binary_length_nonzero_components

Exact expanded first-order arithmetic statement

forall n ell. ~(n = 0) -> ((((n) = 0 /\ (ell) = 1) \/ exists ff_exponent_bl_pc_length_components_source ff_lower_bl_pc_length_components_source ff_upper_bl_pc_length_components_source. (((ell) = S ff_exponent_bl_pc_length_components_source) /\ ((exists ff_positive_bl_pc_length_components_source. ff_positive_bl_pc_length_components_source + 1 = (n)) /\ ((exists pa_b_bl_pc_length_components_source_lower pa_c_bl_pc_length_components_source_lower. ((forall pa_i_bl_pc_length_components_source_lower_repeat. (exists pa_lt_bl_pc_length_components_source_lower_repeat_bound. pa_lt_bl_pc_length_components_source_lower_repeat_bound + S pa_i_bl_pc_length_components_source_lower_repeat = ff_exponent_bl_pc_length_components_source) -> (((exists pa_h_bl_pc_length_components_source_lower_repeat_decoded. pa_h_bl_pc_length_components_source_lower_repeat_decoded + S (2) = S ((S (pa_i_bl_pc_length_components_source_lower_repeat)) * pa_c_bl_pc_length_components_source_lower)) /\ exists pa_q_bl_pc_length_components_source_lower_repeat_decoded. pa_b_bl_pc_length_components_source_lower = pa_q_bl_pc_length_components_source_lower_repeat_decoded * S ((S (pa_i_bl_pc_length_components_source_lower_repeat)) * pa_c_bl_pc_length_components_source_lower) + (2)))) /\ (exists pa_u_bl_pc_length_components_source_lower_product pa_v_bl_pc_length_components_source_lower_product. ((((exists pa_h_bl_pc_length_components_source_lower_product_start. pa_h_bl_pc_length_components_source_lower_product_start + S (1) = S ((S (0)) * pa_v_bl_pc_length_components_source_lower_product)) /\ exists pa_q_bl_pc_length_components_source_lower_product_start. pa_u_bl_pc_length_components_source_lower_product = pa_q_bl_pc_length_components_source_lower_product_start * S ((S (0)) * pa_v_bl_pc_length_components_source_lower_product) + (1))) /\ ((((exists pa_h_bl_pc_length_components_source_lower_product_terminal. pa_h_bl_pc_length_components_source_lower_product_terminal + S (ff_lower_bl_pc_length_components_source) = S ((S (ff_exponent_bl_pc_length_components_source)) * pa_v_bl_pc_length_components_source_lower_product)) /\ exists pa_q_bl_pc_length_components_source_lower_product_terminal. pa_u_bl_pc_length_components_source_lower_product = pa_q_bl_pc_length_components_source_lower_product_terminal * S ((S (ff_exponent_bl_pc_length_components_source)) * pa_v_bl_pc_length_components_source_lower_product) + (ff_lower_bl_pc_length_components_source))) /\ forall pa_i_bl_pc_length_components_source_lower_product. (exists pa_lt_bl_pc_length_components_source_lower_product_bound. pa_lt_bl_pc_length_components_source_lower_product_bound + S pa_i_bl_pc_length_components_source_lower_product = ff_exponent_bl_pc_length_components_source) -> exists pa_p_bl_pc_length_components_source_lower_product pa_r_bl_pc_length_components_source_lower_product pa_s_bl_pc_length_components_source_lower_product. ((((exists pa_h_bl_pc_length_components_source_lower_product_factor. pa_h_bl_pc_length_components_source_lower_product_factor + S (pa_p_bl_pc_length_components_source_lower_product) = S ((S (pa_i_bl_pc_length_components_source_lower_product)) * pa_c_bl_pc_length_components_source_lower)) /\ exists pa_q_bl_pc_length_components_source_lower_product_factor. pa_b_bl_pc_length_components_source_lower = pa_q_bl_pc_length_components_source_lower_product_factor * S ((S (pa_i_bl_pc_length_components_source_lower_product)) * pa_c_bl_pc_length_components_source_lower) + (pa_p_bl_pc_length_components_source_lower_product))) /\ ((((exists pa_h_bl_pc_length_components_source_lower_product_partial. pa_h_bl_pc_length_components_source_lower_product_partial + S (pa_r_bl_pc_length_components_source_lower_product) = S ((S (pa_i_bl_pc_length_components_source_lower_product)) * pa_v_bl_pc_length_components_source_lower_product)) /\ exists pa_q_bl_pc_length_components_source_lower_product_partial. pa_u_bl_pc_length_components_source_lower_product = pa_q_bl_pc_length_components_source_lower_product_partial * S ((S (pa_i_bl_pc_length_components_source_lower_product)) * pa_v_bl_pc_length_components_source_lower_product) + (pa_r_bl_pc_length_components_source_lower_product))) /\ ((((exists pa_h_bl_pc_length_components_source_lower_product_successor. pa_h_bl_pc_length_components_source_lower_product_successor + S (pa_s_bl_pc_length_components_source_lower_product) = S ((S (S pa_i_bl_pc_length_components_source_lower_product)) * pa_v_bl_pc_length_components_source_lower_product)) /\ exists pa_q_bl_pc_length_components_source_lower_product_successor. pa_u_bl_pc_length_components_source_lower_product = pa_q_bl_pc_length_components_source_lower_product_successor * S ((S (S pa_i_bl_pc_length_components_source_lower_product)) * pa_v_bl_pc_length_components_source_lower_product) + (pa_s_bl_pc_length_components_source_lower_product))) /\ pa_s_bl_pc_length_components_source_lower_product = pa_r_bl_pc_length_components_source_lower_product * pa_p_bl_pc_length_components_source_lower_product)))))))) /\ ((exists pa_b_bl_pc_length_components_source_upper pa_c_bl_pc_length_components_source_upper. ((forall pa_i_bl_pc_length_components_source_upper_repeat. (exists pa_lt_bl_pc_length_components_source_upper_repeat_bound. pa_lt_bl_pc_length_components_source_upper_repeat_bound + S pa_i_bl_pc_length_components_source_upper_repeat = ell) -> (((exists pa_h_bl_pc_length_components_source_upper_repeat_decoded. pa_h_bl_pc_length_components_source_upper_repeat_decoded + S (2) = S ((S (pa_i_bl_pc_length_components_source_upper_repeat)) * pa_c_bl_pc_length_components_source_upper)) /\ exists pa_q_bl_pc_length_components_source_upper_repeat_decoded. pa_b_bl_pc_length_components_source_upper = pa_q_bl_pc_length_components_source_upper_repeat_decoded * S ((S (pa_i_bl_pc_length_components_source_upper_repeat)) * pa_c_bl_pc_length_components_source_upper) + (2)))) /\ (exists pa_u_bl_pc_length_components_source_upper_product pa_v_bl_pc_length_components_source_upper_product. ((((exists pa_h_bl_pc_length_components_source_upper_product_start. pa_h_bl_pc_length_components_source_upper_product_start + S (1) = S ((S (0)) * pa_v_bl_pc_length_components_source_upper_product)) /\ exists pa_q_bl_pc_length_components_source_upper_product_start. pa_u_bl_pc_length_components_source_upper_product = pa_q_bl_pc_length_components_source_upper_product_start * S ((S (0)) * pa_v_bl_pc_length_components_source_upper_product) + (1))) /\ ((((exists pa_h_bl_pc_length_components_source_upper_product_terminal. pa_h_bl_pc_length_components_source_upper_product_terminal + S (ff_upper_bl_pc_length_components_source) = S ((S (ell)) * pa_v_bl_pc_length_components_source_upper_product)) /\ exists pa_q_bl_pc_length_components_source_upper_product_terminal. pa_u_bl_pc_length_components_source_upper_product = pa_q_bl_pc_length_components_source_upper_product_terminal * S ((S (ell)) * pa_v_bl_pc_length_components_source_upper_product) + (ff_upper_bl_pc_length_components_source))) /\ forall pa_i_bl_pc_length_components_source_upper_product. (exists pa_lt_bl_pc_length_components_source_upper_product_bound. pa_lt_bl_pc_length_components_source_upper_product_bound + S pa_i_bl_pc_length_components_source_upper_product = ell) -> exists pa_p_bl_pc_length_components_source_upper_product pa_r_bl_pc_length_components_source_upper_product pa_s_bl_pc_length_components_source_upper_product. ((((exists pa_h_bl_pc_length_components_source_upper_product_factor. pa_h_bl_pc_length_components_source_upper_product_factor + S (pa_p_bl_pc_length_components_source_upper_product) = S ((S (pa_i_bl_pc_length_components_source_upper_product)) * pa_c_bl_pc_length_components_source_upper)) /\ exists pa_q_bl_pc_length_components_source_upper_product_factor. pa_b_bl_pc_length_components_source_upper = pa_q_bl_pc_length_components_source_upper_product_factor * S ((S (pa_i_bl_pc_length_components_source_upper_product)) * pa_c_bl_pc_length_components_source_upper) + (pa_p_bl_pc_length_components_source_upper_product))) /\ ((((exists pa_h_bl_pc_length_components_source_upper_product_partial. pa_h_bl_pc_length_components_source_upper_product_partial + S (pa_r_bl_pc_length_components_source_upper_product) = S ((S (pa_i_bl_pc_length_components_source_upper_product)) * pa_v_bl_pc_length_components_source_upper_product)) /\ exists pa_q_bl_pc_length_components_source_upper_product_partial. pa_u_bl_pc_length_components_source_upper_product = pa_q_bl_pc_length_components_source_upper_product_partial * S ((S (pa_i_bl_pc_length_components_source_upper_product)) * pa_v_bl_pc_length_components_source_upper_product) + (pa_r_bl_pc_length_components_source_upper_product))) /\ ((((exists pa_h_bl_pc_length_components_source_upper_product_successor. pa_h_bl_pc_length_components_source_upper_product_successor + S (pa_s_bl_pc_length_components_source_upper_product) = S ((S (S pa_i_bl_pc_length_components_source_upper_product)) * pa_v_bl_pc_length_components_source_upper_product)) /\ exists pa_q_bl_pc_length_components_source_upper_product_successor. pa_u_bl_pc_length_components_source_upper_product = pa_q_bl_pc_length_components_source_upper_product_successor * S ((S (S pa_i_bl_pc_length_components_source_upper_product)) * pa_v_bl_pc_length_components_source_upper_product) + (pa_s_bl_pc_length_components_source_upper_product))) /\ pa_s_bl_pc_length_components_source_upper_product = pa_r_bl_pc_length_components_source_upper_product * pa_p_bl_pc_length_components_source_upper_product)))))))) /\ ((exists ff_lower_gap_bl_pc_length_components_source. ff_lower_gap_bl_pc_length_components_source + (ff_lower_bl_pc_length_components_source) = (n)) /\ (exists ff_upper_gap_bl_pc_length_components_source. ff_upper_gap_bl_pc_length_components_source + S (n) = (ff_upper_bl_pc_length_components_source))))))))) -> exists e v w. ell = S e /\ ((exists pa_b_pc_length_components_lower_power pa_c_pc_length_components_lower_power. ((forall pa_i_pc_length_components_lower_power_repeat. (exists pa_lt_pc_length_components_lower_power_repeat_bound. pa_lt_pc_length_components_lower_power_repeat_bound + S pa_i_pc_length_components_lower_power_repeat = e) -> (((exists pa_h_pc_length_components_lower_power_repeat_decoded. pa_h_pc_length_components_lower_power_repeat_decoded + S (2) = S ((S (pa_i_pc_length_components_lower_power_repeat)) * pa_c_pc_length_components_lower_power)) /\ exists pa_q_pc_length_components_lower_power_repeat_decoded. pa_b_pc_length_components_lower_power = pa_q_pc_length_components_lower_power_repeat_decoded * S ((S (pa_i_pc_length_components_lower_power_repeat)) * pa_c_pc_length_components_lower_power) + (2)))) /\ (exists pa_u_pc_length_components_lower_power_product pa_v_pc_length_components_lower_power_product. ((((exists pa_h_pc_length_components_lower_power_product_start. pa_h_pc_length_components_lower_power_product_start + S (1) = S ((S (0)) * pa_v_pc_length_components_lower_power_product)) /\ exists pa_q_pc_length_components_lower_power_product_start. pa_u_pc_length_components_lower_power_product = pa_q_pc_length_components_lower_power_product_start * S ((S (0)) * pa_v_pc_length_components_lower_power_product) + (1))) /\ ((((exists pa_h_pc_length_components_lower_power_product_terminal. pa_h_pc_length_components_lower_power_product_terminal + S (v) = S ((S (e)) * pa_v_pc_length_components_lower_power_product)) /\ exists pa_q_pc_length_components_lower_power_product_terminal. pa_u_pc_length_components_lower_power_product = pa_q_pc_length_components_lower_power_product_terminal * S ((S (e)) * pa_v_pc_length_components_lower_power_product) + (v))) /\ forall pa_i_pc_length_components_lower_power_product. (exists pa_lt_pc_length_components_lower_power_product_bound. pa_lt_pc_length_components_lower_power_product_bound + S pa_i_pc_length_components_lower_power_product = e) -> exists pa_p_pc_length_components_lower_power_product pa_r_pc_length_components_lower_power_product pa_s_pc_length_components_lower_power_product. ((((exists pa_h_pc_length_components_lower_power_product_factor. pa_h_pc_length_components_lower_power_product_factor + S (pa_p_pc_length_components_lower_power_product) = S ((S (pa_i_pc_length_components_lower_power_product)) * pa_c_pc_length_components_lower_power)) /\ exists pa_q_pc_length_components_lower_power_product_factor. pa_b_pc_length_components_lower_power = pa_q_pc_length_components_lower_power_product_factor * S ((S (pa_i_pc_length_components_lower_power_product)) * pa_c_pc_length_components_lower_power) + (pa_p_pc_length_components_lower_power_product))) /\ ((((exists pa_h_pc_length_components_lower_power_product_partial. pa_h_pc_length_components_lower_power_product_partial + S (pa_r_pc_length_components_lower_power_product) = S ((S (pa_i_pc_length_components_lower_power_product)) * pa_v_pc_length_components_lower_power_product)) /\ exists pa_q_pc_length_components_lower_power_product_partial. pa_u_pc_length_components_lower_power_product = pa_q_pc_length_components_lower_power_product_partial * S ((S (pa_i_pc_length_components_lower_power_product)) * pa_v_pc_length_components_lower_power_product) + (pa_r_pc_length_components_lower_power_product))) /\ ((((exists pa_h_pc_length_components_lower_power_product_successor. pa_h_pc_length_components_lower_power_product_successor + S (pa_s_pc_length_components_lower_power_product) = S ((S (S pa_i_pc_length_components_lower_power_product)) * pa_v_pc_length_components_lower_power_product)) /\ exists pa_q_pc_length_components_lower_power_product_successor. pa_u_pc_length_components_lower_power_product = pa_q_pc_length_components_lower_power_product_successor * S ((S (S pa_i_pc_length_components_lower_power_product)) * pa_v_pc_length_components_lower_power_product) + (pa_s_pc_length_components_lower_power_product))) /\ pa_s_pc_length_components_lower_power_product = pa_r_pc_length_components_lower_power_product * pa_p_pc_length_components_lower_power_product)))))))) /\ ((exists pa_b_pc_length_components_upper_power pa_c_pc_length_components_upper_power. ((forall pa_i_pc_length_components_upper_power_repeat. (exists pa_lt_pc_length_components_upper_power_repeat_bound. pa_lt_pc_length_components_upper_power_repeat_bound + S pa_i_pc_length_components_upper_power_repeat = ell) -> (((exists pa_h_pc_length_components_upper_power_repeat_decoded. pa_h_pc_length_components_upper_power_repeat_decoded + S (2) = S ((S (pa_i_pc_length_components_upper_power_repeat)) * pa_c_pc_length_components_upper_power)) /\ exists pa_q_pc_length_components_upper_power_repeat_decoded. pa_b_pc_length_components_upper_power = pa_q_pc_length_components_upper_power_repeat_decoded * S ((S (pa_i_pc_length_components_upper_power_repeat)) * pa_c_pc_length_components_upper_power) + (2)))) /\ (exists pa_u_pc_length_components_upper_power_product pa_v_pc_length_components_upper_power_product. ((((exists pa_h_pc_length_components_upper_power_product_start. pa_h_pc_length_components_upper_power_product_start + S (1) = S ((S (0)) * pa_v_pc_length_components_upper_power_product)) /\ exists pa_q_pc_length_components_upper_power_product_start. pa_u_pc_length_components_upper_power_product = pa_q_pc_length_components_upper_power_product_start * S ((S (0)) * pa_v_pc_length_components_upper_power_product) + (1))) /\ ((((exists pa_h_pc_length_components_upper_power_product_terminal. pa_h_pc_length_components_upper_power_product_terminal + S (w) = S ((S (ell)) * pa_v_pc_length_components_upper_power_product)) /\ exists pa_q_pc_length_components_upper_power_product_terminal. pa_u_pc_length_components_upper_power_product = pa_q_pc_length_components_upper_power_product_terminal * S ((S (ell)) * pa_v_pc_length_components_upper_power_product) + (w))) /\ forall pa_i_pc_length_components_upper_power_product. (exists pa_lt_pc_length_components_upper_power_product_bound. pa_lt_pc_length_components_upper_power_product_bound + S pa_i_pc_length_components_upper_power_product = ell) -> exists pa_p_pc_length_components_upper_power_product pa_r_pc_length_components_upper_power_product pa_s_pc_length_components_upper_power_product. ((((exists pa_h_pc_length_components_upper_power_product_factor. pa_h_pc_length_components_upper_power_product_factor + S (pa_p_pc_length_components_upper_power_product) = S ((S (pa_i_pc_length_components_upper_power_product)) * pa_c_pc_length_components_upper_power)) /\ exists pa_q_pc_length_components_upper_power_product_factor. pa_b_pc_length_components_upper_power = pa_q_pc_length_components_upper_power_product_factor * S ((S (pa_i_pc_length_components_upper_power_product)) * pa_c_pc_length_components_upper_power) + (pa_p_pc_length_components_upper_power_product))) /\ ((((exists pa_h_pc_length_components_upper_power_product_partial. pa_h_pc_length_components_upper_power_product_partial + S (pa_r_pc_length_components_upper_power_product) = S ((S (pa_i_pc_length_components_upper_power_product)) * pa_v_pc_length_components_upper_power_product)) /\ exists pa_q_pc_length_components_upper_power_product_partial. pa_u_pc_length_components_upper_power_product = pa_q_pc_length_components_upper_power_product_partial * S ((S (pa_i_pc_length_components_upper_power_product)) * pa_v_pc_length_components_upper_power_product) + (pa_r_pc_length_components_upper_power_product))) /\ ((((exists pa_h_pc_length_components_upper_power_product_successor. pa_h_pc_length_components_upper_power_product_successor + S (pa_s_pc_length_components_upper_power_product) = S ((S (S pa_i_pc_length_components_upper_power_product)) * pa_v_pc_length_components_upper_power_product)) /\ exists pa_q_pc_length_components_upper_power_product_successor. pa_u_pc_length_components_upper_power_product = pa_q_pc_length_components_upper_power_product_successor * S ((S (S pa_i_pc_length_components_upper_power_product)) * pa_v_pc_length_components_upper_power_product) + (pa_s_pc_length_components_upper_power_product))) /\ pa_s_pc_length_components_upper_power_product = pa_r_pc_length_components_upper_power_product * pa_p_pc_length_components_upper_power_product)))))))) /\ ((exists pc_le_length_components_lower. pc_le_length_components_lower + (v) = (n)) /\ (exists pc_lt_length_components_upper. pc_lt_length_components_upper + S (n) = (w)))))