binary_length_exists_unique

Exact expanded first-order arithmetic statement

forall n. exists l. (((((n) = 0 /\ (l) = 1) \/ exists ff_exponent_bl_length ff_lower_bl_length ff_upper_bl_length. (((l) = S ff_exponent_bl_length) /\ ((exists ff_positive_bl_length. ff_positive_bl_length + 1 = (n)) /\ ((exists pa_b_bl_length_lower pa_c_bl_length_lower. ((forall pa_i_bl_length_lower_repeat. (exists pa_lt_bl_length_lower_repeat_bound. pa_lt_bl_length_lower_repeat_bound + S pa_i_bl_length_lower_repeat = ff_exponent_bl_length) -> (((exists pa_h_bl_length_lower_repeat_decoded. pa_h_bl_length_lower_repeat_decoded + S (2) = S ((S (pa_i_bl_length_lower_repeat)) * pa_c_bl_length_lower)) /\ exists pa_q_bl_length_lower_repeat_decoded. pa_b_bl_length_lower = pa_q_bl_length_lower_repeat_decoded * S ((S (pa_i_bl_length_lower_repeat)) * pa_c_bl_length_lower) + (2)))) /\ (exists pa_u_bl_length_lower_product pa_v_bl_length_lower_product. ((((exists pa_h_bl_length_lower_product_start. pa_h_bl_length_lower_product_start + S (1) = S ((S (0)) * pa_v_bl_length_lower_product)) /\ exists pa_q_bl_length_lower_product_start. pa_u_bl_length_lower_product = pa_q_bl_length_lower_product_start * S ((S (0)) * pa_v_bl_length_lower_product) + (1))) /\ ((((exists pa_h_bl_length_lower_product_terminal. pa_h_bl_length_lower_product_terminal + S (ff_lower_bl_length) = S ((S (ff_exponent_bl_length)) * pa_v_bl_length_lower_product)) /\ exists pa_q_bl_length_lower_product_terminal. pa_u_bl_length_lower_product = pa_q_bl_length_lower_product_terminal * S ((S (ff_exponent_bl_length)) * pa_v_bl_length_lower_product) + (ff_lower_bl_length))) /\ forall pa_i_bl_length_lower_product. (exists pa_lt_bl_length_lower_product_bound. pa_lt_bl_length_lower_product_bound + S pa_i_bl_length_lower_product = ff_exponent_bl_length) -> exists pa_p_bl_length_lower_product pa_r_bl_length_lower_product pa_s_bl_length_lower_product. ((((exists pa_h_bl_length_lower_product_factor. pa_h_bl_length_lower_product_factor + S (pa_p_bl_length_lower_product) = S ((S (pa_i_bl_length_lower_product)) * pa_c_bl_length_lower)) /\ exists pa_q_bl_length_lower_product_factor. pa_b_bl_length_lower = pa_q_bl_length_lower_product_factor * S ((S (pa_i_bl_length_lower_product)) * pa_c_bl_length_lower) + (pa_p_bl_length_lower_product))) /\ ((((exists pa_h_bl_length_lower_product_partial. pa_h_bl_length_lower_product_partial + S (pa_r_bl_length_lower_product) = S ((S (pa_i_bl_length_lower_product)) * pa_v_bl_length_lower_product)) /\ exists pa_q_bl_length_lower_product_partial. pa_u_bl_length_lower_product = pa_q_bl_length_lower_product_partial * S ((S (pa_i_bl_length_lower_product)) * pa_v_bl_length_lower_product) + (pa_r_bl_length_lower_product))) /\ ((((exists pa_h_bl_length_lower_product_successor. pa_h_bl_length_lower_product_successor + S (pa_s_bl_length_lower_product) = S ((S (S pa_i_bl_length_lower_product)) * pa_v_bl_length_lower_product)) /\ exists pa_q_bl_length_lower_product_successor. pa_u_bl_length_lower_product = pa_q_bl_length_lower_product_successor * S ((S (S pa_i_bl_length_lower_product)) * pa_v_bl_length_lower_product) + (pa_s_bl_length_lower_product))) /\ pa_s_bl_length_lower_product = pa_r_bl_length_lower_product * pa_p_bl_length_lower_product)))))))) /\ ((exists pa_b_bl_length_upper pa_c_bl_length_upper. ((forall pa_i_bl_length_upper_repeat. (exists pa_lt_bl_length_upper_repeat_bound. pa_lt_bl_length_upper_repeat_bound + S pa_i_bl_length_upper_repeat = l) -> (((exists pa_h_bl_length_upper_repeat_decoded. pa_h_bl_length_upper_repeat_decoded + S (2) = S ((S (pa_i_bl_length_upper_repeat)) * pa_c_bl_length_upper)) /\ exists pa_q_bl_length_upper_repeat_decoded. pa_b_bl_length_upper = pa_q_bl_length_upper_repeat_decoded * S ((S (pa_i_bl_length_upper_repeat)) * pa_c_bl_length_upper) + (2)))) /\ (exists pa_u_bl_length_upper_product pa_v_bl_length_upper_product. ((((exists pa_h_bl_length_upper_product_start. pa_h_bl_length_upper_product_start + S (1) = S ((S (0)) * pa_v_bl_length_upper_product)) /\ exists pa_q_bl_length_upper_product_start. pa_u_bl_length_upper_product = pa_q_bl_length_upper_product_start * S ((S (0)) * pa_v_bl_length_upper_product) + (1))) /\ ((((exists pa_h_bl_length_upper_product_terminal. pa_h_bl_length_upper_product_terminal + S (ff_upper_bl_length) = S ((S (l)) * pa_v_bl_length_upper_product)) /\ exists pa_q_bl_length_upper_product_terminal. pa_u_bl_length_upper_product = pa_q_bl_length_upper_product_terminal * S ((S (l)) * pa_v_bl_length_upper_product) + (ff_upper_bl_length))) /\ forall pa_i_bl_length_upper_product. (exists pa_lt_bl_length_upper_product_bound. pa_lt_bl_length_upper_product_bound + S pa_i_bl_length_upper_product = l) -> exists pa_p_bl_length_upper_product pa_r_bl_length_upper_product pa_s_bl_length_upper_product. ((((exists pa_h_bl_length_upper_product_factor. pa_h_bl_length_upper_product_factor + S (pa_p_bl_length_upper_product) = S ((S (pa_i_bl_length_upper_product)) * pa_c_bl_length_upper)) /\ exists pa_q_bl_length_upper_product_factor. pa_b_bl_length_upper = pa_q_bl_length_upper_product_factor * S ((S (pa_i_bl_length_upper_product)) * pa_c_bl_length_upper) + (pa_p_bl_length_upper_product))) /\ ((((exists pa_h_bl_length_upper_product_partial. pa_h_bl_length_upper_product_partial + S (pa_r_bl_length_upper_product) = S ((S (pa_i_bl_length_upper_product)) * pa_v_bl_length_upper_product)) /\ exists pa_q_bl_length_upper_product_partial. pa_u_bl_length_upper_product = pa_q_bl_length_upper_product_partial * S ((S (pa_i_bl_length_upper_product)) * pa_v_bl_length_upper_product) + (pa_r_bl_length_upper_product))) /\ ((((exists pa_h_bl_length_upper_product_successor. pa_h_bl_length_upper_product_successor + S (pa_s_bl_length_upper_product) = S ((S (S pa_i_bl_length_upper_product)) * pa_v_bl_length_upper_product)) /\ exists pa_q_bl_length_upper_product_successor. pa_u_bl_length_upper_product = pa_q_bl_length_upper_product_successor * S ((S (S pa_i_bl_length_upper_product)) * pa_v_bl_length_upper_product) + (pa_s_bl_length_upper_product))) /\ pa_s_bl_length_upper_product = pa_r_bl_length_upper_product * pa_p_bl_length_upper_product)))))))) /\ ((exists ff_lower_gap_bl_length. ff_lower_gap_bl_length + (ff_lower_bl_length) = (n)) /\ (exists ff_upper_gap_bl_length. ff_upper_gap_bl_length + S (n) = (ff_upper_bl_length))))))))) /\ forall L. ((((n) = 0 /\ (L) = 1) \/ exists ff_exponent_bl_other_length ff_lower_bl_other_length ff_upper_bl_other_length. (((L) = S ff_exponent_bl_other_length) /\ ((exists ff_positive_bl_other_length. ff_positive_bl_other_length + 1 = (n)) /\ ((exists pa_b_bl_other_length_lower pa_c_bl_other_length_lower. ((forall pa_i_bl_other_length_lower_repeat. (exists pa_lt_bl_other_length_lower_repeat_bound. pa_lt_bl_other_length_lower_repeat_bound + S pa_i_bl_other_length_lower_repeat = ff_exponent_bl_other_length) -> (((exists pa_h_bl_other_length_lower_repeat_decoded. pa_h_bl_other_length_lower_repeat_decoded + S (2) = S ((S (pa_i_bl_other_length_lower_repeat)) * pa_c_bl_other_length_lower)) /\ exists pa_q_bl_other_length_lower_repeat_decoded. pa_b_bl_other_length_lower = pa_q_bl_other_length_lower_repeat_decoded * S ((S (pa_i_bl_other_length_lower_repeat)) * pa_c_bl_other_length_lower) + (2)))) /\ (exists pa_u_bl_other_length_lower_product pa_v_bl_other_length_lower_product. ((((exists pa_h_bl_other_length_lower_product_start. pa_h_bl_other_length_lower_product_start + S (1) = S ((S (0)) * pa_v_bl_other_length_lower_product)) /\ exists pa_q_bl_other_length_lower_product_start. pa_u_bl_other_length_lower_product = pa_q_bl_other_length_lower_product_start * S ((S (0)) * pa_v_bl_other_length_lower_product) + (1))) /\ ((((exists pa_h_bl_other_length_lower_product_terminal. pa_h_bl_other_length_lower_product_terminal + S (ff_lower_bl_other_length) = S ((S (ff_exponent_bl_other_length)) * pa_v_bl_other_length_lower_product)) /\ exists pa_q_bl_other_length_lower_product_terminal. pa_u_bl_other_length_lower_product = pa_q_bl_other_length_lower_product_terminal * S ((S (ff_exponent_bl_other_length)) * pa_v_bl_other_length_lower_product) + (ff_lower_bl_other_length))) /\ forall pa_i_bl_other_length_lower_product. (exists pa_lt_bl_other_length_lower_product_bound. pa_lt_bl_other_length_lower_product_bound + S pa_i_bl_other_length_lower_product = ff_exponent_bl_other_length) -> exists pa_p_bl_other_length_lower_product pa_r_bl_other_length_lower_product pa_s_bl_other_length_lower_product. ((((exists pa_h_bl_other_length_lower_product_factor. pa_h_bl_other_length_lower_product_factor + S (pa_p_bl_other_length_lower_product) = S ((S (pa_i_bl_other_length_lower_product)) * pa_c_bl_other_length_lower)) /\ exists pa_q_bl_other_length_lower_product_factor. pa_b_bl_other_length_lower = pa_q_bl_other_length_lower_product_factor * S ((S (pa_i_bl_other_length_lower_product)) * pa_c_bl_other_length_lower) + (pa_p_bl_other_length_lower_product))) /\ ((((exists pa_h_bl_other_length_lower_product_partial. pa_h_bl_other_length_lower_product_partial + S (pa_r_bl_other_length_lower_product) = S ((S (pa_i_bl_other_length_lower_product)) * pa_v_bl_other_length_lower_product)) /\ exists pa_q_bl_other_length_lower_product_partial. pa_u_bl_other_length_lower_product = pa_q_bl_other_length_lower_product_partial * S ((S (pa_i_bl_other_length_lower_product)) * pa_v_bl_other_length_lower_product) + (pa_r_bl_other_length_lower_product))) /\ ((((exists pa_h_bl_other_length_lower_product_successor. pa_h_bl_other_length_lower_product_successor + S (pa_s_bl_other_length_lower_product) = S ((S (S pa_i_bl_other_length_lower_product)) * pa_v_bl_other_length_lower_product)) /\ exists pa_q_bl_other_length_lower_product_successor. pa_u_bl_other_length_lower_product = pa_q_bl_other_length_lower_product_successor * S ((S (S pa_i_bl_other_length_lower_product)) * pa_v_bl_other_length_lower_product) + (pa_s_bl_other_length_lower_product))) /\ pa_s_bl_other_length_lower_product = pa_r_bl_other_length_lower_product * pa_p_bl_other_length_lower_product)))))))) /\ ((exists pa_b_bl_other_length_upper pa_c_bl_other_length_upper. ((forall pa_i_bl_other_length_upper_repeat. (exists pa_lt_bl_other_length_upper_repeat_bound. pa_lt_bl_other_length_upper_repeat_bound + S pa_i_bl_other_length_upper_repeat = L) -> (((exists pa_h_bl_other_length_upper_repeat_decoded. pa_h_bl_other_length_upper_repeat_decoded + S (2) = S ((S (pa_i_bl_other_length_upper_repeat)) * pa_c_bl_other_length_upper)) /\ exists pa_q_bl_other_length_upper_repeat_decoded. pa_b_bl_other_length_upper = pa_q_bl_other_length_upper_repeat_decoded * S ((S (pa_i_bl_other_length_upper_repeat)) * pa_c_bl_other_length_upper) + (2)))) /\ (exists pa_u_bl_other_length_upper_product pa_v_bl_other_length_upper_product. ((((exists pa_h_bl_other_length_upper_product_start. pa_h_bl_other_length_upper_product_start + S (1) = S ((S (0)) * pa_v_bl_other_length_upper_product)) /\ exists pa_q_bl_other_length_upper_product_start. pa_u_bl_other_length_upper_product = pa_q_bl_other_length_upper_product_start * S ((S (0)) * pa_v_bl_other_length_upper_product) + (1))) /\ ((((exists pa_h_bl_other_length_upper_product_terminal. pa_h_bl_other_length_upper_product_terminal + S (ff_upper_bl_other_length) = S ((S (L)) * pa_v_bl_other_length_upper_product)) /\ exists pa_q_bl_other_length_upper_product_terminal. pa_u_bl_other_length_upper_product = pa_q_bl_other_length_upper_product_terminal * S ((S (L)) * pa_v_bl_other_length_upper_product) + (ff_upper_bl_other_length))) /\ forall pa_i_bl_other_length_upper_product. (exists pa_lt_bl_other_length_upper_product_bound. pa_lt_bl_other_length_upper_product_bound + S pa_i_bl_other_length_upper_product = L) -> exists pa_p_bl_other_length_upper_product pa_r_bl_other_length_upper_product pa_s_bl_other_length_upper_product. ((((exists pa_h_bl_other_length_upper_product_factor. pa_h_bl_other_length_upper_product_factor + S (pa_p_bl_other_length_upper_product) = S ((S (pa_i_bl_other_length_upper_product)) * pa_c_bl_other_length_upper)) /\ exists pa_q_bl_other_length_upper_product_factor. pa_b_bl_other_length_upper = pa_q_bl_other_length_upper_product_factor * S ((S (pa_i_bl_other_length_upper_product)) * pa_c_bl_other_length_upper) + (pa_p_bl_other_length_upper_product))) /\ ((((exists pa_h_bl_other_length_upper_product_partial. pa_h_bl_other_length_upper_product_partial + S (pa_r_bl_other_length_upper_product) = S ((S (pa_i_bl_other_length_upper_product)) * pa_v_bl_other_length_upper_product)) /\ exists pa_q_bl_other_length_upper_product_partial. pa_u_bl_other_length_upper_product = pa_q_bl_other_length_upper_product_partial * S ((S (pa_i_bl_other_length_upper_product)) * pa_v_bl_other_length_upper_product) + (pa_r_bl_other_length_upper_product))) /\ ((((exists pa_h_bl_other_length_upper_product_successor. pa_h_bl_other_length_upper_product_successor + S (pa_s_bl_other_length_upper_product) = S ((S (S pa_i_bl_other_length_upper_product)) * pa_v_bl_other_length_upper_product)) /\ exists pa_q_bl_other_length_upper_product_successor. pa_u_bl_other_length_upper_product = pa_q_bl_other_length_upper_product_successor * S ((S (S pa_i_bl_other_length_upper_product)) * pa_v_bl_other_length_upper_product) + (pa_s_bl_other_length_upper_product))) /\ pa_s_bl_other_length_upper_product = pa_r_bl_other_length_upper_product * pa_p_bl_other_length_upper_product)))))))) /\ ((exists ff_lower_gap_bl_other_length. ff_lower_gap_bl_other_length + (ff_lower_bl_other_length) = (n)) /\ (exists ff_upper_gap_bl_other_length. ff_upper_gap_bl_other_length + S (n) = (ff_upper_bl_other_length))))))))) -> l = L)