bpr_index_b5cblonbp_middle_prefix) = x) -> exists bpr_value_b5cblonbp_middle_prefix. ((((exists bpr_height_b5cblonbp_middle_prefix_decoded. bpr_height_b5cblonbp_middle_prefix_decoded + S (bpr_value_b5cblonbp_middle_prefix) = S ((S (bpr_index_b5cblonbp_middle_prefix)) * bpr_scale_b5cblonbp_middle)) /\ exists bpr_quotient_b5cblonbp_middle_prefix_decoded. bpr_code_b5cblonbp_middle = bpr_quotient_b5cblonbp_middle_prefix_decoded * S ((S (bpr_index_b5cblonbp_middle_prefix)) * bpr_scale_b5cblonbp_middle) + (bpr_value_b5cblonbp_middle_prefix))) /\ (((((~(S (s + bpr_index_b5cblonbp_middle_prefix) = 1) /\ forall bpr_left_b5cblonbp_middle_prefix_choice_prime bpr_right_b5cblonbp_middle_prefix_choice_prime. S (s + bpr_index_b5cblonbp_middle_prefix) = bpr_left_b5cblonbp_middle_prefix_choice_prime * bpr_right_b5cblonbp_middle_prefix_choice_prime -> bpr_left_b5cblonbp_middle_prefix_choice_prime = 1 \/ bpr_right_b5cblonbp_middle_prefix_choice_prime = 1)) /\ exists bpr_choice_exponent_b5cblonbp_middle_prefix_choice. ((((exists bpr_le_gap_b5cblonbp_middle_prefix_choice_valuation_selected_bound. bpr_le_gap_b5cblonbp_middle_prefix_choice_valuation_selected_bound + (bpr_choice_exponent_b5cblonbp_middle_prefix_choice) = (C)) /\ (exists bpr_power_value_b5cblonbp_middle_prefix_choice_valuation_selected. ((exists bpr_power_code_b5cblonbp_middle_prefix_choice_valuation_selected_power bpr_power_scale_b5cblonbp_middle_prefix_choice_valuation_selected_power. ((forall bpr_power_index_b5cblonbp_middle_prefix_choice_valuation_selected_power. (exists bpr_gap_b5cblonbp_middle_prefix_choice_valuation_selected_power_repeat_bound. bpr_gap_b5cblonbp_middle_prefix_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_b5cblonbp_middle_prefix_choice_valuation_selected_power) = bpr_choice_exponent_b5cblonbp_middle_prefix_choice) -> (((exists bpr_height_b5cblonbp_middle_prefix_choice_valuation_selected_power_repeat_entry. bpr_height_b5cblonbp_middle_prefix_choice_valuation_selected_power_repeat_entry + S (S (s + bpr_index_b5cblonbp_middle_prefix)) = S ((S (bpr_power_index_b5cblonbp_middle_prefix_choice_valuation_selected_power)) * bpr_power_scale_b5cblonbp_middle_prefix_choice_valuation_selected_power)) /\ exists bpr_quotient_b5cblonbp_middle_prefix_choice_valuation_selected_power_repeat_entry. bpr_power_code_b5cblonbp_middle_prefix_choice_valuation_selected_power = bpr_quotient_b5cblonbp_middle_prefix_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_b5cblonbp_middle_prefix_choice_valuation_selected_power)) * bpr_power_scale_b5cblonbp_middle_prefix_choice_valuation_selected_power) + (S (s + bpr_index_b5cblonbp_middle_prefix))))) /\ (exists ff_u_b5cblonbp_middle_prefix_choice_valuation_selected_power_product ff_v_b5cblonbp_middle_prefix_choice_valuation_selected_power_product. ((((exists ff_h_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_start. ff_h_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_start. ff_u_b5cblonbp_middle_prefix_choice_valuation_selected_power_product = ff_q_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_terminal. ff_h_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_terminal + S (bpr_power_value_b5cblonbp_middle_prefix_choice_valuation_selected) = S ((S (bpr_choice_exponent_b5cblonbp_middle_prefix_choice)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_terminal. ff_u_b5cblonbp_middle_prefix_choice_valuation_selected_power_product = ff_q_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_b5cblonbp_middle_prefix_choice)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_selected_power_product) + (bpr_power_value_b5cblonbp_middle_prefix_choice_valuation_selected))) /\ forall ff_i_b5cblonbp_middle_prefix_choice_valuation_selected_power_product. (exists ff_lt_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_bound. ff_lt_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_bound + S ff_i_b5cblonbp_middle_prefix_choice_valuation_selected_power_product = bpr_choice_exponent_b5cblonbp_middle_prefix_choice) -> exists ff_p_b5cblonbp_middle_prefix_choice_valuation_selected_power_product ff_r_b5cblonbp_middle_prefix_choice_valuation_selected_power_product ff_s_b5cblonbp_middle_prefix_choice_valuation_selected_power_product. ((((exists ff_h_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_factor. ff_h_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_factor + S (ff_p_b5cblonbp_middle_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_b5cblonbp_middle_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_b5cblonbp_middle_prefix_choice_valuation_selected_power)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_factor. bpr_power_code_b5cblonbp_middle_prefix_choice_valuation_selected_power = ff_q_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_factor * S ((S (ff_i_b5cblonbp_middle_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_b5cblonbp_middle_prefix_choice_valuation_selected_power) + (ff_p_b5cblonbp_middle_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_partial. ff_h_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_partial + S (ff_r_b5cblonbp_middle_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_b5cblonbp_middle_prefix_choice_valuation_selected_power_product)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_partial. ff_u_b5cblonbp_middle_prefix_choice_valuation_selected_power_product = ff_q_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_partial * S ((S (ff_i_b5cblonbp_middle_prefix_choice_valuation_selected_power_product)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_selected_power_product) + (ff_r_b5cblonbp_middle_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_successor. ff_h_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_successor + S (ff_s_b5cblonbp_middle_prefix_choice_valuation_selected_power_product) = S ((S (S ff_i_b5cblonbp_middle_prefix_choice_valuation_selected_power_product)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_successor. ff_u_b5cblonbp_middle_prefix_choice_valuation_selected_power_product = ff_q_b5cblonbp_middle_prefix_choice_valuation_selected_power_product_successor * S ((S (S ff_i_b5cblonbp_middle_prefix_choice_valuation_selected_power_product)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_selected_power_product) + (ff_s_b5cblonbp_middle_prefix_choice_valuation_selected_power_product))) /\ ff_s_b5cblonbp_middle_prefix_choice_valuation_selected_power_product = ff_r_b5cblonbp_middle_prefix_choice_valuation_selected_power_product * ff_p_b5cblonbp_middle_prefix_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_b5cblonbp_middle_prefix_choice_valuation_selected_divides. C = (bpr_power_value_b5cblonbp_middle_prefix_choice_valuation_selected) * bpr_divides_quotient_b5cblonbp_middle_prefix_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_b5cblonbp_middle_prefix_choice_valuation. (exists bpr_le_gap_b5cblonbp_middle_prefix_choice_valuation_candidate_bound. bpr_le_gap_b5cblonbp_middle_prefix_choice_valuation_candidate_bound + (bpr_valuation_candidate_b5cblonbp_middle_prefix_choice_valuation) = (C)) -> (exists bpr_power_value_b5cblonbp_middle_prefix_choice_valuation_candidate. ((exists bpr_power_code_b5cblonbp_middle_prefix_choice_valuation_candidate_power bpr_power_scale_b5cblonbp_middle_prefix_choice_valuation_candidate_power. ((forall bpr_power_index_b5cblonbp_middle_prefix_choice_valuation_candidate_power. (exists bpr_gap_b5cblonbp_middle_prefix_choice_valuation_candidate_power_repeat_bound. bpr_gap_b5cblonbp_middle_prefix_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_b5cblonbp_middle_prefix_choice_valuation_candidate_power) = bpr_valuation_candidate_b5cblonbp_middle_prefix_choice_valuation) -> (((exists bpr_height_b5cblonbp_middle_prefix_choice_valuation_candidate_power_repeat_entry. bpr_height_b5cblonbp_middle_prefix_choice_valuation_candidate_power_repeat_entry + S (S (s + bpr_index_b5cblonbp_middle_prefix)) = S ((S (bpr_power_index_b5cblonbp_middle_prefix_choice_valuation_candidate_power)) * bpr_power_scale_b5cblonbp_middle_prefix_choice_valuation_candidate_power)) /\ exists bpr_quotient_b5cblonbp_middle_prefix_choice_valuation_candidate_power_repeat_entry. bpr_power_code_b5cblonbp_middle_prefix_choice_valuation_candidate_power = bpr_quotient_b5cblonbp_middle_prefix_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_b5cblonbp_middle_prefix_choice_valuation_candidate_power)) * bpr_power_scale_b5cblonbp_middle_prefix_choice_valuation_candidate_power) + (S (s + bpr_index_b5cblonbp_middle_prefix))))) /\ (exists ff_u_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product ff_v_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_start. ff_h_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_start. ff_u_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product = ff_q_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_terminal. ff_h_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_b5cblonbp_middle_prefix_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_b5cblonbp_middle_prefix_choice_valuation)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_terminal. ff_u_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product = ff_q_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_b5cblonbp_middle_prefix_choice_valuation)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product) + (bpr_power_value_b5cblonbp_middle_prefix_choice_valuation_candidate))) /\ forall ff_i_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product. (exists ff_lt_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_bound. ff_lt_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_bound + S ff_i_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product = bpr_valuation_candidate_b5cblonbp_middle_prefix_choice_valuation) -> exists ff_p_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product ff_r_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product ff_s_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_factor. ff_h_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_factor + S (ff_p_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_b5cblonbp_middle_prefix_choice_valuation_candidate_power)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_factor. bpr_power_code_b5cblonbp_middle_prefix_choice_valuation_candidate_power = ff_q_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_factor * S ((S (ff_i_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_b5cblonbp_middle_prefix_choice_valuation_candidate_power) + (ff_p_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_partial. ff_h_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_partial + S (ff_r_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_partial. ff_u_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product = ff_q_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_partial * S ((S (ff_i_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product) + (ff_r_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_successor. ff_h_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_successor + S (ff_s_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product) = S ((S (S ff_i_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_successor. ff_u_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product = ff_q_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product)) * ff_v_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product) + (ff_s_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product))) /\ ff_s_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product = ff_r_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product * ff_p_b5cblonbp_middle_prefix_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_b5cblonbp_middle_prefix_choice_valuation_candidate_divides. C = (bpr_power_value_b5cblonbp_middle_prefix_choice_valuation_candidate) * bpr_divides_quotient_b5cblonbp_middle_prefix_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_b5cblonbp_middle_prefix_choice_valuation_candidate_below. bpr_le_gap_b5cblonbp_middle_prefix_choice_valuation_candidate_below + (bpr_valuation_candidate_b5cblonbp_middle_prefix_choice_valuation) = (bpr_choice_exponent_b5cblonbp_middle_prefix_choice))) /\ (exists bpr_power_code_b5cblonbp_middle_prefix_choice_power bpr_power_scale_b5cblonbp_middle_prefix_choice_power. ((forall bpr_power_index_b5cblonbp_middle_prefix_choice_power. (exists bpr_gap_b5cblonbp_middle_prefix_choice_power_repeat_bound. bpr_gap_b5cblonbp_middle_prefix_choice_power_repeat_bound + S (bpr_power_index_b5cblonbp_middle_prefix_choice_power) = bpr_choice_exponent_b5cblonbp_middle_prefix_choice) -> (((exists bpr_height_b5cblonbp_middle_prefix_choice_power_repeat_entry. bpr_height_b5cblonbp_middle_prefix_choice_power_repeat_entry + S (S (s + bpr_index_b5cblonbp_middle_prefix)) = S ((S (bpr_power_index_b5cblonbp_middle_prefix_choice_power)) * bpr_power_scale_b5cblonbp_middle_prefix_choice_power)) /\ exists bpr_quotient_b5cblonbp_middle_prefix_choice_power_repeat_entry. bpr_power_code_b5cblonbp_middle_prefix_choice_power = bpr_quotient_b5cblonbp_middle_prefix_choice_power_repeat_entry * S ((S (bpr_power_index_b5cblonbp_middle_prefix_choice_power)) * bpr_power_scale_b5cblonbp_middle_prefix_choice_power) + (S (s + bpr_index_b5cblonbp_middle_prefix))))) /\ (exists ff_u_b5cblonbp_middle_prefix_choice_power_product ff_v_b5cblonbp_middle_prefix_choice_power_product. ((((exists ff_h_b5cblonbp_middle_prefix_choice_power_product_start. ff_h_b5cblonbp_middle_prefix_choice_power_product_start + S (1) = S ((S (0)) * ff_v_b5cblonbp_middle_prefix_choice_power_product)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_power_product_start. ff_u_b5cblonbp_middle_prefix_choice_power_product = ff_q_b5cblonbp_middle_prefix_choice_power_product_start * S ((S (0)) * ff_v_b5cblonbp_middle_prefix_choice_power_product) + (1))) /\ ((((exists ff_h_b5cblonbp_middle_prefix_choice_power_product_terminal. ff_h_b5cblonbp_middle_prefix_choice_power_product_terminal + S (bpr_value_b5cblonbp_middle_prefix) = S ((S (bpr_choice_exponent_b5cblonbp_middle_prefix_choice)) * ff_v_b5cblonbp_middle_prefix_choice_power_product)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_power_product_terminal. ff_u_b5cblonbp_middle_prefix_choice_power_product = ff_q_b5cblonbp_middle_prefix_choice_power_product_terminal * S ((S (bpr_choice_exponent_b5cblonbp_middle_prefix_choice)) * ff_v_b5cblonbp_middle_prefix_choice_power_product) + (bpr_value_b5cblonbp_middle_prefix))) /\ forall ff_i_b5cblonbp_middle_prefix_choice_power_product. (exists ff_lt_b5cblonbp_middle_prefix_choice_power_product_bound. ff_lt_b5cblonbp_middle_prefix_choice_power_product_bound + S ff_i_b5cblonbp_middle_prefix_choice_power_product = bpr_choice_exponent_b5cblonbp_middle_prefix_choice) -> exists ff_p_b5cblonbp_middle_prefix_choice_power_product ff_r_b5cblonbp_middle_prefix_choice_power_product ff_s_b5cblonbp_middle_prefix_choice_power_product. ((((exists ff_h_b5cblonbp_middle_prefix_choice_power_product_factor. ff_h_b5cblonbp_middle_prefix_choice_power_product_factor + S (ff_p_b5cblonbp_middle_prefix_choice_power_product) = S ((S (ff_i_b5cblonbp_middle_prefix_choice_power_product)) * bpr_power_scale_b5cblonbp_middle_prefix_choice_power)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_power_product_factor. bpr_power_code_b5cblonbp_middle_prefix_choice_power = ff_q_b5cblonbp_middle_prefix_choice_power_product_factor * S ((S (ff_i_b5cblonbp_middle_prefix_choice_power_product)) * bpr_power_scale_b5cblonbp_middle_prefix_choice_power) + (ff_p_b5cblonbp_middle_prefix_choice_power_product))) /\ ((((exists ff_h_b5cblonbp_middle_prefix_choice_power_product_partial. ff_h_b5cblonbp_middle_prefix_choice_power_product_partial + S (ff_r_b5cblonbp_middle_prefix_choice_power_product) = S ((S (ff_i_b5cblonbp_middle_prefix_choice_power_product)) * ff_v_b5cblonbp_middle_prefix_choice_power_product)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_power_product_partial. ff_u_b5cblonbp_middle_prefix_choice_power_product = ff_q_b5cblonbp_middle_prefix_choice_power_product_partial * S ((S (ff_i_b5cblonbp_middle_prefix_choice_power_product)) * ff_v_b5cblonbp_middle_prefix_choice_power_product) + (ff_r_b5cblonbp_middle_prefix_choice_power_product))) /\ ((((exists ff_h_b5cblonbp_middle_prefix_choice_power_product_successor. ff_h_b5cblonbp_middle_prefix_choice_power_product_successor + S (ff_s_b5cblonbp_middle_prefix_choice_power_product) = S ((S (S ff_i_b5cblonbp_middle_prefix_choice_power_product)) * ff_v_b5cblonbp_middle_prefix_choice_power_product)) /\ exists ff_q_b5cblonbp_middle_prefix_choice_power_product_successor. ff_u_b5cblonbp_middle_prefix_choice_power_product = ff_q_b5cblonbp_middle_prefix_choice_power_product_successor * S ((S (S ff_i_b5cblonbp_middle_prefix_choice_power_product)) * ff_v_b5cblonbp_middle_prefix_choice_power_product) + (ff_s_b5cblonbp_middle_prefix_choice_power_product))) /\ ff_s_b5cblonbp_middle_prefix_choice_power_product = ff_r_b5cblonbp_middle_prefix_choice_power_product * ff_p_b5cblonbp_middle_prefix_choice_power_product)))))))))) \/ (~((~(S (s + bpr_index_b5cblonbp_middle_prefix) = 1) /\ forall bpr_left_b5cblonbp_middle_prefix_choice_prime bpr_right_b5cblonbp_middle_prefix_choice_prime. S (s + bpr_index_b5cblonbp_middle_prefix) = bpr_left_b5cblonbp_middle_prefix_choice_prime * bpr_right_b5cblonbp_middle_prefix_choice_prime -> bpr_left_b5cblonbp_middle_prefix_choice_prime = 1 \/ bpr_right_b5cblonbp_middle_prefix_choice_prime = 1)) /\ bpr_value_b5cblonbp_middle_prefix = 1))))) /\ (exists ff_u_b5cblonbp_middle_product ff_v_b5cblonbp_middle_product. ((((exists ff_h_b5cblonbp_middle_product_start. ff_h_b5cblonbp_middle_product_start + S (1) = S ((S (0)) * ff_v_b5cblonbp_middle_product)) /\ exists ff_q_b5cblonbp_middle_product_start. ff_u_b5cblonbp_middle_product = ff_q_b5cblonbp_middle_product_start * S ((S (0)) * ff_v_b5cblonbp_middle_product) + (1))) /\ ((((exists ff_h_b5cblonbp_middle_product_terminal. ff_h_b5cblonbp_middle_product_terminal + S (v) = S ((S (x)) * ff_v_b5cblonbp_middle_product)) /\ exists ff_q_b5cblonbp_middle_product_terminal. ff_u_b5cblonbp_middle_product = ff_q_b5cblonbp_middle_product_terminal * S ((S (x)) * ff_v_b5cblonbp_middle_product) + (v))) /\ forall ff_i_b5cblonbp_middle_product. (exists ff_lt_b5cblonbp_middle_product_bound. ff_lt_b5cblonbp_middle_product_bound + S ff_i_b5cblonbp_middle_product = x) -> exists ff_p_b5cblonbp_middle_product ff_r_b5cblonbp_middle_product ff_s_b5cblonbp_middle_product. ((((exists ff_h_b5cblonbp_middle_product_factor. ff_h_b5cblonbp_middle_product_factor + S (ff_p_b5cblonbp_middle_product) = S ((S (ff_i_b5cblonbp_middle_product)) * bpr_scale_b5cblonbp_middle)) /\ exists ff_q_b5cblonbp_middle_product_factor. bpr_code_b5cblonbp_middle = ff_q_b5cblonbp_middle_product_factor * S ((S (ff_i_b5cblonbp_middle_product)) * bpr_scale_b5cblonbp_middle) + (ff_p_b5cblonbp_middle_product))) /\ ((((exists ff_h_b5cblonbp_middle_product_partial. ff_h_b5cblonbp_middle_product_partial + S (ff_r_b5cblonbp_middle_product) = S ((S (ff_i_b5cblonbp_middle_product)) * ff_v_b5cblonbp_middle_product)) /\ exists ff_q_b5cblonbp_middle_product_partial. ff_u_b5cblonbp_middle_product = ff_q_b5cblonbp_middle_product_partial * S ((S (ff_i_b5cblonbp_middle_product)) * ff_v_b5cblonbp_middle_product) + (ff_r_b5cblonbp_middle_product))) /\ ((((exists ff_h_b5cblonbp_middle_product_successor. ff_h_b5cblonbp_middle_product_successor + S (ff_s_b5cblonbp_middle_product) = S ((S (S ff_i_b5cblonbp_middle_product)) * ff_v_b5cblonbp_middle_product)) /\ exists ff_q_b5cblonbp_middle_product_successor. ff_u_b5cblonbp_middle_product = ff_q_b5cblonbp_middle_product_successor * S ((S (S ff_i_b5cblonbp_middle_product)) * ff_v_b5cblonbp_middle_product) + (ff_s_b5cblonbp_middle_product))) /\ ff_s_b5cblonbp_middle_product = ff_r_b5cblonbp_middle_product * ff_p_b5cblonbp_middle_product)))))))) /\ x2 = u * v)
0035specialize central_binom_factorization_small n
0036specialize central_binom_factorization_small s
0037specialize central_binom_factorization_small q
0038specialize central_binom_factorization_small r
0039specialize central_binom_factorization_small C
0040specialize central_binom_factorization_small x
0041specialize central_binom_factorization_small x1
0042specialize central_binom_factorization_small x2
0043apply central_binom_factorization_small
0044exact hexclusion
0045exact hpositive
0046exact hfloor
0047exact hdivision
0048exact hcentral
0049exact hgaps_witness_witness_left
0050exact hgaps_witness_witness_right
0051exact hproduct_witness_left
0052cases hfactorization
0053cases hfactorization_witness
0054cases hfactorization_witness_witness
0055cases hfactorization_witness_witness_right
0056have hsmall : exists bcf_le_gap_b5cblonbp_x_a. bcf_le_gap_b5cblonbp_x_a + (x3) = A
0057specialize no_bertrand_small_contribution_product_le_power n
0058specialize no_bertrand_small_contribution_product_le_power s
0059specialize no_bertrand_small_contribution_product_le_power q
0060specialize no_bertrand_small_contribution_product_le_power r
0061specialize no_bertrand_small_contribution_product_le_power C
0062specialize no_bertrand_small_contribution_product_le_power x3
0063specialize no_bertrand_small_contribution_product_le_power A
0064apply no_bertrand_small_contribution_product_le_power
0065exact hexclusion
0066exact hpositive
0067exact hfloor
0068exact hdivision
0069exact hcentral
0070exact hfactorization_witness_witness_left
0071exact hpower_a
0072have hmiddle : exists bcf_le_gap_b5cblonbp_y_b. bcf_le_gap_b5cblonbp_y_b + (x4) = B
0073specialize no_bertrand_middle_contribution_interval_le_four_pow n
0074specialize no_bertrand_middle_contribution_interval_le_four_pow s
0075specialize no_bertrand_middle_contribution_interval_le_four_pow q
0076specialize no_bertrand_middle_contribution_interval_le_four_pow r
0077specialize no_bertrand_middle_contribution_interval_le_four_pow C
0078specialize no_bertrand_middle_contribution_interval_le_four_pow x
0079specialize no_bertrand_middle_contribution_interval_le_four_pow x4
0080specialize no_bertrand_middle_contribution_interval_le_four_pow B
0081apply no_bertrand_middle_contribution_interval_le_four_pow
0082exact hexclusion
0083exact hpositive
0084exact hfloor
0085exact hdivision
0086exact hcentral
0087exact hgaps_witness_witness_left
0088exact hfactorization_witness_witness_right_left
0089exact hpower_b
0090have hproduct_bound : exists bcf_le_gap_b5cblonbp_product_bound. bcf_le_gap_b5cblonbp_product_bound + (x3 * x4) = A * B
0091specialize mul_le_mul x3
0092specialize mul_le_mul A
0093specialize mul_le_mul x4
0094specialize mul_le_mul B
0095apply mul_le_mul
0096exact hsmall
0097exact hmiddle
0098rewrite hproduct_witness_right
0099rewrite hfactorization_witness_witness_right_right
0100exact hproduct_bound